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Home , Aliquot sequence, Aliquot sum, Alternating factorial, Automorphic number, Bell number, Casting out nines

0_11 The Eleven Set Contents

1 1 1 1.1 Etymology ...... 1 1.2 As a ...... 1 1.3 As a digit ...... 1 1.4 ...... 1 1.4.1 Table of basic calculations ...... 3 1.5 In technology ...... 3 1.6 In science ...... 3 1.6.1 In astronomy ...... 3 1.7 In philosophy ...... 3 1.8 In literature ...... 4 1.9 In comics ...... 4 1.10 In sports ...... 4 1.11 In other fields ...... 6 1.12 See also ...... 6 1.13 References ...... 6 1.14 External ...... 6

2 2 (number) 7 2.1 In mathematics ...... 7 2.1.1 List of basic calculations ...... 8 2.2 Evolution of the glyph ...... 8 2.3 In science ...... 8 2.3.1 Astronomy ...... 8 2.4 In technology ...... 9 2.5 In ...... 9 2.5.1 ...... 9 2.6 Numerological significance ...... 9 2.7 In sports ...... 10 2.8 In other fields ...... 10 2.9 See also ...... 11 2.10 References ...... 11 2.11 External links ...... 11

i ii CONTENTS

3 3 (number) 12 3.1 Evolution of the glyph ...... 12 3.1.1 Flat top 3 ...... 12 3.2 In mathematics ...... 12 3.2.1 In systems ...... 13 3.2.2 List of basic calculations ...... 13 3.3 In science ...... 13 3.3.1 In protoscience ...... 14 3.3.2 In astronomy ...... 14 3.3.3 In pseudoscience ...... 14 3.4 In philosophy ...... 14 3.5 In religion ...... 14 3.5.1 In ...... 14 3.5.2 In Judaism ...... 15 3.5.3 In ...... 15 3.5.4 In ...... 15 3.5.5 In ...... 15 3.5.6 In ...... 15 3.5.7 In ...... 15 3.5.8 In ...... 16 3.5.9 In other ...... 16 3.5.10 In esoteric tradition ...... 16 3.5.11 As a lucky or unlucky number ...... 16 3.6 In sports ...... 16 3.7 See also ...... 17 3.8 References ...... 17 3.9 External links ...... 18

4 4 (number) 19 4.1 In mathematics ...... 19 4.2 List of basic calculations ...... 20 4.3 Origins ...... 20 4.4 In religion ...... 20 4.5 In politics ...... 22 4.6 In computing ...... 22 4.7 In science ...... 22 4.7.1 In astronomy ...... 22 4.7.2 In biology ...... 22 4.7.3 In chemistry ...... 22 4.7.4 In physics ...... 22 4.8 In logic and philosophy ...... 23 4.9 In technology ...... 24 CONTENTS iii

4.10 In transport ...... 24 4.11 In sports ...... 25 4.12 In other fields ...... 26 4.13 In music ...... 27 4.14 Groups of four ...... 27 4.15 References ...... 28 4.16 External links ...... 28

5 5 (number) 29 5.1 In mathematics ...... 29 5.1.1 List of basic calculations ...... 29 5.2 Evolution of the glyph ...... 29 5.3 Science ...... 30 5.3.1 Astronomy ...... 30 5.3.2 Biology ...... 30 5.3.3 Computing ...... 30 5.4 Religion and culture ...... 30 5.4.1 Christian ...... 30 5.4.2 ...... 30 5.4.3 Islamic ...... 30 5.4.4 Jewish ...... 31 5.4.5 Sikh ...... 31 5.4.6 Other religions and cultures ...... 31 5.5 Art, entertainment, and media ...... 31 5.5.1 Events ...... 31 5.5.2 Fictional entities ...... 31 5.5.3 Films ...... 31 5.5.4 Music ...... 32 5.5.5 Television ...... 32 5.5.6 Literature ...... 33 5.6 Sports ...... 33 5.7 Technology ...... 34 5.8 Miscellaneous fields ...... 35 5.9 See also ...... 36 5.10 References ...... 36 5.11 External links ...... 36

6 6 (number) 37 6.1 In mathematics ...... 37 6.1.1 In numeral systems ...... 38 6.1.2 List of basic calculations ...... 38 6.2 Greek and word parts ...... 38 iv CONTENTS

6.2.1 Hexa ...... 38 6.2.2 The prefix sex- ...... 38 6.3 Evolution of the glyph ...... 39 6.4 In music ...... 39 6.4.1 In artists ...... 39 6.4.2 In instruments ...... 39 6.4.3 In music theory ...... 39 6.4.4 In works ...... 40 6.5 In religion ...... 40 6.6 In science ...... 40 6.6.1 Astronomy ...... 40 6.6.2 Biology ...... 40 6.6.3 Chemistry ...... 41 6.6.4 Medicine ...... 41 6.6.5 Physics ...... 41 6.7 In sports ...... 41 6.8 In technology ...... 42 6.9 In calendars ...... 43 6.10 In the arts and entertainment ...... 43 6.11 In other fields ...... 43 6.12 References ...... 44 6.13 External links ...... 44

7 7 (number) 45 7.1 Mathematics ...... 45 7.1.1 Numeral systems ...... 46 7.1.2 Basic calculations ...... 46 7.2 Evolution of the glyph ...... 46 7.3 Automotive and transportation ...... 47 7.4 Classical world ...... 47 7.4.1 Classical antiquity ...... 47 7.5 Commerce and business ...... 47 7.6 Food and beverages ...... 47 7.7 Media and entertainment ...... 47 7.7.1 Film ...... 47 7.7.2 Games ...... 48 7.7.3 Literature ...... 48 7.7.4 Music ...... 50 7.7.5 Sports ...... 50 7.7.6 Television ...... 52 7.8 Places ...... 53 7.9 ...... 53 CONTENTS

7.9.1 Old Testament ...... 53 7.9.2 New Testament ...... 54 7.9.3 Hinduism ...... 54 7.9.4 ...... 54 7.9.5 Judaism ...... 55 7.9.6 Astrology ...... 55 7.9.7 Others ...... 55 7.10 Science ...... 56 7.10.1 Astronomy ...... 56 7.10.2 Biology ...... 56 7.10.3 Chemistry ...... 56 7.10.4 Physics ...... 57 7.10.5 Psychology ...... 57 7.11 Software ...... 57 7.12 Temporal, seasonal and holidays ...... 57 7.13 See also ...... 58 7.14 Notes ...... 58 7.15 References ...... 58 7.16 External links ...... 58

8 8 (number) 59 8.1 In mathematics ...... 59 8.1.1 In numeral systems ...... 60 8.1.2 List of basic calculations ...... 60 8.2 Etymology ...... 60 8.3 Glyph ...... 60 8.4 In science ...... 60 8.4.1 Physics ...... 60 8.4.2 Astronomy ...... 61 8.4.3 Chemistry ...... 61 8.4.4 Geology ...... 61 8.4.5 Biology ...... 61 8.5 In technology ...... 61 8.5.1 In measurement ...... 62 8.6 In culture ...... 62 8.6.1 Architecture ...... 62 8.6.2 In religion, folk and ...... 62 8.6.3 In music and dance ...... 63 8.6.4 In film and television ...... 64 8.6.5 In sports and other games ...... 64 8.6.6 In foods ...... 65 8.6.7 In literature ...... 65 vi CONTENTS

8.6.8 In slang ...... 66 8.7 See also ...... 66 8.8 References ...... 66 8.9 External links ...... 66

9 9 (number) 67 9.1 and codes ...... 67 9.2 Commerce ...... 67 9.3 Culture and mythology ...... 67 9.3.1 Indian culture ...... 67 9.3.2 ...... 67 9.3.3 ...... 67 9.3.4 European culture ...... 68 9.3.5 ...... 68 9.4 Evolution of the glyph ...... 68 9.5 Idioms and popular phrases ...... 68 9.6 Internet ...... 68 9.7 Literature ...... 68 9.8 Mathematics ...... 69 9.8.1 List of basic calculations ...... 70 9.8.2 Numeral systems ...... 70 9.8.3 Probability ...... 70 9.9 Organizations ...... 70 9.10 Places and thoroughfares ...... 70 9.11 Religion and philosophy ...... 70 9.12 Science ...... 71 9.12.1 Astronomy ...... 71 9.12.2 Chemistry ...... 71 9.12.3 Physiology ...... 71 9.13 Sports ...... 71 9.13.1 ...... 71 9.13.2 Baseball ...... 71 9.13.3 Billiards ...... 72 9.13.4 Rugby ...... 72 9.13.5 Soccer ...... 72 9.13.6 All sports ...... 72 9.14 Technology ...... 72 9.15 Other fields ...... 73 9.16 See also ...... 73 9.17 References ...... 74 9.18 Further reading ...... 74 CONTENTS vii

10 10 (number) 75 10.1 Common usage and derived terms ...... 75 10.2 In mathematics ...... 75 10.2.1 In numeral systems ...... 75 10.2.2 List of basic calculations ...... 76 10.3 In science ...... 76 10.3.1 Astronomy ...... 76 10.4 In religion and philosophy ...... 76 10.5 In money ...... 77 10.6 In music ...... 77 10.7 In sports and games ...... 77 10.8 In technology ...... 78 10.9 In other fields ...... 78 10.10See also ...... 78 10.11References ...... 79 10.12External links ...... 79

11 11 (number) 80 11.1 Name ...... 80 11.2 In languages ...... 80 11.2.1 Grammar ...... 80 11.3 In mathematics ...... 80 11.3.1 List of basic calculations ...... 81 11.4 In numeral systems ...... 81 11.5 In science ...... 81 11.5.1 Astronomy ...... 81 11.6 In religion ...... 82 11.6.1 Christianity ...... 82 11.6.2 ...... 82 11.6.3 Babylonian ...... 82 11.7 In music ...... 82 11.8 In sports ...... 82 11.9 In the military ...... 83 11.10In computing ...... 83 11.11In Canada ...... 84 11.12In other fields ...... 84 11.13See also ...... 84 11.14References ...... 84 11.15External links ...... 85

12 0 86 12.1 Etymology ...... 86 viii CONTENTS

12.2 History ...... 86 12.2.1 Ancient Near East ...... 86 12.2.2 Pre-Columbian Americas ...... 86 12.2.3 Classical antiquity ...... 87 12.2.4 ...... 87 12.2.5 and Southeast Asia ...... 88 12.2.6 Middle Ages ...... 88 12.3 Mathematics ...... 89 12.3.1 Elementary ...... 89 12.3.2 Other branches of mathematics ...... 90 12.3.3 Related mathematical terms ...... 90 12.4 Physics ...... 90 12.5 Chemistry ...... 90 12.6 Astronomy ...... 91 12.7 science ...... 91 12.8 Other fields ...... 91 12.9 Symbols and representations ...... 91 12.10Year label ...... 92 12.11See also ...... 92 12.12References ...... 92 12.13Bibliography ...... 94 12.14External links ...... 94

13 95 13.1 See also ...... 95 13.2 References ...... 95

14 96 14.1 Examples ...... 96 14.2 History ...... 96 14.3 Even perfect ...... 96 14.4 Odd perfect numbers ...... 98 14.5 Minor results ...... 98 14.6 Related concepts ...... 98 14.7 See also ...... 99 14.8 Notes ...... 99 14.9 References ...... 100 14.10Further reading ...... 100 14.11External links ...... 100

15 101 15.1 Significance and formula of the digital root ...... 101 CONTENTS ix

15.2 Abstract of digital roots ...... 101 15.3 Formal definition ...... 101 15.3.1 Example ...... 101 15.3.2 Proof that a constant value exists ...... 102 15.4 Congruence formula ...... 102 15.5 Some properties of digital roots ...... 102 15.6 In other bases ...... 103 15.7 See also ...... 103 15.8 References ...... 104 15.9 External links ...... 104 15.10Text and image sources, contributors, and licenses ...... 105 15.10.1 Text ...... 105 15.10.2 Images ...... 115 15.10.3 Content license ...... 119 Chapter 1

1

“One”and “No. 1”redirect here. For other uses, see 1.3 As a digit One (disambiguation).

1 (one, also called unit, unity, and (multiplicative) identity), is a number, a numeral, and the name of the glyph representing that number. It represents a single en- tity, the unit of or measurement. For example, a line segment of unit length is a line segment of length 1. The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, 1.1 Etymology traces its roots back to the Indians, who wrote 1 as a hor- izontal line, much like the Chinese character 一. The Gupta wrote it as a curved line, and the Nagari sometimes The word one can be used as a noun, an adjective and a added a small on the left (rotated a quarter turn to pronoun.*[1] the right, this 9-look-alike became the present day nu- It comes from the English word an,*[1] which comes meral 1 in the Gujarati and Punjabi scripts). The Nepali from the Proto-Germanic root *ainaz.*[1] The Proto- also rotated it to the right but kept the circle small.*[3] Germanic root *ainaz comes from the Proto-Indo- This eventually became the top serif in the modern nu- European root *oi-no-.*[1] meral, but the occasional short horizontal line at the bot- Compare the Proto-Germanic root *ainaz to Old Frisian tom probably originates from similarity with the Roman an, Gothic ains, Danish een, Dutch een, German eins numeral I. In some countries, the little serif at the top is and einn. sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with Compare the Proto-Indo-European root *oi-no- (which the glyph for seven in other countries. Where the 1 is * means one, single [1]) to Greek oinos (which means written with a long upstroke, the number 7 has a horizon- * * “ace”on dice [1]), Latin unus (one [1]), Old Persian tal stroke through the vertical line. aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one*[1]). While the shape of the 1 character has an ascender in most modern , in typefaces with text figures, the character usually is of -height, as, for example, in

1.2 As a number . Many older typewriters do not have a separate symbol for One, sometimes referred to as unity,*[2] is the first non- 1 and use the lowercase instead. It is possible to zero . It is thus the before two and find cases when the uppercase is used, while it may be after zero, and the first positive odd number. for decorative purposes. Any number multiplied by one is that number, as one is the identity for multiplication. As a result, 1 is its own , its own square, its own , and so on. One 1.4 Mathematics is also the result of the empty , as any number multiplied by one is itself. It is also the only natural num- Mathematically, 1 is: ber that is neither composite nor prime with respect to , but instead considered a unit. • in (algebra) and calculus, the natural

1 2 CHAPTER 1. 1

and 0 are also present. When such a has characteristic not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields. 1 is the first figurate number of every kind, such as , and centered , to name just a few. In many mathematical and engineering equations, nu- meric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Because of the multiplicative identity, if (x) is a , then f(1) must equal 1. It is also the first and second number in the se- quence (0 is the zeroth) and is the first number in many other mathematical . As a matter of conven- tion, Sloane' early Handbook of Integer Sequences added The 24-hour tower clock in Venice, using J as a symbol for 1. an initial 1 to any that did not already have it and considered these initial 1s in its - number that follows 0 and precedes 2 and the multi- ing. Sloane's later Encyclopedia of Integer Sequences and plicative identity element of the , real num- its Web counterpart, the On-Line Encyclopedia of Integer bers and complex numbers; Sequences, ignore initial 1s in their lexicographic ordering of sequences, because such initial 1s often correspond to • more generally, in algebra, the multiplicative iden- trivial cases. tity (also called unity), usually of a group or a ring. 1 is neither a nor a , but a unit, like −1 and, in the Gaussian integers, i and 1 cannot be used as the base of a positional numeral sys- −i. The fundamental theorem of arithmetic guarantees tem, as the only digit that would be permitted in such a unique over the integers only up to units. system would be 0. (Sometimes tallying is referred to (For example, 4 = 22, but if units are included, is also as “base 1”, since only one mark —the tally itself —is equal to, say, (−1)6 × 123 × 22, among infinitely many needed, but this is not a .) similar “”.) Since the base 1 (1*x) always equals The definition of a field requires that 1 must not be equal 1, its inverse does not exist (which would be called the to 0. Thus, there are no fields of characteristic 1. Nev- base 1 if it did exist). ertheless, abstract algebra can consider the field with one There are two ways to write the 1 as a element, which is not a singleton and is not a set at all. recurring : as 1.000..., and as 0.999.... There is 1 is the only positive integer divisible by exactly one pos- only one way to represent the real number 1 as a Dedekind itive integer (whereas prime numbers are divisible by ex- cut: actly two positive integers, composite numbers are divis- ible by more than two positive integers, and zero is divis- ible by all positive integers). 1 was formerly considered ({x|x < 1} , {x|x ≥ 1}) prime by some mathematicians, using the definition that Formalizations of the natural numbers have their own a prime is divisible only by 1 and itself. However, this representations of 1: complicates the fundamental theorem of arithmetic, so modern definitions exclude units. • in the Peano axioms, 1 is the successor of 0; 1 is one of three possible values of the Möbius function: it takes the value 1 for square-free integers with an even • in Principia Mathematica, 1 is defined as the set of number of distinct prime factors. all singletons (sets with one element); 1 is the only odd number in the range of Euler's totient • in the Von Neumann cardinal assignment of natural function φ(x), in the cases x = 1 and x = 2. numbers, 1 is defined as the set {0}. 1 is the only 1-perfect number (see multiply perfect num- In a multiplicative group or monoid, the identity element ber). is sometimes denoted 1, but (from the German Einheit, By definition, 1 is the magnitude, absolute value, or norm “unity”) is also traditional. However, 1 is especially com- of a unit , unit vector, and a unit mon for the multiplicative identity of a ring, i.e., when an 1.6. IN SCIENCE 3

(more usually called an identity matrix). Note that the • In many physical devices, 1 represents the value term unit matrix is sometimes used to mean something for “on”, which means that electricity is flow- quite different. ing.*[5]*[6]

By definition, 1 is the probability of an event that is almost • The numerical value of true in many programming certain to occur. languages. 1 is the most common leading digit in many sets of data, a consequence of Benford's law. The ancient Egyptians represented all (with the 1.6 In science exception of 2/3) in terms of sums of fractions with numerator 1 and distinct denominators. For example, 2/5 • 1 is the atomic number of hydrogen, and the atomic = 1/3 + 1/15. Such representations are popularly known mass of its most common isotope. as Egyptian Fractions or Unit Fractions. • 1 is the ASCII code of "Start of Header". The that has all coefficients 1 is given by • 1 is the electric charge of positrons and protons. • Group 1 of the periodic table consists of the alkali 2 3 1/1 − x = 1 + x + x + x + … metals.

• Period 1 of the periodic table consists of just two This power converges and has finite value if and elements, hydrogen and helium. only if, |x| < 1.

1.6.1 In astronomy 1.4.1 Table of basic calculations • 1 is the Saros number of the solar eclipse series 1.5 In technology which began on June 4, 2872 BC and ended on July 11, 1592 BC. The duration of Saros series 1 was 1280.14 years, and it contained 72 solar eclipses.*[7]

• The Saros number of the lunar eclipse series which began on March 14, 2570 BC and ended on April 30, 1272 BC. The duration of Saros series 1 was 1298.17 years, and it contained 73 lunar eclipses.*[7]

• The dwarf planet Ceres has the minor planet desig- nation 1 Ceres because it was the first asteroid to be discovered.

• M1 is the Messier designation of the Crab Nebula.

• NGC 1 is the designation of a distant galaxy.

• The Roman numeral I stands for supergiant in the Yerkes spectral classification scheme.

1 as a resin identification code, used in recycling • The Roman numeral I often stands for the first- discovered satellite of a planet or minor planet (such as Neptune I, a..a. Triton). For some earlier dis- • The resin identification code used in recycling to coveries, the originally reflected identify polyethylene terephthalate.*[4] increasing distance from the primary instead.

• the ITU country code for the North American Num- bering Plan area, which includes the United States, Canada, and parts of the Caribbean 1.7 In philosophy

• A binary code is a sequence of 1 and 0 that is used In the philosophy of Plotinus and a number of other in for representing any kind of data. neoplatonists, The One is the ultimate reality and source 4 CHAPTER 1. 1 of all existence. Philo of Alexandria (20 BC – AD 50) • 1 point is awarded for a successful free throw. regarded the number one as 's number, and the basis • In the 3×3 variant of the game, shots made “ ” for all numbers ( De Allegoriis Legum, ii.12 [i.66]). from inside the “three-point”arc are also worth 1 point. (Shots from outside the arc are worth 2 points.) 1.8 In literature • The number 1 is used to designate the point guard position. • Number One is a character in the book series Lorien Legacies by Pittacus Lore. • In association football (soccer) the number 1 is often given to the goalkeeper • Number 1 is also a character in the series“Artemis Fowl”by Eoin Colfer. • In Gaelic football, hurling and camogie, a “point” , with a scoring value of 1, is awarded when the at- • In a 1968 song by Three Dog's Night, the number tacking team legally sends the ball over the oppo- “ ” one is identified as the loneliest number . nent's crossbar (above the goal).

• In gridiron football codes, one point is awarded un- 1.9 In comics der the following circumstances: • In almost all leagues, for a successful place • A character in the Italian comic book Alan Ford (au- kick after a touchdown. In American football, thors Max Bunker and Magnus), very old disabled the score is formally known as a “try”, al- man, the supreme leader of the group TNT. though the terms“extra point”,“conversion” ,“PAT”(for“point after touchdown”), and “point after”are far more commonly used. In 1.10 In sports Canadian football, the score is formally and popularly called a “convert”. Conversions • 1 is the lowest number permitted for use by play- can also be scored by the now rare drop kick; ers of the National Hockey League (NHL), as the in standard American and Canadian football, league has banned 00 and 0. (The highest number such a conversion is worth 1 point, while most permitted is 98.) forms of indoor football, including the Football League, award 2 points for a drop- • Many sports use 1 as their standard scoring incre- kicked conversion. ment—examples include goals in a large number of • In college football, if a point after “try”is sports, runs in baseball and , and points in vol- blocked, if the blocked ball stayed in the field leyball. Examples where 1 is a non-standard incre- of play a defender may pick up and run the ball ment, or used for one of several possible classes of to his end zone at the other end of the field for scores, are listed below by sport. a one-point safety. • In Australian rules football, 1 point is awarded to the • In six-man football, one point is awarded for attacking team for a behind, scored when: a successful conversion from scrimmage af- ter a touchdown. Note that in standard 11- • The ball is kicked by the attacking team and man (American) or 12-man (Canadian) foot- passes between a goal post (taller post) and the ball, place kicks are worth 1 point and con- nearest behind post (shorter post) on the de- versions from scrimmage worth 2; this is re- fensive side of the field without touching the versed in six-man because the reduced num- behind post. ber of players makes kicked conversions much • The ball passes between the defending team's more difficult. goal posts, but either (1) was not kicked by the • In Canadian football only, a single or“rouge” attacking team or (2) hit a goal post. is awarded when the ball is legally kicked into • The defending team deliberately forces the the opponent's end zone (except for a success- ball between any two of its own posts. This ful field goal), and the receiving team does not particular score is officially called a “rushed return, or kick, the ball out of its end zone. (In behind”. American football, the same play would result in a touchback and no points.) • In baseball scoring, the number 1 is assigned to the • pitcher. Some forms of indoor football in the U.S. award a “single”, similar to the Canadian • In basketball: score, on kickoffs only. 1.10. IN SPORTS 5

• In rugby league: • The Seattle SuperSonics, for Gus Williams. The team has since relocated • A drop goal is worth 1 point. to become the Oklahoma City Thunder, • In most competitions (though not the Euro- but the Thunder have yet to issue any pean Super League, which uses static squad number retired by the franchise in numbering), the starting fullback wears jersey Seattle. number 1. • The Utah , for Frank Layden, who • In : served the team first as head coach and then as president. • The starting loosehead prop wears the jersey • number 1. In the NFL: • In the early years of the sport (prior to 1890), • The New York Giants, for Hall of Famer conversions, penalties, drop goals, and goals Ray Flaherty. from mark were all worth 1 point. At that time, • The Tennessee Titans, for Hall of Famer a try was worth no points, only giving the at- Warren Moon, who played for the team in tacking team the right to attempt a conversion. its past incarnation as the Houston Oilers. In 1890–91, tries were rewarded with 1 point, while all other scores were increased in value. • In the NHL: After that time, all scores have been worth at • The Chicago Blackhawks, for Hall of least 2 points (the goal from mark was abol- Famer Glenn Hall. ished in 1977). • The Detroit Red Wings, for Hall of • The jersey number 1 has been retired by several Famer Terry Sawchuk. North American sports teams in honor of past play- • The Montreal Canadiens, for Hall of ing greats or other key figures (or, in one case, a Famer Jacques Plante. team's fans): • The New York Rangers, for Hall of • In Major League Baseball: Famer Eddie Giacomin. • The Boston Red Sox, for Hall of Famer • The Minnesota Wild, for their fans. Bobby Doerr. • The Philadelphia Flyers, for Hall of • The Cincinnati Reds, for manager Fred Famer Bernie Parent. Hutchinson. • The Toronto Maple Leafs have a pol- • The Los Angeles Dodgers, for Hall of icy of not retiring numbers unless the Famer Pee Wee Reese. player honoured either died or suffered a • The New York Yankees, for Billy Mar- career-ending incident while a member of tin, who both played for and managed the the team. Other players whose numbers team. would otherwise be retired instead have • The Philadelphia Phillies, for Hall of their numbers enshrined by the team as Famer Richie Ashburn. “Honoured Numbers”, which remain in • The Pittsburgh Pirates, for manager Billy circulation for future players. The num- Meyer. ber 1 is currently honoured for Hall of • The St. Louis Cardinals, for Hall of Famers Johnny Bower and Turk Broda. Famer Ozzie Smith. • In F1: • In the NBA: • The previous year's world champion is al- • The Boston Celtics, for founding owner lowed to use the number 1. Also, it is one Walter Brown, a member of the Hall of of two numbers from 1–99 that F1 drivers Fame as a contributor. cannot use, the other being 17, which • The Milwaukee Bucks, for Hall of Fame has been retired after Jules Bianchi's ac- player Oscar Robertson. cident. • The Portland Trail Blazers, for found- • ing owner Larry Weinberg. Unlike most In NASCAR numbers so honored, this number re- • The number of a car in the Sprint Cup Se- mains in circulation for players. ries originally owned by Dale Earnhardt • The Sacramento Kings, for Hall of Fame Inc. (1989–2007) and since 2008 by Chip player Nate Archibald, honoring the Ganassi Racing (when DEI merged into number's retirement when the team was Ganassi Racing). The car, a Chevrolet, is known as the Kansas City Kings. currently driven by Jamie McMurray. 6 CHAPTER 1. 1

1.11 In other fields

• 1 is the value of an ace in many playing card games, such as cribbage. • List of highways numbered 1 • List of public transport routes numbered 1 • 1 is often used to denote the Gregorian calendar month of January. • 1 CE, the first year of the Common Era • 01, the former dialing code for Greater • PRS One, a German paraglider design

1.12 See also

• −1 • +1 (disambiguation) • One (word) • Root of unity

1.13 References

[1] “Online Etymology Dictionary”. etymonline.com. Dou- glas Harper.

[2] Skoog, Douglas. Principles of Instrumental Analysis. Brooks/Cole, 2007, p. 758.

[3] Ifrah, Georges; et al. (1998). The Universal History of Numbers: From Prehistory to the Invention of the Com- puter. Translated by Bellos, David. yes. London: The Harvill Press. p. 392, Fig. 24.61.

[4] “Plastic Packaging Resins”(PDF). American Chemistry Council.

[5] Woodford, Chris (2006), Digital Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9

[6] Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8

[7] “Saros Series 1”. Saros Series Catalog of Solar Eclipses. NASA. Retrieved September 29, 2016.

1.14 External links

• The Number 1 • The Positive Integer 1 • Prime curiosities: 1 Chapter 2

2 (number)

“II”,“Two”, and“Number 2”redirect here. For other all other primes, because 2 and 5 are factors of ten, the uses, see II (disambiguation), Two (disambiguation), and decimal base. Number 2 (disambiguation). Two is the base of the simplest in which natural numbers can be written concisely, being the 2 (Two; *i/ˈtuː/) is a number, numeral, and glyph. It is length of the number a logarithm of the value of the num- the natural number following 1 and preceding 3. ber (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers. 2.1 In mathematics For any number x:

x + x = 2 ·x addition to multiplication The number two has many properties in mathematics.*[1] 2 An integer is called even if it is divisible by 2. For integers x ·x = x multiplication to written in a numeral system based on an even number, x*x = x↑↑2 exponentiation to such as decimal and , divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then In general: the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, hyper(x,n,x) = hyper(x,(n + 1),2) 4, 6, or 8. In numeral systems based on an odd number, divisibility by 2 can be tested by having a digital root that Two also has the unique property that 2 + 2 = 2 ·2 = 22 is even. = 2↑↑2 = 2↑↑↑2, and so on, no matter how high the level Two is the smallest and the first prime number, and the of the is. only even prime number*[2] (for this reason it is some- Two is the only number x such that the sum of the recip- times called“the oddest prime”).*[3] The next prime is rocals of the powers of x equals itself. In symbols three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first , the first Lucas prime, the first Ramanujan ∞ ∑ 1 1 1 1 1 prime,*[4] and the first Smarandache-Wellin prime. It = 1 + + + + + ··· = 2. 2k 2 4 8 16 is an Eisenstein prime with no imaginary part and real k=0 part of the form 3n − 1. It is also a Stern prime,*[5] a ,*[6] the first , and a Markov This comes from the fact that: number—appearing in infinitely many solutions to the Markov involving odd-indexed Pell ∞ ∑ 1 1 numbers. = 1 + for all n ∈ > 1. nk n − 1 It is the third , and the second and k=0 * fourth Perrin numbers. [7] Powers of two are central to the concept of Mersenne Despite being prime, two is also a superior highly com- primes, and important to . Two is the posite number, because it is a natural number which has first exponent. more than any other number scaled relative to * Taking the of a number is such a common the number itself. [8] The next superior highly compos- mathematical , that the spot on the root ite number is six. where the exponent would normally be written for cubic Vulgar fractions with only 2 or 5 in the denominator do roots and other such roots, is left blank for square roots, not yield infinite decimal expansions, as is the case with as it is considered tacit.

7 8 CHAPTER 2. 2 (NUMBER)

The was the first known irrational num- ber. The smallest field has two elements. In the set-theoretical construction of the natural numbers, 2 is identified with the set {{∅},∅}. This latter set is im- portant in category theory: it is a subobject classifier in the category of sets. and made its bottom end curve towards the center of the Two is a , as well as its own factorial. Two of- bottom line. Apparently for speed, the Nagari started ten occurs in numerical sequences, such as the Fibonacci making the top line more like a curve and connecting to number sequence, but not quite as often as one does. Two the bottom line. The Ghubar Arabs made the bottom line is also a ,*[9] a ,*[10] an all- completely vertical, and now the glyph looks like a dot- , a meandric number, a semi-meandric less closing question mark. Restoring the bottom line to number, and an open meandric number. its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our Two is the number of n queens problem solutions for n modern glyph.*[11] = 4. With one exception, all known solutions to Znám's problem start with 2. In fonts with text figures, 2 usually is of x-height, for ex- Two also has the unique property such that ample, .

n∑−1 2k = 2n − 1 2.3 In science k=0

and also • The number of polynucleotide strands in a DNA double helix.

n∑−1 a∑−1 • The first magic number. 2k = 2n − 2k − 1 k=a k=0 • The atomic number of helium.

for a not equal to zero • The atomic mass of deuterium, an isotope of The number of domino tilings of a 2×2 checkerboard is hydrogen. 2. • The ASCII code of "Start of Text". In n-dimensional space for any n, any two distinct points determine a line. • Group 2 in the Periodic table of the elements con- sists of the alkaline metals whose usual valence For any polyhedron homeomorphic to a sphere, the Euler is +2. characteristic is χ = V − E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number • Period 2 in the Periodic table consists of the eight of faces. elements lithium through neon. As of June 2015, there are only two known Wieferich primes in base 2. 2.3.1 Astronomy With the exception of the sequence 3, 5, 7, the maximum number of consecutive odd numbers that are prime is two. • 2 Pallas, a large asteroid in the main belt and the second asteroid ever to be discovered.

2.1.1 List of basic calculations • Messier object M2, a magnitude 6.5 globular cluster in the Aquarius. 2.2 Evolution of the glyph • The New General Catalogue object NGC 2, a magnitude 14.2 spiral galaxy in the constellation The glyph used in the modern Western world to represent . the number 2 traces its roots back to the Brahmin Indi- ans, who wrote “2”as two horizontal lines. The mod- • The Saros number of the solar eclipse series which ern Chinese and Japanese languages still use this method. began on May 4, 2861 BC and ended on June 21, The Gupta rotated the two lines 45 degrees, making them 1563 BC. The duration of Saros series 2 was 1298.1 diagonal, and sometimes also made the top line shorter years, and it contained 73 solar eclipses. 2.6. NUMEROLOGICAL SIGNIFICANCE 9

2 as a resin identification code, used in recycling.

2.5.1 Judaism

The number 2 is important in Judaism, with one of the earliest references being that God ordered Noah to put two of every unclean animal (Gen. 7:2) in his (see Noah's Ark). Later on, the Ten Commandments were given in the form of two tablets. The number also has ceremonial importance, such as the two candles that are traditionally kindled to usher in the Shabbat, recalling the two different ways Shabbat is referred to in the two times the Ten Commandments are recorded in the Torah. שמור וזכור These two expressions are known in Hebrew as “( guard”and“remember”), as in“Guard the Shabbat day to sanctify it”(Deut. 5:12) and “Remember the Shab- • The Saros number of the lunar eclipse series which bat day to sanctify it”(Ex. 20:8). Two challahs (lechem began on February 21, 2541 BC and ended on April mishneh) are placed on the table for each Shabbat meal 22, 1225 BC. The duration of Saros series 2 was and a blessing made over them, to commemorate the dou- 1316.2 years, and it contained 74 lunar eclipses. ble portion of manna which fell in the desert every Friday to cover that day's meals and the Shabbat meals. • The Roman numeral II stands for bright giant in the In Jewish law, the testimony of two witnesses are required Yerkes spectral classification scheme. to verify and validate events, such as marriage, divorce, • The Roman numeral II (usually) stands for the and a crime that warrants capital punishment. second-discovered satellite of a planet or minor “Second-Day Yom Tov”(Yom Tov Sheini Shebegaliyot) is planet (e.. Pluto II or (87) Sylvia II Remus). a rabbinical enactment that mandates a two-day celebra- • A binary is a stellar system consisting of two tion for each of the one-day Jewish festivals (i.e., the first orbiting around their center of mass. and seventh day of Passover, the day of Shavuot, the first day of Sukkot, and the day of Shemini Atzeret) outside the Land of Israel. 2.4 In technology

• The resin identification code used in recycling to 2.6 Numerological significance identify high-density polyethylene. The most common philosophical dichotomy is perhaps the one of good and evil, but there are many others. See 2.5 In religion dualism for an overview. In Hegelian dialectic, the pro- cess of synthesis creates two perspectives from one. 10 CHAPTER 2. 2 (NUMBER)

• In Association football, the scoring of two goals by one individual in a single match is referred to as a brace. • The successor of a brace is the "hat-trick", three goals scored by one player.

Basketball:

• In standard basketball, the value of any made shot taken from inside the three-point arc in normal play is 2 points. • The twos of all four suits in playing cards In the half-court 3x3 variant, made shots taken from outside the “three-point”arc are worth 2 points.

The ancient language of India, does not only Rugby union: have a singular and plural form for nouns, as do many other languages, but instead has, a singular (1) form, a • In both rugby union and its sevens variant: dual (2) form, and a plural (everything above 2) form, for • all nouns, due to the significance of 2. It is viewed as Conversion kicks following a try are worth 2 important because of the anatomical significance of 2 (2 points. hands, 2 nostrils, 2 eyes, 2 legs, etc.) • The starting hooker wears number 2. Two (二, èr) is a good number in Chinese culture. There • In sevens, a yellow card results in the offender being is a Chinese saying, “good things come in pairs”. It is required to leave the field for 2 minutes of play. common to use double symbols in product brandnames, e.g. double happiness, double coin, double elephants etc. Ice hockey: people like the number two because it sounds the same as the word “easy”(易) in Cantonese. • In Ice hockey, a minor penalty is two minutes in In Finland, two candles are lit on Independence Day. length. Putting them on the windowsill invokes the symbolical meaning of division, and thus independence. In pre-1972 Indonesian and Malay orthography, 2 was 2.8 In other fields shorthand for the reduplication that forms plurals: orang “person”, orang-orang or orang2 “people”. Groups of two: In Astrology, Taurus is the second sign of the Zodiac. • Lists of pairs • list of twins 2.7 In sports • The name of several fictional characters: Number Two. Baseball: • 2 CE, the second year of the Common Era. • • In baseball scorekeeping, 2 is the position of the The designation of the Trans-Canada Highway in catcher. most of the province of New Brunswick. • Interstate 2, the lowest-numbered highway in the • Former New York Yankees player Derek Jeter wore U.S. Interstate Highway System, which runs through the number 2 the lower Rio Grande Valley in South Texas.

Football: • #2 Pencils mean that the pencil point will make a mark darker than a #3 pencil, but a lighter mark than a #1 pencil. • In American football, a safety has a two-point value. Also, a two-point conversion is a point after touch- • U.S. Route 2, two separated highways in the north- down (PAT) attempt where the ball the goal ern tier of the United States, the western seg- line via run or pass. (In six-man football, how- ment connecting Everett, Washington to St. Ignace, ever, the traditional PAT kick is worth two points, Michigan and the eastern route connecting Rouses whereas a PAT via pass or run is only one point.) Point, New York to Houlton, Maine. 2.11. EXTERNAL LINKS 11

• The lowest channel of television in the United States, [11] Georges Ifrah, The Universal History of Numbers: From Canada, and Mexico on which television Prehistory to the Invention of the Computer transl. David signals are broadcast. Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.62 • Brace is also used in hunting to refer to a pair. For example, “He shot a brace of pheasants”. [12] For a typical example, see the University of Oklahoma grading regulations. •“Two turtle doves”is the gift on the second day of Christmas in the carol "The Twelve Days of Christ- mas" 2.11 External links

In North American educational systems, the number 2.00 • The Number 2 denotes a grade-point average of“”, which in some col- • The Positive Integer 2 leges and universities is the minimum required for good academic standing at the undergraduate level.*[12] • Prime curiosities: 2

2.9 See also

• Square (algebra) – (2 superscript)

2.10 References

[1] Wells, . The Penguin Dictionary of Curious and Interest- ing Numbers London: Penguin Group. (1987): 41–44

[2] Bryan Bunch, The Kingdom of Infinite Number. New York: . . Freeman & Company (2000): 31

[3] & Richard K. Guy, The Book of Numbers. New York: Springer (1996): 25. ISBN 0- 387-97993-X.“Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all.”

[4] “Sloane's A104272 : Ramanujan primes”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[5] “Sloane's A042978 : Stern primes”. The On-Line En- cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01.

[6] “Sloane's A000129 : Pell numbers”. The On-Line En- cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01.

[7] “Sloane's A001608 : Perrin sequence”. The On-Line En- cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01.

[8]“Sloane's A002201 : Superior highly composite numbers” . The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.

[9] “Sloane's A001006 : Motzkin numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[10] “Sloane's A000110 : Bell or exponential numbers”. The On-Line Encyclopedia of Integer Sequences. OEIS Foun- dation. Retrieved 2016-06-01. Chapter 3

3 (number)

This article is about the number. For the year, see AD 3. 3.1.1 Flat top 3 For other uses, see 3 (disambiguation). A common variant of the digit 3 has a flat top, similar to the character Ʒ (). This form is sometimes used 3 (three; /ˈθriː/) is a number, numeral, and glyph. It is to prevent people from fraudulently changing a 3 into an the natural number following 2 and preceding 4. 8. It is usually found on UPC-A barcodes and standard 52-card decks.

3.1 Evolution of the glyph 3.2 In mathematics

3 is:

• a rough approximation of π (3.1415...) and a very rough approximation of e (2.71828..) when doing quick estimates.

• the first odd prime number,*[2] and the second smallest prime.

Three is the largest number still written with as many lines • the first Fermat prime (2*2*n + 1). as the number represents. (The Ancient Romans usually wrote 4 as IIII, but this was almost entirely replaced by • the first Mersenne prime (2*n − 1). the subtractive notation IV in the Middle Ages.) To this day 3 is written as three lines in Roman and Chinese nu- • the only number that is both a Fermat prime and a merals. This was the way the Brahmin Indians wrote it, Mersenne prime. and the Gupta made the three lines more curved. The Na- • the first lucky prime. gari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventu- • the first super-prime. ally they made these strokes connect with the lines be- low, and evolved it to a character that looks very much • the first due to the properties of its like a modern 3 with an extra stroke at the bottom as ३. reciprocal. It was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. (The “ex- • the second Sophie Germain prime. tra”stroke, however, was very important to the Eastern • the second Mersenne prime exponent. Arabs, and they made it much larger, while rotating the strokes above to lie along a horizontal axis, and to this day • the second factorial prime (2! + 1). Eastern Arabs write a 3 that looks like a mirrored 7 with .1] • the second Lucas prime]*٣ :(ridges on its top line

While the shape of the 3 character has an ascender in • the second Stern prime.*[3] most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in • the second triangular number and it is the only prime triangular number. . In some French text-figure typefaces, though, it has an ascender instead of a descender. • the third Heegner number.*[4]

12 3.3. IN SCIENCE 13

• both the zeroth and third Perrin numbers in the Per- the sum of all the terms below it, and the only num- rin sequence.*[5] ber whose sum with those below equals the product of them and itself.*[6] • the fourth Fibonacci number. • Three was the first number, according to the • the fourth open meandric number. Pythagoreans, and the first male number. • the aliquot sum of 4. • Three was the first number, according to , i.e. n2 is greater than 2n. • the smallest number of sides that a simple (non-self- intersecting) polygon can have. • The trisection of the angle was one of the three fa- mous problems of antiquity. • the only prime which is one less than a perfect square. Any other number which is n2 − 1 for some • Gauss proved that every integer is the sum of at most integer n is not prime, since it is (n − 1)(n + 1). This 3 triangular numbers. is true for 3 as well (with n = 2), but in this case the • smaller factor is 1. If n is greater than 2, both n − 1 Gauss proved that for any prime number p (with the and n + 1 are greater than 1 so their product is not sole exception of 3) the product of its primitive roots prime. is ≡ 1 (mod p). • * • the number of non-collinear points needed to deter- Any number not in the form of 4 n(8m + 7) is the mine a and a circle. sum of 3 squares. • Each real number belongs to only 1 of the following Also: 3 categories: positive, negative, and zero.

• Vulgar fractions with 3 in the denominator have 3.2.1 In numeral systems a single digit repeating sequences in their decimal expansions: 0.000..., 0.333..., 0.666... (See also There is some evidence to suggest that early man may 0.999..., which = 1.) have used counting systems which consisted of “One, • A natural number is divisible by three if the sum of Two, Three”and thereafter“Many”to describe counting its digits in base 10 is divisible by 3. For example, limits. Early peoples had a word to describe the quanti- the number 21 is divisible by three (3 times 7) and ties of one, two, and three but any quantity beyond was the sum of its digits is 2 + 1 = 3. Because of this, the simply denoted as“Many”. This is most likely based on reverse of any number that is divisible by three (or the prevalence of this phenomena among people in such indeed, any of its digits) is also divisi- disparate regions as the deep Amazon and Borneo jun- ble by three. For instance, 1368 and its reverse 8631 gles, where western civilization's explorers have histori- are both divisible by three (and so are 1386, 3168, cal records of their first encounters with these indigenous * 3186, 3618, etc.). See also . This people. [7] works in base 10 and in any positional numeral sys- tem whose base divided by three leaves a of one (bases 4, 7, 10, etc.). 3.2.2 List of basic calculations • Three of the five regular polyhedra have triangu- 3.3 In science lar faces – the , the , and the icosahedron. Also, three of the five regular • The Roman numeral III stands for giant star in the polyhedra have vertices where three faces meet – Yerkes spectral classification scheme. the tetrahedron, the hexahedron (cube), and the dodecahedron. Furthermore, only three different • Three is the atomic number of lithium. types of polygons comprise the faces of the five regular polyhedra – the triangle, the quadrilateral, • Three is the ASCII code of "End of Text". and the pentagon. • Three is the number of that can • There are only three distinct 4×4 panmagic squares. perceive. Humans perceive the universe to have three spatial dimensions, but some theories, such as • Only three tetrahedral numbers are also perfect string theory, suggest there are more. squares. • The triangle, a polygon with three edges and three • According to Pythagoras and the Pythagorean vertices, is the most stable physical shape. For this school, the number 3, which they called triad, is the reason it is widely utilized in construction, engineer- noblest of all digits, as it is the only number to equal ing and design.*[8] 14 CHAPTER 3. 3 (NUMBER)

• The ability of the eye to distinguish colors is • the Hindu Trimurti based upon the varying sensitivity of different cells • in the retina to light of different wavelengths. Hu- the Hindu Tridevi mans being trichromatic, the retina contains three • the Three Jewels of Buddhism types of color receptor cells, or cones. • the Three Pure Ones of Taoism 3.3.1 In protoscience • the Christian Holy Trinity • • In European alchemy, the three primes (Latin: tria the Triple of Wicca prima) were salt ( ), sulfur ( ) and mercury ( ).*[9]*[10]

• The three doshas (weaknesses) and their antidotes are the basis of Ayurvedic medicine in India.

3.3.2 In astronomy

• The Saros number of the solar eclipse series which began on April 24, 2814 BC and ended on June 1, 1534 BC. The duration of Saros series 3 was 1280.14 years, and it contained 72 solar eclipses.

• The Saros number of the lunar eclipse series which began on December 30, 2567 BC and ended on March 21, 1214 BC. The duration of Saros se- ries 3 was 1352.26 years, and it contained 76 lunar eclipses. The Shield of the Trinity is a diagram of the Christian doctrine of the Trinity 3.3.3 In pseudoscience

• Three is the symbolic representation for Mu (lost 3.5.1 In Christianity continent), Augustus Le Plongeon's and James Churchward's lost continent.*[11] • The threefold office of Christ is a Christian doctrine that Christ performs the functions of prophet, priest, and king.

3.4 In philosophy • The ministry of lasted approximately three years (27-30 AD). Main article: Trichotomy (philosophy) • During the Agony in the Garden, Christ asked three times for the chalice to be taken from his lips. • Philosophers such as Aquinas, Kant, Hegel, and • Jesus rose from the dead on the third day after his C. S. Peirce have made threefold divisions, or death (Sunday April 9, 30 AD). trichotomies, which have been important in their work. • The devil tempted Jesus three times.

• Hegel's dialectic of Thesis + Antithesis = Synthesis • Saint Peter thrice denied Jesus and thrice affirmed creates three-ness from two-ness. his in Jesus • The Magi - Zoroaster priests as- tronomers/astrologers from Persia - gave Jesus 3.5 In religion three gifts. • See also: Triple There are three Synoptic Gospels and three epistles of John.

Many contain triple or concepts of • Paul the Apostle went blind for three days after his trinity, including: conversion to Christianity. 3.5. IN RELIGION 15

3.5.2 In Judaism 3.5.4 In Shinto

• Noah had three sons: Ham, Shem and Japheth • The Regalia of Japan of the sword, mirror, • The Three Patriarchs: Abraham, Isaac and Jacob and jewel. • The prophet Balaam beat his donkey three times. • The prophet Jonah spent three days and nights in the 3.5.5 In Taoism belly of a large fish • The Three Treasures (Chinese: 三寶; : sān- • Three divisions of the Written Torah: Torah (Five bǎo; Wade–Giles: san-pao), the basic virtues in Book of Moses), Nevi'im (Prophets), Ketuvim Taoism. (Writings)*[12] • Three divisions of the Jewish people: Kohen, Levite, Yisrael 3.5.6 In Hinduism • Three daily : Shacharit, Mincha, Maariv • Three Shabbat meals • Shabbat ends when three stars are visible in the night *[13] • Three Pilgrimage Festivals: Passover, Shavuot, Sukkot • Three matzos on the Passover Seder table*[14] • The Three Weeks, a period of mourning bridging the fast days of Seventeenth of Tammuz and Tisha 'Av • Three cardinal sins for which a Jew must die rather than transgress: idolatry, murder, sexual immoral- ity*[15] • Upsherin, a Jewish boy's first haircut at age 3*[16] • A Beth din is composed of three members • Potential converts are traditionally turned away three The "Om" symbol, in is also written ओ३म् (ō̄m [õːːm]), where is (dirgha,"three times as long”) times to test their sincerity*[17] ३ दीघ셍 • In the Jewish mystical tradition of the Kabbalah, it is believed that the consists of three parts, with • The Trimurti: Brahma the Creator, Vishnu the Pre- “ ” the highest being neshamah ( breath ), the middle server, and Shiva the Destroyer. being ruach (“wind”or “spirit”) and the lowest being nefesh (“repose”).*[18] Sometimes the two • The three Gunas underlie action, in the Vedic system elements of Chayah“( ”or“animal”) and Yechi- of knowledge. dah (“unit”) are additionally mentioned. • In the Kabbalah, the Tree of Life (Hebrew: Etz • The Three Yogas in the context of monotheistic Hin- refers to a latter 3-pillar di- duism are three religious paths for the human spirit (עץ החיים ,ha-Chayim agrammatic representation of its central mystical to achieve union (yoga) with Ishvara, Supreme Be- symbol, known as the 10 Sephirot. ing, i.e. God. They are Karma Yoga, Bhakti Yoga and Jnana Yoga. 3.5.3 In Buddhism 3.5.7 In Zoroastrianism • The Triple Bodhi (ways to understand the end of birth) are Budhu, Pasebudhu, and Mahaarahath. • The three virtues of Humata, Hukhta and Huvarshta • The Three Jewels, the three things that Buddhists (Good Thoughts, Good Words and Good Deeds) are take refuge in. a basic tenet in Zoroastrianism. 16 CHAPTER 3. 3 (NUMBER)

3.5.8 In Norse mythology three people in it; it is professed that the person in the middle will die first. Three is a very significant number in Norse mythology, There is another that it is unlucky to take a along with its powers 9 and 27. third light, that is, to be the third person to light a cigarette from the same match or lighter. This superstition is some- • Prior to Ragnarök, there will be three hard winters times asserted to have originated among soldiers in the without an intervening summer, the Fimbulwinter. trenches of the First World War when a sniper might see • Odin endured three hardships upon the World Tree the first light, take aim on the second and fire on the third. in his quest for the runes: he hanged himself, The phrase "Third time's the charm" refers to the super- wounded himself with a spear, and suffered from stition that after two failures in any endeavor, a third at- hunger and thirst. tempt is more likely to succeed. This is also sometimes seen in reverse, as in “third man [to do something, pre- • Bor had three sons, Odin, Vili, and Vé. sumably forbidden] gets caught”. , especially bad luck, is often said to “come in 3.5.9 In other religions threes”.*[19]

• The Wiccan Rule of Three. • The Triple Goddess: Maiden, Mother, Crone; the 3.6 In sports three fates. • In association football in almost all leagues, and in • The sons of Cronus: Zeus, Poseidon, and Hades. the group phases of most international competitions, three competition points are awarded for a win. 3.5.10 In esoteric tradition • In Gaelic football, hurling and camogie, a “goal”, • The Theosophical Society has three conditions of with a scoring value of three, is awarded when the membership. attacking team legally sends the ball into the oppo- nent's goal. • Gurdjieff's Three Centers and the Law of Three. • In baseball, three is the number of strikes before the • Liber AL vel Legis, the central scripture of the re- batter is out and the number of outs per side per in- ligion of Thelema, consists of three chapters, cor- ning; in scorekeeping,“3”denotes the first baseman. responding to three divine narrators respectively: Nuit, Hadit and Ra-Hoor-Khuit. • In basketball:

• The Triple Greatness of Hermes Trismegistus is an • A shot made from behind the three-point arc is important theme in Hermeticism. worth three points (except in the 3x3 variant, in which it is worth two points). 3.5.11 As a lucky or unlucky number • A potential “three-point play”exists when a player is fouled while successfully completing Three (三, formal writing: 叁, pinyin sān, Cantonese: a two-point field goal, thus being awarded one saam1) is considered a good number in Chinese culture additional free throw attempt. because it sounds like the word“alive”(生 pinyin shēng, • On offense, the "three-second rule" states that 1 Cantonese: saang ), compared to four (四, pinyin: sì, an offensive player cannot remain in the op- 1 Cantonese: sei ), which sounds like the word “death” ponent's free-throw lane for more than three 2 (死 pinyin sǐ, Cantonese: sei ). seconds while his team is in possession of the Counting to three is common in situations where a group ball and the clock is running. of people wish to perform an action in synchrony: Now, • In the NBA only, the defensive three-second on the count of three, everybody pull! Assuming the violation, also known as “illegal defense”, counter is proceeding at a uniform rate, the first two states that a defensive player cannot remain in counts are necessary to establish the rate, and the count of his own free-throw lane for more than three “three”is predicted based on the timing of the“one”and seconds unless he is actively guarding an of- “two”before it. Three is likely used instead of some other fensive player. number because it requires the minimal amount counts • “ ” while setting a rate. The 3 position is the small forward. In East and Southeast Asia, there is a widespread super- • In gridiron football, a field goal is a scoring play in stition that considers it inauspicious to take a photo with which a kicks the football from behind the 3.8. REFERENCES 17

line of scrimmage through the goalposts on the op- [2] Bryan Bunch, The Kingdom of Infinite Number. New ponent's side of the field. This play, if successful, is York: W. H. Freeman & Company (2000): 39 worth three points. [3] “Sloane's A042978 : Stern primes”. The On-Line En- • A hat trick in sports is associated with succeeding at cyclopedia of Integer Sequences. OEIS Foundation. Re- anything three times in three consecutive attempts, trieved 2016-06-01. as well as when any player in ice hockey or soccer “ ” scores three goals in one game (whether or not in [4] Sloane's A003173 : Heegner numbers . The On- Line Encyclopedia of Integer Sequences. OEIS Founda- succession). In cricket, if a bowler takes three wick- tion. Retrieved 2016-06-01. ets in a row it is called a hat trick. [5] “Sloane's A001608 : Perrin sequence”. The On-Line En- • A"threepeat" is a term for winning three consecu- cyclopedia of Integer Sequences. OEIS Foundation. Re- tive championships. trieved 2016-06-01. • A triathlon consists of three events: swimming, bi- [6] Priya Hemenway (2005), Divine Proportion: Phi In Art, cycling, and running. Nature, and Science, Sterling Publishing Company Inc., pp. 53–54, ISBN 1-4027-3522-7 • A pin (professional wrestling) in professional wrestling is when one's shoulders are held the oppo- [7] Big Numbers. ISBN 1840464313. nent's shoulders against the mat for a count of three. [8] "Most stable shape- triange". Maths in the city. Retrieved • In the NASCAR Sprint Cup Series, 3 is the num- February 23, 2015. ber of a car owned by Richard Childress Racing. It was originally driven by team founder and owner [9] Eric John Holmyard. Alchemy. 1995. p.153 Richard Childress, but became most famous as the car driven by seven-time Cup Series champion Dale [10] Walter J. Friedlander. The golden wand of medicine: a Earnhardt, first in 1981 and then from 1984 until history of the caduceus symbol in medicine. 1992. p.76- his death at the 2001 Daytona 500. Childress did 77 not use the number again in the Cup Series until [11] Churchward, James (1931).“The Lost Continent of Mu - 2014; since then, his oldest grandson Austin Dillon Symbols, Vignettes, Tableaux and Diagrams”. Biblioteca has driven the #3 car. Pleyades. Retrieved 2016-03-15. • In , 3 is used by Australian driver, [12] Marcus, Rabbi Yossi (2015). “Why are many things in Daniel Ricciardo. Judaism done three times?". Ask Moses. Retrieved 16 March 2015. • In many sports a competitor or team is said to win a Triple Crown if they win three particularly presti- [13] “Shabbat”. Judaism 101. 2011. Retrieved 16 March gious competitions. 2015.

• In association football a team that wins three tro- [14] Kitov, Eliyahu (2015).“The Three Matzot”. Chabad.org. phies in a season is said to have won a treble. Retrieved 16 March 2015.

• In bowling, three strikes bowled consecutively is [15] Kaplan, Rabbi Aryeh (28 August 2004). “Judaism and known as a “turkey”. Martyrdom”. Aish.com. Retrieved 16 March 2015.

[16] “The Basics of the Upsherin: A Boy's First Haircut”. 3.7 See also Chabad.org. 2015. Retrieved 16 March 2015. [17] “The Conversion Process”. Center for Conversion to • Cube (algebra) – (3 superscript) Judaism. Retrieved 16 March 2015.

• Third [18] Kaplan, Aryeh. "The Soul". Aish. From The Handbook of Jewish Thought (Vol. 2, Maznaim Publishing. Reprinted • Triad with permission.) September 4, 2004. Retrieved Febru- ary 24, 2015. 3.8 References [19] See "bad" in the Oxford Dictionary of Phrase and Fable, 2006, via Encyclopedia.com.

[1] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David • Wells, D. The Penguin Dictionary of Curious and In- Bellos et al. London: The Harvill Press (1998): 393, Fig. teresting Numbers London: Penguin Group. (1987): 24.63 46–48 18 CHAPTER 3. 3 (NUMBER)

3.9 External links

• Tricyclopedic Book of Threes by Michael Eck

• Threes in Human Anatomy by Dr. John A. McNulty • Grime, James. “3 is everywhere”. Numberphile. Brady Haran.

• The Number 3 • The Positive Integer 3

• Prime curiosities: 3 Chapter 4

4 (number)

“Four”redirects here. For other uses, see 4 (disam- sometimes also called a tetragon.A circle divided by 4 biguation). makes right angles and four quadrants. Because of it, four (4) is the base number of plane (mathematics). Four 4 (four; /ˈfɔər/) is a number, numeral, and glyph. It is the cardinal directions, four seasons, system, and natural number following 3 and preceding 5. Four is the system are based on four. only cardinal numeral in the that has the A solid figure with four faces as well as four vertices is same number of letters as its number value. a tetrahedron, and 4 is the smallest possible number of faces (as well as vertices) of a polyhedron. The regular tetrahedron is the simplest .A tetrahedron, which can also be called a 3-, has four triangular 4.1 In mathematics faces and four vertices. It is the only self-dual regular polyhedron. Four is the smallest composite number, its proper divisors Four-dimensional space is the highest-dimensional space being 1 and 2. Four is also a . featuring more than three convex regular figures: The next highly composite number is 6. Four is the second , the second centered • Two-dimensional: infinitely many convex regular triangular number. polygons. 4 is the smallest squared prime (p2) and the only even • Three-dimensional: five convex regular polyhedra number in this form. It has an aliquot sum of 3 which is (the five Platonic Solids). itself prime. The of 4 has 4 members • (4, 3, 1, 0) and is accordingly the first member of the 3- Four-dimensional: six convex regular polychora. aliquot tree. • Five-dimensional and every higher-dimensional: A number is a of 4 if its last two digits are a mul- three regular convex polytopes (regular , tiple of 4. For example, 1092 is a multiple of 4 because hypercubes, -polytopes). 92 = 4 × 23. Four-dimensional differential manifolds have some Only one number has an aliquot sum of 4 and that is unique properties. There is only one differential struc- squared prime 9. ture on ℝ*n except when n = 4, in which case there are Four is the smallest composite number that is equal to the uncountably many. sum of its prime factors. (As a consequence of this, it is * The smallest non-cyclic group has four elements; it is the the smallest ). [1] However, it is the only Klein four-group. Four is also the order of the smallest composite number n for which (n − 1)! ≡ 0 (mod n) is non-trivial groups that are not simple. false. Four is the only natural integer n for which the (non triv- It is also a Motzkin number.*[2] ial) alternating group An is not simple. In bases 6 and 12, 4 is a 1-. Four is the maximum number of dimensions of a In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pat- real associative division algebra (the quaternions), by a tern in Knuth's up- notation, 2 ↑↑ 2 = 2 ↑↑↑ 2 = 4, theorem of Ferdinand Georg Frobenius. and so on, for any number of up arrows. (That is, 2 [n] The four-color theorem states that a planar graph (or, 2 = 4 for every positive integer n, where a [n] b is the equivalently, a flat map of two-dimensional regions such hyperoperation.) as countries) can be colored using four colors, so that A four-sided plane figure is a quadrilateral (quadran- adjacent vertices (or regions) are always different col- gle) which include kites, rhombi, rectangles and squares, ors.*[3] Three colors are not, in general, sufficient to

19 20 CHAPTER 4. 4 (NUMBER)

guarantee this. The largest planar complete graph has four vertices. Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares. Four is the first positive non-Fibonacci number. Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2 − z2. Four is an all-Harshad number and a semi-meandric num- ber. Four is the highest degree general polynomial equation for which there is a solution in radicals.

4.2 List of basic calculations

4.3 Origins

Representing 1, 2 and 3 in as many lines as the number represented worked well. The Brahmin Indians simplified 4 by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the numeral, and the Kshatrapa and Pallava evolved the numeral to a point where speed of writing 4.4 In religion was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was Buddhism made in one stroke by connecting the “western”end to the“northern”end; the“eastern”end was finished off • Four Noble Truths with a curve. The Europeans dropped the finishing curve and gradually made the numeral less , ending up • Dukkha – The Noble Truth of Suffering * with a glyph very close to the original Brahmin cross. [4] • Samudaya – The Noble Truth of the Cause of While the shape of the 4 character has an ascender in Suffering most modern typefaces, in typefaces with text figures the • Nirodha – The Noble Truth of the Cessation character usually has a descender, as, for example, in of Suffering • . Magga – The Noble Truth of the Path leading to the Cessation of Suffering On the seven-segment displays of pocket and digital watches, as well as certain optical character recog- • Four sights – observations which affected Prince nition fonts, 4 is seen with an open top. Siddhartha deeply and made him realize the suffer- Television stations that operate on channel 4 have occa- ings of all beings, and compelled him to begin his — sionally made use of another variation of the “open 4” spiritual journey an old man, a sick man, a dead , with the open portion being on the side, rather than the man, and an ascetic top. This version resembles the Canadian Aboriginal syl- • Four Great Elements – earth, , fire, and wind labics letter . The magnetic ink character recognition “CMC-7”font also uses this variety of “4”. • Four Heavenly Kings 4.4. IN RELIGION 21

• Four Foundations of Mindfulness – contemplation • The four primary castes or strata of society: of the body, contemplation of feelings, contempla- Brahmana (priest/teacher), Kshatriya (war- tion of mind, contemplation of mental objects rior/politician), Vaishya (landowner/entrepreneur) and Shudra (servant/manual labourer). • Four Right Exertions Islam • Four Bases of Power • • Four jhānas Eid al-Adha lasts for four days, from the 10th to the 14th of Dhul Hijja. • Four arūpajhānas • There are four Rashidun or Rightly Guided Caliphs: • Four Divine Abidings – loving-kindness, Abu Bakr, Umar ibn al-Khattab, Uthman ibn Affan compassion, sympathetic joy, and equanimity and Ali ibn Abi Talib. • The Four Arch Angels in Islam are: Jibraeel • Four stages of enlightenment – stream-enterer, (Gabriel), Mikaeel (Michael), Izraeel (Azrael), and once-returner, non-returner, and arahant Israfil (Raphael) • Four main pilgrimage sites – Lumbini, Bodh Gaya, • There are four Sacred Months in Islam: Muharram, Sarnath, and Kusinara Rajab, Dhu al-Qi'dah and Dhu al-Hijjah. • Judeo-Christian symbolism There are four Sunni schools of fiqh: Hanafi, Shafi`i, Maliki and Hanbali.

• The Tetragrammaton is the four-letter name of God. • There are four major sunni Imams: Abū Ḥanīfa, Muhammad ibn Idris ash-Shafi`i, Malik ibn Anas • The four Matriarchs (foremothers) of Judaism are and Ahmad ibn Hanbal. Sarah, Rebekah, Leah, and Rachel. • There are four books in Islam: Torah, Zaboor, In- • The Four Species (lulav, hadass, aravah and etrog) jeel, Quran. are taken as one of the mitzvot on the Jewish holiday • Waiting for four months is ordained for those who of Sukkot.(Judaism) take an oath for abstention from their wives. • The Four Cups of Wine to drink on the Jewish hol- • The waiting period of the woman whose husband iday of Passover.(Judaism) dies, is four months and ten days. • The Four Questions to be asked on the Jewish holi- • When Abraham said:“My Lord, show me how You day of Passover.(Judaism) give life to the dead,”Allah said: “Why! Do you have no faith?”Abraham replied: “Yes, but in or- • The Four Sons to be dealt with on the Jewish holiday der that my heart be at rest.”He said: “Then take of Passover.(Judaism) four birds, and tame them to yourself, then put a part of them on every hill, and summon them; they will • The Four Expressions of Redemption to be said on come to you flying. [Al-Baqara 2:260] the Jewish holiday of Passover.(Judaism) • The respite of four months was granted to give time • The four Gospels: Matthew, Mark, Luke, and John. to the mushriks in Surah At-Tawba so that they (Christianity) should consider their position carefully and decide whether to make preparation for war or to emigrate • The Four Horsemen of the Apocalypse ride in the from the country or to accept Islam. Book of Revelation.(Christianity) • Those who accuse honourable women (of un- Hinduism chastity) but do not produce four witnesses, flog them with eighty lashes, and do not admit their tes- timony ever after. They are indeed transgressors. • There are 4 Vedas: Rigveda, Samaveda, Yajurveda [An-Noor 24:4] and Atharvaveda.

• In Puruṣārtha, there are four aims of human life: Other Dharma, Artha, Kāma, Moksha. • In a more general sense, numerous mythological and • The four stages of life Brahmacharya (student life), cosmogonical systems consider Four corners of the Grihastha (household life), Vanaprastha (retired world as essentially corresponding to the four points life) and Sannyasa (renunciation). of the compass. 22 CHAPTER 4. 4 (NUMBER)

• Four is the sacred number of the Zia, an indigenous 4.7.2 In biology tribe located in the U.S. state of New Mexico. • Four is the number of nucleobase types in DNA and • The Chinese, Vietnamese, the Korean and the RNA – adenine, guanine, cytosine, thymine (uracil Japanese are superstitious about the number four be- in RNA). cause it is a homonym for“death”in their languages. • Many chordates have four feet, legs or leglike ap- pendages (Tetrapods).

4.5 In politics • The mammalian heart consists of four chambers. • • Four Freedoms Many mammals (Carnivora, Ungulata) use four fin- gers for movement.

• The fourth finger of a human (on the left hand 4.6 In computing – ring-finger) is moved when the little finger moves.

• All insects with wings except flies have four wings. • 0x04 is the ASCII code of the character End of Transmission, which is abbreviated to EOT. • Insects of the superorder Endopterygota, also known as Holometabola, such as butterflies, ants, bees, • UTF-32 uses exactly 4 per . beetles, fleas, flies, moths, and wasps, undergo holometabolism – complete metamorphism in four stages – from (1) embryo (ovum, egg), to (2) larva 4.7 In science (such as grub, caterpillar), then (3) pupa (such as the chrysalis), and finally (4) the imago. • A tetramer is a thing formed out of four sub-units. • In the common ABO blood group system, there are 4 blood types (A, B, , AB).

4.7.1 In astronomy • Humans have four canines and four wisdom teeth.

• Four terrestrial (or rocky) planets in the Solar sys- • The cow's stomach is divided in four digestive com- tem: Mercury, Venus, Earth, and Mars. partments: reticulum, rumen, omasum and aboma- sum. • Four giant gas/ice planets in the Solar system: Jupiter, Saturn, Uranus, and Neptune. 4.7.3 In chemistry • Four of Jupiter's moons (the Galilean moons) are readily visible from Earth. • Valency of carbon (that is basis of life on the Earth) • Messier object M4, a magnitude 7.5 globular cluster is four. Also because of its tetrahedral crystal bond in the constellation Scorpius. structure, diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring • The New General Catalogue object NGC 4, a faint material. It is also the valence of silicon, whose galaxy in the constellation Pisces compounds form the majority of the mass of the Earth's crust. • The Roman numeral IV stands for subgiant in the Yerkes spectral classification scheme. • The atomic number of beryllium • • The Roman numeral IV (usually) stands for the There are four basic states of matter: solid, liquid, fourth-discovered satellite of a planet or minor gas, and plasma. planet (e.g. Jupiter IV)

• The Saros number of the solar eclipse series which 4.7.4 In physics began on May 6, 2731 BC and ended on June 13, 1451 BC. The duration of Saros series 4 was • Special relativity and general relativity treat nature 1280.14 years, and it contained 72 solar eclipses. as four-dimensional: 3D regular space and one- dimensional time are treated together and called • The Saros number of the lunar eclipse series which . Also, any event E has a light cone com- began on October 6, 2646 BC and ended on April posed of four zones of possible communication and 2, 1131 BC. The duration of Saros series 4 was cause and effect (outside the light cone is strictly in- 1514.53 years, and it contained 78 lunar eclipses. communicado). 4.8. IN LOGIC AND PHILOSOPHY 23

• An alpha particle (helium nucleus, also called a he- • Immanuel Kant expounded a table of judgments in- lion) consists of four hadrons. volving four three-way alternatives, in regard to (1) Quantity, (2) Quality, (3) , (4) Modality, • There are four fundamental forces and, based thereupon, a table of four categories, (electromagnetism, gravitation, the weak nuclear named by the terms just listed, and each with three force, and the strong nuclear force). subcategories.

• In statistical mechanics, the four functions inequal- • Arthur Schopenhauer's doctoral thesis was On the ity is an inequality for four functions on a finite Fourfold Root of the Principle of Sufficient Reason. distributive lattice. • Franz Brentano held that any major philosophical period has four phases: (1) Creative and rapidly pro- gressing with scientific interest and results; then de- 4.8 In logic and philosophy clining through the remaining phases, (2) practical, (3) increasingly skeptical, and (4) literary, mysti- • The symbolic meanings of the number four are cal, and scientifically worthless – until philosophy linked to those of the cross and the square. “Al- is renewed through a new period's first phase. (See most from prehistoric times, the number four was Brentano's essay “The Four Phases of Philosophy employed to signify what was solid, what could be and Its Current State”1895, tr. by Mezei and Smith touched and felt. Its relationship to the cross (four 1998.) points) made it an outstanding symbol of wholeness • C. S. Peirce, usually a trichotomist, discussed four and universality, a symbol which drew all to itself”. methods for overcoming troublesome uncertainties Where lines of latitude and longitude intersect, they and achieving secure beliefs: (1) the method of divide the earth into four proportions. Throughout tenacity (policy of sticking to initial belief), (2) the world kings and chieftains have been called“lord the method of authority, (3) the method of con- of the four suns”..."lord of the four quarters of the gruity (following a fashionable paradigm), and (4) earth”...*[5] by which is understood to the extent of the fallibilistic, self-correcting method of science their powers both territorially and in terms of total (see "The Fixation of Belief", 1877); and four - control of their subjects' doings. riers to inquiry, barriers refused by the fallibilist: (1) assertion of absolute certainty; (2) maintaining • The Square of Opposition, in both its Aristotelian that something is absolutely unknowable; (3) main- version and its Boolean version, consists of four taining that something is absolutely inexplicable be- forms: A (“All S is R"), I (“Some S is R"), E ( cause absolutely basic or ultimate; (4) holding that “No S is R"), and O (“Some S is not R"). perfect exactitude is possible, especially such as to • In regard to whether two given propositions can have quite preclude unusual and anomalous phenomena the same truth value, there are four separate logical (see "F.R.L." [First Rule of Logic], 1899). possibilities: the propositions are subalterns (possi- • Paul Weiss built a system involving four modes of bly both are true, and possibly both are false); sub- being: Actualities (substances in the sense of sub- contraries (it is possible that both are true, but not stantial, spatio-temporally finite beings), Ideality or that both are false); contraries (it is possible that both Possibility (pure normative form), Existence (the are false, but not that both are true); or contradicto- dynamic field), and God (unity). (See Weiss's Modes ries (it is not possible that both are true, and it is not of Being, 1958). possible that both are false). • Karl Popper outlined a tetradic schema to describe • Aristotle held that there are basically four causes in the growth of theories and, via generalization, also nature: the efficient cause, the matter, the end, and the emergence of new behaviors and living organ- the form. isms: (1) problem, (2) tentative theory, (3) (at- tempted) error-elimination (especially by way of • The Stoics held with four basic categories, all viewed critical discussion), and (4) new problem(s). (See as bodies (substantial and insubstantial): (1) sub- Popper's Objective Knowledge, 1972, revised 1979.) stance in the sense of substrate, primary formless matter; (2) quality, matter's organization to differen- • John Boyd (military strategist) made his key con- tiate and individualize something, and coming down cept the decision cycle or OODA loop, consisting to a physical ingredient such as pneuma, breath; (3) of four stages: (1) observation (data intake through somehow holding (or disposed), as in a posture, state, the senses), (2) orientation (analysis and synthesis of shape, size, action, and (4) somehow holding (or dis- data), (3) decision, and (4) action. Boyd held that his posed) toward something, as in relative location, fa- decision cycle has philosophical generality, though milial relation, and so forth. for strategists the point remains that, through swift 24 CHAPTER 4. 4 (NUMBER)

decisions, one can disrupt an opponent's decision cy- • The four color process (CMYK) is used for printing. cle. • Wide use of rectangles (with four angles and four • Richard McKeon outlined four classes (each with sides) because they have effective form and capabil- four subclasses) of modes of philosophical inquiry: ity for close adjacency to each other (houses, rooms, (1) Modes of Being (Being); (2) Modes of Thought tables, bricks, sheets of paper, screens, film frames). (That which is); (3) Modes of Fact (Existence); • In the Rich Text Format specification, language code (4) Modes of Simplicity (Experience) – and, cor- 4 is for the . Codes for regional responding to them, four classes (each with four variants of Chinese are congruent to 4 mod 256. subclasses) of philosophical semantics: Principles, Methods, Interpretations, and Selections. (See • Credit card machines have four-twelve function McKeon's“Philosophic Semantics and Philosophic keys. Inquiry”in Freedom and History and Other Essays, • 1989.) On most phones, the 4 key is associated with the letters G, H, and I, but on the BlackBerry cellular • Jonathan Lowe (E.J. Lowe) argues in The Four- phone, it is the key for D and F. Category Ontology, 2006, for four categories: kinds • “ ” (substantial universals), attributes (relational uni- On many computer keyboards, the 4 key may also versals and property-universals), objects (substantial be used to type the dollar sign ($) if the shift key is particulars), and modes (relational particulars and held down. property-particulars, also known as "tropes"). (See • It is the number of in a , equivalent to half Lowe's “Recent Advances in Metaphysics,”2001, a Eprint) • In internet slang, “4”can replace the word “for” • Four opposed camps of the morality and nature of (as “four”and “for”are pronounced similarly). evil: moral absolutism, amoralism, moral relativism, For example, typing “4u”instead of “for you”. and moral . • In Leetspeak,“4”may be used to replace the letter “A”. 4.9 In technology • The TCP/IP stack consists of four layers.

4.10 In transport

4 as a resin identification code, used in recycling.

• The resin identification code used in recycling to identify low-density polyethylene. • • Most furniture has four legs – tables, chairs, etc. Many internal combustion engines are called four- stroke engines because they complete one thermo- • Four horses (quadriga) is the maximal number of dynamic cycle in four distinct steps: Intake, com- horses in one row for carriage. pression, power, and exhaust. 4.11. IN SPORTS 25

• Most vehicles, including motor vehicles, and partic- • In baseball scoring, number 4 is assigned to the ularly cars/automobiles and light commercial vehi- second baseman. Also, four is the most number of cles have four road wheels. runs that can be scored on any single at bat, whereby all three baserunners and the batter score (the most • "Quattro", meaning four in the , is common being via a grand slam). used by Audi as a trademark to indicate that four- wheel drive (4WD) technologies are used on Audi- • In gridiron football codes, branded cars. The word“Quattro”was initially used • by Audi in 1980 in its original 4WD coupé, the Audi In American football, the offensive team has Quattro. Audi also has a privately held subsidiary four downs (plays) to advance the ball 10 yards company called quattro GmbH. otherwise loss of possession occurs. • four points are awarded in a handful of leagues • List of highways numbered 4 for rarely attempted types of field goals; an example is in six-man football. Because of the difficulties of getting a successful kick due 4.11 In sports to the few players on the field, a field goal is worth four points. Also, in Arena Football, a • In cricket, a four is a specific type of scoring event, successful dropkicked field goal attempt scores whereby the ball crosses the boundary after touch- four points. ing the ground at least one time, scoring four runs. • the “four hole”in offense terminology is the Taking four in four consecutive balls is typ- space between the right guard and the right ically referred to as a double hat trick (two consec- tackle on the offensive line utive, overlapping hat tricks). • the“four back”is an extra running back (out- • In basketball, the number four is used to designate side the fullback and halfback, often referred the Power Forward position, often referred to as to as an H-back) in the backfield; e.g. a play “The four spot”. call for a“44 lead”indicates the H-back will follow the fullback into the hole between the • Also in basketball, there is a term called the four right guard and the right tackle. point play, when a fouled player attempts a 3-point field goal and makes it, and makes the resulting free • In rugby union: throw. • One of the two starting locks wears the jersey • The number of quarters in an NBA or NFL game. number 4. • The basketball term Final Four refers to: • At several points in the history of the sport, 4 points were awarded for the following scores: • in the United States, the last four teams re- • A goal from mark from 1891 to 1904 (3 maining in the NCAA playoff tournament, points after that time; abolished in 1977). each of which is the winner of its respective • region (for collegiate hockey the term Frozen A drop goal from 1891 to 1947 (now 3 Four is used). points). • • In Europe, the last four teams remaining in A try from 1971 to 1991 (now 5 points). many national and transnational club compe- • In most league competitions, as well as the titions, most notably the Euroleague. pool phases of the Rugby and many other worldwide championships, 4 competi- • In roller skating, roller skates typically have four tion points are awarded for a win (see rugby wheels. When the wheels are arranged in a sin- union bonus points system). gle line, they are called “in-line”skates. When the wheels are arranged in a four-point configuration • In rugby league: (two in front, two in back), they are called “quad” • skates. A try is worth 4 points. • In most competitions (though not the Euro- • In rowing, a four refers to a boat for four rowers, pean Super League, which uses static squad with or without coxswain. In rowing nomenclature numbering), the starting left centre wears jer- 4− represents a coxless four and 4+ represents a sey number 4. coxed four. • The jersey number 4 has been retired by several • There are four bases in the game of baseball: first North American sports teams in honor of past play- base, second base, third base, and home plate; to ing greats or other key figures: score a run, an offensive player must complete, in the sequence shown, a circuit of those four bases. • In Major League Baseball: 26 CHAPTER 4. 4 (NUMBER)

• The Baltimore Orioles, for Hall of Fame “Honoured Numbers”, which remain in manager Earl Weaver. circulation for future players. The num- • The Boston Red Sox, for Hall of Famer ber 4 is currently honoured for Hall of Joe Cronin. Famer Red Kelly. • The Chicago White Sox, for Hall of • In the NFL: Famer Luke Appling. • The New York Giants, for Hall of Famer • The Los Angeles Dodgers, for Hall of Tuffy Leemans. Famer Duke Snider. • The Green Bay Packers, for Hall of • The Milwaukee Brewers, for Hall of Famer Brett Favre. Famer Paul Molitor. • The Dallas Cowboys, for Dak Prescott. • The New York Giants, for Hall of Famer • The Oakland Raiders, for Derek Carr. Mel Ott; the team has continued to honor • the number in its current home of San In F1: Francisco. • Max Chilton uses the number 4 on his car. • The New York Yankees, for Hall of • In Professional boxing: Famer Lou Gehrig. Notably, he was • Bouts featuring novices are usually lim- the first player in major North American ited to four rounds each. sports to be so honored by a U.S.-based team (he was preceded in this honor by Ace Bailey, whose #6 was retired by the Toronto Maple Leafs five years earlier). 4.12 In other fields • The Pittsburgh Pirates, for Hall of Famer Ralph Kiner. • In the NBA: • The Chicago Bulls, for Jerry Sloan, who later served as the team's head coach and went on to a Hall of Fame career in that role (mainly with the Utah Jazz). • The Detroit Pistons, for Hall of Fame player Joe Dumars, who later served as team president. • The Milwaukee Bucks, for Sidney Mon- crief. • The New York Nets, for Wendell Ladner, who died in a 1975 plane crash. • The Sacramento Kings, for Chris Web- ber. • The Utah Jazz, for Hall of Fame player Adrian Dantley. • In the NHL: International maritime signal flag for 4 • The Boston Bruins, for Hall of Famer Bobby Orr. See also 4 (disambiguation). • The Montreal Canadiens, for Hall of Famer Jean Beliveau. • The phrase "four-letter word" is used to describe • The New Jersey Devils, for Hall of Famer many swear words in the English language. Scott Stevens. • Four is the only number whose name in English has • The Philadelphia Flyers, for Barry Ash- the same number of letters as its value. bee. • The Toronto Maple Leafs have a pol- • Four (四, formal writing: 肆, pinyin sì) is con- icy of not retiring numbers unless the sidered an unlucky number in Chinese, Korean, player honoured either died or suffered a Vietnamese and Japanese cultures because it sounds career-ending incident while a member of like the word“death”(死, pinyin sǐ). To avoid com- the team. Other players whose numbers plaints from people with , many num- would otherwise be retired instead have bered product lines skip the “four": e.g. their numbers enshrined by the team as cell phones (there is no series beginning with a 4), 4.13. IN MUSIC 27

Palm PDAs, etc. Some buildings skip floor 4 or re- 4.13 In music place the number with the letter “F”, particularly in heavily Asian areas. See tetraphobia and Numbers • In written music, common time is constructed of in Chinese culture. four beats per measure and a quarter-note receives • The number of characters in a canonical four- one beat. character idiom. • In popular or modern music, the most common time • In the NATO phonetic , the digit 4 is called signature is also founded on four beats, i.e., 4/4 hav- “fower”. ing four quarter note beats. • In Astrology, Cancer is the 4th astrological sign of • The common major scale is built on two sets of four the Zodiac. notes (e.g., CDEF, GABC), where the first and last notes create an octave interval (a pair-of-four rela- tionship).

• The interval of a perfect fourth is a foundational ele- ment of many genres of music, represented in music theory as the tonic and subdominant relationship. Four is also embodied within the circle of fifths (also known as circle of fourths), which reveals the inter- val of four in more active harmonic contexts.

• The typical number of movements in a symphony.

• The number of completed, numbered symphonies by Johannes Brahms.

• The number of strings on a violin, a viola, a , double bass, a cuatro and a ukulele, and the number of string pairs on a .

•“Four calling birds”is the gift on the fourth day of Christmas in the carol "The Twelve Days of Christ- mas".

• Four, British rock band Bloc Party's fourth full- length .

• Four, British-Irish boy band One Direction's fourth full-length album.

• 4, American artist Beyoncé's fourth album. playing cards for 4 4.14 Groups of four • In Tarot, The Emperor is the fourth trump or Major Arcana card. • Four rules: addition, , multiplication, division. • In Nintendo's Mario Kart series, four is the number of courses that each cup holds. • Greek classical elements (fire, air, water, earth).

• In Tetris, a game named for the Greek word for 4, • Four seasons: spring, summer, autumn, winter. every shape in the game is formed of 4 blocks each. • A leap year occurs every four years. • 4 represents the number of Justices on the Supreme Court of the United States necessary to grant a writ • Four is the minimal number of contemporary of certiorari (i.e., agree to hear a case; it is one less continents: Americas, Eurafrasia, Sahul, Antarctica. than the number necessary to render a majority de- • cision). Approximately four weeks (4 times 7 days) to a lu- nar month (synodic month = 29.53 days). Thus the • Number Four is a character in the book series Lorien number four is universally an integral part of prim- Legacies itive sacred calendars. 28 CHAPTER 4. 4 (NUMBER)

• Four cardinal directions: north, south, east, west. • There are four limbs on the human body.

• Four Temperaments: sanguine, choleric, • Four Houses of Hogwarts in the Harry Potter series: melancholic, phlegmatic. Gryffindor, Hufflepuff, Ravenclaw, Slytherin.

• Four Humors: blood, yellow bile, black bile, • Four known continents of the world in the A Song phlegm. of Ice and Fire series: Westeros, Essos, Sothoryos, Ulthos. • Four Great Ancient Capitals of China. • Four corner method. • Cardinal principles. 4.15 References

• Four cardinal virtues: justice, prudence, temper- [1] “Sloane's A006753 : Smith numbers”. The On-Line En- ance, fortitude. cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01. • Four suits of playing cards: hearts, diamonds, clubs, spades. [2] “Sloane's A001006 : Motzkin numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- • Four nations of the United Kingdom: England, tion. Retrieved 2016-06-01. Wales, , Northern Ireland. [3] Bryan Bunch, The Kingdom of Infinite Number. New • Four provinces of Ireland: Munster, Ulster, York: W. H. Freeman & Company (2000): 48 Leinster, Connacht. [4] Georges Ifrah, The Universal History of Numbers: From • Four Noble Truths in the Buddhist religion. Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. • Four estates: politics, administration, judiciary, 24.64 journalism. Especially in the expression "Fourth - tate", which means journalism. [5] Chevalier, Jean and Gheerbrant, Alain (1994), The Dic- tionary of Symbols. The quote beginning “Almost from • Four Corners is the only location in the United States prehistoric times...”is on p. 402. where four states come together at a single point: Colorado, Utah, New Mexico, and Arizona. • Wells, D. The Penguin Dictionary of Curious and In- teresting Numbers London: Penguin Group. (1987): • The Fantastic Four: Mr. Fantastic, The Invisible 55–58 Woman, The Human , The Thing. • The Interesting Four: Seiko, The Human Stapler, Weather Woman, and Mr. Wonderful. Parody su- 4.16 External links perhero team featured on NBC’s Saturday Night Live. • Marijn.Org on Why is everything four? • “ The Beatles were also known as the Fab Four": • A few thoughts on the number four, by Penelope John Lennon, Ringo Starr, George Harrison, Paul Merritt at samuel-beckett.net McCartney. • The Number 4 • Gang of Four is a British post-punk rock band formed in the late 1970s. • The Positive Integer 4

• Typically there are four string players in a classical • Prime curiosities: 4 string quartet, usually two violinists, a violist, and a cellist. • Four rivers in the Garden of Eden (Genesis 2:10– 14): Pishon (perhaps the Jaxartes or Syr Darya), Gihon (perhaps the Oxus or Amu Darya), Hiddekel (Tigris), and P'rat (Euphrates). • There are also four years in a single Olympiad (du- ration between the Olympic Games). Many major international sports competitions follow this cycle, among them the FIFA World Cup and its women's version, the FIBA World Championships for men and women, and the . Chapter 5

5 (number)

This article is about the number five. For the year, see Five is the second Sierpinski number of the first kind, and AD 5. For other uses of 5, see 5 (disambiguation) and can be written as S2=(22)+1 The Five (disambiguation). While polynomial equations of degree 4 and below can be solved with radicals, equations of degree 5 and higher 5 (five /ˈfaɪv/) is a number, numeral, and glyph. It is the cannot generally be so solved. This is the Abel–Ruffini natural number following four and preceding six. theorem. This is related to the fact that the symmetric group Sn is a solvable group for n ≤ 4 and not solvable for n ≥ 5. While all graphs with 4 or fewer vertices are planar, there 5.1 In mathematics exists a graph with 5 vertices which is not planar: K5, the complete graph with 5 vertices. Five is the third prime number. Because it can be written Five is also the number of Platonic solids.*[3] as 2*21 + 1, five is classified as a Fermat prime; there- A polygon with five sides is a pentagon. Figurate num- fore a regular polygon with 5 sides (a regular pentagon) is bers representing pentagons (including five) are called constructible with compass and unmarked straightedge. pentagonal numbers. Five is also a square pyramidal 5 is the third Sophie Germain prime, the first safe prime, number. the third , and the third Mersenne prime exponent. Five is the first and the third Five is the only prime number to end in the digit 5, be- factorial prime, also an alternating factorial. Five is the cause all other numbers written with a 5 in the ones-place first .*[1] It is an Eisenstein prime with no under the decimal system are multiples of five. As a con- imaginary part and real part of the form 3n − 1. It is sequence of this, 5 is in base 10 a 1-automorphic number. also the only number that is part of more than one pair of Vulgar fractions with 5 or 2 in the denominator do not * twin primes. Five is a . [2] yield infinite decimal expansions, unlike expansions with Five is conjectured to be the only odd untouchable num- all other prime denominators, because they are prime fac- ber and if this is the case then five will be the only odd tors of ten, the base. When written in the decimal system, prime number that is not the base of an aliquot tree. all multiples of 5 will end in either 5 or 0. Five is also the only prime that is the sum of two consec- There are five Exceptional Lie groups. utive primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 5.1.1 List of basic calculations 3. 5 is also a Pell number and a , appear- ing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... 5.2 Evolution of the glyph ( A030452 lists Markov numbers that appear in solu- tions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers. 5 is the length of the hypotenuse of the smallest integer- sided . In bases 10 and 20, 5 is a 1-automorphic number. The evolution of the modern Western glyph for the nu- 5 and 6 form a Ruth–Aaron pair under either definition. meral five cannot be traced back to the Indian system as There are five solutions to Znám's problem of length 6. for the numbers 1 to 4. The Kushana and Gupta empires

29 30 CHAPTER 5. 5 (NUMBER) in what is now India had among themselves several dif- • The Saros number of the solar eclipse series which ferent glyphs which bear no resemblance to the modern began on April 4, 2720 BC and ended on May 24, glyph. The Nagari and Punjabi took these glyphs and all 1422 BC. The duration of Saros series 5 was 1298.1 came up with glyphs that are similar to a lowercase “h” years, and it contained 73 solar eclipses. rotated 180°. The Ghubar Arabs transformed the glyph in several different ways, producing glyphs that were more • There are five Lagrangian points in a two-body sys- similar to the numbers 4 or 3 than to the number 5.*[4] tem. It was from those characters that Europeans finally came up with the modern 5, though from purely graphical evi- dence, it would be much easier to conclude that the mod- 5.3.2 Biology ern symbol came from the Khmer. The Khmer glyph • develops from the Kushana/Ândhra/Gupta numeral, its Perception is conceived to occur through five senses. shape looking like the modern version with an extended • swirled 'tail' *[5] Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each While the shape of the 5 character has an ascender in extremity.*[6] most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in 5.3.3 Computing . • 5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ. 5.3 Science

• The atomic number of boron. 5.4 Religion and culture • The number of appendages on most starfish, which 5.4.1 Christian exhibit pentamerism. • • The most destructive known hurricanes rate as There are traditionally Five Wounds of Jesus Christ Category 5 on the Saffir–Simpson hurricane wind in Christianity: the Scourging at the Pillar, the scale. Crowning with Thorns, the wounds in Christ's hands, the wounds in Christ's feet, and the Side • The most destructive known tornadoes rate an F-5 Wound of Christ. on the Fujita scale or EF-5 on the Enhanced Fujita scale. 5.4.2 Discordianism

5.3.1 Astronomy • In Discordianism, 5 is seen as a very important num- ber. This is demonstrated in the Law of Fives, as • Messier object M5, a magnitude 7.0 globular cluster well as in the Pentabarf, which contains five rules. in the constellation Serpens. • Each page of the Principia Discordia—the primary • The New General Catalogue object NGC 5, a religious document in Discordianism —is labeled magnitude 13 spiral galaxy in the constellation with five digits. Andromeda.

• The Roman numeral V stands for dwarfs (main se- quence stars) in the Yerkes spectral classification 5.4.3 Islamic scheme. • The Five Pillars of Islam • The Roman numeral V (usually) stands for the fifth- discovered satellite of a planet or minor planet (e.g. • Muslims pray to Allah five times a day Jupiter V). • In Islam, particularly , the Panjetan or • The Saros number of the lunar eclipse series which the Five Holy Purified Ones are the members of began on October 8, 2581 BC and ended on March Muhammad's family: Muhammad, Ali, Fatimah, 24, 1084 BC. The duration of Saros series 5 was Hasan, and Husayn and is often symbolically rep- 1496.5 years, and it contained 84 lunar eclipses. resented by an image of the Khamsa. 5.5. ART, ENTERTAINMENT, AND MEDIA 31

5.4.4 Jewish Western, Chinese-Buddhist, and Japanese names for each weekday. • The Torah contains five books—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—which are • Members of The Nation of and , a pri- collectively called the Five Books of Moses, the Pen- marily African American religious organization, call tateuch (Greek for “five containers,”referring to themselves the“Five-Percenters”because they be- the scroll cases in which the books were kept), or lieve that only 5% of mankind is truly enlightened. .(”Hebrew for “fifth ,חומש) Humash • The book of Psalms is arranged into five books, par- 5.5 Art, entertainment, and media alleling the Five Books of Moses. • The Khamsa, an ancient symbol shaped like a hand 5.5.1 Events with four fingers and one thumb, is used as a protec- tive by Jews; that same symbol is also very • November 5 is now known as Guy Fawkes Night or popular in Arabic culture, known to protect from Guy Fawkes Day in the UK. envy and the .

5.5.2 Fictional entities 5.4.5 Sikh • James the Red Engine, a fictional character num- • The five sacred Sikh symbols prescribed by Guru bered 5. Gobind Singh are commonly known as panj kakars or the 'Five ' because they start with letter K • Johnny 5 is the lead character in the film Short Cir- representing kakka (ਕ) in the 's cuit (1986) script. They are: kesh (unshorn hair), • kangha (the comb), kara (the bracelet), kach- Number Five is a character in Lorien Legacies hehra (the soldiers shorts), and kirpan (the sword) • Sankara Stones, five magical rocks in Indiana Jones (in Gurmukhi: ਕੇਸ, ਕੰਘਾ, ਕੜਾ, ਕਛਹਰਾ, ਕਿਰਪਾਨ). and the Temple of Doom that are sought by the Also, there are five deadly evils: kam (lust), krodh Thuggees for evil purposes (anger), moh (attachment), lobh (greed), and ankhar (ego). • The Mach Five (マッハ号 Mahha-gō?), the racing car (Go Mifune in the Japanese ver- sion) drives in the anime series of the same name 5.4.6 Other religions and cultures (known as “Mach Go! Go! Go!" in Japan)

• According to ancient Greek philosophers such as • In the works of J.R.R. Tolkien, five wizards Aristotle, the universe is made up of five classical (Saruman, Gandalf, Radagast, Alatar and Pallando) elements: water, earth, air, fire, and ether. This are sent to Middle-Earth to aid against the threat of concept was later adopted by Medieval alchemists the Dark Lord Sauron and more recently by practitioners of -Pagan re- ligions such as Wicca. • In the A Song of Ice and Fire series, the War of the Five Kings is fought between different • The pentagram, or five-pointed star, bears religious claimants to the Iron Throne of Westeros, as well significance in various including Baha'i, to the thrones of the individual regions of West- Christianity, Freemasonry, Satanism, Taoism, eros (Joffrey Baratheon, Stannis Baratheon, Renly Thelema, and Wicca. Baratheon, Robb Stark and Balon Greyjoy) • In Cantonese, “five”sounds like the word “not” (character: 唔). When five appears in front of a 5.5.3 Films , e.g. “58”, the result is considered unlucky. • Towards the end of the film Monty Python and the • In East Asian tradition, there are five elements: Holy Grail (1975), the character of King Arthur re- (water, fire, earth, wood, and metal). The Japanese peatedly confuses the number five with the number names for the days of the week, Tuesday through three. Saturday, come from these elements via the identi- • Five Go Mad in Dorset (1982) was the first of the fication of the elements with the five planets visible long-running series of Comic Strip Presents... tele- with the naked eye. Also, the traditional Japanese vision comedy films calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining • The Fifth Element (1997), a science fiction film 32 CHAPTER 5. 5 (NUMBER)

• Fast Five (2011), the fifth installment of the Fast and Other uses Furious film series. • A perfect fifth is the most , and • V for Vendetta (2005), produced by Warner Bros., is the basis for most western tuning systems. directed by James McTeigue, and adapted from • Alan Moore's graphic novel V for Vendetta promi- Modern musical notation uses a musical staff made nently features number 5 and Roman Numeral V; of five horizontal lines. the story is based on the historical event in which • In harmonics – the fifth partial (or 4th overtone) of a a group of men attempted to destroy Parliament on fundamental has a frequency ratio of 5:1 to the fre- November 5, 1605 quency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure 5.5.4 Music third. A major triad chord when played in just in- tonation (most often the case in a cappella vocal Groups ensemble singing), will contain such a pure major third. • Five (band), a U.K. Boy band • The number of completed, numbered concer- • The Five (composers), 19th Russian com- tos of Ludwig van Beethoven, Sergei Prokofiev, and posers Camille Saint-Saëns. • • 5 Seconds of Summer, pop band that originated in Using the Latin root, five musicians are called a Sydney, Australia quintet. • A scale with five notes per octave is called a • Five Americans, American rock band active 1965– pentatonic scale. 1969 • Five is the lowest possible number that can be the • Five Man Electrical Band, Canadian rock group top number of a time signature with an asymmetric billed (and active) as the Five Man Electrical Band, meter. 1969 – 1975

• Maroon 5, American pop rock band that originated 5.5.5 Television in Los Angeles, California Stations • MC5, American punk rock band • Channel 5, a television channel that broadcasts in the • Pentatonix, a Grammy-winning a cappella group United Kingdom originated in Arlington, Texas

• The 5th , American pop vocal group, ac- Series tive 1977–present • Babylon 5, a science fiction television series created, • The Five, a.k.a. DC5, an English pop produced and largely written by J. Michael Straczyn- rock group comprising Dave Clark, Lenny David- ski son, Rick Huxley, Denis Payton, and Mike Smith; active 1958–1970 • The number 5 features in the television series Battlestar Galactica in regards to the Final Five • The Jackson 5, American pop rock group featuring cylons and the Temple of Five various members of the Jackson family; they were • billed (and active) as The Jackson 5, 1966–1975 Hi-5 (Australian TV series), a television series from Australia • We Five: American folk rock group active 1965– • Hi-5 (British TV series), a television show from the 1967 and 1968–1977 United Kingdom

• Grand Master Flash and the Furious Five: American • Odyssey 5, a 2002 science fiction television series rap group, 1970-80's • The Five: Fox News Channel roundtable current • Fifth Harmony, Biggest in America. events television show, premiered 2011. So named 2012-present for its panel of five commentators. 5.6. SPORTS 33

• Yes! Pretty Cure 5 is a 2007 anime series which fol- • The Florida Marlins retired the number lows the adventures of Nozomi and her friends. It is for their first president Carl Barger, who also followed by the 2008 sequel Yes! Pretty Cure 5 died in December 1992, four months be- GoGo! fore the team's first game. The num- ber was chosen because DiMaggio was Barger's favorite player. When the team 5.5.6 Literature renamed itself the Miami Marlins in ad- vance of its 2012 move to a new stadium, • The Famous Five is a series of children's books by it decided to honor Barger with a plaque British writer Enid Blyton at the new park and placed the number • The Power of Five is a series of children's books by into circulation. British writer and screenwriter Anthony Horowitz • In the NBA: • • The Fall of Five is a book written under the collec- The Phoenix Suns, for Dick Van Arsdale. tive pseudonym Pittacus Lore in the series Lorien • In the NHL: Legacies • The Boston Bruins, for Hall of Famer Dit • The Book of Five Rings is a text on kenjutsu and Clapper. the martial arts in general, written by the swordsman • The Detroit Red Wings, for Nicklas Lid- Miyamoto Musashi circa 1645 strom. • The Montreal Canadiens, for Hall of • Slaughterhouse-Five is a book by Kurt Vonnegut Famer Bernie Geoffrion. about World War II • The New York Islanders, for Hall of Famer Denis Potvin. 5.6 Sports • The Toronto Maple Leafs, for Bill Bar- ilko. The Leafs have a unique policy of not retiring numbers unless the player • The Olympic Games have five interlocked rings as honoured either died or suffered a career- their symbol, representing the number of inhabited ending incident while a member of the continents represented by the Olympians (Europe, team. Barilko disappeared while on a Asia, Africa, Australia and Oceania, and the Amer- fishing trip in 1951; his presumed death icas). was confirmed when the wreckage of the • Five-a-side football is a variation of association foot- plane he was on was discovered in a re- ball in which each team fields five players. mote section of Ontario in 1962. • The Washington Capitals, for Hall of • The jersey number 5 has been retired by several Famer Rod Langway. North American sports teams in honor of past play- • ing greats or other key figures: In the NFL: • The Chicago Bears, for Hall of Famer • In Major League Baseball: George McAfee. • The Baltimore Orioles, for Hall of Famer • Brooks Robinson. In baseball scorekeeping, the number 5 represents the third baseman's position. • The Cincinnati Reds have retired the number twice. The first was in 1940 • In basketball: for Willard Hershberger, who commit- ted suicide during the season. The num- • The number 5 is used to represent the position ber was returned to service in 1942, and of center. was later retired a second time for Hall of • The number of players of a basketball team on Famer Johnny Bench. the court at a given time. Thus, the phrase five • The Cleveland Indians, for Hall of Famer on five is commonly used to describe standard Lou Boudreau. competitive basketball. • The Detroit Tigers, for Hall of Famer • The“5-second rule”refers to several related Hank Greenberg. rules designed to promote continuous play. • The Houston Astros, for Jeff Bagwell. In all cases, violation of the rule results in a • The Kansas City Royals, for Hall of turnover. Famer George Brett. • In all major rulesets, a team has 5 seconds • The New York Yankees, for Hall of to release the ball toward the court on a Famer Joe DiMaggio. throw-in. 34 CHAPTER 5. 5 (NUMBER)

• In all major rulesets except that of the • The number of points awarded for a try. NBA, a “closely guarded”player has 5 • The number of the starting lock forward who seconds to shoot, pass, or penetrate past usually jumps at number 4 in the line-out in the defender. rugby union. • In the NBA only, a player cannot drib- • The playing field contains two lines that are ble with his back or side to the basket for each 5 metres from the try line. These are sig- more than 5 seconds. nificant because no scrum can be set between • Under FIBA rules, a player has 5 seconds this line and the try line. to attempt a free throw. (North American rulesets allow 10 seconds.) • In Formula One racing, the number 5 & 6 cars tradi- 5.7 Technology tionally belonged to the Williams team, until the end of the 1995 Formula One season. It was most syn- onymous with Nigel Mansell in the late 1980s and early 1990s. • Sebastian Vettel chose this number for his car, although he first used it in 2015 as he was 2013 world champion. • In hockey, the area between the goaltender's legs is known as the five-hole. • In the NASCAR Sprint Cup Series, the #5 car has been owned by Hendrick Motorsports since 1984. Drivers who have raced in this car during this period include: • Geoff Bodine (1984–1989) • Ricky Rudd (1990–1993) • Terry Labonte (1994–2004, including a Cup Series title in 1996) 5 as a resin identification code, used in recycling. • Ron Hornaday, Jr. drove #5 in one race in 2000 that Labonte missed due to injury. • 5 is the most common number of gears for automo- • Todd Bodine, younger brother of Geoff, drove biles with manual transmission. #5 in another race in 2000 that Labonte missed • In radio communication, the term "Five by five" is due to injury. used to indicate perfect signal strength and clarity. • Kyle Busch (2005–2007) • On almost all devices with a numeric keypad such • Casey Mears (2008) as telephones, computers, etc., the 5 key has a raised • Mark Martin (2009–2011) dot or raised bar to make dialing easier. Persons who • Kasey Kahne (2012–). are blind or have low vision find it useful to be able to feel the keys of a telephone. All other numbers • In professional wrestling, if a wrestler grabs the can be found with their relative position around the ropes when he is in a submission hold, the attack- 5 button (on computer keyboards, the 5 key of the ing wrestler has up to a 5 count to break the hold numpad has the raised dot or bar, but the 5 key that until a disqualification is made. This is also the case shifts with % does not). for choking. • On most telephones, the 5 key is associated with the • In rugby league: letters J, K, and L, but on some of the BlackBerry phones, it is the key for G and H. • In most competitions, the starting left wing wears this number. An exception is the Eu- • The Pentium, coined by Intel Corporation, is a fifth- ropean Super League, which uses static squad generation x86 architecture microprocessor. numbering. • The resin identification code used in recycling to • The number of tackles the attacking team has identify polypropylene. to score a try before the handover. • A pentamer is an oligomer composed of five sub- • In rugby union: units. 5.8. MISCELLANEOUS FIELDS 35

5.8 Miscellaneous fields • Quintessence, meaning 'fifth element', refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth)

• The designation of an Interstate Highway (Interstate 5) that runs from San Diego, California to Blaine, Washington. In addition, all major north-south In- terstate Highways in the United States end in 5.

• The five basic tastes are sweet, salty, sour, bitter, and umami.

• In the computer game Riven, 5 is considered a holy number, and is a recurring theme throughout the International maritime signal flag for 5 game, appearing in hundreds of places, from the number of islands in the game, to the number of bolts on pieces of machinery.

The fives of all four suits in playing cards

• The Garden of Cyrus 1658 by Sir Thomas Browne St. Petersburg Metro, Line 5 is a Pythagorean Discourse based upon the number 5. See also: 5 (disambiguation) and Take Five (disambigua- tion) • The holy number of Discordianism, as dictated by the Law of Fives.

Five can refer to: • The number of Justices on the Supreme Court of the United States necessary to render a majority de- •“Give me five”is a common phrase used preceding cision. a High five. • The number of dots in a quincunx. • An informal term for the British Security Service, MI5. • The number of permanent members with veto power on the United Nations Security Council. • Five babies born at one time are quintuplets. The most famous set of quintuplets were the Dionne • The number of sides and the number of angles in a quintuplets born in the 1930s. pentagon.

• In the United States legal system, the Fifth Amend- • The number of points in a pentagram. ment to the United States Constitution can be re- ferred to in court as“pleading the fifth”, absolving • The number of Korotkoff sounds when measuring the defendant from self-incrimination. blood pressure • Pentameter is verse with five repeating feet per line; • The drink Five Alive is named for its five ingredi- iambic pentameter was the most popular form in ents. The drink punch derives its name after the Shakespeare. Sanskrit पञ्च (pañc) for having five ingredients. 36 CHAPTER 5. 5 (NUMBER)

• The Keating Five were five United States Senators 5.11 External links accused of corruption in 1989. • • The Inferior Five: Merryman, Awkwardman, The The Number 5 Blimp, White Feather, and Dumb Bunny. DC • The Positive Integer 5 Comics parody superhero team. • Prime curiosities: 5 • No. 5 is the name of the iconic fragrance created by Coco Chanel. • The Committee of Five was delegated to draft the United States Declaration of Independence. • The 5th U.S. President was James Monroe. • The five-second rule is a commonly used rule of thumb for dropped food. • 5 is a character in the Peanuts comic strip. • Number Five/#00.05 is a character from the comic book series The Umbrella Academy

5.9 See also

• Five Families • Five Nations

5.10 References

[1] “Sloane's A028388 : Good primes”. The On-Line En- cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01. [2] “Sloane's A003273 : Congruent numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01. [3] Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61 [4] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.65 [5] Ifrah, Georges (1998). The universal history of numbers : from prehistory to the invention of the computer. trans- lated from the French by David Bellos ... [et al.] London: Harvill Press. ISBN 978-1-86046-324-2. [6] Kisia, S. . (2010), Vertebrates: Structures and Functions, Biological Systems in Vertebrates, CRC Press, p. 106, ISBN 9781439840528, The typical limb of tetrapods is the limb (Gr. penta, five) that has five toes. Tetrapods evolved from an ancestor that had limbs with five toes. ... Even though the number of digits in different vertebrates may vary from five, vertebrates develop from an embryonic five-digit stage.

• Wells, D. The Penguin Dictionary of Curious and In- teresting Numbers London: Penguin Group. (1987): 58–67 Chapter 6

6 (number)

This article is about the mathematical number. For other 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, uses, see 6 (disambiguation). 94, 96, 97, 98. (sequence A228499 in the OEIS) Six is a ,*[7] a harmonic 6 (six /ˈsɪxs/) is the natural number following 5 and pre- number*[8] and a superior highly composite number, the ceding 7. last to also be a primorial. The next superior highly com- The SI prefix for 10006 is exa- (E), and for its reciprocal posite number is 12. The next primorial is 30. atto- (a). 5 and 6 form a Ruth-Aaron pair under either definition. There are no Graeco-Latin squares with order 6. If n is a natural number that is not 2 or 6, then there is a Graeco- 6.1 In mathematics Latin square with order n. The smallest non-abelian group is the symmetric group 6 is the smallest positive integer which is neither a square S3 which has 3! = 6 elements. number nor a prime number. Six is the second smallest S , with 720 elements, is the only finite symmetric group composite number; its proper divisors are 1, 2 and 3. 6 which has an outer automorphism. This automorphism Since six equals the sum of its proper divisors, six is allows us to construct a number of exceptional mathe- the smallest perfect number, Granville number, and S - matical objects such as the S(5,6,12) Steiner system, the perfect number.*[1]*[2] projective plane of order 4 and the Hoffman-Singleton As a perfect number: graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construc- tion of n isomorphic objects on an n-set A, invariant un- • 1 2 6 is related to the Mersenne prime 3, since 2 (2 − der all of A, but not naturally in one-to-one 1) = 6. (The next perfect number is 28.) correspondence with the elements of A. This can also be expressed category theoretically: consider the category • 6 is the only even perfect number that is not the sum * whose objects are the n element sets and whose arrows of successive odd cubes. [3] are the bijections between the sets. This category has a non-trivial functor to itself only for n = 6. • As a perfect number, 6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one num- 6 similar coins can be arranged around a central coin of ber; the square number, 25. the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 Six is the only number that is both the sum and the prod- the answer to the two-dimensional kissing number prob- uct of three consecutive positive numbers.*[4] lem. The densest sphere packing of the plane is obtained Unrelated to 6 being a perfect number, a Golomb ruler by extending this pattern to the hexagonal lattice in which of length 6 is a “perfect ruler.”*[5] Six is a congruent each circle touches just six others. number.*[6] 6 is the largest of the four all-Harshad numbers. Six is the first discrete biprime (2 × 3) and the first mem- A six-sided polygon is a hexagon, one of the three regu- ber of the (2 × ) discrete biprime family. lar polygons capable of tiling the plane. Figurate num- Six is the smallest natural number that can be written as bers representing hexagons (including six) are called the sum of two( positive) ( rational) cubes which are not in- hexagonal numbers. Because 6 is the product of a power 17 3 37 3 1 tegers: 6 = 21 + 21 . Others up to 100 are: 7, 9, of 2 (namely 2 ) with nothing but distinct Fermat primes 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, (specifically 3), a regular hexagon is a constructible poly- 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 65, 67, 68,

37 38 CHAPTER 6. 6 (NUMBER)

A cube has 6 faces gon. Six is also an .*[9] It is a triangular X-ray of a polydactyl human hand with six fingers number and so is its square (36). There are six basic trigonometric functions. • A hexahedron is a polyhedron with six faces, with a cube being a special case There are six convex regular polytopes in four dimensions. • Hexameter is a poetic form consisting of six feet per Six is the four- binary complement of number nine (6 line = 01102 and 9 = 10012). • A “hex nut”is a nut with six sides, and a hex The six exponentials theorem guarantees (given the right has a six-sided head conditions on the exponents) the transcendence of at least one of a set of exponentials. • The prefix "hexa-" also occurs in the systematic All primes above 3 are of the form 6n ± 1 for n ≥ 1. name of many chemical compounds, such as hexane which has 6 carbon atoms (C 6H 6.1.1 In numeral systems 14).

In bases 10, 15 and 30, 6 is a 1-automorphic number. 6.2.2 The prefix sex- 6.1.2 List of basic calculations Sex- is a Latin prefix meaning “six”. Thus:

6.2 Greek and Latin word parts • is the ordinal adjective meaning “sixth”

6.2.1 Hexa • People with sexdactyly have six fingers on each hand

Hexa is classical Greek for “six”. Thus: • The measuring instrument called a sextant got its name because its shape forms one-sixth of a whole • "Hexadecimal" combines hexa- with the Latinate circle decimal to name a number base of 16 • A group of six musicians is called a sextet • A hexagon is a regular polygon with six sides • Six babies delivered in one birth are sextuplets • L’Hexagone is a French nickname for the con- tinental part of Metropolitan France for its re- • pairs - Prime pairs differing by six are semblance to a regular hexagon sexy, because sex is the Latin word for six.*[10] 6.4. IN MUSIC 39

6.3 Evolution of the glyph

The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numer- als. Our modern 6 can be traced back to the Brahmins of India, who wrote it in one stroke like a cursive lower- case e rotated 90 degrees clockwise. Gradually, the up- per part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Ghubar Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.*[11] On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a 'b' is not practical. Just as in most modern typefaces, in typefaces with text figures the 6 character usually has an ascender, as, for example, in . A standard guitar has 6 strings This numeral resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, • #6 is the pseudonym of American musician Shawn the 6 has often been underlined, both in handwriting and Crahan, when performing with the band Slipknot on printed labels.

6.4.2 In instruments 6.4 In music • A standard guitar has 6 strings 6.4.1 In artists • Most woodwind instruments have 6 basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxo- • “ ” Les Six ( The Six in English) was a group consist- phone); these holes or keys are usually not given ing of the French composers Georges Auric, Louis numbers or letters in the fingering charts Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s

• Bands with the number six in their name include Six 6.4.3 In music theory Organs of Admittance, 6 O'Clock Saints, Electric Six, Eve 6, Los Xey (sei is Basque for “six”), Out • There are 6 whole tones in an octave. On Blue Six, Six In Six, Sixpence None the Richer, Slant 6, Vanity 6, and You Me At Six • There are 6 semitones in a tritone. 40 CHAPTER 6. 6 (NUMBER)

6.4.4 In works • The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan •“Six geese a-laying”were given as a present on • the sixth day in the popular Christmas carol,"The Seraphs have six wings. Twelve Days of Christmas.” • In Islam: • Divided in six arias, Hexachordum Apollinis is gen- • There are Six articles of belief erally regarded as one of the pinnacles of Johann • Fasting six days of Shawwal, together with the Pachelbel's oeuvre. month of Ramadan, is equivalent to fasting the • The theme of the sixth album by Dream Theater, whole year Six Degrees Of Inner Turbulence, was the number • In Hindu , a trasarenu is the combination of six: the album has six songs, and the sixth song — 6 celestial paramānus (atoms) that is, the complete second disc —explores the sto- ries of six individuals suffering from various mental See also 666. illnesses. • Aristotle gave six elements of tragedy, the first of which is Mythos. 6.6 In science

6.6.1 Astronomy 6.5 In religion • Messier object M6, a magnitude 4.5 open clus- ter in the constellation Scorpius, also known as the Butterfly Cluster • The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda • The Saros number of the solar eclipse series which began on March 16, 2691 BC and ended on May 3, 1393 BC. The duration of Saros series 6 was 1298.1 years, and it contained 73 solar eclipses • The Saros number of the lunar eclipse series which began on July 25, 2642 BC and ended on February 10, 1091 BC. The duration of Saros series 6 was 1550.6 years, and it contained 87 lunar eclipses • The Roman numeral VI: • Stands for subdwarfs in the Yerkes spectral classification scheme • (Usually) stands for the sixth-discovered satel- lite of a planet or minor planet (e.g. Jupiter VI)

6.6.2 Biology • There are 6: • The cells of a beehive are 6-sided • Points on a Star of David • Insects have 6 legs • Orders of the Mishnah • Six kingdoms in the taxonomic rank below domain • Symbolic foods placed on the Passover Seder (biology); Animalia, Plantae, Fungi, Protista, Plate Archaea/Archaeabacteria, and Bacteria/Eubacteria. See Kingdom (biology). • Yahweh took 6 days to create the world in the Old Testament Book of Genesis; humankind was cre- • The six elements most common in biomolecules ated on day 6. In the City of God, Augustine of are called the CHNOPS elements; the letters stand Hippo suggested (book 11, chapter 30) that God's for the chemical abbreviations of carbon, hydrogen, creation of the world took six days because 6 is a nitrogen, oxygen, phosphorus, and sulfur. See perfect number. CHON. 6.7. IN SPORTS 41

6.6.3 Chemistry

A molecule of benzene has a ring of 6 carbon atoms

• A benzene molecule has a ring of 6 carbon atoms • 6 is the atomic number of carbon In the Standard Model of particle physics, there are 6 types of quarks and 6 types of leptons • The sixfold symmetry of snowflakes arises from the hexagonal crystal structure of ordinary ice 6.7 In sports • A hexamer is an oligomer made of six subunits • 6 entities in a grouping:

• The National Basketball Association and Ma- jor League Baseball have six divisions (the Na- tional Hockey League had six divisions until realigning into four divisions for the 2013-14 season). • The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit. They are the oldest remaining teams in the league, though not nec- essarily the first six; they comprised the entire league from 1942 to 1967

• Number of players:

The cells of a beehive are 6-sided • Six-man football is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the tradi- 6.6.4 Medicine tional 11-man (American) or 12-man (Cana- dian) squad • There are 6 tastes in traditional Indian Medicine • In football (soccer), the number of substitutes called Ayurveda: sweet, sour, salty, bitter, pungent, combined by both teams, that are allowed in and astringent. These tastes are used to suggest a the game. diet based on the symptoms of the body • In ice hockey, the number of players per team, • Phase 6 is one of six pandemic influenza phases including the goaltender, that are on the ice at any one time, excluding penalty situations. • In volleyball, 6 players from each team on each 6.6.5 Physics side play against each other

• In the Standard Model of particle physics, there are • Scoring and related terminology: 6 types of quarks and 6 types of leptons • In some sports, 6 goals is known as a double- • In statistical mechanics, the six-vertex model has 6 hat-trick, but is very hard to accomplish possible configurations of arrows at each vertex • In American and Canadian football, a • There are six colors in the RGB color wheel: (pri- touchdown earns 6 points mary) red, blue, green, (secondary) cyan, magenta, • In Australian Rules football, six points are re- and yellow. (See Tertiary color) ceived for a goal 42 CHAPTER 6. 6 (NUMBER)

• In baseball scorekeeping, the number 6 repre- • The San Diego Padres, for Steve Gar- sents the shortstop's position. vey • In cricket (See cricket terminology): • In the NBA: • a“six”or“sixer”is a shot in which the • The Boston Celtics, for Hall of ball clears the boundary without bounc- Famer Bill Russell ing, scoring six runs • The Orlando Magic, for their fans • there are six balls to an over (the “sixth man”) • • Jersey number 6 and car number 6: The Philadelphia 76ers, for Hall of Famer Julius Erving • In football/soccer: • The Phoenix Suns, for Walter Davis • Is the number of AC Milan's centre back • The Sacramento Kings, also for their and Franco Baresi; the shirt was fans retired in 1997 • In the NFL: • In NASCAR: • The Kansas City Chiefs, for Warren • The number 6 is currently owned by McVea Roush Fenway Racing. Since the 2007 • In the NHL: season, the first year in which Roush Rac- • ing was merged with the Fenway Sports The Detroit Red Wings, for Larry Group that owns the Boston Red Sox, the Aurie Cup Series version of the car has been • The Pittsburgh Penguins, for Ian driven by David Ragan. From 1988 to Ackerman 2006, Mark Martin drove the #6 in the • The Toronto Maple Leafs, for Hall of Cup Series for what was then Roush Rac- Famer Ace Bailey. The Leafs have ing. a unique policy of not retiring num- • In rugby: bers unless the player honoured ei- ther died or suffered a career-ending • In most rugby league competitions (but incident while a member of the team. not the European Super League, which Bailey suffered a fractured skull dur- uses static squad numbering), the jer- ing a game in 1933; while he recov- sey number 6 is worn by the starting ered and lived for nearly 60 years five-eighth (Southern Hemisphere term) after the incident, he never played or stand-off half (Northern Hemisphere again. The Leafs would issue the term). number to Ron Ellis in 1968 at Bai- • In rugby union, the starting blindside ley's personal request, and Ellis wore flanker wears jersey number 6. (Some it until his own retirement in 1981. teams use“left”and“right”flankers in- “ ” “ ” stead of openside and blindside , with • In Formula One both Rosbergs used this number. 6 being worn by the starting left flanker.) Keke Rosberg used when his • The jersey number 6 has been retired by sev- in 1982. In 2014, Nico Rosberg uses this number eral North American sports teams in honor of until present. past playing greats or other key figures (or, in two cases, a team's fans): • In Major League Baseball: 6.8 In technology • The Atlanta Braves, for manager Bobby Cox • On most phones, the 6 key is associated with the • The Boston Red Sox, for Johnny letters M, N, and O, but on the BlackBerry it is the Pesky key for J and K, and on the BlackBerry 8700 series • The Detroit Tigers, for Hall of Famer and Curve 8900 with full keyboard, it is the key for Al Kaline F • The Minnesota Twins, for Tony Oliva • The "6-meter band" in amateur radio includes the • The New York Yankees, for manager frequencies from 50 to 54 MHz Joe Torre • The St. Louis Cardinals, for Hall of • 6 is the resin identification code used in recycling to Famer Stan Musial identify polystyrene 6.11. IN OTHER FIELDS 43

• Six is a character in the television series Blossom played by Jenna von Oÿ.

• Six is the nickname of Kal Varrik, a central char- acter in the television series Dark Matter, played by Roger Cross.

6.11 In other fields

6 as a resin identification code, used in recycling.

6.9 In calendars

• In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August

• Sextidi was the sixth day of the in the French Revolutionary calendar

International maritime signal flag for 6 6.10 In the arts and entertainment

• The name of the smallest group of Cub Scouts and • The - 1966 animated cartoon series featuring Guiding's equivalent Brownies, traditionally consist- six different super-powered heroes. ing of six people and is led by a 'sixer'. • The Bionic Six are the heroes of the eponymous an- • imated series Six pack is a common form of packaging for six bot- tles or cans of drink (especially beer), and by exten- • The number of sides on a cube, hence the highest sion, other assemblages of six items. number on a standard die • The fundamental flight instruments lumped together • The highest number on one end of a standard on a cockpit display are often called the Basic Six or domino six-pack.

• The Power of Six is a book written by Pittacus Lore, • A coffin is traditionally buried six feet under the and the second in the Lorien Legacies series. ground; thus, the phrase “six feet under”means that a person (or thing, or concept) is dead • Number Six (Tricia Helfer), is a family of fictional characters from the reimagined science fiction tele- • In Western astrology, Virgo is the 6th astrological vision series, Battlestar Galactica sign of the Zodiac

• Number 6 is a character in the book series Lorien • The number of dots in a cell. Legacies • There are said to be no more than six degrees of • Number 6 (Teresa Palmer) is a character in the separation between any two people on Earth. movie I Am Number Four (2011) • See also Six degrees (disambiguation). • Number 6, the main protagonist in The Prisoner played by Patrick McGoohan, portrayed by Jim • is sometimes called the Caviezel in the remake. “sixth sense” 44 CHAPTER 6. 6 (NUMBER)

• Six human physical needs: breathe, urination, • The Odd Number 6, Todd, Math. Proc. Camb. defecation, water, food, and sex Phil. Soc. 41 (1945) 66—68

• The Six Dynasties form part of Chinese history • A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and • 6 is a lucky number in Chinese culture their Links ISBN 0-521-42385-6

• The Birmingham Six were a British miscarriage of • Wells, D. The Penguin Dictionary of Curious and In- justice, held in prison for 16 years teresting Numbers London: Penguin Group. (1987): 67 - 69 •“Six”is used as an informal slang term for the British Secret Intelligence Service, MI6

• Six Flags, an American company running amuse- 6.13 External links ment parks and theme parks in the US, Canada, and Mexico • The Number 6 • Sinister Six is a group of super villains who appear in • The Positive Integer 6 American comic books published by Marvel Comics • Prime curiosities: 6

6.12 References

[1] Higgins, Peter (2008). Number Story: From Counting to . New York: Copernicus. p. 11. ISBN 978- 1-84800-000-1.

[2]“Granville number”. OeisWiki. The Online Encyclopedia of Integer Sequences. Archived from the original on 29 March 2011. Retrieved 27 March 2011.

[3] David Wells, The Penguin Dictionary of Curious and In- teresting Numbers. London: Penguin Books (1987): 67

[4] Peter Higgins, Number Story. London: Copernicus Books (2008): 12

[5] Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72

[6] “Sloane's A003273 : Congruent numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[7] “Sloane's A002827 : Unitary perfect numbers”. The On-Line Encyclopedia of Integer Sequences. OEIS Foun- dation. Retrieved 2016-06-01.

[8] “Sloane's A001599 : Harmonic or Ore numbers”. The On-Line Encyclopedia of Integer Sequences. OEIS Foun- dation. Retrieved 2016-06-01.

[9] “Sloane's A005900 : Octahedral numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[10] Chris K. Caldwell; G. L. Honaker Jr. (2009). Prime Cu- rios!: The Dictionary of Prime Number Trivia. CreateS- pace Independent Publishing Platform. p. 11. ISBN 978- 1448651702.

[11] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.66 Chapter 7

7 (number)

“Seven”redirects here. For other uses, see Seven the same six-digit repeating sequence after the dec- (disambiguation). imal point, but the sequence can start with any of those six digits.*[7] For example, 1/7 = 0.142857 7 (seven; /ˈsɛvən/) is the natural number following 6 and 142857... and 2/7 = 0.285714 285714.... preceding 8. In fact, if one sorts the digits in the number 142857 in ascending order, 124578, it is pos- sible to know from which of the digits the dec- 7.1 Mathematics imal part of the number is going to begin with. The remainder of dividing any number by 7 • Seven, the fourth prime number, is not only a will give the position in the sequence 124578 Mersenne prime (since 23 − 1 = 7) but also a that the decimal part of the resulting number double Mersenne prime since the exponent, 3, is will start. For example, 628 ÷ 7 = 89 5/7; here itself a Mersenne prime. It is also a Newman– 5 is the remainder, and would correspond to Shanks–Williams prime,*[1] a Woodall prime,*[2] number 7 in the ranking of the ascending se- a factorial prime,*[3] a lucky prime,*[4] a happy quence. So in this case, 628 ÷ 7 = 89.714285. number (happy prime),*[5] a safe prime (the only Another example, 5238 ÷ 7 = 748 2/7, hence Mersenne safe prime), and the fourth Heegner num- the remainder is 2, and this corresponds to ber.*[6] number 2 in the sequence. In this case, 5238 ÷ 7 = 748.285714. • Seven is the lowest natural number that cannot be represented as the sum of the squares of • A seven-sided shape is a heptagon. The regular n- three integers. (See Lagrange's four-square theo- gons for n ≤ 6 can be constructed by compass and rem#Historical development.) straightedge alone, but the regular heptagon cannot. • Seven is the aliquot sum of one number, the cubic Figurate numbers representing heptagons (including number 8 and is the base of the 7-aliquot tree. seven) are called heptagonal numbers. Seven is also a centered hexagonal number.*[8] • n = 7 is the first natural number for which the • next statement does not hold: “Two nilpotent Seven is the first integer reciprocal (multiplicative endomorphisms from C*n with the same minimal inverse) with infinitely repeating repre- polynomial and the same rank are similar.” sentation. • There are seven frieze groups, the groups consist- • 7 is the only number D for which the equation 2*n ing of symmetries of the plane whose group of − D = x2 has more than two solutions for n and x is isomorphic to the group of integers. natural. In particular, the equation 2*n − 7 = x2 is known as the Ramanujan–Nagell equation. • There are seven fundamental types of catastrophes. • 7 is the only dimension, besides the familiar 3, in which a vector cross product can be defined. • When rolling two standard six-sided dice, seven has a 6 in 36 (or 1/6) probability of being rolled (1–6, 6– • 7 is the lowest dimension of a known exotic sphere, 1, 2–5, 5–2, 3–4, or 4–3), the greatest of any num- although there may exist as yet unknown exotic ber. smooth structures on the 4-dimensional sphere. • The Millennium Prize Problems are seven problems • 999,999 divided by 7 is exactly 142,857. Therefore, in mathematics that were stated by the Clay Mathe- when a vulgar with 7 in the denominator matics Institute in 2000. Currently, six of the prob- is converted to a decimal expansion, the result has lems remain unsolved.

45 46 CHAPTER 7. 7 (NUMBER)

p(S) 1_ upstroke in the glyph for 1. 0.16 6 0.14 5__ 36 0.12 1_ 0.10 9 1__ 1 2 0.08 12 0.06 1__ 18 0.04 1__ 0.02 36

2 3 4 5 6 7 8 9101112 S

Graph of the of the sum of 2 six-sided dice

7.1.1 Numeral systems

In quaternary, 7 is the smallest prime with a composite sum of digits.

7.1.2 Basic calculations

7.2 Evolution of the glyph

In the beginning, various wrote 7 more or less in On the seven-segment displays of pocket calculators and one stroke as a curve that looks like an uppercase J verti- digital watches, 7 is the number with the most common cally inverted. The western Ghubar Arabs' main contri- glyph variation (1, 6 and 9 also have variant glyphs). Most bution was to make the longer line diagonal rather than calculators use three line segments, but on Sharp, Casio, straight, though they showed some tendencies to making and a few other brands of calculators, 7 is written with the character more rectilinear. The eastern Arabs de- four line segments because, in Japan, Korea and veloped the character from a 6-look-alike into an upper- 7 is written as ① in the illustration to the right. case V-look-alike. Both modern Arab forms influenced the European form, a two-stroke character consisting of While the shape of the 7 character has an ascender in a horizontal upper line joined at its right to a line go- most modern typefaces, in typefaces with text figures the ing down to the bottom left corner, a line that is slightly character usually has a descender, as, for example, in curved in some font variants. As is the case with the European glyph, the Cham and Khmer glyph for 7 also . evolved to look like their glyph for 1, though in a differ- Most people in Continental Europe*[10] and increasingly ent way, so they were also concerned with making their 7 in the UK and Ireland as well as write 7 more different. For the Khmer this often involved adding with a line in the middle (“7”), sometimes with the top a horizontal line above the glyph.*[9] This is analogous line crooked. The line through the middle is useful to to the horizontal stroke through the middle that is some- clearly differentiate the character from the number one, times used in handwriting in the Western world but which as these can appear similar when written in certain styles is almost never used in computer fonts. This horizontal of handwriting. This glyph is used in official handwrit- stroke is, however, important to distinguish the glyph for ing rules for primary school in Russia, Ukraine, Poland, seven from the glyph for one in writings that use a long other Slavic countries,*[11] as well as in France, Belgium, 7.5. COMMERCE AND BUSINESS 47

• Seven colors in the rainbow • Seven Seas • Seven Continents • Seven against Thebes • Seven Kings of Rome • Seven Emperors (and period; Rome, history) • Julius Caesar, Augustus, Galba, Hadrian, Nerva, Sallust, Vespasian • Seven hills of Istanbul • Seven hills of Rome • Seven Liberal Arts • Seven Sages of the Bamboo Grove in China • Seven Sages of Greece • Saptarishi Seven Sages concept in Ancient India • Seven Wise Masters, a cycle of medieval stories • Seven Wonders of the ancient world • 7 was considered a God number in ancient Egypt. The usually ordered things in groups of Finland,*[12] Romania, Germany and Hungary.*[13] multiples of 7. For a time, 7 was not even used in writings for the people of Egypt. • 7 pure notes in the diatonic scale 7.3 Automotive and transportation

See also: List of highways numbered 7 7.5 Commerce and business

• 7-Eleven is the trading name of a chain of conve- • 7 ( Subway service), a service of the nience stores based in the U.S. New York City Subway • Seven Jeans, also known as Seven '7' for all • BMW 7 Series, a flagship range of luxury vehicles Mankind, a brand of designer jeans. from German automobile manufacturer, BMW. • The Lotus Seven, a sports car produced by Lotus 7.6 Food and beverages • Each airplane in Boeing's line of passenger jets is named with a 3-digit number beginning and ending • 7 Up is the name of a popular soft drink. in 7.

7.4 Classical world 7.7 Media and entertainment

7.4.1 Classical antiquity 7.7.1 Film • Seven Arts Productions • Seven deadly sins • • Seven Classical Planets (“7 Luminaires”) Seven Arts Pictures • • Seven Heavens Se7en, also just called Seven is a 1995 movie about a serial killer basing his murders and victims on the • Seven days in the week seven deadly sins. 48 CHAPTER 7. 7 (NUMBER)

Characters • Seven is the sum of any two opposite sides on a stan- dard six-sided dice. • James Bond's secret agent number is 007 • In the game of craps, a dice roll of 7 on the come- • Star Trek: features a character named out constitutes a “natural”and the Pass line wins Seven of Nine. immediately. However, if the shooter (dice thrower) rolls a 4, 5, 6, 8, 9, or 10 on the come-out roll, that • The Star Trek: The Original Series episode number becomes his “point”. Now, the shooter "Assignment: Earth" features a character named must repeat his point before he throws a seven. If Gary Seven. he repeats his point, the Pass line wins. If he rolls a seven, not only does the Pass line lose, the shooter “ ” Titles loses control of the dice. This is called a seven-out . Films Video games • Seven Chances (1925) starring Buster Keaton • The 7th Guest, produced by Trilobyte and released • The Disney film Snow White and the Seven Dwarfs by Virgin Games in 1993, is an FMV-based puzzle (1937) horror video game. • Seven Sinners (1940) directed by Tay Garnett, star- •“The 7”are a group of deadly mercenaries that kid- ring Marlene Dietrich nap the protagonist's family in the video game Kane • The musical film Seven Brides for Seven Brothers & Lynch: Dead Men. (1954) • In the Killer7 game, the number of individual per- • The Seven (1954) by Akira Kurosawa star- sonalities serving Harman Smith as part of the group ring Toshiro Mifune of assassins called the Killer7 • • The Marilyn Monroe film The Seven Year Itch In 999: Nine Hours, Nine Persons, Nine Doors, one “ ” (1955) which is famous for her white dress billowing of the characters goes by the code name Seven . up from a grate • In the Fallen London story Seeking Mr. Eaten's • The Swedish film The Seventh Seal (1957) directed Name, the number 7 is a recurring theme. by Ingmar Bergman

• The American western film The Magnificent Seven 7.7.3 Literature (1960) starring Yul Brynner • As You Like It by William Shakespeare's contains • "7 Faces of Dr. Lao" (1964) starring Tony Randall. Shakespeare's 7 Ages of Man theory.

• "Seven Days in May" (1964( starring Burt Lan- • In Beloved, the novel by Toni Morrison, the number caster, Kirk Douglas and Fredric March. seven seems to have recurring significance, Beloved has seven letters, 124 adds up to seven, each chapter • The Seventh Sign (1988), directed by Carl Schultz begins on a number containing seven, and the second and starring Jürgen Prochnow and Demi Moore chapter contains seven parts. • Seven (1995) by David Fincher and starring Brad • The original British version of Anthony Burgess' Pitt and Morgan Freeman novel A Clockwork Orange is split into three sections of seven chapters. • Seven Years in Tibet (1997), directed by Jean- Jacques Annaud • In the first of the three Deltora Quest series of books by Emily Rodda, the protagonist must collect the • Seven Pounds (2008), starring Will Smith seven gems of the fabled Belt of Deltora: topaz, ruby, opal, lapis lazuli, emerald, amethyst, and di- 7.7.2 Games amond. Each of the seven gems are guarded by one of seven guardians located in one of the seven • Lotto Super 7, a Canadian-lottery game that had its locations around Deltora: The of Silence, final drawing September 18, 2009. Lake of Tears, City of the Rats, The Shifting Sands, Dread Mountain, Maze of the Beast, and Valley of • Sevens, a card game the Lost. 7.7. MEDIA AND ENTERTAINMENT 49

• In the Dragon Ball series, there are 7 crystalline • In the One Piece series, the Shichibukai (literally, spheres known as the Dragon balls. Each Dragon “Seven Military Seas”, known in the English ver- ball has a certain number of stars on it displaying its sions as the“Seven Warlords of the Sea”) are seven respective number in the set. powerful pirates who work for the government.

• The manga Eureka Seven, based on the anime series • The Saga of Seven Suns, a series of seven novels by Kevin J. Anderson. • In Fist of the North Star (Hokuto no Ken), the main character, Kenshiro, is also known as“the man with • In Neil Gaiman's Sandman series, there are 7 End- seven wounds”for the seven scars on his body in the less: Destiny, Death, Dream, Destruction, Despair, shape of the Big Dipper. Desire and Delirium. • • In the Fushigi Yuugi manga series, the four beast The Secret Seven, name of a children's adventure gods each have seven warriors, the Genbu Seven, the book by Enid Blyton Byakko Seven, the Seiryuu Seven, and the Suzaku • In the Septimus Heap series, Septimus Is the seventh Seven. son of a seventh son, 7 is believed to be the most • In the Harry Potter series of novels by J.K. Rowling, magykal number. There will also be 7 books in the seven is said to be the most powerfully magical num- series, along with many other things relating to the ber. There are a number of references to seven in number 7. Harry Potter: there are a total of seven books in the • Aleksis Kivi's "Seven Brothers", the most famous series; Ginny Weasley is the seventh child and only book of Finnish literature daughter of the Weasley family; Harry Potter was born in July, the seventh month of the year; Wizard- • The Seven Dials Mystery by Agatha Christie ing students must complete seven years of school at Hogwarts; There are seven secret passages leading • The Seven Habits of Highly Effective People, a book out of Hogwarts; Seven is also the number on the by Stephen Covey back of Harry Potter's Quidditch robes (Prisoner of • Azkaban movie). Professor Moody was also hidden Seven Pillars of Wisdom, a book written by .E. in a chest with an underground cubby, which was Lawrence locked with seven locks Harry Potter and the Gob- • The book Seven Years in Tibet let of Fire. In Quidditch each team consists of seven players. Chapter four of Book 7 is titled“The Seven • The novel Seven Days in May, written by Fletcher Potters”. Lord Voldemort planned on making six Knebel Horcruxes (objects containing parts of his soul), so that his soul would exist in seven parts, but made an • In The Silmarillion, also a Tolkien work, there are extra one, making a total of seven Horcruxes. seven Lords of the Valar, and seven Ladies of the Valar; Fëanor, a central character, has seven sons: • In the InuYasha series, the Shichinintai are, as their Maedhros, Maglor, Celegorm, Curufin, Caranthir, name says, a band of seven mercenaries. Amrod, and Amras.

• The Seven Rings of the Vongola Family in Katekyo • In George R.R. Martin's A Song of Ice and Fire, Hitman Reborn: Sky, Storm, Rain, , Thunder, seven gods, the Mother, the Father, the Warrior, the Mist, and Cloud. Crone, the Smith, the Maiden, and the Stranger, are worshipped in the land of Westeros, which is made • In The Lost Years of Merlin series, a book is titled up of seven former kingdoms and often called the “The Seven Songs of Merlin” Seven Kingdoms. The Seven are the primary deities of Westeros. • The character 7 (Nana) and the organization Septi- mal Moon in Loveless (manga) • In W. E. B. Du Bois's book The of Black Folk, he refers to “the Negro”as “the seventh son”. • In the "Nana manga series, there are many recurrent instances of the number seven (“nana”is Japanese • In The Tales of Alvin Maker, an alternate history fan- for“seven”), including the name of both main char- tasy series by Orson Scott Card, seventh sons are im- acters. bued with special powers, referred to as“knacks”, over the physical world. The powers of a seventh son • Team 7 consists of the main characters in Naruto: of a seventh son, the birthright of the series' main Naruto Uzumaki, Sasuke Uchiha, Kakashi Hatake, character, Alvin Smith, are especially pronounced. and Sakura Haruno. Later in the series, Sai joined Men who wield these powers to create are referred team 7 after Sasuke left Konohagakure. to as “Makers.” 50 CHAPTER 7. 7 (NUMBER)

• In Toaru Majutsu no Index, there are a total of seven • "7" is a song by Prince and The New Power Gener- Level 5 espers: Accelerator, Kakine Teitoku, Mis- ation, from the 1992 Love Symbol Album. aka Mikoto, Mugino Shizuri, Shokuhou Misaki, an • unknown esper and Sogiita Gunha. 7 is a 2003 Enrique Iglesias album. • • In The Two Towers, the second volume of J. R. 7 (Mick Thompson) is one of the guitarists for pop- R. Tolkien's The Lord of the Rings, the following ular metal band Slipknot. poem appears: “Tall ships and tall kings/Three • Sweet 7 is an album by times three/What brought they from the foundered land/Over the flowing sea?/Seven stars and seven • The number of dastgahs in Persian traditional music stones/And one white tree.”The“foundered land” • is Númenor, and the “Seven Stars”are unknown, 'Seventh son of a Seventh son' is the 7th album of but they may refer to a constellation, the Sickle of the English heavy metal band Iron Maiden, brought the Valar. The Seven Stones are the palantíri, the out in 1988 seeing-stones. The number seven was also signif- • Sevens is the 7th album by Country singer Garth icant to Dwarvish culture as portrayed by Tolkien: Brooks, released in 1997, consisting of 14 songs(2 there were seven fathers of the Dwarves (as re- sets of 7). counted in The Silmarillion) and the Dwarves were later given seven rings by Sauron (according to The •“Seven”is a song by Megadeth on their 1999 album Lord of the Rings). Risk

• David Eastis authored two books about the popular • UK-based alt-rock band, Take The Seven (TTVII). number seven both with the title “7”. •“The Munificent Seven”is the appended parentheti- • Jackie Leo wrote “Seven”. cal title for the song“Love Is Stronger Than Justice” on Sting's 1993 album Ten Summoner's Tales • We are Seven by William Wordsworth • Seven is a song on David Bowie's album Hours • In The Owl Keeper]], the owl keeper (by Christine Brodien-Jones) is said to be born on the 7th day of the 7th month at 7 past midnight. 7.7.5 Sports

• Marina or Number Seven is a character in the book • In the NHL, MLB, and NBA, the maximum number series The Lorien Legacies. of games played in a playoff series.

• The Revenge of Seven by Pittacus Lore • The jersey number 7 has been retired by several North American sports teams in honor of past play- • In Rick Riordan's series The Heroes of Olympus ing greats or other key figures: there are seven heroes of the Great : Percy Jackson, Annabeth Chase, Jason Grace, Piper • Major League Baseball McLean, Leo Valdez, Hazel Levesque, and Frank • The Houston Astros, for Hall of Famer Zhang. Craig Biggio. • The New York Yankees, for Hall of 7.7.4 Music Famer Mickey Mantle. • National Football League: • The number of notes in the traditional Western dia- • The Chicago Bears, for Hall of Fame tonic scale (major or minor) player, coach, and owner George Halas. • In music, the Roman numeral vii is the leading • The Denver Broncos, for Hall of Famer tone (vii°) or subtonic (VII) scale degree, chord John Elway. (leading-tone seventh chord or secondary leading- • The Detroit Lions, for Hall of Famer tone chord), or diatonic function, when distin- Dutch Clark. guished V = major and v = minor. • The Los Angeles Rams, for Hall of Famer • The original diameter in inches of the 45rpm format Bob Waterfield. gramophone record. • The New York Giants, for Hall of Famer Mel Hein. • There are seven musicians in a septet or a septuor. • National Basketball Association • There are seven basic swaras [saptaswaras] in In- • The Cleveland Cavaliers, for Bingo dian Carnatic music Smith. 7.7. MEDIA AND ENTERTAINMENT 51

• The New Orleans Hornets, for Hall of • In American and Canadian football, a team scores 7 Famer Pete Maravich. He died in January points by scoring a touchdown and kicking the extra 1988, before the franchise began as the point. Charlotte Hornets, but his number was re- • tired based on his years in the city with the In American and Canadian football, the minimum New Orleans Jazz. number of players who, by rule, must be lined up at the line of scrimmage on offense (otherwise the • The Phoenix Suns, for Kevin Johnson. team commits the foul of illegal formation) • The Utah Jazz, also for Maravich. He briefly played with the team while it was Frisbee and Ultimate Frisbee in Utah, and played five seasons with the team in its previous location of New Or- leans. • In Ultimate Frisbee, the number of players on the field for each team • National Hockey League • The Boston Bruins, for Hall of Famer Phil Handball Esposito. • The Buffalo Sabres, for Rick Martin. • In team handball, the number of players on each • The Dallas Stars, for Neal Broten. team including the goalkeeper • The Detroit Red Wings, for Hall of Famer Ted Lindsay. Motor sports • The Edmonton Oilers, for Hall of Famer Paul Coffey. • In Formula 1 the #7 is owned by 2007 World Cham- • The Montreal Canadiens, for Hall of pion Kimi Räikkönen Famer Howie Morenz. • • The New York Rangers, for Hall of In motorcycling, the racing number for British Famer Rod Gilbert. World Champion Barry Sheene • The Philadelphia Flyers, for Hall of • In NASCAR: Famer Bill Barber. • • The Washington Capitals, for Yvon In the Sprint Cup Series, the #7 car is currently Labre. owned by Robby Gordon Motorsports. Team owner Robby Gordon, plus other team drivers, • The Toronto Maple Leafs have a pol- race a part-time schedule in this car. icy of not retiring numbers unless the player honoured either died or suffered a • In the Nationwide Series, the #7 car is cur- career-ending incident while a member of rently owned by JR Motorsports. Danica the team. Other players whose numbers Patrick, who began racing in that car late in would otherwise be retired instead have the 2011 season, took over the full-time ride their numbers enshrined by the team as in the 2012 season. “Honoured Numbers”, which remain in • In the IndyCar Series, the #7 car is currently owned circulation for future players. The num- by Andretti Autosport. Patirck drove full-time in ber 7 is currently honoured for Hall of this car through the 2011 season before moving to Famers King Clancy and Tim Horton. NASCAR.

Baseball Rugby Rugby league • The jersey number 7 is worn by the starting scrum-half/halfback • In high school baseball, a regulation game lasts seven in most competitions (but not in the Eu- innings. ropean Super League, which uses static • In baseball, the 7th Inning stretch attributed to the squad numbering). 27th U.S. President, William Howard Taft. • 7, The number of consecutive State of Origin Series wins (2006-2012) by • In scoring baseball, seven represents the left fielder's Queensland against New South Wales in position. Australia. The highest since the incep- tion of the State of Origin Series concept Football and soccer which commenced in 1980. Rugby union • The jersey number 7 is worn • In association football, the right midfielder tradition- by the starting openside flanker. (Some ally wears the number 7 teams use “left”and “right”flankers 52 CHAPTER 7. 7 (NUMBER)

instead of “openside”and “blindside” Programs , with 7 being worn by the starting right flanker.) • 7 Days, a comedy gameshow based on news stories • A converted try is worth 7 points. This from New Zealand. number is also significant because the • The name of a television show 7th Heaven. standard bonus points system used in most competitions awards 1 bonus point • The name of a The Adventures of Super Mario Bros. to a team that loses by 7 points or less. 3 cartoon episode, "7 Continents for 7 Koopas". • The IRB Sevens World Series, an annual competition for national teams in rugby • Blake's 7 was a BBC science fiction series that ran union sevens (see below) from 1978 to 1981 • The Rugby World Cup Sevens, the World • Eureka Seven, a science fiction and romance anime. Cup for the union version of Sevens Cross-code • , a variant of union • In Red Dwarf II episode " Universe", Holly and league that contains only seven play- forgets the number 7 in a countdown for the Holly ers per side instead of the standard 15 Hop Drive. After Rimmer reminds him, he admits (union) or 13 (league) he has “a blind spot for sevens”.

Softball • Seven, an episodic character from the sitcom Married... with Children. • A regulation game in softball consists of seven in- • In the Seinfeld episode "The Seven", Seven is the nings. name George Costanza desires to give his first-born, Track and field having allegedly promised this to the widow of base- ball great Mickey Mantle (whose uniform number • In cross country running, a team traditionally con- was 7). George is enraged, however, when his fi- sists of seven athletes ance's cousin and spouse copy the idea and give the name to their child. • A heptathlon is a combined event of seven running, jumping and throwing disciplines for both men's and • Seven Days, a science fiction show about the NSA women's athletics. using time travel. • Water polo The name of one of the monsters in the cartoon tele- vision show Seven Little Monsters. • Each team is allowed to have a maximum of seven • In Star Trek: Voyager, Seven of Nine (also called players, one of them a goalkeeper, in the playing Seven for short) is one of the crewmembers area of the pool at any given time. • The Ultra Series has two heroes with “Seven”in their name: Ultra Seven and Ultra Seven-21. 7.7.6 Television • Ultra Seven, a television series aired in 1967, and Networks and stations the superhero of the same name. He is the honorary seventh member of the Ultra Garrison • The circle 7 logo used by numerous ABC network O&O stations in the U.S. (In the late 1940s, ABC • In House of Anubis (on Nickelodeon), the“chosen arranged for its five original, founding O&O TV sta- one”has to be born on 7/7 (July 7) at 7:00am. tions to broadcast on channel 7.) • George Carlin in what is perhaps his best-known • Maximum number of VHF broadcast channels routine,“Seven Words You Can Never Say on Tele- available in any TV market in the United States of vision”. America; they are channels 2, 4, 5, 7, 9, 11, and 13 in the Greater New York City and Los Angeles- • Lucky 7: Short-lived ABC TV series (premiered Long Beach SMSAs. 2013) about seven people who win the lottery. • ProSieben (sieben being German for seven), a tele- • The last episode of American Horror Story: Coven vision network in Germany is named The Seven Wonders, after the seven great powers a witch should perform to become the leader • Seven Network, a television network in Australia. or“Supreme”of the generation: Telekinesis, Con- • Trans7 (formerly TV7 before taken over by Trans cilium, Descensum, Divination, Transmutation, Vi- Corporation), a television network in . talum Vitalis and Pyrokinesis. 7.9. RELIGION AND MYTHOLOGY 53

• In the anime series Yu-Gi-Oh! Duel Monsters, there are seven magical ancient Egyptian artefacts, known as the Millenium Items (Puzzle, Eye, Ring, Key, Scale, Rod and Necklace), which play a significant role in the storyline of the show throughout its dura- tion.

• The number 7 is the racing number of 's girl- friend, Giselle, in the movie Herbie Goes to Monte Carlo

7.8 Places

• The traditional count of Basque provinces as ex- pressed in the slogan Zazpiak Bat, translates as“the seven [are] one”and refers to the seven Basque tra- ditional provinces..

• The figurative number of seas

• The Seven Virgins mountain range in Sri Lanka, which was the scene of an air disaster on Decem- ber 4, 1974, involving a DC-8 Series 55F passenger jetliner operated by the charter company Martinair, which left 191 dead.

• “ ” The Sevens (sometimes typeset as 7he Sevens ), Seven Days of Creation - 1765 book a rugby union stadium in , UAE and home to the Dubai Sevens tournament • In regards to the sin sacrifice, the anointed priest was to sprinkle the bullock's blood seven times before the 7.9 Religion and mythology lord (Leviticus 4:6) • Seven days of the feast of Passover (Exodus 13:3– 7.9.1 Old Testament 10) • Seven-day week and the pattern concerning distri- • Seven Hebrew words of Bereishit 1:1 / Genesis 1:1 bution and use of manna (Exodus 16) (28 letters). is a seven-branched,( מנורה :The Menorah (Hebrew • candelabrum lit by olive oil in the Tabernacle and The number seven in the seven days of Creation is the Temple in Jerusalem (Exodus 25) typological and the number seven appears commonly elsewhere in the Bible. These include: • Seven year cycle around the years of Jubilee (Leviti- cus 25) • Seven days of Creation (Genesis 1) e.g., God rested • Jericho's walls fall on the seventh day after seven on and sanctified the seventh day (Sabbath) priests with seven trumpets march around the city seven times (Joshua 6:8) • Anyone who dares to kill Cain 'will suffer vengeance seven times over' (Genesis 4:15) • King David had seven older brothers (1 Samuel 16)

• Lamech in his “Song of the Sword”claims that 'if • The child sneezed seven times after Elisha raised Cain shall be avenged sevenfold', he himself shall be him from the dead (2 Kings 4:35) 'seventy-sevenfold' (Genesis 4:24) • Seven things that are detestable to the LORD • Noah is commanded to bring seven pairs of every (Proverbs 6:16–19) clean animal onto the ark (Genesis 7:2) • Seven Pillars of the House of Wisdom (Proverbs 9:1) • Seven years of plenty and seven years of famine in Pharaoh's dream (Genesis 41) • The woman with seven sons in 2 Maccabees. 54 CHAPTER 7. 7 (NUMBER)

7.9.2 New Testament 7.9.3 Hinduism

• Seven loaves multiplied into seven basketfuls of sur- • The Sanskrit word sapta refers to number seven. plus (Matthew 15:32–37) • Indian Music has "Saptak Swaras,”seven octats (sa • Peter asked Jesus if he should forgive those who re ga ma pa dha ni) which are basics of music, using sinned against him up to 7 times; Jesus responded by which hundreds of Ragas are composed. saying to forgive them 'seventy times seven times', • Celestial group of seven stars (7 Classical Planets) remembering so the of Cain and the song of are named as "Sapta Rishi" based on the seven great Lamech in Genesis 4.*[14] saints. • Seven demons were driven out of Mary Magdelene. • Seven Promises [Saptapadi], Seven Rounds in (Luke 8:2). Hindu Wedding and Seven • The seven last sayings of Jesus on the cross. • According to Hinduism, there are seven worlds in • Seven men of honest report, full of the Holy the universe, seven seas in the world and seven and wisdom (Acts 6:3) Rishies (seven gurus) called sapta rishis. • In the Book of Revelation, seven is a central figure • Seven hills at tirumala also known as Yedu Kon- of quantities: Seven Spirits of God, Seven Churches dalavadu [in Telugu], ezhu malaiyan [in Tamil] (to which the book is addressed); means Sevenhills God • Seven of the following appear in Revelation: • There are 7 Chakras in the basic model used in var- golden lampstands(1:12), stars(1:16), torches of ious eastern traditions and philosophies. fire(4:5) Seven Seals(5:1), angels and their trum- pets(8:2), last plagues(15:1), golden bowls(15:7), thunders(10:3), horns and eyes(5:6), diadems(12:3) 7.9.4 Islam and kings(17:10). See Category:Seven in the Book • The number of ayat in surat al-Fatiha of Revelation. • The number of layers of the Earth in Islamic religion Other sevens in Christian knowledge and practice include: is seven • • The Seven Corporal Acts of Mercy and Seven Spir- The number of in Islamic religion is seven itual Acts of Mercy of Roman Catholic, Anglican, • The number of circumambulations (Tawaf) that are and other traditions made around the Kaaba • The Seven deadly sins: lust, gluttony, greed, sloth, • The number of walks between Al-Safa and Al- wrath, envy, and pride Marwah mountains —that is travelling back and • Seven Gifts of the Holy Spirit forth—seven times during the pilgrimages of Hajj and Umrah • The Seven Joys of the Virgin Mary, of Roman • Catholic, Anglican, and other traditions The number of heavens in Islamic religion, i.e. levels in heaven. • The Seven Sacraments in the Catholic faith (though • some traditions assign a different number) The number of hells in hell is also seven i.e. levels in hell. • The Seven Sorrows of the Virgin Mary, of Roman • Catholic, Anglican, and other traditions The number of doors to hell is seven (for heaven the number of doors is eight). • There are seven suicides mentioned in the Bible (OT • and NT).*[15] In Verse 12:46 (see Islamic view of Joseph) of the Quran, Joseph (Yusef) is asked to interpret the • The seven terraces of Mount Purgatory (one per King's dream where seven fat cows were dreamt to deadly sin) have been devoured by seven skinny cows and seven green spikes, and others shrivelled. • The Seven Virtues: chastity, temperance, charity, diligence, kindness, patience, and humility • The number of the big sins or vices is seven which are from a Hadith of the prophet Mohamed :“Avoid • In the genealogy in the Gospel of Luke, Jesus is 77th the seven sins polytheism, witchcraft, the killing in a direct line of the soul which Allah has forbidden except by • The number of heads of the three beasts (7 × 10 × right, consuming riba, consuming the wealth of the 7 + 7 × 10 × 10 + 7 × 10 = 1260) of the Book of orphan, to escape from the battles and slandering Revelation, and the seven seals chaste women” 7.9. RELIGION AND MYTHOLOGY 55

• A naming ceremony is held for babies on their sev- enth day of life

7.9.5 Judaism

• Shiv`a (another pronunciation of the Hebrew word The Seven Lucky Gods in seven”)), is the number“ ;שבעה :for 7—(Hebrew of days of mourning. Hence, one sits Shiva. As in • In Buddhism, Buddha walked 7 steps at his birth. Shiva (Judaism) • In Khasi mythology, the seven divine women who • The weekly Torah portion is divided into seven were left behind on earth and became the ances- aliyahs, and seven Jewish men (or boys over the tresses of all humankind. age of 13 who are considered men; Bar Mitzvah) are called up for the reading of these aliyahs during • The number of sleeping men in the Christian Shabbat morning services. of the "Seven Sleepers.”

• Seven blessings are recited under the chuppah dur- • The number of sages in ; their ing a Jewish wedding ceremony. wives are the deities referred to as the“Seven Moth- ers.” • A Jewish bride and groom are feted with seven days * of festive meals after their wedding, known as Sheva • In Iran, [16] German, Spanish, and other cultures Berachot (“Seven Blessings”). that speak Romance Languages, cats are said to have 7 as opposed to English, where cats are said to • The number of Ushpizzin (also known as the“Seven have 9 lives. Shepherds”) who visit the sukkah during the holiday • of Sukkot: Abraham, Isaac, Jacob, Joseph, Moses, In , the hero Cúchulainn is asso- Aaron, and David. ciated with the number 7. He has 7 fingers on each hand, 7 toes on each foot, and 7 pupils in each eye. • The number of nations God told the Israelites they In the Irish epic Táin Bó Cúailnge, Cúchulainn is 7 would displace when they entered the land of Israel years old when he receives his first weapons and de- (Deut. 7:1): the Hittite, the Girgashite, the Amorite, feats the armies of the Ulaidh and his son Connla is the Canaanite, the Perizzite, the Hivite, and the 7 years old when he is slain by Cúchulainn in“The Jebusite. Death of Aife's Only Son.” • In Breslov tradition, the seven orifices of the face (2 • In Galician , a seventh son will be a eyes, 2 nostrils, 2 ears, and the mouth) are called werewolf. In other , after six daughters, the “The Seven Candles.” seventh child is to be a son and a werewolf. In other European folklores, the seventh son of a seventh son • The 7 Sephirot of primary conscious emotion that will be a child with special powers of healing and are attributes of the creator. clairvoyant seeing, and in other cultures that seventh son of a seventh son would be a . 7.9.6 Astrology • In Guaraní mythology, the number of prominent legendary monsters. • The number of daughters of Atlas in the Pleiades (also called the "Seven Sisters") • In the eponymous British folk tale, Thomas the Rhymer went to live in the faerie kingdom for 7 • The number of saints appearing in a constellation years. called“Saptharishi Mandalam”in Indian astronomy. • Mahatma Gandhi's list of the destructive Seven Blunders of the World that cause violence: Wealth 7.9.7 Others without work, pleasure without conscience, knowl- edge without character, commerce without moral- Deity, being, or character ity, science without humanity, religion without sac- rifice and politics without principle. • * • The Seven Lucky Gods refer to the seven gods of The cosmogony of Urantia [17] gives an explana- good fortune in Japanese mythology. tion to the sacredness of the number 7 in some re- ligions, including those based on a triple deity: 7 • The number of archangels according to some sys- indeed is the number of all the possible combina- tems. tions of three elements taken one by one, two by 56 CHAPTER 7. 7 (NUMBER)

two, or by three. It therefore expresses all the as- 7.10 Science sociative possibilities of the three fundamental as- pects of the absolute (Christian Trinity, for exam- ple, or Trimurti in Hinduism) which organizes the 7.10.1 Astronomy Creation. The seventh of these combinations being the one that combines the three aspects, 7 therefore • The number of celestial objects in the solar system also expresses spiritual achievement. visible to the naked eye from Earth – the Sun, the Moon and the five classical naked eye planets: Mars, • In other traditions and teachings, seven is seen as the Mercury, Jupiter, Venus, and Saturn. perfect number that holds creation and the universe * together. [18] From the 7 deadly sins to 7 chakras, 7 • Messier object M7, a magnitude 3.5 open cluster in heavens and 7 planes of creation, this number holds the constellation Scorpius. great significance. • The New General Catalogue object NGC 7, a Place 14th magnitude spiral galaxy in the constellation Sculptor. • The number of main islands of mythological Atlantis. • The number of main stars in the asterism of the Big Dipper and the constellation of Orion. • The number of gateways traversed by Inanna during her descent into the underworld. • The Saros number of the solar eclipse series which • Cibola was one of the legendary Seven Cities of began on April 8, 2590 BC and ended on May Gold the Spanish thought existed. 16, 1310 BC. The duration of Saros series 2 was 1280.14 years, and it contained 72 solar eclipses. • In the Bahá'í faith, the text The Seven Valleys, by the Prophet-Founder Bahá'u'lláh, relates the journey of • The Saros number of the lunar eclipse series which the soul through the seven“valleys”of Search, Love, began on July 15, 2595 BC and ended on February Knowledge, Unity, Contentment, Wonderment, and 22, 1008 BC. The duration of Saros series 7 was finally True Poverty and Absolute Nothingness. 1586.6 years, and it contained 89 lunar eclipses. Scripture • The Roman numeral VII stands for white dwarfs in the Yerkes spectral classification scheme. • Circle Seven Koran, the holy scripture of the Moor- ish Science Temple of America • The Roman numeral VII (usually) stands for the seventh-discovered satellite of a planet or minor Thing, concept, or symbol planet (e.g., Jupiter VII).

• The Seven-Branched Sword in Japanese mythology. • The Theosophical teachings of Alice A. Bailey di- 7.10.2 Biology vide the human race into seven psychological types called the Seven Rays, which she calls“the basis of • Almost all mammals have 7 cervical vertebrae psychology". • The number of spots on a seven-spot ladybird • The minor symbol number of yang from the Taoist yin-yang. • There are 7 types of virus according to the Baltimore • The number of palms in an Egyptian Sacred Cubit. classification • The number of ranks in . • The number seven is of particular significance 7.10.3 Chemistry within Cherokee cosmology. • A neutral pH value between acidity and alkalinity Time • The atomic number of nitrogen • In British folklore, every 7 years the Queen of the Fairies pays a tithe to Hell (or possibly Hel) in the • Heptane is an alkane with 7 carbon atoms in each tale of Tam Lin. molecule 7.11. SOFTWARE 57

• Isaac Newton identified 7 colors of the rainbow – red, orange, yellow, green, blue, indigo, and violet

7.10.5 Psychology

• The Magical Number Seven, Plus or Minus Two

7.11 Software

Heptane is an alkane with 7 carbon atoms per molecule • 7 is the calling code of Russia. Only the North American Numbering Plan (+1) and Russia (+7) have a single-digit code.

• There are seven layers in the OSI model.

• Windows 7, one of the previous releases of Windows series of operating systems from Microsoft.

• Video game developer Bungie, famed for the Halo series of games, often references the number seven in their products.

• The number of periods, or horizontal rows of elements, in the periodic table 7.12 Temporal, seasonal and holi- • The number of principal levels for electrons days around the nucleus • The number of crystal systems: triclinic, mono- • The seven days of the week clinic, orthorhombic, tetragonal, trigonal, hexago- • nal, and cubic. The Kulin people of Australia living near the Dande- nong Ranges traditionally have seven seasons. Some • The number of crystal lattice systems: cubic (or iso- of the Native Americans of Montana also have seven metric), hexagonal, tetragonal, rhombohedral, or- seasons: chinook season, muddy spring, green sum- thorhombic, monoclinic and triclinic. mer, gold summer (or dry summer),“Indian”Sum- mer, late fall, and cold winter.

7.10.4 Physics • The Heptarchy, from the (Greek for seven realm), is the name applied by historians to the period (500– 850 CE) in English history after the Anglo-Saxon conquest of England, derived from the seven king- doms of Northumbria, Mercia, East Anglia, Essex, Kent, Sussex, and Wessex, which eventually merged to become the Kingdom of England during the early 10th century.

• September was the seventh month in the ancient Roman calendar, as its name indicates. After the re- form that led to the current order, the seventh month Isaac Newton's 7 colors of the rainbow is July. • Septidi was the seventh day of the decade in the • There are 7 SI base units.*[19] These are a coher- French Revolutionary Calendar. ent set of units defined to measure 7 basic phys- ical properties: metre, kilogram, second, ampere, • The 7th glyph of the Maya Calendar is Blue Hand, it kelvin, mole, candela. The 7 SI base units are used represents the days in creation and is associated with to define all other SI units, which are known as SI creative perfection. This is the glyph of the last day derived units.*[20] of their calendar that ends on December 21, 2012. 58 CHAPTER 7. 7 (NUMBER)

7.13 See also [13] “Example of teaching materials for pre-schoolers” (French) • Septenary (numeral system) [14] 21Then Peter came to Jesus and asked, 'Lord, how many times shall I forgive my brother when he sins against me? • Septenary (Theosophy) Up to seven times?' 22Jesus answered, 'I tell you, not seven • Seven climes times, but seventy times seven.' [15] “Sermon Illustrations”. Bible.org. Retrieved 2012-09- • Year Seven (School) 07. • Se7en (disambiguation) [16] "''Encyclopædia Britannica''“Number Symbolism"". Bri- tannica.com. Retrieved 2012-09-07. • Luck [17] Urantia Fondation website • One-seventh area triangle [18] Nunitus, Septem (2016). 777: Messages from the Elder • Diatonic scale Gods (777 Series Book 1). Elder Gods Press. p. 5. ASIN B01I0EUQGA

[19] “SI brochure, The seven SI base units”. Retrieved 2009- 7.14 Notes 09-11.

[1] “Sloane's A088165 : NSW primes”. The On-Line En- [20] “SI brochure, SI derived units”. Retrieved 2009-09-11. cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01.

[2] “Sloane's A050918 : Woodall primes”. The On-Line En- 7.15 References cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01. • Wells, D. The Penguin Dictionary of Curious and In- teresting Numbers London: Penguin Group (1987): [3] “Sloane's A088054 : Factorial primes”. The On-Line En- 70–71 cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01. [4] “Sloane's A031157 : Numbers that are both lucky and 7.16 External links prime”. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01. • The Significance of The Number Seven by Hillel ben [5] “Sloane's A035497 : Happy primes”. The On-Line En- David (Greg Killian) cyclopedia of Integer Sequences. OEIS Foundation. Re- • trieved 2016-06-01. The Number Seven in the Holy Bible by Ed F. Val- lowe [6] “Sloane's A003173 : Heegner numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- • The Bible Wheel: The Number Seven and Scripture tion. Retrieved 2016-06-01. • The Number 7 [7] Bryan Bunch, The Kingdom of Infinite Number. New • York: W. H. Freeman & Company (2000): 82 The Positive Integer 7 • [8] “Sloane's A003215 : Hex (or centered hexagonal) num- Prime curiosities: 7 bers”. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.

[9] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.67

[10] Eeva Törmänen (September 8, 2011). “Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista” . Tekniikka & Talous (in Finnish).

[11] “Education writing numerals in grade 1.”(Russian)

[12] Elli Harju (August 6, 2015). ""Nenosen seiska”teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?". Iltalehti (in Finnish). Chapter 8

8 (number)

“8th”, “Eight”, and “Eighth”redirect here. For All powers of 2 (2*x), have an aliquot sum of one less other uses, see 8 (disambiguation). than themselves. A number is divisible by 8 if its last 3 digits, when written 8 (eight /ˈeɪt/) is the natural number following 7 and pre- in decimal, are also divisible by 8, or its last 3 digits are ceding 9. 0 when written in binary. 8 and 9 form a Ruth–Aaron pair under the second defini- tion in which repeated prime factors are counted as often 8.1 In mathematics as they occur. There are a total of eight convex deltahedra. 8 is: A polygon with eight sides is an . Figurate num- bers representing (including eight) are called • a composite number, its proper divisors being 1, 2, octagonal numbers. and 4. It is twice 4 or four times 2. A polyhedron with eight faces is an octahedron.A • a , being 23 (two cubed), and is the first cuboctahedron has as faces six equal squares and eight number of the form p3, p being an integer greater equal regular triangles. than 1. A cube has eight vertices. • the first number which is neither prime nor Sphenic numbers always have exactly eight divisors. . The number 8 is involved with a number of interesting • the base of the number system, which is mostly mathematical phenomena related to the notion of Bott used with computers. In octal, one digit represents periodicity. For example, if O(∞) is the direct of 3 bits. In modern computers, a byte is a grouping of the inclusions of real orthogonal groups eight bits, also called an octet.

• a Fibonacci number, being 3 plus 5. The next Fi- O(1) ,→ O(2) ,→ ...,→ O(k) ,→ ... bonacci number is 13. 8 is the only positive Fi- bonacci number, aside from 1, that is a perfect then cube.*[1] ∼ • the only nonzero that is one less than πk+8(O(∞)) = πk(O(∞)) another perfect power, by Mihăilescu's Theorem. Clifford also display a periodicity of 8. For ex- • the order of the smallest non-abelian group all of ample, the algebra Cl(p + 8,q) is isomorphic to the al- whose subgroups are normal. gebra of 16 by 16 matrices with entries in Cl(p,q). We also see a period of 8 in the K-theory of spheres and in • the dimension of the octonions and is the highest the representation theory of the rotation groups, the lat- possible dimension of a normed division algebra. ter giving rise to the 8 by 8 spinorial chessboard. All of these properties are closely related to the properties of the • the first number to be the aliquot sum of two num- octonions. bers other than itself; the discrete biprime 10, and the square number 49. The spin group Spin(8) is the unique such group that ex- hibits the phenomenon of triality. It has an aliquot sum of 7 in the 4 member aliquot se- The lowest-dimensional even unimodular lattice is the 8- quence (8,7,1,0) being the first member of 7-aliquot tree. dimensional E8 lattice. Even positive definite unimodular

59 60 CHAPTER 8. 8 (NUMBER)

lattice exist only in dimensions divisible by 8. A figure 8 is the common name of a geometric shape, of- ten used in the context of sports, such as skating. Figure- eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.

8.1.1 In numeral systems Evolution of the numeral 8 from the Indians to the Europeans

8.1.2 List of basic calculations glyph used in India in the early centuries of the Common Era developed considerable variation, and in some cases 8.2 Etymology took the shape of a single wedge, which was adopted into and also gave rise to the) ٨ the Perso-Arabic tradition as English eight, from eahta, æhta, Proto- later Devanagari numeral ८; the alternative curved glyph Germanic *ahto is a direct continuation of Proto-Indo- also existed as a variant in Perso-Arabic tradition, where European *oḱtṓ(w)-, and as such cognate with Greek it came to look similar to our glyph 5. ὀκτώ and Latin octo-, both of which stems are reflected The numerals as used in Al-Andalus by the 10th century by the English prefix oct(o)-, as in the ordinal adjective were a distinctive western variant of the glyphs used in octaval or octavary, the distributive adjective is octonary. the Arabic-speaking world, known as ghubār numerals The adjective octuple (Latin octu-plus) may also be used (ghubār translating to "sand table"). In these numerals, as a noun, meaning“a set of eight items"; the diminutive the line of the 5-like glyph used in Indian manuscripts for octuplet is mostly used to refer to eight sibling delivered eight came to be formed in ghubār as a closed loop, which in one birth. was the 8-shape that became adopted into European use The Semitic numeral is based on a root *θmn-, whence in the 10th century.*[4] Akkadian smn-, Arabic ṯmn-, Hebrew šmn- etc. The Just as in most modern typefaces, in typefaces with text Chinese numeral (pinyin bā) is from an figures the 8 character usually has an ascender, as, for *priāt-, ultimately from Sino-Tibetan b-r-gyat or b-g-ryat which also yielded Tibetan brgyat. It has been argued that, as the cardinal number seven example, in . is the highest number of item that can universally be The infinity symbol ∞, described as “sideways figure cognitively processed as a single set, the etymology of the eight”is unrelated to the 8 glyph in origin; it is first used numeral eight might be the first to be considered compos- (in the mathematical meaning “infinity”) in the 17th ite, either as “twice four”or as “two short of ten”, century, and it may be derived from the Roman numeral or similar. The Turkic words for “eight”are from a for “one thousand”CIƆ, or alternatively from the final Proto-Turkic stem *sekiz, which has been suggested as Greek letter, ω. originating as a negation of eki “two”, as in “with- out two fingers”(i.e., “two short of ten; two fingers are The numeral eight in , developed in not being held up”);*[2] this same principle is found in Classical Greece by the 5th century BC, was written as Finnic *kakte-ksa, which conveys a meaning of “two Η, the eighth letter of the . before (ten)". The Proto-Indo-European reconstruction The Chinese numeral eight is written in two strokes, 八; *oḱtṓ(w)- itself has been argued as representing an old the glyph is also the 12th Kangxi radical. dual, which would correspond to an original meaning of “twice four”. Proponents of this “quaternary hypothe- sis”adduce the numeral nine, which might be built on the stem new-, meaning“new”(indicating the beginning of a 8.4 In science “new set of numerals”after having counted to eight).*[3] 8.4.1 Physics 8.3 Glyph • In nuclear physics, the second magic number. The modern 8 glyph, like all modern Hindu-Arabic nu- • merals (other than zero) originates with the Brahmi nu- In particle physics, the eightfold way is used to clas- merals. The Brahmi numeral for eight by the 1st century sify sub-atomic particles. was written in one stroke as a curve └┐ looking like an uppercase H with the bottom half of the left line and the • In statistical mechanics, the eight-vertex model has upper half of the right line removed. However the eight 8 possible configurations of arrows at each vertex. 8.5. IN TECHNOLOGY 61

8.4.2 Astronomy • Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, • Messier object M8, a magnitude 5.0 nebula in the each plate representing a row of large modified cilia. constellation of Sagittarius. • The eight-spotted forester (genus Alypia, family • The New General Catalogue object NGC 8, a double Zygaenidae) is a diurnal moth having black wings star in the constellation Pegasus. with brilliant white spots.

• The Saros number of the solar eclipse series which • The ascus in fungi of the class Ascomycetes, fol- began on 2579 BCE 7 March and ended on 1281 lowing nuclear fusion, bears within it typically eight BCE 26 April. The duration of Saros series 8 was ascospores. 1298.1 years, and it contained 73 solar eclipses. • Herbs of genus Coreopsis (tickseed) have showy • The Saros number of the lunar eclipse series which flower heads with involucral bracts in two distinct began on 2512 BCE 27 July and ended on 961 BCE series of eight each. 13 February. The duration of Saros series 7 was 1550.6 years, and it contained 87 lunar eclipses. • Timothy Leary identified a hierarchy of eight levels of consciousness. • Since the demotion of Pluto to a dwarf planet on August 24, 2006, in our Solar System, eight of the • In human adult dentition there are eight teeth in each bodies orbiting the Sun are considered to be planets. quadrant. The eighth tooth is the so-called wisdom tooth.

8.4.3 Chemistry • There are eight cervical nerves on each side in man and most mammals. • The atomic number of oxygen. • The number of allotropes of carbon. 8.5 In technology • The most stable allotrope of a sulfur molecule is made of eight sulfur atoms arranged in a rhombic form.

• The maximum number of electrons that can occupy a valence shell.

• The red pigment lycopene consists of eight isoprene units.

8.4.4 Geology

• A disphenoid crystal is bounded by eight scalene tri- angles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal.

8.4.5 Biology

• All spiders, and more generally all arachnids, have NATO signal flag for 8 eight legs. Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes. • A byte is eight bits • The octopus and its cephalopod relatives in genus Argonauta have eight arms (tentacles). • Many (mostly historic) computer architectures are eight-bit, among them the Nintendo Entertainment • Compound coelenterates of the subclass or order System Alcyonaria have polyps with eight-branched tenta- cles and eight septa. • Standard-8 and Super-8 are 8 mm film formats

• Sea anemones of genus Edwardsia have eight • Video8, Hi8 and Digital8 are related 8 mm video mesenteries. formats 62 CHAPTER 8. 8 (NUMBER)

• On most phones, the 8 key is associated with the letters T, U, and V, but on the BlackBerry it is the key for B, N, and X.

• An eight may refer to an eight-cylinder engine or au- tomobile. A V8 engine is an internal combustion engine with eight cylinders configured in two banks (rows) of four forming a “V”when seen from the end.

• A figure-eight knot (so named for its configuration) is a kind of stopper knot.

• The number eight written in parentheses is the code for the musical note in Windows Live Messenger

• On a seven-segment display, when an 8 is illumi- nated, all the display bulbs are on.

In Buddhism, the 8-spoked Dharmacakra represents the Noble 8.5.1 In measurement Eightfold Path

• The SI prefix for 10008 is yotta (), and for its re- ciprocal, yocto (y). • The Dharmacakra, a Buddhist symbol, has eight spokes. The Buddha's principal teaching—the Four • In liquid measurement (United States customary Noble Truths—ramifies as the Noble Eightfold Path units), there are eight fluid ounces in a cup, eight and the Buddha emphasizes the importance of the pints in a gallon and eight tablespoonfuls in a gill. eight attainments or jhanas.

• There are eight furlongs in a mile. • In Buddhism, the branches of the Eightfold Path are embodied by the Eight Great • The clove, an old English unit of weight, was equal Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, to eight pounds when measuring cheese. Maitreya, Ksitigarbha, Nivaranavishkambhi, • An eight may be an article of clothing of the eighth Akasagarbha, and Samantabhadra). These are size. later (controversially) associated with the Eight Consciousnesses according to the Yogacara school • Force eight is the first wind strength attributed to of thought: consciousness in the five senses, a gale on the Beaufort scale when announced on a thought-consciousness, self-consciousness, and Shipping Forecast. unconsciousness-'consciousness' or 'store-house consciousness' (alaya-vijñana). The 'irreversible' state of enlightenment, at which point a Bodhisattva goes on 'autopilot', is the Eight Ground or bhūmi. 8.6 In culture In general, 'eight' seems to be an auspicious number for Buddhists, e.g., the 'eight auspicious symbols' 8.6.1 Architecture (the jewel-encrusted parasol; the goldfish (always shown as a pair, e.g., the glyph of Pisces); the • Various types of buildings are usually eight-sided self-replenishing amphora; the white kamala lotus- (octagonal), such as single-roomed gazebos and flower; the white conch; the eternal (Celtic-style, multi-roomed pagodas (descended from stupas; see infinitely looping) knot; the banner of imperial religion section below). victory; the eight-spoked wheel that guides the ship of state, or that symbolizes the Buddha's teaching). • Eight caulicoles rise out of the leafage in a Similarly, Buddha's birthday falls on the 8th day of Corinthian capital, ending in leaves that support the the 4th month of the . volutes. Judaism 8.6.2 In religion, folk belief and divination • The religious rite of brit milah (commonly known as Buddhism circumcision) is held on a baby boy's eighth day of life. 8.6. IN CULTURE 63

• Hanukkah is an eight-day Jewish holiday that starts • Eight (⼋ hachi, ya) is also considered a lucky num- on the 25th day of Kislev. ber in Japan, but the reason is different from that in Chinese culture. Eight gives an idea of growing • Shemini Atzeret (Hebrew: “Eighth Day of Assem- prosperous, because the letter (⼋) broadens gradu- bly”) is a one-day Jewish holiday immediately fol- ally. lowing the seven-day holiday of Sukkot. • The Japanese thought eight (や ya) as a holy number Christianity in the ancient times. The reason is less well under- stood, but it is thought that it is related to the fact • The spiritual Eighth Day, because the number 7 they used eight to express vaguely refers to the days of the week (which repeat them- such as manyfold (やえはたえ Yae Hatae) (liter- selves). ally, eightfold and twentyfold), many clouds (やく • The number of Beatitudes. も Yakumo) (literally, eight clouds), millions and millions of Gods (やおよろずのかみ Yaoyorozu • 1 Peter 3:20 states that there were eight people on no Kami) (literally, eight millions of Gods), etc. It Noah's Ark. is also guessed that the ancient Japanese gave im- • The Antichrist is the eighth king in the Book of Rev- portance to pairs, so some researchers guess twice よ elation.*[5] as four ( yo), which is also guessed to be a holy number in those times because it indicates the world Islam (north, south, east, and west) might be considered a very holy number. • In Islam, eight is the number of angels carrying the • In , 8 is the number of building, and in throne of Allah in heaven. some theories, also the number of destruction. • The number of gates of heaven. In astrology Other • In astrology, Scorpio is the 8th astrological sign of • The Eight Immortals are Chinese demigods. the Zodiac. • In Wicca, there are eight Sabbats, festivals, seasons, • In the Middle Ages, 8 was the number of “unmov- or spokes in the Wheel of the Year. ing”stars in the sky, and symbolized the perfection • In Hinduism, it is the number of wealth, abundance. of incoming planetary energy. The Goddess Lakshmi has eight forms. There are eight nidhi, or seats of wealth. There are also eight Guardians of the directions. 8.6.3 In music and dance

• In Taoism and Chinese cosmology, the eight tri- • A note played for one-eighth the duration of a whole grams of the Bagua.“Bagua”literally means“eight note is called an eighth note, or quaver. symbols”. • An octave, the interval between two musical notes • In Ancient , the Ogdoad repre- with the same letter name (where one has double the sents the eight primordial deities of creation. frequency of the other), is so called because there • In Scientology there are 8 dynamics of existence are eight notes between the two on a standard major or minor diatonic scale, including the notes them- selves and without chromatic deviation. The ecclesi- As a lucky number astical modes are ascending diatonic musical scales of eight notes or tones comprising an octave. • The number eight is considered to be a lucky num- ber in Chinese and other Asian cultures.*[6] Eight • There are eight notes in the octatonic scale. (八; accounting 捌; pinyin bā) is considered a lucky • number in Chinese culture because it sounds like the There are eight musicians in a double quartet or an word meaning to generate wealth (發 (T) 发 (S); octet. Both terms may also refer to a musical com- Pinyin: fā). Property with the number 8 may be position for eight voices or instruments. valued greatly by Chinese. For example, a Hong • Caledonians is a square dance for eight, resembling Kong number plate with the number 8 was sold the quadrille. for $640,000.*[7] The opening ceremony of the Summer Olympics in started at 8 seconds • with the number eight in their title include and 8 minutes past 8 pm (local time) on 8 August 8 by the Swedish band Arvingarna and The Meaning 2008.*[8] of 8 by Minnesota indie rock band Cloud Cult 64 CHAPTER 8. 8 (NUMBER)

• Dream Theater's eighth album Octavarium contains 8.6.5 In sports and other games many different references to the number 8, includ- ing the number of songs and various aspects of the music and cover artwork •“Eight maids a-milking”is the gift on the eighth day of Christmas in the carol "The Twelve Days of Christmas" • The 8-track cartridge is a musical recording format • "#8”is the stagename of Slipknot vocalist Corey Taylor •“Too Many Eights”is a song by Athens, Georgia's Supercluster. • Eight Seconds, a Canadian musical group popular in the 1980s with their most notable song “Kiss You (When It's Dangerous)" An 8-ball in billiards • "Eight Days a Week" is a #1 single for the music • group The Beatles. Eight-ball pocket billiards is played with a cue ball and 15 numbered balls, the black ball numbered 8 • Figure 8 is the fifth studio album by singer- being the middle and most important one, as the Elliott Smith, released in the year 2000. winner is the player or side that legally pockets it after first pocketing its numerical group of 7 object balls (for other meanings see Eight ball (disambigua- 8.6.4 In film and television tion)). • 8 Guys is a 2003 short film written and directed by • Balklines divide a billiards table into eight outside Dane Cook compartments or divisions called balks. In balkline • 8 Man (or Eightman): 1963 Japanese manga and billiards the table also has eight anchor spaces. anime superhero. • In the 2008 Games of the XXIX Olympiad, the of- • 8 Mile is a 2002 film directed by Curtis Hanson ficial opening was on 08/08/08 at 8:8:08 p.m. local time in Beijing, China. • 8mm is a 1999 film directed by Joel Schumacher • In football, the number 8 has historically been the • 8 Women (Original French title: 8 femmes) is a 2002 number of the Central Midfielder. film directed by François Ozon • in baseball scorekeeping, the center fielder is desig- • Eight Below is a 2006 film directed by Frank Mar- nated as number 8. shall • In rugby union, the only position without a proper • Eight Legged Freaks is a 2002 film directed by Ellory name is the Number 8, a forward position. Elkayem • In most rugby league competitions (though not the • Eight Men Out is a 1988 film directed by John Sayles European Super League, which uses static squad • Jennifer Eight, also known as Jennifer 8, is a 1992 numbering), one of the two starting props wears the film written and directed by Bruce Robinson number 8. • Eight Is Enough is an American television comedy- • In rowing an “eight”refers to a sweep-oar racing drama series boat with a crew of eight rowers plus a coxswain. • In Stargate SG-1 and Stargate Atlantis, dialing an 8- • In chess, each side has eight pawns and the board chevron address will open a wormhole to another is made of 64 squares arranged in an eight by eight galaxy. lattice. The eight queens puzzle is a challenge to ar- range eight queens on the board so that none can • The Hateful Eight is a 2015 American western mys- capture any of the others. tery film written and directed by Quentin Tarantino. • In the game of eights or Crazy Eights, each succes- • Plus 8 is an American reality television show sive player must play a card either of the same suit • The Weather Channel's segment Local on the 8s fea- or of the same rank as that played by the preceding tures daily and weekly forecasts for major cities in player, or may play an eight and call for any suit. The the United States object is to get rid of all one's cards first. 8.6. IN CULTURE 65

• In poker, a "Dead Man's Hand" consists of two pairs; • The New Orleans Saints have never for- aces and eights. While playing poker in a saloon in mally retired the number, but have not Deadwood, Dakota Territory (now South Dakota), reissued #8 since Archie Manning left the Wild Bill Hickok held this hand when he was shot team during the 1982 season. from behind and killed. • In the NHL: • The throw of eight in craps is called (in rhyming • The Anaheim Ducks retired Teemu slang) an eighter from Decatur. Selännes number 8 in 2015.*[9] • The Boston Bruins, for Hall of Famer • While racing for DEI, Dale Earnhardt, Jr. raced in Cam Neely.*[10] the #8 car in the NASCAR Sprint Cup Series . • The Minnesota North Stars, for Bill • Ralph Earnhardt, Dale Earnhardt, Jr.'s grandfather Goldsworthy. The franchise, now the drove #8 in the NASCAR Sportsman Division and Dallas Stars, continues to honor the num- NASCAR Grand National Series ber. • • In Israeli Football Premiere league, Maccabi Tel The Ottawa Senators, for Frank Finnigan, Aviv legendary star- Avi Nimni wore the number 8 the final surviving player from the original and it retired with him. Senators team that won the 1927 Stanley Cup, and an ambassador for the group • Nintendo's Mario Kart series: that returned the NHL to Ottawa. • • The number of characters required to compete The St. Louis Blues, for Barclay Plager. in each race. • The Quebec Nordiques retired the num- ber for Marc Tardif. However, the • The number of cups since Mario Kart . team moved to Denver as the Colorado • The number on the logo for Mario Kart 8 Avalanche after the 1994–95 season, • The jersey number 8 has been retired by several and the Avalanche placed all of the North American sports teams in honor of past play- Nordiques' retired numbers back into cir- ing greats or other key figures: culation. • In the NBA: • In Major League Baseball: • The Los Angeles Lakers, once worn by • The Baltimore Orioles, for Hall of Famer Kobe Bryant. Cal Ripken, Jr. • In Formula One the number 8 is used by • The Boston Red Sox, for Hall of Famer Romain Grosjean for Haas F1 Team Carl Yastrzemski. • The Cincinnati Reds, for Hall of Famer Joe Morgan. 8.6.6 In foods • The Montreal Expos, for Hall of Famer Gary Carter. The franchise continues to • Nestlé sells a brand of chocolates filled with honor the number in its current incar- peppermint-flavoured cream called After Eight, re- nation as the Washington Nationals (al- ferring to the time 8 p.m. though it initially planned to all of • There are eight vegetables in V8 juice the Expos' retired numbers). • The New York Yankees, for Hall of • In cooking recipes, there are approximately 8 Famers Yogi Berra and Bill Dickey. pinches to a teaspoon • The Pittsburgh Pirates, for Hall of Famer Willie Stargell. 8.6.7 In literature • In the NFL: • The St. Louis Cardinals, for Hall of • Eights may refer to octosyllabic, usually iambic, Famer Larry Wilson. The franchise, now lines of verse. the Arizona Cardinals, continues to honor • the number. The drott-kvaett, an Old Icelandic verse, consisted of a stanza of eight regular lines. • The San Francisco 49ers, for Hall of Famer Steve Young. • In Terry Pratchett's Discworld series, eight is a magi- • The Dallas Cowboys have a policy of not cal number and is considered . Eight is not safe retiring numbers. However, they have not to be said by wizards on the Discworld and is the issued #8 since the retirement of Hall of number of Bel-Shamharoth. Also, there are eight Famer Troy Aikman after the 2000 sea- days in a Disc week and eight colours in a Disc spec- son. trum, the eighth one being octarine. 66 CHAPTER 8. 8 (NUMBER)

• Lewis Carroll's poem The Hunting of the Snark has [3] the hypothesis is discussed critically (and rejected as 8 “fits”(cantos), which is noted in the full name “without sufficient support”) by Werner Winter, 'Some “The Hunting of the Snark - An Agony, in Eight Fits thought about Indo-European numerals' in: Jadranka Gvozdanović (ed.), Indo-European Numerals, Walter de • 8 apparitions appear to Macbeth in Act 4 scene 1 of Gruyter, 1992, 14f. Shakespeare's Macbeth as representations of the 8 [4] Georges Ifrah, The Universal History of Numbers: From descendants of Banquo Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.68. 8.6.8 In slang [5] “Life Application New Testament Commentary”, Bruce • An “eighth”is a common measurement of B. Barton. Tyndale House Publishers, Inc., 2001. ISBN marijuana, meaning an eighth of an ounce. It is also 0-8423-7066-8, ISBN 978-0-8423-7066-0. p. 1257 a common unit of sale for psilocybin mushrooms. [6] Ang, Swee Hoon (1997). “Chinese consumers' Also, an eighth of an ounce of cocaine is commonly perception of alpha-numeric brand names”. Jour- * referred to as an “8-ball.” [11] nal of Consumer Marketing. 14 (3): 220–233. doi:10.1108/07363769710166800. • Referring to the shape of the numeral, eight was for- merly represented in bingo slang as“One Fat Lady” [7] Steven C. Bourassa; Vincent S. Peng (1999). “Hedonic . Eighty-eight was “Two Fat Ladies”. Prices and House Numbers: The Influence of Feng Shui” (PDF). International Real Estate Review. 2 (1): 79–93. • The numeral “8”is sometimes used in informal Archived from the original (PDF) on 13 April 2015. writing and Internet slang to represent the syllable [8]“Patriot games: China makes its point with greatest show” “ate”, as in writing “H8”for “hate”, or “con- by Richard Williams, , published 9 August ” “ ” gratul8ions for congratulations . Avril Lavigne's 2008 song "Sk8er Boi" uses this convention in the title. [9] “Anaheim Ducks retire Teemu Selanne's No. 8”. •“Section 8”is common U.S. slang for “crazy”, NHL.com. based on the U.S. military's Section 8 discharge for mentally unfit personnel [10] “A to Encyclopaedia of Ice Hockey - Ne”. [11] “Cocaine - Frequently Asked Questions”. thegooddrugs- • The Housing Choice Voucher Program, operated by guide.com. the United States Department of Housing and Urban Development, is commonly referred to as the Sec- tion 8 program, as this was the original section of the Act which instituted the program 8.9 External links

• In Colombia and Venezuela, “volverse un ocho” • The Octonions, John C. Baez (meaning to tie oneself in a figure 8) refers to getting in trouble or contradicting one's self.

• In China, '8' is used in chat speak as a term for part- ing. This is due to the closeness in pronunciation of '8' (bā) and the English word 'bye'.

8.7 See also

• The Magical Number Seven, Plus or Minus Two

8.8 References

[1] Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 88

[2] Etymological Dictionary of Turkic Languages: Common Turkic and Interturkic stems starting with letters «L», «M», «N», «P», «S», Vostochnaja Literatura RAS, 2003, 241f. (altaica.ru) Chapter 9

9 (number)

This article is about the number. For other uses, see 9 • In Indian aesthetics, there are nine kinds of Rasa. (disambiguation). 9.3.2 Chinese culture 9 (nine /ˈnaɪn/) is the natural number following 8 and pre- ceding 10. • Nine (九 pinyin jiǔ) is considered a good number in Chinese culture because it sounds the same as the word “long-lasting”(久 pinyin jiǔ). 9.1 Alphabets and codes • Nine is strongly associated with the Chinese dragon, • In the NATO phonetic alphabet, the digit 9 is called a symbol of magic and power. There are nine forms “Niner”. of the dragon, it is described in terms of nine at- tributes, and it has nine children. It has 117 scales • Five-digit produce PLU codes that begin with 9 are – 81 yang (masculine, heavenly) and 36 yin (femi- organic. nine, earthly). All three numbers are multiples of 9 (9 × 13 = 117, 9 × 9 = 81, 9 × 4 = 36)*[1] as well as having the same digital root of 9.

9.2 Commerce • The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the • Common terminal digit in psychological pricing. .

• The circular altar platform (Earthly Mount) of the 9.3 Culture and mythology Temple of Heaven has one circular plate in the center, surrounded by a ring of nine plates, then by a ring of 18 plates, and so on, for a total of nine 9.3.1 Indian culture rings, with the outermost having 81 = 9 × 9 plates.

Nine is a number that appears often in Indian Culture and • The name of the area called Kowloon in mythology. Some instances are enumerated below. literally means: nine dragons.

• The nine-dotted line (Chinese: 南 海 九 段 线; • Nine influencers are attested in Indian astrology. pinyin: nánhǎi jiǔduàn xiàn; literally: “Nine- • In the Vaisheshika of , segment line of the South China Sea”) delimits cer- there are nine universal substances or elements: tain island claims by China in the South China Sea. Earth, Water, Air, Fire, Ether, Time, Space, Soul, • The nine-rank system was a civil service nomination and Mind. system used during certain Chinese dynasties. • Navaratri is a nine-day festival dedicated to the nine forms of Durga. 9.3.3 Ancient Egypt • , meaning “9 jewels”may also refer to Navaratnas - accomplished courtiers, Navratan - a • The nine bows is a term used in Ancient Egypt to kind of dish, or a form of architecture. represent the traditional enemies of Egypt. • According to Yoga, the human body has nine doors • The Ennead is a group of 9 Egyptian deities, who, - two eyes, two ears, the mouth, two nostrils, and the in the some versions of the Osiris myth, judged openings for defecation and procreation. whether Horus or Set should inherit Egypt.

67 68 CHAPTER 9. 9 (NUMBER)

9.3.4 European culture character usually has a descender, as, for example, in

• The Nine Worthies are nine historical, or semi- . legendary figures who, in the Middle Ages, were be- lieved to personify the ideals of chivalry. This numeral resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, • In Norse mythology, the universe is divided into the 9 is often underlined, as is done for the 6. Another nine worlds which are all connected by the world tree distinction from the 6 is that it is sometimes handwritten Yggdrasil. with a straight stem, resembling a raised lower-case letter q.

9.3.5 Greek mythology 9.5 Idioms and popular phrases • The nine Muses in Greek mythology are Calliope (epic poetry), Clio (history), Erato (erotic po- •“to go the whole nine yards-" etry), Euterpe (lyric poetry), Melpomene (tragedy), • Polyhymnia (song), Terpsichore (dance), Thalia “A cat-o'-nine-tails suggests perfect punishment and (comedy), and Urania (astronomy). atonement.”--Robert Ripley. •“A cat has nine lives” • It takes nine days (for an anvil) to fall from heaven to earth, and nine more to fall from earth to Tartarus •“to be on cloud nine” —a place of torment in the underworld. •“A stitch in time saves nine” • Leto labored for nine days and nine nights for •“found true 9 out of 10 times” Apollo, according to the Homeric Hymn to Delian Apollo. •“possession is nine tenths of the law” • The word“K-9”pronounces the same as canine and is used in many U.S. police departments to denote 9.4 Evolution of the glyph the police dog unit. Despite not sounding like the of the word canine in other languages, See also: Hindu-Arabic numeral system many police and military units around the world use the same designation.

According to Georges Ifrah, the origin of the 9 integers • Someone dressed “to the nines”is dressed up as can be attributed to the ancient Indian civilization, and much as they can be. was adopted by subsequent civilizations in conjunction • “ ” * In North American urban culture, nine is a slang with the 0. [2] word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.

9.6 Internet

• The 9 on Yahoo!, hosted by Maria Sansone, was a In the beginning, various Indians wrote 9 similar to the daily video compilation show, or vlog, on Yahoo! modern closing question mark without the bottom dot. featuring the nine top “web finds”of the day. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the 9.7 Literature @ character encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued • There are nine of Hell in Dante's Divine Com- beyond the circle downwards, as the 3-look-alike became edy. smaller. Soon, all that was left of the 3-look-alike was a • The Nine Bright Shiners, characters in Garth Nix's squiggle. The Arabs simply connected that squiggle to the Old Kingdom trilogy. The Nine Bright Shiners was downward stroke at the middle and subsequent European a 1930s book of poems by Anne Ridler*[3] and a change was purely cosmetic. 1988 fiction book by Anthea Fraser;*[4] the name While the shape of the 9 character has an ascender in derives from“a very curious old semi-pagan, semi- most modern typefaces, in typefaces with text figures the Christian”song.*[5] 9.8. MATHEMATICS 69

• The Nine Tailors is a 1934 mystery novel by British In bases 12, 18 and 24, nine is a 1-automorphic number writer Dorothy L. Sayers, her ninth featuring sleuth and in base 6 a 2-automorphic number (displayed as '13'). Lord Peter Wimsey. A polygon with nine sides is called a nonagon or en- * • Nine Unknown Men are, in occult legend, the cus- neagon. [9] A group of nine of anything is called an en- todians of the sciences of the world since ancient nead. times. In base 10 a positive number is divisible by nine if and only if its digital root is 9.*[10] That is, if any natural • In J.R.R. Tolkien's Middle-earth, there are nine number is multiplied by nine, and repeatedly add the dig- rings of power given to men, and consequently, nine its of the answer until it is just one digit, the sum will be ringwraiths. nine: • Additionally, The Fellowship of the Ring con- • 2 × 9 = 18 (1 + 8 = 9) sists of nine companions, representing the free races and also as a positive mirror of the nine • 3 × 9 = 27 (2 + 7 = 9) ring wraiths. • 9 × 9 = 81 (8 + 1 = 9) • In Lorien Legacies there are nine Garde sent to • Earth. 121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9) • 234 × 9 = 2106 (2 + 1 + 0 + 6 = 9) • Number Nine is a character in Lorien Legacies. • 578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = • In the A Song of Ice and Fire series, there are nine 27; 2 + 7 = 9) regions of Westeros (the Crownlands, the North, the Riverlands, the Westerlands, the Reach, the • 482729235601 × 9 = 4344563120409 (4 + 3 + 4 + Stormlands, the Vale of Arryn, the Iron Islands and 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = Dorne). 9) • Additionally, there is a group of nine city- There are other interesting patterns involving multiples of states in western Essos known collectively as nine: the Free Cities (Braavos, Lorath, Lys, Myr, Norvos, Pentos, Qohor, Tyrosh and Volantis). • 12345679 × 9 = 111111111 • 12345679 × 18 = 222222222 9.8 Mathematics • 12345679 × 81 = 999999999

Nine is a composite number, its proper divisors being 1 This works for all the multiples of 9. n = 3 is the only and 3. It is 3 times 3 and hence the third square number. * other n > 1 such that a number is divisible by n if and Nine is a Motzkin number. [6] It is the first composite only if its digital root is n. In base N, the divisors of N − lucky number, along with the first composite odd number. 1 have this property. Another consequence of 9 being 10 9 is the only positive perfect power that is one more than − 1, is that it is also a . another positive perfect power, by Mihăilescu's Theorem. The difference between a base-10 positive integer and the Nine is the highest single-digit number in the decimal sys- sum of its digits is a whole multiple of nine. Examples: tem. It is the second non-unitary square prime of the form 2 (p ) and the first that is odd. All subsequent squares of • The sum of the digits of 41 is 5, and 41 − 5 = 36. The this form are odd. It has a unique aliquot sum 4 which is digital root of 36 is 3 + 6 = 9, which, as explained itself a square prime. Nine is; and can be, the only square above, demonstrates that it is divisible by nine. prime with an aliquot sum of the same form. The aliquot sequence of nine has 5 members (9,4,3,1,0) this num- • The sum of the digits of 35967930 is 3 + 5 + 9 + 6 + ber being the second composite member of the 3-aliquot 7 + 9 + 3 + 0 = 42, and 35967930 − 42 = 35967888. tree. It is the aliquot sum of only one number the discrete The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + semiprime 15. 8 + 8 + 8 = 54, 5 + 4 = 9. There are nine Heegner numbers.*[7] Subtracting two base-10 positive integers that are trans- * 1 * Since 9 = 3 2 , 9 is an . [8] positions of each other yields a number that is a whole 8 and 9 form a Ruth-Aaron pair under the second defi- multiple of nine. Examples: nition that counts repeated prime factors as often as they occur. • 41 − 14 = 27 (2 + 7 = 9) 70 CHAPTER 9. 9 (NUMBER)

• 36957930 − 35967930 = 990000, a multiple of downtime of no more than five minutes per year − typi- nine. cally a very high degree of reliability; but never 100%.

This works regardless of the number of digits that are transposed. For example, the largest transposition of 9.9 Organizations 35967930 is 99765330 (all digits in descending order) and its smallest transposition is 03356799 (all digits in • Divine Nine—The National Pan-Hellenic Council ascending order); subtracting pairs of these numbers pro- (NPHC) is a collaborative organization of nine his- duces: torically African American, international Greek let- tered fraternities and sororities. • 99765330 − 35967930 = 63797400; 6 + 3 + 7 + 9 + 7 + 4 + 0 + 0 = 36; 3 + 6 = 9. • 99765330 − 03356799 = 96408531; 9 + 6 + 4 + 0 9.10 Places and thoroughfares + 8 + 5 + 3 + 1 = 36; 3 + 6 = 9. • 35967930 − 03356799 = 32611131; 3 + 2 + 6 + 1 • List of highways numbered 9 + 1 + 1 + 3 + 1 = 18; 1 + 8 = 9. • Ninth Avenue is a major avenue in Manhattan.

Casting out nines is a quick way of testing the calculations • Outside the Norwegian coast, east of the Barents sea of sums, differences, products, and of integers, number 9, in a 260 meters depth. known as long ago as the 12th Century.*[11] Every prime in a of the first kind with a length of 4 or greater is congruent to 9 mod 10 (the only 9.11 Religion and philosophy exception being the chain 2, 5, 11, 23, 47). Six recurring nines appear in the decimal places 762 through 767 of π, see Six nines in . If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.*[12] If dividing a number by the amount of 9s correspond- ing to its number of digits, the number is turned into a . (e.g. 274/999 = 0.274274274274...) It appears in the , preceded by the terms 4, 5, 7 (it is the sum of the first two of these).*[13] Nine is the binary complement of number six: 9 = 1001 6 = 0110

9.8.1 List of basic calculations

9.8.2 Numeral systems

9.8.3 Probability A nine-pointed star

In probability, the nine is a logarithmic measure of prob- ability of an event, defined as the negative of the base-10 • Nine, as the highest single-digit number (in base logarithm of the probability of the event's complement. ten), symbolizes completeness in the Bahá'í Faith. For example, an event that is 99% likely to occur has In addition, the word Bahá' in the Abjad notation an unlikelihood of 1% or 0.01, which amounts to −log10 has a value of 9, and a 9-pointed star is used to 0.01 = 2 nines of probability. Zero probability gives zero symbolize the religion. nines (−log10 1 = 0). A 100% probability is considered to be impossible in most circumstances: that results in • The number 9 is revered in Hinduism and consid- infinite improbability. The effectivity of processes and ered a complete, perfected and divine number be- the availability of systems can be expressed (as a rule of cause it represents the end of a cycle in the decimal thumb, not explicitly) as a series of“nines”. For exam- system, which originated from the Indian subconti- ple, “five nines”(99.999%) availability implies a total nent as early as 3000 BC. 9.13. SPORTS 71

• In Buddhism, Gautama Buddha, was believed to 9.12.3 Physiology have nine virtues, which he was (1) Accomplished, (2) Perfectly Enlightened, (3) Endowed with knowl- A human pregnancy normally lasts nine months, the basis edge and Conduct or Practice, (4) Well-gone or of Naegele's rule. Well-spoken, (5) the Knower of worlds, (6) the Guide Unsurpassed of men to be tamed, (7) the Teacher of gods and men, (8) Enlightened, and (9) 9.13 Sports Blessed. • Important Buddhist usually involve nine monks. • The first nine days of the Hebrew month of Av are collectively known as“The Nine Days”(Tisha HaYamim), and are a period of semi-mourning lead- ing up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed. • Nine is a significant number in Norse Mythology. Odin hung himself on an ash tree for nine days to learn the runes. • The Fourth Way Enneagram is one system of knowl- edge which shows the correspondence between the 9 integers and the circle. Billiards: A Nine-ball rack with the no. 9 ball at the center • In the Christian angelic hierarchy there are 9 choirs of angels. 9.13.1 Auto racing • Ramadan, the month of fasting and , is the ninth month of the Islamic calendar. • A car in the Sprint Cup Series currently owned by Richard Petty Motorsports. The number was most notably borne by the car that Bill Elliott drove 9.12 Science to the Cup Series title in 1988 with Melling Rac- ing. Evernham Motorsports, the predecessor team 9.12.1 Astronomy to Richard Petty Motorsports, acquired the number in 2001 when Elliott joined that team after a brief • Before 2006 (when Pluto was officially designated stint as a driver-owner. Elliott used this number as a non-planet), there were nine planets in the solar again through the 2003 season. Kasey Kahne has system. driven the 9 car since 2004. Currently, the 9 car is not used in NASCAR. • Messier object M9 is a magnitude 9.0 globular clus- ter in the constellation Ophiuchus. • In Formula One the number 9 is used by Marcus Ericsson for Sauber. • The New General Catalogue object NGC 9, a spiral galaxy in the constellation Pegasus • The Saros number of the lunar eclipse series which 9.13.2 Baseball began on June 26, 2501 BC and ended on September 5, 1167 BC. The duration of Saros series 9 was • In baseball, nine represents the right fielder's posi- 1334.23 years, and it contained 75 lunar eclipses. tion. • The Saros number of the solar eclipse series which • The number of innings in a regulation, non-tied began on February 6, 2568 BC and ended on April 4, game of baseball. 1252 BC. The duration of Saros series 9 was 1316.2 • years, and it contained 74 solar eclipses. The number of players on the field including the pitcher.

9.12.2 Chemistry • The number worn by Roy Hobbs in the movie The Natural. • The purity of chemicals (see Nine (purity)) • The score of a forfeit in a league where the game • Nine is the atomic number of fluorine. would ordinarily last nine innings is 9-0. 72 CHAPTER 9. 9 (NUMBER)

• NINE: A Journal of Baseball History and Culture, • The Chicago Blackhawks, for Hall of Famer published by the University of Nebraska Press*[14] Bobby Hull. • The Dallas Stars, for Mike Modano. • 9.13.3 Billiards The Detroit Red Wings, for Hall of Famer Gordie Howe. • Nine-ball is the standard professional pocket • The Edmonton Oilers, for Hall of Famer billiards variant played in the United States. Glenn Anderson. • The Montreal Canadiens, for Hall of Famer Maurice Richard. 9.13.4 Rugby • The New York Islanders, for Hall of Famer • In rugby league, the jersey number assigned to the Clark Gillies. hooker. • The New York Rangers, for Hall of Famer Andy Bathgate and Adam Graves. • In rugby union, the number worn by the starting • The Toronto Maple Leafs have a policy of not scrum-half. retiring numbers unless the player honoured either died or suffered a career-ending inci- 9.13.5 Soccer dent while a member of the team. Other play- ers whose numbers would otherwise be retired instead have their numbers enshrined by the • In association football (soccer) the centre- team as“Honoured Numbers”, which remain forward/striker traditionally (since at least the in circulation for future players. The number fifties) wears the number 9 shirt. 9 is currently honoured for Hall of Famers Ted Kennedy and Charlie Conacher. 9.13.6 All sports • The first NHL incarnation of the Winnipeg Jets, also for Hull. Although the Jets moved The jersey number 9 has been retired by several North from Winnipeg to become the Phoenix Coy- American sports teams in honor of past playing greats (or otes, the Coyotes continue to honor all num- in one case, an owner): bers retired by the Jets. The Coyotes briefly took the number out of retirement for Hull's • In Major League Baseball: son Brett Hull, also a Hall of Famer, in 2005– 06 until the younger Hull retired five games • The Boston Red Sox, for Hall of Famer Ted into that season. The current Winnipeg Jets Williams. have yet to officially retire any numbers, but • The Chicago White Sox, for Minnie Miñoso. Evander Kane received Bobby Hull's blessing to wear the number. • The New York Yankees, for Roger Maris. • No NFL team has yet retired #9. • The Oakland Athletics, for Hall of Famer Reggie Jackson. • The Pittsburgh Pirates, for Hall of Famer Bill 9.14 Technology Mazeroski. • The St. Louis Cardinals, for Hall of Famer • ISO 9 is the ISO's standard for the of Enos Slaughter. Cyrillic characters into Latin characters • In the NBA: • In the Rich Text Format specification, 9 is the lan- guage code for the English language. All codes for • The Atlanta Hawks, for Hall of Famer Bob regional variants of English are congruent to 9 mod Pettit. 256. • The Phoenix Suns, for Dan Majerle. • The seven-segment display allows the number 9 to • The Utah Jazz, for owner Larry Miller. be constructed two ways, either with a at the end of its stem or without one. Most LCD calcula- • In the NHL: tors use the former, but some VFD models use the • The Boston Bruins, for Hall of Famer Johnny latter. Bucyk. • The9 Limited (owner of the9.com) is a company in • The Calgary Flames, for Hall of Famer Lanny the video-game industry, including former ties to the McDonald. extremely popular MMORPG World of Warcraft 9.16. SEE ALSO 73

9.15 Other fields

• Nine justices sit on the United States Supreme Court.

• Stanines, a method of scaling test scores, range from 1 to 9.

• 9 square feet in a square yard.

• 9 semitones in a Major 6th interval in music.

9.16 See also

• 9 (disambiguation) International maritime signal flag for 9 • 0.999...

• Cloud Nine 74 CHAPTER 9. 9 (NUMBER)

9.18 Further reading

• Cecil Balmond,“Number 9, the search for the sigma code”1998, Prestel 2008, ISBN 3-7913-1933-7, ISBN 978-3-7913-1933-9

Playing cards showing the 9 of all four suits

9.17 References

[1] Donald Alexander Mackenzie (2005). of China And Japan. Kessinger. ISBN 1-4179-6429-4.

[2] Georges Ifrah (1985). From One to Zero: A Universal History of Numbers. Viking. ISBN 0-670-37395-8.

[3] Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0-415-13095-6.

[4] Anthea Fraser (1988). The Nine Bright Shiners. Double- day. ISBN 0-385-24323-5.

[5] Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898-1902. Spottiswoode.

[6] “Sloane's A001006 : Motzkin numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[7] Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93

[8] “Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foun- dation. Retrieved 2016-06-01.

[9] Robert Dixon, Mathographics. New York: Dover Publications: 24

[10] , A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155

[11] Cajori, Florian (1991, 5e) A , AMS. ISBN 0-8218-2102-4. p.91

[12] Eyob Delele Yirdaw, "Proving Touchard's Theorem from Euler's Form" ArXiv preprint.

[13] “Sloane's A000931 : Padovan sequence”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-11.

[14] “Web site for NINE: A Journal of Baseball History & Culture”. Retrieved 20 February 2013. Chapter 10

10 (number)

This article is about the number. For the year, see AD the first discrete semiprime to be in deficit. All subse- 10. For other uses of “ten”, see Ten (disambiguation). quent discrete are in deficit. The aliquot se- For other uses of the ordinal 10th, see Tenth (disam- quence for 10 comprises five members (10,8,7,1,0) with biguation). this number being the second composite member of the 7-aliquot tree. 10 (ten *i/ˈtɛn/) is an even natural number following Ten is the smallest semiprime that is the sum of all the 9 and preceding 11. Ten is the base of the decimal distinct prime numbers from its lower factor through its numeral system, by far the most common system of de- higher factor (10 = 2 + 3 + 5 = 2 . 5) Only three other noting numbers in both spoken and written language. The small semiprimes (39, 155, and 371) share this attribute. reason for the choice of ten is assumed to be that humans It is the aliquot sum of only one number the discrete have ten fingers (digits). semiprime 14. Ten is a semi-meandric number. 10.1 Common usage and derived Ten is the sum of the first three prime numbers, of the four first numbers (1 + 2 + 3 + 4), of the square of the terms two first odd numbers and also of the first four (0! + 1! + 2! + 3!). Ten is the eighth , • A collection of ten items (most often ten years) is preceded in the sequence by 5, 5, 7. called a decade. A polygon with ten sides is a decagon, and 10 is a * • . [2] Because 10 is the product of The ordinal adjective is decimal; the distributive ad- 1 jective is denary. a power of 2 (namely 2 ) with nothing but distinct Fermat primes (specifically 5), a regular decagon is a • Increasing a quantity by one constructible polygon. is most widely understood to mean multiplying the Ten is also a triangular number, a centered triangular quantity by ten. number,*[3] and a .*[4]

• To reduce something by one tenth is to decimate. (In Ten is the number of n queens problem solutions for n = ancient Rome, the killing of one in ten soldiers in a 5. cohort was the punishment for cowardice or mutiny; Ten is the smallest number whose status as a possible or, one-tenth of the able-bodied men in a village as is unknown. a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.) 10.2.1 In numeral systems

10.2 In mathematics Decimal system

Ten is a composite number, its proper divisors being 1, Main article: Decimal 2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer * As is the case for any base in its system, ten is the first and the total number of coprimes below it. [1] two-digit number in decimal and thus the lowest number Ten is the second discrete semiprime (2 × 5) and the where the position of a numeral affects its value. Any second member of the (2 × q) discrete semiprime fam- integer written in the decimal system can be multiplied ily. Ten has an aliquot sum σ(n) of 8 and is accordingly by ten by adding a zero to the end (e.g. 855 × 10 = 8550).

75 76 CHAPTER 10. 10 (NUMBER)

Roman numerals • The Saros number of the lunar eclipse series which began on June 17, 2454 BC and ended on August The Roman numeral for ten is X (which looks like two 15, 1138 BC. The duration of Saros series 10 was V's [the Roman numeral for 5] put together); it is thought 1316.2 years, and it contained 74 lunar eclipses. that the V for five is derived from an open hand (five digits • displayed), and X for ten from both hands. Incidentally, Messier object M10, a magnitude 6.4 globular clus- the Chinese word numeral for ten, is also a cross: 十. ter in the constellation Ophiuchus.

Positional numeral systems other than decimal 10.4 In religion and philosophy The digit '1' followed by '0' is how the value of p is written in base p. (E.g. 16 in hexadecimal is 10.) Properties:

• Ten is a Harshad number in bases 2, 3, 5, 6, 9 and all others above. • In bases 15, 18 and 30, ten is a 1-automorphic num- ber (displayed as the numeral 'A')..

10.2.2 List of basic calculations

10.3 In science

The SI prefix for 10 is “deca-". The meaning “10”is part of the following terms: The tetractys

• decapoda, an order of crustaceans with ten feet • References in the Bible, Judaism and Christianity: • decane, a hydrocarbon with 10 carbon atoms • The Ten Commandments of Exodus*[5] and Deuteronomy*[6] are considered a corner- Also, the number 10 plays a role in the following: stone of Judaism and Christianity. • • The atomic number of neon. People traditionally tithed one-tenth of their produce. The practice of tithing is still com- • The number of hydrogen atoms in butane, a hydro- mon in Christian churches today, though it is carbon. disputed in some circles as to whether or not it is required of Christians. • The number of spacetime dimensions in some superstring theories. • In Deuteronomy 26:12, the Torah commands Jews to give one-tenth of their produce to the The metric system is based on the number 10, so con- poor (Maaser Ani). From this verse and from verting units is done by adding or removing zeros (e.g. 1 an earlier verse (Deut. 14:22) there derives a centimeter = 10 millimeters, 1 decimeter = 10 centime- practice for Jews to give one-tenth of all earn- * ters, 1 meter = 100 centimeters, 1 dekameter = 10 meters, ings to the poor. [7] 1 kilometer = 1,000 meters). • Ten Plagues were inflicted on Egypt in Exodus 7-12 • 10.3.1 Astronomy Jews observe the annual Ten Days of Repen- tance beginning on Rosh Hashanah and ending • The New General Catalogue object NGC 10, a on Yom Kippur. magnitude 12.5 spiral galaxy in the constellation • In Jewish liturgy, Ten Martyrs are singled out Sculptor. as a group. • • The Saros number of the solar eclipse series which There are said to be Ten Lost Tribes of Israel began on February 28, 2467 BC and ended on April (those other than Judah and Benjamin) 18, 1169 BC. The duration of Saros series 10 was • There are Ten Sephirot in the Kabbalistic Tree 1298.1 years, and it contained 73 solar eclipses. of Life 10.7. IN SPORTS AND GAMES 77

• In Judaism, ten men are the required quorum, • In association football, the number 10 is traditionally called a minyan, for prayer services. worn by the team's advanced playmaker. This use has led to “Number 10”becoming a synonym for • Interpretations of Genesis in Talmudic and the player in that particular role, even if he or she Midrashic teachings suggest that on the first does not wear that number.*[8] day, God drew forth ten primal elements from the abyss in order to construct all of Creation: • In Australian rules football, considered the break Heaven (or Fire), Earth, Chaos, Void, Light, even amount of games won in a regular 22 game sea- Darkness, Wind (or Spirit), Water, Day, and son. Night. See also Bereshit (parsha). • • Jesus tells the Parable of the Ten Virgins in In baseball, 10 is the minimum number of players Matthew 25:1-13. on the field at any given time during play (including the batter). • In , the number 10 played an impor- • tant role and was symbolized by the tetractys. In basketball: • The top of the rim (goal) is 10 feet from the • In Hinduism, Lord Vishnu appeared on the earth in floor. 10 incarnations. • In standard full-court basketball, there are 10 players on the court (5 on each team).

10.5 In money • In blackjack, the Ten, Jack, Queen and King are all worth 10 points. Most countries issue coins and bills with a denomination • of 10 (See e.g. 10 dollar note). Of these, the U.S. dime, In boxing, if the referees counts to 10 whether the with the value of ten cents, or one tenth of a dollar, de- boxer is unconscious or not, it will declare a winner rives its name from the meaning“one-tenth”− see Dime by knockout. (United States coin)#Denomination history and etymol- • In cricket, 10 is the number of wickets required to ogy. be taken by the bowling side for the side to be bowled out. 10.6 In music • In gridiron football, 10 is the number of yards the offense must advance to maintain possession in a single set of downs—four in American and three in • The interval of a major tenth is an octave plus a ma- Canadian. jor third. • In most rugby league competitions, the number 10 • The interval of a minor tenth is an octave plus a mi- is worn by one of the two starting props. (One ex- nor third. ception to this rule is the European Super League, which uses static squad numbering.) • The title of quite a few albums. See Ten (album). • In rugby union, the starting fly-half wears the 10 •“Ten lords a-leaping”is the gift on the tenth day of shirt. Christmas in the carol "The Twelve Days of Christ- mas" • In ten-pin bowling, 10 pins are arranged in a trian- gular pattern and there are 10 frames per game. 10.7 In sports and games • The highest score possible in Olympics gymnastics competitions.

The meaning “10”is part of the following terms: • Driving a racing car at ten-tenths is driving as fast as possible, on the limit. • decathlon, a combined event in athletics consisting • Danica Patrick's car number in the NASCAR Sprint of ten track and field events. Cup Series.

Also, the number 10 plays a role in the following: The jersey number 10 has been retired by several North American sports teams in honor of past playing greats or • In American football, the end zones are 10 yards other key figures, including Major League Baseball, the deep. NBA, the NFL, and the NHL. 78 CHAPTER 10. 10 (NUMBER)

10.8 In technology • A 1977 short documentary film Powers of Ten de- picts the relative scale of the Universe in factors of • Ten-codes are commonly used on emergency service ten (orders of magnitude). radio systems. • CBS has a game show called , where • Ten refers to the "meter band" on the radio spectrum the player's prize goes up and down by either the between 28 and 29.7 MHz, used by amateur radio. previous or next power of ten.

• ASCII and code point for line feed •“Ten Chances”is one the pricing games on The Price is Right. • In MIDI, Channel 10 is reserved for unpitched percussion instruments. • There are ten official inkblots in the Rorschach inkblot test. • In the Rich Text Format specification, all language codes for regional variants of the • The traditional Snellen chart uses 10 different let- are congruent to 10 mod 256. ters.

• In macOS, the F10 function key tiles all the windows • Ten is an Australian television network. The Syd- of the current application and grays the windows of ney member of the network has the three-letter call- other applications. sign TEN and used to broadcast in analogue on VHF Channel 10. • The IP addresses in the range 10.0.0.0/8 (meaning the interval between 10.0.0.0 and 10.255.255.255) • Number Ten (also called Ella) is a character in the is reserved for use by Private networks by RFC book series Lorien Legacies. The sixth book, The 1918. Fate of Ten, is named after her.

• The ten-code may be used in law-enforcement 10.9 In other fields brevity communications.

Ten is:

• The number of kingdoms in Five Dynasties and Ten Kingdoms period

• The house number of 10 Downing Street.

• The number of Provinces in Canada.

• The designation of United States Interstate 10, a freeway that runs from California to Florida.

• Number of dots in a tetractys.

10 playing cards of all four suits • The number of the French department Aube. • The number of regions in Ghana. • Blake Edwards' 1979 movie 10. • The state number of Virginia. • Series on HBO entitled 1st & Ten which aired be- • The number in tarot decks that corresponds to either tween December 1984 and January 1991. Chance, Fortune, or the Wheel of Fortune depend- • Series on ESPN and ESPN2 entitled 1st and 10 ing on the deck variant which launched on ESPN in October 2003 to 2008 and moved to ESPN2 from 2008 to present. 10.10 See also • In astrology, Capricorn is the 10th astrological sign of the Zodiac.

• In Chinese astrology, the 10 Heavenly Stems, refer to a system that is used also for time reckoning. • Mathematics portal 10.12. EXTERNAL LINKS 79

10.11 References

[1] “Sloane's A005278 : Noncototients”. The On-Line En- cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01.

[2] “Sloane's A001107 : 10-gonal (or decagonal) numbers” . The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.

[3] “Sloane's A005448 : Centered triangular numbers”. The On-Line Encyclopedia of Integer Sequences. OEIS Foun- dation. Retrieved 2016-06-01.

[4] “Sloane's A000292 : Tetrahedral numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[5] Exodus 20:2-13

[6] Deuteronomy 5:6-17

[7] Archived February 23, 2006, at the Wayback Machine.

[8] Khalil Garriot (21 June 2014).“Mystery solved: Why do the best soccer players wear No. 10?". Yahoo. Retrieved 19 May 2015.

10.12 External links Chapter 11

11 (number)

This article is about the number. For other uses, see 11 11.3 In mathematics (disambiguation). "Ⅺ" redirects here. For other uses, see XI (disambigua- tion) and xi (disambiguation). 11 is the 5th smallest prime number. It is the smallest two-digit prime number in the decimal base; as well as, of course, in undecimal (where it is the smallest two- 11 is the natural number following 10 and preceding 12. digit number). It is also the smallest three-digit prime In English, it is the smallest positive integer requiring in ternary, and the smallest four-digit prime in binary, three syllables and the largest prime number with a single- but a single-digit prime in bases larger than 11, such as morpheme name. duodecimal, hexadecimal, vigesimal and sexagesimal. 11 is the fourth Sophie Germain prime,*[5] the third safe prime,*[6] the fourth Lucas prime,*[7] the first prime,*[8] and the second good prime.*[9] Although it is 11.1 Name necessary for n to be prime for 2*n − 1 to be a Mersenne prime, the converse is not true: 211 − 1 = 2047 which is Eleven derives from the Old English ęndleofon which 23 × 89. The next prime is 13, with which it comprises is first attested in Bede's late 9th-century Ecclesiastical a . 11 is an Eisenstein prime with no imagi- History of the English People.*[1]*[3] It has cognates in nary part and real part of the form 3n − 1. Displayed on a every Germanic language (for example, German elf), calculator, 11 is a strobogrammatic prime and a dihedral whose Proto-Germanic ancestor has been reconstructed prime because it reads the same whether the calculator is as *ainlif, from the prefix *aino- (adjectival "one") and turned upside down or reflected on a mirror, or both. suffix *-lif- of uncertain meaning.*[3] It is sometimes If a number is divisible by 11, reversing its digits will compared with the Lithuanian vënólika, although -lika is result in another multiple of 11. As long as no two ad- used as the suffix for all numbers from 11 to 19 (analo- jacent digits of a number added together exceed 9, then gous to "-teen”).*[3] multiplying the number by 11, reversing the digits of the The Old English form has closer cognates in Old Frisian, product, and dividing that new number by 11, will yield Saxon, and Norse, whose ancestor has been reconstructed a number that is the reverse of the original number. (For as *ainlifun. This has formerly been considered derived example: 142,312 × 11 = 1,565,432 → 2,345,651 ÷ 11 from Proto-Germanic *tehun ("ten");*[3]*[4] it is now = 213,241.) sometimes connected with *leiq or *leip (“left; remain- Because it has a reciprocal of unique period length among ing”), with the implicit meaning that“one is left”after primes, 11 is the second unique prime.*[10] 11 goes into having already counted to ten.*[3] 99 exactly 9 times, so vulgar fractions with 11 in the denominator have two digit repeating sequences in their decimal expansions. Multiples of 11 by one-digit num- bers all have matching double digits: 00 (=0), 11, 22, 11.2 In languages 33, 44, etc. Bob Dorough, in his Schoolhouse Rock song “The Good Eleven”, called them “Double-digit doo- gies”(soft g). 11 is the Aliquot sum of one number, the 11.2.1 Grammar discrete semiprime 21 and is the base of the 11-aliquot tree. While, as mentioned above, 11 has its own name in Ger- As 11 is the smallest factor of the first 11 terms of the manic languages such as English and German, it is the first * compound number in many other languages, e.g. Italian –Mullin sequence, it is the 12th term. [11] ùndici (but in Spanish and Portuguese, 16, and in French, An 11-sided polygon is called a hendecagon or un- 17 is the first compound number), Japanese 十一 jūichi. decagon.

80 11.4. IN NUMERAL SYSTEMS 81

In both base 6 and base 8, the smallest prime with a com- In base 13 and higher bases (such as hexadecimal), 11 is posite sum of digits is 11. represented as B, where ten is A. In duodecimal, however, 11 is sometimes represented as E and ten as T or X. Any number b + 1 is written as “11b" in base b, so 11 is trivially a in base 10. However 11 is a strictly 11 is a Størmer number,*[14] a Heegner number,*[15] non-.*[12] It is the only palindromic and a Mills prime.*[16] prime with an even number of digits. There are 11 orthogonal curvilinear coordinate systems In base 10, there is a simple test to determine if an integer (to within a conformal symmetry) in which the 3-variable is divisible by 11: take every digit of the number located Helmholtz equation can be solved using the separation of in odd position and add them up, then take the remaining variables technique. digits and add them up. If the difference between the two See also 11-cell. sums is a multiple of 11, including 0, then the number is divisible by 11.*[13] For instance, if the number is 65,637 11 of the thirty-five hexominoes can be folded to form then (6 + 6 + 7) - (5 + 3) = 19 - 8 = 11, so 65,637 is cubes. 11 of the sixty-six octiamonds can be folded to divisible by 11. This technique also works with groups of form octahedra. digits rather than individual digits, so long as the number The partition numbers (sequence A000041 in the OEIS) of digits in each group is odd, although not all groups have contain much more multiples of 11 than the one-eleventh to have the same number of digits. For instance, if one one would expect. uses three digits in each group, one gets from 65,637 the calculation (065) - 637 = −572, which is divisible by 11. According to David A. Klarner, a leading researcher and contributor to the study of polyominoes, it is possible to Another test for divisibility is to separate a number into cut a rectangle into an odd number of congruent, non- groups of two consecutive digits (adding a rectangular polyominoes. 11 is the smallest such number, if there is an odd number of digits), and then add up the the only such number that is prime, and the only such numbers so formed; if the result is divisible by 11, the number that is not a multiple of three. number is divisible by 11. For instance, if the number is 65,637, 06 + 56 + 37 = 99, which is divisible by 11, so 11 raised to the nth power is the nth row of Pascal's Tri- 65,637 is divisible by eleven. This also works by adding angle. (This works for any base, but the number eleven a instead of a leading one: 65 + 63 + 70 = must be changed to the number represented as 11 in that 198, which is divisible by 11. This also works with larger base; for example, in duodecimal this must be done using groups of digits, providing that each group has an even thirteen.) number of digits (not all groups have to have the same number of digits). 11.3.1 List of basic calculations An easy way of multiplying numbers by 11 in base 10 is: If the number has: 11.4 In numeral systems • 1 digit - Replicate the digit (so 2 x 11 becomes 22). 11.5 In science • 2 digits - Add the 2 digits together and place the re- sult in the middle (so 47 x 11 becomes 4 (11) 7 or 4 (10+1) 7 or (4+1) 1 7 or 517). • 11 is the atomic number of the element sodium.

• 3 digits - Keep the first digit in its place for the re- • In chemistry, Group 11 of the Periodic Table of the sult's first digit, add the first and second digits to- Elements (IUPAC numbering) consists of the three gether to form the result's second digit, add the sec- coinage metals copper, silver, and gold known from ond and third digits together to form the result's third antiquity, and roentgenium, a recently synthesized digit, and keep the third digit as the result's fourth superheavy element. digit. For any resulting numbers greater than 9, the 1 to the left. Example 1: 123 x 11 be- • The number of spacetime dimensions in M-theory. comes 1 (1+2) (2+3) 3 or 1353. Example 2: 481 x 11 becomes 4 (4+8) (8+1) 1 or 4 (10+2) 9 1 or (4+1) 2 9 1 or 5291. 11.5.1 Astronomy

• 4 or more digits - Follow the same pattern as for 3 • Apollo 11 was the first manned spacecraft to land on digits. the Moon.

In base 10, 11 is the smallest integer that is not a • The approximate periodicity of a sunspot cycle is 11 Nivenmorphic number. years. 82 CHAPTER 11. 11 (NUMBER)

• Messier object M11, a magnitude 7.0 open cluster • The interval of an octave and a fourth is an 11th. in the constellation Scutum, also known as the Wild A complete 11th chord has almost every note of a Duck Cluster. diatonic scale.

• The New General Catalogue object NGC 11, a spiral • The number of thumb keys on a bassoon, not count- galaxy in the constellation Andromeda ing the whisper key. (A few bassoons have a 12th thumb key.) • The Saros number of the solar eclipse series which began on −2511 December 26 and ended on −1158 • In the mockumentary This Is Spinal Tap, Spinal March 18. The duration of Saros series 11 was Tap's amplifiers go up to eleven. 1352.2 years, and it contained 76 solar eclipses. • In Igor Stravinsky's The Rite of Spring, there are 11 • The Saros number of the lunar eclipse series which consecutive repetitions of the same chord. began on −2389 June 19 and ended on −1037 • September 8. The duration of Saros series 11 was In Tool's song Jimmy, and in Negativland's song 1352.2 years, and it contained 76 lunar eclipses. Time Zones the number 11 is heard numerous times in the lyrics. • The 11th moon of Jupiter is Himalia. •“Eleven pipers piping”is the gift on the 11th day of Christmas in the carol "The Twelve Days of Christ- 11.6 In religion mas" • In Green Grow the Rushes, O, Eleven is for “the 11.6.1 Christianity eleven who went to heaven” • The Eleven is a song by The Grateful Dead. After Judas Iscariot was disgraced, the remaining apostles of Jesus were sometimes described as “the • In “Time Enough For Rocking When We're Old” Eleven”(Mark 16:11; Luke 24:9 and 24:33); this oc- by The Magnetic Fields, a lyric references “when curred even after Matthias was added to bring the number our pheromones go up to eleven.” to twelve, as in Acts 2:14: Peter stood up with the eleven (New International Version). The New Living Transla- • Eleven Records is the of Jason Web- tion says Peter stepped forward with the eleven other apos- ley, and many of Webley's works feature the number * tles, making clear that the number of apostles was now 11. [19] twelve. Saint Ursula is said to have been martyred in the third or fourth century in Cologne with a number of compan- 11.8 In sports ions, whose reported number“varies from five to eleven” .*[17] A legend that Ursula died with eleven thousand vir- • There are 11 players on a soccer team on the field at gin companions *[18] has been thought to appear from a time as well as in a cricket team. Within a school misreading XI. M. V. (Latin abbreviation for “Eleven or college, the phrase first eleven (or first XI) - often martyr virgins”) as “Eleven thousand virgins”. “first football XI”and “first cricket XI”- gener- ally refers to the first (best) team currently playing. Other teams are often referred to as “the second 11.6.2 Thelema XI”etc.

11 is a spiritually significant number in Thelema. • Also in soccer, in the (and others like Italian -“gli undici metri”-, countries that pre- dominantly use the metric system) a penalty kick is 11.6.3 Babylonian referred to as “Elfmeter”because the penalty spot is approximately 11m (precisely 12 yards) from the In the Enûma Eliš the goddess Tiamat creates eleven goal line. Historically, in the Pyramid formation that monsters to take revenge for the death of her husband, position names are taken from, a left wing-forward Apsû. in football wears number 11. In the modern game, especially using the 4-4-2 formation, it is worn by a left-sided midfielder. Less commonly a striker will 11.7 In music wear the shirt. • There are 11 players in a field hockey team. The See also: Eleven (disambiguation) § Music player wearing 11 will usually play on the left-hand side, as in soccer. 11.9. IN THE MILITARY 83

• An American football team also has 11 players on • The New York Giants, for Phil Simms. the field at one time during play. 11 is also worn by • In the NHL: quarterbacks, kickers, punter and wide receivers in • American football's NFL. The Buffalo Sabres, for Hall of Famer Gilbert Perreault. • In most rugby league competitions (but not the Eu- • The Edmonton Oilers and New York ropean Super League, which uses static squad num- Rangers, for Hall of Famer Mark bering), one of the starting second-row forwards Messier. wears the number 11. • The St. Louis Blues, for Brian Sutter. • In rugby union, the starting left wing wears the 11 • The Washington Capitals, for Hall of shirt. Famer Mike Gartner. • In cricket, the 11th batsman is usually the weakest • In racing: batsman, at the end of the tail. He is primarily in • 11 is the number of a race car in the the team for his bowling abilities. NASCAR Sprint Cup Series owned by Joe Gibbs Racing and currently driven by • The jersey number 11 has been retired by several Denny Hamlin. North American sports teams in honor of past play- ing greats or other key figures: • In Major League Baseball: 11.9 In the military • The Chicago White Sox, for Hall of Famer Luis Aparicio. In 2010 and • The number of guns in a gun salute to U.S. Army, 2011, Aparicio allowed fellow Venezue- Air Force and Marine Corps Brigadier Generals, lan Omar Vizquel to wear the number. and to Navy and Coast Guard Rear Admirals Lower • The Cincinnati Reds, for Hall of Famer Half. Barry Larkin. • The Military Occupational Specialty (MOS) desig- • The Detroit Tigers, for Hall of Fame nator given to US Army Infantry Officer as well as manager Sparky Anderson. to enlisted personnel (AKA 11 MOS Series, or 11B, • The Los Angeles Angels of Anaheim, for 11C, 11D, 11H, 11M, etc.) Jim Fregosi, who played for the team in its former incarnations as the Los Ange- • The number of General Orders for Sentries in the les Angels and California Angels, and also Marine Corps and United States Navy. managed the California Angels. • • The Pittsburgh Pirates, for Hall of Famer A page in the Service Record Book of an enlisted Paul Waner. Marine for writing down disciplinary actions. • The San Francisco Giants, for Hall of • World War I ended with an Armistice on November Famer Carl Hubbell, honoring the num- 11, 1918, which went into effect at 11:00 am—the ber's retirement when the team was 11th hour on the 11th day of the 11th month of the known as the New York Giants. year. Armistice Day is still observed on November • In the NBA: 11 of each year, although it is now called Veterans • The Cleveland Cavaliers, for Zydrunas Il- Day in the United States and Remembrance Day in gauskas.*[20] the Commonwealth of Nations and parts of Europe. • The Detroit Pistons, for Hall of Famer Isiah Thomas. • The Sacramento Kings, for Hall of Famer 11.10 In computing Bob Davies, honoring the number's re- tirement when the team was known as the • In Mozilla , , for KDE, Rochester Royals. Chrome and for Windows, • The Washington Wizards, for Hall of the function key F11 key toggles full screen viewing Famer Elvin Hayes, who played for the mode. In macOS, F11 hides all open windows. team in its past incarnations as the Bal- • The windowing system for computers is known timore, Capital, and Washington Bullets, as X11. and won the franchise's only NBA Cham- pionship with the 1977-1978 Washington • Computers of the PDP-11 series from Digital Bullets. Equipment Corporation were informally referred to • In the NFL: as “elevens”. 84 CHAPTER 11. 11 (NUMBER)

11.11 In Canada • 's Eleven is the name of two American films. • In the anime series Code Geass, Japan is known as • The stylized maple leaf on the Flag of Canada has Area 11 of the Brittanian Empire. 11 points. • Eleven is the name of a character in the 2016 Netflix • The loonie is a hendecagon, an 11-sided polygon. original series Stranger Things. • Clocks depicted on Canadian currency, like the • In bingo 11 is referred to as chicken legs. Canadian fifty-dollar bill, show 11:00. • Eleven denominations of Canadian currency are produced in large quantities. 11.13 See also • Due to Canada's federal nature, eleven legally • distinct Crowns effectively exist in the country, 11:11 with the Monarch represented separately in each • 11:11 (numerology) province, and at the federal level.

11.14 References 11.12 In other fields [1] Specifically, in the line Osred ðæt rice hæfde endleofan • Sector 11 in the North American Industry Classifi- wintra.*[2] cation System is the code for Agriculture, Forestry, Fishing and Hunting industries. [2] Bede, Eccl. Hist., Bk. V, . xviii. • The number 11 is important in numerology, as it is [3] Oxford English Dictionary, 1st ed. “eleven, adj. and n." the first of the Master Numbers. Oxford University Press (Oxford), 1891. [4] Dantzig, Tobias (1930), Number: The Language of Sci- • Being only one hour before 12:00, the eleventh hour ence. means the last possible to take care of something, and often implies a situation of urgent [5] “Sloane's A005384 : Sophie Germain primes”. The danger or emergency (see Doomsday clock). On-Line Encyclopedia of Integer Sequences. OEIS Foun- dation. Retrieved 2016-06-01. • In Astrology, Aquarius is the 11th astrological sign of the Zodiac. [6]“Sloane's A005385 : Safe primes”. The On-Line Encyclo- pedia of Integer Sequences. OEIS Foundation. Retrieved • In Basque, hamaika (“eleven”) has the double 2016-06-01. meaning of "infinite", probably from amaigabe, [7] “Sloane's A005479 : Prime Lucas numbers”. The On- “endless”, as in Hamaika aldiz etortzeko esan dizut! Line Encyclopedia of Integer Sequences. OEIS Founda- “ ( I told you infinite/eleven times to come!"). tion. Retrieved 2016-06-01. • English-speaking surveyors have developed several [8] “Sloane's A004022 : Primes of the form (10^n - 1)/9” slang terms for 11 to distinguish it from its rhyme . The On-Line Encyclopedia of Integer Sequences. OEIS * “seven": “punk,”“top,”& “railroad” [21] Foundation. Retrieved 2016-06-01.

• American Airlines flight 11, a Boston-Los Angeles [9] “Sloane's A028388 : Good primes”. The On-Line En- flight which was reported to have crashed into the cyclopedia of Integer Sequences. OEIS Foundation. Re- North Tower of the World Trade Center in New trieved 2016-06-01. York City, New York on September 11, 2001. [10] “Sloane's A040017 : Unique period primes”. The On- • The number 11 bus is a low-cost way of sightseeing Line Encyclopedia of Integer Sequences. OEIS Founda- in London. tion. Retrieved 2016-06-01. • In the game of blackjack, an Ace can be counted as [11] “Sloane's A000945 : Euclid-Mullin sequence”. The On-Line Encyclopedia of Integer Sequences. OEIS Foun- either one or 11, whichever is more advantageous for dation. Retrieved 2016-06-01. the player. “ ” • [12] Sloane's A016038 : Strictly non-palindromic numbers 11 is the number of the French department Aude. . The On-Line Encyclopedia of Integer Sequences. OEIS • Three films -- Ben-Hur (1959), Titanic (1997), and Foundation. Retrieved 2016-06-01. The Lord of the Rings: The Return of the King (2003) [13] Higgins, Peter (2008). Number Story: From Counting to -- have each won 11 Academy Awards, including Cryptography. New York: Copernicus. p. 47. ISBN 978- Best Picture of their respective years. 1-84800-000-1. 11.15. EXTERNAL LINKS 85

[14] “Sloane's A005528 : Størmer numbers”. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.

[15] “Sloane's A003173 : Heegner numbers”. The On- Line Encyclopedia of Integer Sequences. OEIS Founda- tion. Retrieved 2016-06-01.

[16] “Sloane's A051254 : Mills primes”. The On-Line En- cyclopedia of Integer Sequences. OEIS Foundation. Re- trieved 2016-06-01.

[17] Ursulines of the Roman Union, Province of Southern Africa, St. Ursula and Companions, accessed 10 July 2016

[18] Four scenes from the life of St Ursula, accessed 10 July 2016

[19] Corazon, Billy (July 1, 2009). “Imaginary Interview: Ja- son Webley”. Three Imaginary Girls. Archived from the original on 2012-04-04. Retrieved 2012-09-06.

[20] “Cavs Announce Zydrunas Ilgauskas' Jersey (#11) to be Retired”. THE OFFICIAL SITE OF THE CLEVELAND CAVALIERS.

[21] “Surveying Units and Terms”. Directlinesoftware.com. 2012-07-30. Retrieved 2012-08-20.

11.15 External links

• Grimes, James. “Eleven”. Numberphile. Brady Haran. Chapter 12

0

This article is about the number and digit 0. It is not to There are different words used for the number or concept be confused with the letter O. of zero depending on the context. For the simple notion “Zero”redirects here. For other uses, see 0 (disam- of lacking, the words nothing and none are often used. biguation) and Zero (disambiguation). Sometimes the words nought, naught and aught*[10] are “Naught”redirects here. For the album by Stolen Babies, used. Several sports have specific words for zero, such see Naught (album). as nil in football, love in tennis and a duck in cricket. It is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, and scratch. 0 (zero; BrE: /ˈzɪərəʊ/ or AmE: /ˈziːroʊ/) is both a * number*[1] and the used to represent that Duck egg and goose egg are also slang for zero. [11] number in numerals. The number 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 12.2 History 0 is used as a placeholder in place value systems. Names for the number 0 in English include zero, nought or (US) 12.2.1 Ancient Near East naught (/ˈnɔːt/), nil, or—in contexts where at least one adjacent digit distinguishes it from the letter“O”—oh or Ancient were base 10. They used o (/ˈoʊ/). Informal or slang terms for zero include zilch hieroglyphs for the digits and were not positional. By and zip.*[2] Ought and aught (/ˈɔːt/),*[3] as well as ci- 1740 BC, the Egyptians had a symbol for zero in ac- pher,*[4] have also been used historically.*[5] counting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the 12.1 Etymology base line as being above or below this line.*[12] By the middle of the 2nd millennium BC, the Babylonian Main articles: Names for the number 0 and Names for mathematics had a sophisticated sexagesimal positional the number 0 in English numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a symbol (two slanted wedges) The word zero came into the English language via French was co-opted as a placeholder in the same Babylonian sys- zéro from Italian zero, Italian contraction of Venetian tem. In a tablet unearthed at Kish (dating from about 700 zevero form of 'Italian zefiro via ṣafira or ṣifr.*[6] In pre- BC), the scribe Bêl-bân-aplu wrote his zeros with three had the meaning (صفر Islamic time the word ṣifr (Arabic hooks, rather than two slanted wedges.*[13] 'empty'.*[7] Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.*[7] The The Babylonian placeholder was not a true zero because first known English use of zero was in 1598.*[8] it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 The Italian mathematician Fibonacci (c.1170–1250), (3×60), 4 and 240 (4×60), looked the same because the who grew up in North Africa and is credited with in- larger numbers lacked a final sexagesimal placeholder. troducing the decimal system to Europe, used the term Only context could differentiate them. zephyrum. This became zefiro in Italian, and was then contracted to zero in Venetian. The Italian word zefiro was already in existence (meaning “west wind”from 12.2.2 Pre-Columbian Americas Latin and Greek zephyrus) and may have influenced the spelling when transcribing Arabic ṣifr.*[9] The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the Modern usage use of zero as a place-holder within its vigesimal (base-

86 12.2. HISTORY 87

12.2.3 Classical antiquity

The ancient Greeks did not have a name for zero and did not use a placeholder.*[17] They seemed unsure about the status of zero as a num- ber. They asked themselves,“How can nothing be some- thing?", leading to philosophical and, by the medieval pe- riod, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.

Example of the early Greek symbol for zero (lower right corner) The back of Olmec stela C from Tres Zapotes, the second oldest from a 2nd-century Long Count date discovered. The numerals 7.16.6.16.18 trans- late to September, 32 BC (Julian). The glyphs surrounding the By 130 AD, Ptolemy, influenced by Hipparchus and the date are thought to be one of the few surviving examples of Epi- Babylonians, was using a symbol for zero (a small circle Olmec script. with a long overbar) in his work on mathematical astron- omy called the Syntaxis Mathematica, also known as the Almagest. The way in which it is used can be seen in his table of chords in that book. Ptolemy's zero was used within a sexagesimal numeral system otherwise using al- phabetic Greek numerals. Because it was used alone, not 20) positional numeral system. Many different glyphs, just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old * including this partial quatrefoil— —were used as a World. [18] However, the positions were usually limited zero symbol for these Long Count dates, the earliest of to the fractional part of a number (called minutes, sec- which (on Stela 2 at Chiapa de Corzo, Chiapas) has a onds, thirds, fourths, etc.)—they were not used for the date of 36 BC.*[14] integral part of a number. In later Byzantine manuscripts of Ptolemy's Almagest, the Hellenistic zero had morphed Since the eight earliest Long Count dates appear out- into the Greek letter omicron (otherwise meaning 70). side the Maya homeland,*[15] it is generally believed that the use of zero in the Americas predated the Maya and Another zero was used in tables alongside Roman numer- was possibly the invention of the .*[16] Many als by 525 (first known use by Dionysius Exiguus), but as * of the earliest Long Count dates were found within the a word, nulla meaning“nothing”, not as a symbol. [19] Olmec heartland, although the Olmec civilization ended When division produced zero as a remainder, nihil, also by the 4th century BC, several centuries before the earli- meaning “nothing”, was used. These medieval zeros est known Long Count dates. were used by all future medieval calculators of Easter. The initial “N”was used as a zero symbol in a table of Although zero became an integral part of , Roman numerals by Bede or his colleagues around 725. with a different, empty tortoise-like “shell shape” used for many depictions of the“zero”numeral, it is assumed to have not influenced Old World numeral systems. 12.2.4 China , a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record The , of unknown date but estimated to accounting and other digital data, is encoded in a base ten be dated from the 1st to 5th centuries CE, and Japanese positional system. Zero is represented by the absence of records dated from the eighteenth century, describe how a knot in the appropriate position. the c4th century BCE Chinese system en- 88 CHAPTER 12. 0

tion of the value of zero divided by zero as zero.*[32]

Epigraphy

This is a depiction of zero expressed in Chinese counting rods, based on the example provided by A History of Mathematics. An empty space is used to represent zero.*[20]

ables one to perform decimal calculations. According to A History of Mathematics, the rods “gave the decimal representation of a number, with an empty space denot- ing zero.”*[20] The counting rod system is considered a positional notation system.*[21] Zero was not treated as a number at that time, but The number 605 in Khmer numerals, from the Sambor inscrip- as a “vacant position”.*[22] Ch'in Chiu-shao's 1247 tion (Saka era 605 corresponds to AD 683). The earliest known Mathematical Treatise in Nine Sections is the oldest sur- material use of zero as a decimal figure. viving Chinese mathematical text using a round symbol for zero.*[23] Chinese authors had been familiar with the There are numerous copper plate inscriptions, with the idea of negative numbers by the (2nd cen- same small o in them, some of them possibly dated to the tury AD), as seen in the The Nine Chapters on the Mathe- 6th century, but their date or authenticity may be open to * matical Art,*[24] much earlier than the fifteenth century doubt. [13] when they became well established in Europe.*[23] A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of“605”in Khmer numerals (a set of numeral 12.2.5 India and Southeast Asia glyphs of the Hindu numerals family). The number is the year of the inscription in the Saka era, corresponding to * The concept of zero as a digit in the decimal place value a date of AD 683. [33] notation was developed in India, presumably as early as The first known use of special glyphs for the decimal dig- during the Gupta period (c. 5th century), with the oldest its that includes the indubitable appearance of a symbol unambiguous evidence dating to the 7th century.*[25] for the digit zero, a small circle, appears on a stone in- The Indian scholar Pingala (c. 200 BC) used binary num- scription found at the Chaturbhuja Temple at Gwalior in * * bers in the form of short and long syllables (the latter India, dated 876. [34] [35] equal in length to two short syllables), a notation simi- * lar to . [26] Pingala used the Sanskrit word 12.2.6 Middle Ages śūnya explicitly to refer to zero.*[27] The earliest text to use a decimal place-value system, in- Transmission to Islamic culture cluding a zero, is the Lokavibhāga, a Jain text surviving in a medieval Sanskrit translation of the Prakrit original, See also: History of the Hindu–Arabic numeral system which is internally dated to AD 458 (Saka era 380). In this text, śūnya (“void, empty”) is also used to refer to * The Arabic-language inheritance of science was largely zero. [28] Greek,*[36] followed by Hindu influences.*[37] In 773, The origin of the modern decimal-based place value at Al-Mansur's behest, translations were made of many notation can be traced to the Aryabhatiya (c. 500), ancient treatises including Greek, Roman, Indian, and which states sthānāt sthānaṁ daśaguṇaṁ syāt “from others. place to place each is ten times the preceding.” * * * * In AD 813, astronomical tables were prepared by [29] [29] [30] [31] a Persian mathematician, Muḥammad ibn Mūsā al- The rules governing the use of zero appeared for the first Khwārizmī, using Hindu numerals;*[37] and about 825, time in the Brahmasputha Siddhanta (7th century). This he published a book synthesizing Greek and Hindu work considers not only zero, but negative numbers, and knowledge and also contained his own contribution to the algebraic rules for the operations of arith- mathematics including an explanation of the use of metic with such numbers. In some instances, his rules zero.*[38] This book was later translated into Latin in the differ from the modern standard, specifically the defini- 12th century under the title Algoritmi de numero Indorum. 12.3. MATHEMATICS 89

This title means “al-Khwarizmi on the Numerals of the chance omitted anything more or less proper Indians”. The word “Algoritmi”was the translator's or necessary, I beg indulgence, since there is Latinization of Al-Khwarizmi's name, and the word“Al- no one who is blameless and utterly provident gorithm”or“Algorism”started meaning any arithmetic in all things. The nine Indian figures are: 9 8 based on .*[37] 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be writ- Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that * * if no number appears in the place of tens in a calculation, ten. [40] [41] a little circle should be used “to keep the rows”. This circle was called ṣifr.*[39] Here Leonardo of Pisa uses the phrase“sign 0”, indicat- ing it is like a sign to do operations like addition or multi- plication. From the 13th century, manuals on calculation Transmission to Europe (adding, multiplying, extracting roots, etc.) became com- mon in Europe where they were called algorismus after The Hindu–Arabic numeral system (base 10) reached the Persian mathematician al-Khwārizmī. The most pop- Europe in the 11th century, via the Iberian Penin- ular was written by Johannes de Sacrobosco, about 1235 sula through Spanish Muslims, the Moors, together and was one of the earliest scientific books to be printed in with knowledge of astronomy and instruments like the 1488. Until the late 15th century, Hindu–Arabic numer- astrolabe, first imported by Gerbert of Aurillac. For this als seem to have predominated among mathematicians, reason, the numerals came to be known in Europe as while merchants preferred to use the Roman numerals. “”. The Italian mathematician Fibonacci In the 16th century, they became commonly used in Eu- or Leonardo of Pisa was instrumental in bringing the sys- rope. tem into European mathematics in 1202, stating:

After my father's appointment by his 12.3 Mathematics homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future See also: of zero usefulness and convenience, had me in my boy- hood come to him and there wanted me to de- 0 is the integer immediately preceding 1. Zero is an even vote myself to and be instructed in the study number,*[42] because it is divisible by 2 with no remain- of calculation for some days. There, follow- der. 0 is neither positive nor negative. By most defini- ing my introduction, as a consequence of mar- tions*[43] 0 is a natural number, and then the only natural velous instruction in the art, to the nine digits number not to be positive. Zero is a number which quan- of the Hindus, the knowledge of the art very tifies a count or an amount of null size. In most cultures, 0 much appealed to me before all others, and was identified before the idea of negative things, or quan- for it I realized that all its aspects were studied tities less than zero, was accepted. in Egypt, Syria, Greece, Sicily, and Provence, The value, or number, zero is not the same as the digit with their varying methods; and at these places zero, used in numeral systems using positional notation. thereafter, while on business. I pursued my Successive positions of digits have higher weights, so in- study in depth and learned the give-and-take of side a numeral the digit zero is used to skip a position disputation. But all this even, and the algorism, and give appropriate weights to the preceding and follow- as well as the art of Pythagoras, I considered ing digits. A zero digit is not always necessary in a posi- as almost a mistake in respect to the method tional number system, for example, in the number 02. In of the Hindus (Modus Indorum). Therefore, some instances, a leading zero may be used to distinguish embracing more stringently that method of the a number. Hindus, and taking stricter pains in its study, while adding certain things from my own un- derstanding and inserting also certain things from the niceties of Euclid's geometric art. I 12.3.1 Elementary algebra have striven to compose this book in its entirety as understandably as I could, dividing it into fif- The number 0 is the smallest non-negative integer. The teen chapters. Almost everything which I have natural number following 0 is 1 and no natural number introduced I have displayed with exact proof, precedes 0. The number 0 may or may not be considered in order that those further seeking this knowl- a natural number, but it is a whole number and hence a edge, with its pre-eminent method, might be and a real number (as well as an algebraic instructed, and further, in order that the Latin number and a complex number). people might not be discovered to be without The number 0 is neither positive nor negative and is usu- it, as they have been up to now. If I have per- ally displayed as the central number in a number line. It is 90 CHAPTER 12. 0

neither a prime number nor a composite number. It can- • In lattice theory, 0 may denote the bottom element not be prime because it has an infinite number of factors of a bounded lattice. and cannot be composite because it cannot be expressed by multiplying prime numbers (0 must always be one of • In category theory, 0 is sometimes used to denote an the factors).*[44] Zero is, however, even. initial object of a category. The following are some basic (elementary) rules for deal- • In theory, 0 can be used to denote the ing with the number 0. These rules apply for any real or Turing degree of the partial computable functions. complex number x, unless otherwise stated.

• Addition: x + 0 = 0 + x = x. That is, 0 is an identity 12.3.3 Related mathematical terms element (or neutral element) with respect to addi- tion. • A zero of a function f is a point x in the domain • Subtraction: x − 0 = x and 0 − x = −x. of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the • · · Multiplication: x 0 = 0 x = 0. function. This is related to zeros of a holomorphic 0 * • Division: ⁄x = 0, for nonzero x. But x⁄0 is function. undefined, because 0 has no multiplicative inverse • (no real number multiplied by 0 produces 1), a con- The zero function (or zero map) on a domain D is the sequence of the previous rule. constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all 0 * • Exponentiation: x = x/x = 1, except that the case x x in D. The zero function is the only function that is = 0 may be left undefined in some contexts. For all both even and odd. A particular zero function is a positive real x, 0*x = 0. zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The 0 The expression ⁄0, which may be obtained in an at- on non-invertible square matrices is a tempt to determine the limit of an expression of the form zero map. * f(x)⁄g(x) as a result of applying the lim operator inde- pendently to both operands of the fraction, is a so-called • Several branches of mathematics have zero ele- "indeterminate form". That does not simply mean that ments, which generalise either the property 0 + x the limit sought is necessarily undefined; rather, it means = x, or the property 0 × x = 0, or both. * that the limit of f(x)⁄g(x), if it exists, must be found by another method, such as l'Hôpital's rule. The sum of 0 numbers is 0, and the product of 0 numbers 12.4 Physics is 1. The factorial 0! evaluates to 1. The value zero plays a special role for many physical 12.3.2 Other branches of mathematics quantities. For some quantities, the zero level is natu- rally distinguished from all other levels, whereas for oth- • In set theory, 0 is the cardinality of the : ers it is more or less arbitrarily chosen. For example, for if one does not have any apples, then one has 0 ap- an absolute temperature (as measured in kelvins) zero is ples. In fact, in certain axiomatic developments of the lowest possible value (negative temperatures are de- mathematics from set theory, 0 is defined to be the fined, but negative-temperature systems are not actually empty set. When this is done, the empty set is the colder). This is in contrast to for example temperatures Von Neumann cardinal assignment for a set with no on the Celsius scale, where zero is arbitrarily defined to be elements, which is the empty set. The cardinality at the freezing point of water. Measuring sound intensity function, applied to the empty set, returns the empty in decibels or phons, the zero level is arbitrarily set at a set as a value, thereby assigning it 0 elements. reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest • Also in set theory, 0 is the lowest ordinal number, possible energy that a quantum mechanical physical sys- corresponding to the empty set viewed as a well- tem may possess and is the energy of the ground state of ordered set. the system. • In propositional logic, 0 may be used to denote the truth value false. • In abstract algebra, 0 is commonly used to denote a 12.5 Chemistry zero element, which is a neutral element for addition (if defined on the structure under consideration) and Zero has been proposed as the atomic number of the the- an absorbing element for multiplication (if defined). oretical element tetraneutron. It has been shown that a 12.8. OTHER FIELDS 91

cluster of four neutrons may be stable enough to be con- A null pointer is a pointer in a computer program that sidered an atom in its own right. This would create an does not point to any object or function. In C, the integer element with no protons and no charge on its nucleus. constant 0 is converted into the null pointer at compile As early as 1926, Andreas von Antropoff coined the term time when it appears in a pointer context, and so 0 is a neutronium for a conjectured form of matter made up of standard way to refer to the null pointer in code. How- neutrons with no protons, which he placed as the chemi- ever, the internal representation of the null pointer may cal element of atomic number zero at the head of his new be any bit pattern (possibly different values for different version of the periodic table. It was subsequently placed data types). as a noble gas in the middle of several spiral representa- In mathematics −0 = +0 = 0; both −0 and +0 represent ex- tions of the periodic system for classifying the chemical actly the same number, i.e., there is no “negative zero” elements. distinct from zero. In some signed number representa- tions (but not the two's complement representation used to represent integers in most computers today) and most 12.6 Astronomy floating point number representations, zero has two dis- tinct representations, one grouping it with the positive • The Saros number*[45] of the solar eclipse series numbers and one with the negatives; this latter represen- which began on May 23, 2955 BC and ended on tation is known as negative zero. June 29, 1675 BC. The duration of Saros series In binary, 0 represents the value for“off”, which means 0 was 1280.14 years, and it contained 72 solar no electricity flow.*[47] eclipses. Zero is the value of false in many programming lan- • The Saros number*[46] of the lunar eclipse series guages. which began on March 1, 2653 BC and ended on In Unix time, zero is midnight before the first of January April 30, 1337 BC. The duration of Saros series 0 1970. This is known as the Unix epoch. was 1316.2 years, and it contained 74 lunar eclipses.

12.7 Computer science 12.8 Other fields • In telephony, pressing 0 is often used for dialling The most common practice throughout human history has out of a company network or to a different city or been to start counting at one, and this is the practice in region, and 00 is used for dialling abroad. In some early classic computer science programming languages countries, dialling 0 places a call for operator assis- such as Fortran and COBOL. However, in the late 1950s tance. LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array • DVDs that can be played in any region are some- subscripts (allowing any positive, negative, or zero integer times referred to as being "region 0" as base for array subscripts), and most subsequent pro- • gramming languages adopted one or other of these po- Roulette wheels usually feature a “0”space (and sitions. For example, the elements of an array are num- sometimes also a “00”space), whose presence is bered starting from 0 in C, so that for an array of n items ignored when calculating payoffs (thereby allowing the sequence of array indices runs from 0 to n−1. This the house to win in the long run). permits an array element's location to be calculated by • In Formula One, if the reigning World Champion adding the index directly to address of the array, whereas no longer competes in Formula One in the year fol- 1 based languages precalculate the array's base address to lowing their victory in the title race, 0 is given to one be the position one element before the first. of the drivers of the team that the reigning champion There can be confusion between 0 and 1 based indexing, won the title with. This happened in 1993 and 1994, for example Java's JDBC indexes parameters from 1 al- with Damon Hill driving car 0, due to the reigning though Java itself uses 0-based indexing. World Champion (Nigel Mansell and Alain Prost re- spectively) not competing in the championship. In databases, it is possible for a field not to have a value. It is then said to have a null value. For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to three- 12.9 Symbols and representations valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including Main article: Symbols for zero a null value delivers a null result. Asking for all records The modern numerical digit 0 is usually written as a circle with value 0 or value not equal 0 will not yield all records, or ellipse. Traditionally, many print typefaces made the since the records with value null are excluded. capital letter O more rounded than the narrower, elliptical 92 CHAPTER 12. 0

12.12 References

[1] Matson, John (21 August 2009). “The Origin of Zero”. Scientific American. Springer Nature. Retrieved 24 April 2016.

[2] Soanes, Catherine; Waite, Maurice; Hawker, Sara, eds. (2001). The Oxford Dictionary, Thesaurus and Word- power Guide (Hardback) (2nd ed.). New York: Oxford University Press. ISBN 978-0-19-860373-3.

[3]“aught, Also ought”in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Mer- digit 0.*[48] Typewriters originally made no distinction in riam. shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into [4]“cipher”, in Webster's Collegiate Dictionary (1927), Third prominence on modern character displays.*[48] Edition, Springfield, MA: G. & C. Merriam. A can be used to distinguish the number from [5] aught at etymonline.com the letter. The digit 0 with a dot in the center seems [6] See: to have originated as an option on IBM 3270 displays and has continued with some modern computer typefaces • Douglas Harper (2011), Zero, Etymology Dictio- such as Andalé Mono, and in some airline reservation nary, Quote="figure which stands for naught in the systems. One variation uses a short vertical bar instead Arabic notation,”also“the absence of all quantity of the dot. Some fonts designed for use with computers considered as quantity,”c.1600, from French zéro made one of the capital-O–digit-0 pair more rounded and or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr “cipher,”translation the other more angular (closer to a rectangle). A further of Sanskrit sunya-m“empty place, desert, naught"; distinction is made in falsification-hindering as • used on German car number plates by slitting open the Menninger, Karl (1992). Number words and num- ber symbols: a cultural history of numbers. Courier digit 0 on the upper right side. Sometimes the digit 0 is Dover Publications. pp. 399–404. ISBN 0-486- used either exclusively, or not at all, to avoid confusion 27096-3.; altogether. • “zero, n.”. OED Online. Oxford University Press. December 2011. Archived from the original on 6 March 2012. Retrieved 2012-03-04. French zéro 12.10 Year label (1515 in Hatzfeld & Darmesteter) or its source Ital- ian zero , for *zefiro , < Arabic çifr

Main article: 0 (year) [7] See: • Smithsonian Institution, Oriental Elements of Cul- In the BC calendar era, the year 1 BC is the first year ture in the Occident, p. 518, at Google Books, An- before AD 1; there is not a year zero. By contrast, in nual Report of the Board of Regents of the Smith- astronomical year numbering, the year 1 BC is numbered sonian Institution; Harvard University Archives, Quote="Sifr occurs in the meaning of “empty” 0, the year 2 BC is numbered −1, and so on.*[49] even in the pre-Islamic time. (...) Arabic sifr in the meaning of zero is a translation of the correspond- ing India sunya.”; 12.11 See also • Jan Gullberg (1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co., ISBN 978- 0393040029, page 26, Quote = ‘‘Zero derives • Zeroth (zero as an ordinal number) from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.’’; • • Robert Logan (2010), The Poetry of Physics and the Physics of Poetry, World Scientific, ISBN 978- • Division by zero 9814295925, page 38, Quote =“The idea of sunya and place numbers was transmitted to the Arabs • Grammatical number who translated sunya or “leave a space”into their language as sifr.” • [8] Zero, Merriam Webster online Dictionary

• Peano axioms [9] Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. • Signed zero ISBN 0-471-39340-1. 12.12. REFERENCES 93

[10] 'Aught' definition, Dictionary.com – Retrieved April [27] Kim Plofker (2009), Mathematics in India, Princeton 2013. University Press, ISBN 978-0691120676, page 54–56. Quote – “In the Chandah-sutra of Pingala, dating per- [11] 'Aught' synonyms, Thesaurus.com – Retrieved April haps the third or second century BC, [...] Pingala's use 2013. of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero.”Kim Plofker (2009), [12] Joseph, George Gheverghese (2011). The Crest of the Mathematics in India, Princeton University Press, ISBN Peacock: Non-European Roots of Mathematics (Third Edi- 978-0691120676, 55–56. “In the Chandah-sutra of Pin- tion). Princeton. p. 86. ISBN 978-0-691-13526-7. gala, dating perhaps the third or second century BC, there [13] Kaplan, Robert. (2000). The Nothing That Is: A Natural are five questions concerning the possible meters for any “ ” 7 History of Zero. Oxford: Oxford University Press. value n . [...] The answer is (2) = 128, as expected, but instead of seven doublings, the process (explained by [14] No long count date actually using the number 0 has been the sutra) required only three doublings and two squarings found before the 3rd century AD, but since the long count – a handy time saver where “n”is large. Pingala’s use system would make no sense without some placeholder, of a zero symbol as a marker seems to be the first known and since Mesoamerican glyphs do not typically leave explicit reference to zero. empty spaces, these earlier dates are taken as indirect ev- idence that the concept of 0 already existed at the time. [28] Ifrah, Georges (2000), p. 416.

[15] Diehl, p. 186 [29] Aryabhatiya of Aryabhata, translated by Walter Eugene Clark. [16] Mortaigne, Véronique (November 28, 2014). “The golden age of Mayan civilisation – exhibition review”. [30] O'Connor, Robertson, J.J., E. F. “Aryabhata the Elder” The Guardian. Archived from the original on 28 Novem- . School of Mathematics and Statistics University of St ber 2014. Retrieved October 10, 2015. Andrews, Scotland. Retrieved 26 May 2013.

[17] Wallin, Nils-Bertil (19 November 2002). “The History [31] William L. Hosch, ed. (15 August 2010). The Britannica of Zero”. YaleGlobal online. The Whitney and Betty Guide to Numbers and Measurement (Math Explained). Macmillan Center for International and Area Studies at books.google.com.my. The Rosen Publishing Group. pp. Yale. Retrieved September 1, 2016. 97–98. ISBN 9781615301089. [18] O'Connor, John J.; Robertson, Edmund F., “A history [32] Algebra with Arithmetic of Brahmagupta and Bhaskara, of Zero”, MacTutor History of Mathematics archive, translated to English by Henry Thomas Colebrooke University of St Andrews. (1817) London [19] “Zero and Fractions”. Know the Romans. Retrieved 21 September 2016. [33] Cœdès, Georges, “A propos de l'origine des chiffres arabes,”Bulletin of the School of Oriental Studies, Uni- [20] Hodgkin, Luke (2 June 2005). A History of Mathematics versity of London, Vol. 6, No. 2, 1931, pp. 323–328. : From Mesopotamia to Modernity: From Mesopotamia to Diller, Anthony,“New Zeros and Old Khmer,”The Mon- Modernity. Oxford University Press. p. 85. ISBN 978-0- Khmer Studies Journal, Vol. 25, 1996, pp. 125–132. 19-152383-0. [34] Casselman, Bill. “All for Nought”. ams.org. University [21] Crossley, Lun. 1999, p.12 “the ancient Chinese system of British Columbia), American Mathematical Society. is a place notation system” [35] Ifrah, Georges (2000), p. 400. [22] Kang-Shen Shen; John N. Crossley; Anthony W. C. Lun; Hui Liu (1999). The Nine Chapters on the Mathemati- [36] Pannekoek, A. (1961). A History of Astronomy. George cal Art: Companion and Commentary. Oxford Univer- Allen & Unwin. p. 165. sity Press. p. 35. ISBN 978-0-19-853936-0. zero was regarded as a number in India... whereas the Chinese em- [37] Will Durant (1950), The Story of Civilization, Volume 4, ployed a vacant position The Age of Faith: Constantine to Dante – A.D. 325– 1300, Simon & Schuster, ISBN 978-0965000758, p. 241, [23] “Mathematics in the Near and Far East” (pdf). gr- Quote = “The Arabic inheritance of science was over- math4.phpnet.us. p. 262. whelmingly Greek, but Hindu influences ranked next. In [24] Struik, Dirk J. (1987). A Concise History of Mathematics. 773, at Mansur's behest, translations were made of the Sid- New York: Dover Publications. pp. 32–33. "In these dhantas – Indian astronomical treatises dating as far back matrices we find negative numbers, which appear here for as 425 BC; these versions may have the vehicle through “ ” the first time in history." which the Arabic numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the [25] Bourbaki, Nicolas Elements of the History of Mathematics Hindu numerals in his astronomical tables.” (1998), p. 46. Britannica Concise Encyclopedia (2007), entry “Algebra” [38] Brezina, Corona (2006). Al-Khwarizmi: The Inventor Of Algebra. The Rosen Publishing Group. ISBN 978-1- [26] “Math for Poets and Drummers” (pdf). people.sju.edu. 4042-0513-0. 94 CHAPTER 12. 0

[39] Will Durant (1950), The Story of Civilization, Volume • Kaplan, Robert (2000) The Nothing That Is: A - 4, The Age of Faith, Simon & Schuster, ISBN 978- ural History of Zero, Oxford: Oxford University 0965000758, p. 241, Quote =“In 976, Muhammad ibn Press. Ahmad, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little • Seife, Charles (2000) Zero: The Biography of a circle should be used“to keep the rows”. This circle the Dangerous Idea, Penguin USA (Paper). ISBN 0-14- Mosloems called ṣifr, “empty”whence our cipher.” 029647-6. [40] Sigler, L., Fibonacci's . English translation, • Bourbaki, Nicolas (1998). Elements of the History Springer, 2003. of Mathematics. Berlin, Heidelberg, and New York: [41] Grimm, R.E.,“The Autobiography of Leonardo Pisano” Springer-Verlag. ISBN 3-540-64767-8. , 11/1 (February 1973), pp. 99–104. • Isaac Asimov (1978). Article“Nothing Counts”in [42] Lemma B.2.2, The integer 0 is even and is not odd, in Pen- Asimov on Numbers. Pocket Books. ner, Robert C. (1999). Discrete Mathematics: Proof Tech- niques and Mathematical Structures. World Scientific. p. • This article is based on material taken from the Free On- 34. ISBN 981-02-4088-0. line Dictionary of Computing prior to 1 November 2008 and incorporated under the “relicensing”terms of the [43] Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathe- GFDL, version 1.3 or later. matics. Courier Dover Publications. pp. 254–255. ISBN • Chris Woodford (2006), Digital Technology, Evans 0-486-13968-9., Extract of pages 254–255 Brothers, ISBN 978-02-3752-725-9 [44] Reid, Constance (1992). From zero to infinity: what makes numbers interesting (4th ed.). Mathematical Association of America. p. 23. ISBN 978-0-88385-505-8. 12.14 External links [45] “Solar Eclipses of Saros 1 to 175”. sun- earth.gsfc.nasa.gov. Archived from the original on • Search for the world's first zero leads to the home of 5 February 2007. Angkor Wat [46] “NASA – Lunar Eclipses of Saros Series 1 to 175”. • A History of Zero sunearth.gsfc.nasa.gov. Archived from the original on 29 February 2008. • Zero Saga [47] Chris Woodford 2006, p. 9. • The [48] Bemer, R. W. (1967). “Towards standards for handwrit- • ten zero and oh: much ado about nothing (and a letter), Edsger W. Dijkstra: Why numbering should start at or a partial dossier on distinguishing between handwrit- zero, EWD831 (PDF of a handwritten manuscript) ten zero and oh”. Communications of the ACM. 10 (8): 513–518. doi:10.1145/363534.363563. • Zero on In Our Time at the BBC.(listen now)

[49] Steel, Duncan (2000). Marking time: the epic quest to in- • Weisstein, Eric W. “0”. MathWorld. vent the perfect calendar. John Wiley & Sons. p. 113. ISBN 0-471-29827-1. In the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came AD 1 Jan- • Texts on Wikisource: uary 1. ... If you object to that no-year-zero scheme, then • don't use it: use the astronomer's counting scheme, with "Zero". Encyclopædia Britannica (11th ed.). negative year numbers. 1911. • "Zero". Encyclopedia Americana. 1920. 12.13 Bibliography

• Amir D. Aczel (2015) Finding Zero, New York City: Palgrave Macmillan. ISBN 978-1-137-27984-2 • Barrow, John D. (2001) The Book of Nothing, Vin- tage. ISBN 0-09-928845-1. • Diehl, Richard A. (2004) The Olmecs: America's First Civilization, Thames & Hudson, London. • Ifrah, Georges (2000) The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley. ISBN 0-471-39340-1. Chapter 13

Aliquot sum

In mathematics, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n. The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively. The aliquot sequence is the sequence obtained by repeat- edly applying the aliquot sum function s. The aliquot sum function is also referred to as the restricted divisor func- tion.*[1] For example, the proper divisors of 15 (that is, the pos- itive divisors of 15 that are not equal to 15) are 1, 3 and 5, so the aliquot sum of 15 is 9 (1 + 3 + 5).

13.1 See also

•

13.2 References

[1] Weisstein, Eric W. “Restricted Divisor Function”. MathWorld.

95 Chapter 14

Perfect number

For the 2012 film, see Perfect Number (film). equal to half the sum of all its positive divisors: ( 1 + 2 + In number theory, a perfect number is a positive inte- 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in the OEIS).

14.2 History

In about 300 BC Euclid showed that if 2*p−1 is prime then (2*p−1)2*p−1 is perfect. The first four perfect num- bers were the only ones known to early Greek mathe- matics, and the mathematician had noted 8128 as early as 100 AD.*[2] Philo of Alexandria in his first-century book “On the creation”mentions per- fect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen,*[3] and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than Demonstration, with Cuisenaire rods, of the perfection of the 10,000. (Commentary on Genesis 1. 14-19).*[4] St Au- number 6 gustine defines perfect numbers in City of God (Part XI, Chapter 30) in the early 5th century AD, repeating the ger that is equal to the sum of its proper positive divisors, claim that God created the world in 6 days because 6 that is, the sum of its positive divisors excluding the num- is the smallest perfect number. The Egyptian mathe- ber itself (also known as its aliquot sum). Equivalently, a matician Ismail ibn Fallūs (1194–1252) mentioned the perfect number is a number that is half the sum of all of next three perfect numbers (33,550,336, 8,589,869,056 its positive divisors (including itself) i.e. σ1(n) = 2n. and 137,438,691,328) and listed a few more which are This definition is ancient, appearing as early as Euclid's now known to be incorrect.*[5] In a manuscript writ- Elements (VII.22) where it is called τέλειος ἀριθμός ten between 1456 and 1461, an unknown mathematician (perfect, ideal, or complete number). Euclid also proved a recorded the earliest European reference to a fifth perfect formation rule (IX.36) whereby q(q+1)/2 is an even per- number, with 33,550,336 being correctly identified for fect number whenever q is a prime of the form 2p − 1 for the first time.*[6]*[7] In 1588, the Italian mathematician prime p —what is now called a Mersenne prime. Much Pietro Cataldi also identified the sixth (8,589,869,056) later, Euler proved that all even perfect numbers are of and the seventh (137,438,691,328) perfect numbers, and this form.*[1] This is known as the Euclid–Euler theo- also proved that every perfect number obtained from Eu- rem. clid's rule ends with a 6 or an 8.*[8]*[9]*[10] It is not known whether there are any odd perfect num- bers, nor whether infinitely many perfect numbers exist. 14.3 Even perfect numbers

14.1 Examples See also: Euclid–Euler theorem

The first perfect number is 6. Its proper divisors are 1, 2, Euclid proved that 2*p−1(2*p − 1) is an even perfect num- and 3, and 1 + 2 + 3 = 6. Equivalently, the number 6 is ber whenever 2*p − 1 is prime (Euclid, Prop. IX.36).

96 14.3. EVEN PERFECT NUMBERS 97

For example, the first four perfect numbers are generated and the 2*p−1th hexagonal number. Furthermore, each by the formula 2*p−1(2*p − 1), with p a prime number, even perfect number except for 6 is the ((2*p + 1)/3) as follows: centered nonagonal number and is equal to the sum of the first 2*(p−1)/2 odd cubes: for p = 2: 21(22 − 1) = 6 for p = 3: 22(23 − 1) = 28 6 = 21(22 − 1) = 1 + 2 + 3, for p = 5: 24(25 − 1) = 496 for p = 7: 26(27 − 1) = 8128. 28 = 22(23 − 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 13 + 33,

4 5 − ··· Prime numbers of the form 2*p − 1 are known as 496 = 2 (2 1) = 1 + 2 + 3 + + 29 + 30 + 31 Mersenne primes, after the seventeenth-century monk = 13 + 33 + 53 + 73, Marin Mersenne, who studied number theory and perfect numbers. For 2*p − 1 to be prime, it is necessary that p 8128 = 26(27 − 1) = 1 + 2 + 3 + ··· + 125 + 126 + 127 itself be prime. However, not all numbers of the form 2*p = 13 + 33 + 53 + 73 + 93 + 113 + 133 + 153, − 1 with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number.*[11] In fact, Mersenne 33550336 = 212(213 − 1) = 1 + 2 + 3 + ··· + 8189 + 8190 + 8191 primes are very rare—of the 9,592 prime numbers p less 3 3 3 3 3 3 than 100,000,*[12] 2*p − 1 is prime for only 28 of them. = 1 + 3 + 5 + ··· + 123 + 125 + 127 . Nicomachus (60-120 AD) conjectured that every per- Even perfect numbers (except 6) are of the form fect number is of the form 2*p−1(2*p − 1) where 2*p − 1 is prime.*[13] Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every even perfect number is of that p − × p form.*[14] It was not until the 18th century that Leonhard (2 2) (2 + 1) T2p−1 = 1 + = 1 + 9 × T(2p−2)/3 Euler proved that the formula 2*p−1(2*p − 1) will yield 2 all the even perfect numbers. Thus, there is a one-to- one correspondence between even perfect numbers and with each resulting triangular number (after subtracting Mersenne primes; each Mersenne prime generates one 1 from the perfect number and dividing the result by 9) even perfect number, and vice versa. This result is often ending in 3 or 5, the sequence starting with 3, 55, 903, * referred to as the Euclid–Euler theorem. As of January 3727815, .... [17] This can be reformulated as follows: 2016, 49 Mersenne primes are known,*[15] and therefore adding the digits of any even perfect number (except 6), 49 even perfect numbers (the largest of which is 274207280 then adding the digits of the resulting number, and re- × (274207281 − 1) with 44,677,235 digits). peating this process until a single digit (called the digital root) is obtained, always produces the number 1. For ex- An exhaustive search by the GIMPS distributed comput- ample, the digital root of 8128 is 1, because 8 + 1 + 2 + ing project has shown that the first 44 even perfect num- 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all * * bers are 2 p−1(2 p − 1) for perfect numbers 2*p−1(2*p − 1) with odd prime p and, in fact, with all numbers of the form 2*m−1(2*m − 1) for p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, odd integer (not necessarily prime) m. 127, 521, 607, 1279, 2203, 2281, 3217, 4253, Owing to their form, 2*p−1(2*p − 1), every even perfect 4423, 9689, 9941, 11213, 19937, 21701, number is represented in binary as p ones followed by p 23209, 44497, 86243, 110503, 132049, − 1 zeros: 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 610 = 1102 and 32582657 (sequence A000043 in the OEIS).*[16] 2810 = 111002

49610 = 1111100002 Five higher perfect numbers have also been discov- ered, namely those for which p = 37156667, 42643801, 812810 = 11111110000002 43112609, 57885161, and 74207281, though there may 3355033610 = be others within this range. It is not known whether there 11111111111110000000000002. are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. As well as having the form 2*p−1(2*p − 1), each even Thus every even perfect number is a pernicious number. perfect number is the (2*p − 1)th triangular number (and Note that every even perfect number is also a practical hence equal to the sum of the integers from 1 to 2*p − 1) number (c.f. Related concepts). 98 CHAPTER 14. PERFECT NUMBER

14.4 Odd perfect numbers 14.5 Minor results

It is unknown whether there is any odd perfect num- All even perfect numbers have a very precise form; odd ber, though various results have been obtained. In 1496, perfect numbers either do not exist or are rare. There are Jacques Lefèvre stated that Euclid's rule gives all perfect a number of results on perfect numbers that are actually numbers,*[18] thus implying that no odd perfect number quite easy to prove but nevertheless superficially impres- exists. More recently, Carl Pomerance has presented a sive; some of them also come under Richard Guy's strong heuristic argument suggesting that indeed no odd perfect law of small numbers: number should exist.*[19] All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as • The only even perfect number of the form x3 + 1 is well that there are no odd Ore's harmonic numbers other 28 (Makowski 1962).*[32] than 1. • 28 is also the only even perfect number that is a sum Any odd perfect number N must satisfy the following of two positive integral cubes (Gallardo 2010).*[33] conditions: • The reciprocals of the divisors of a perfect number • N > 101500.*[20] N must add up to 2 (to get this, take the definition of a perfect number, σ (n) = 2n , and divide both • * 1 N is not divisible by 105. [21] sides by n): • N is of the form N ≡ 1 (mod 12), N ≡ 117 (mod • For 6, we have 1/6 + 1/3 + 1/2 + 1/1 = 2 ; 468), or N ≡ 81 (mod 324).*[22] • For 28, we have 1/28 + 1/14 + 1/7 + 1/4 + • N is of the form 1/2 + 1/1 = 2 , etc. • The number of divisors of a perfect number (whether even or odd) must be even, because N can- α 2e1 ··· 2ek N = q p1 pk , not be a perfect square.*[34] where: • From these two results it follows that every

• q, p1, ..., pk are distinct primes (Euler). perfect number is an Ore's harmonic number. • q ≡ α ≡ 1 (mod 4) (Euler). • The even perfect numbers are not trapezoidal num- • The smallest prime factor of N is less than bers; that is, they cannot be represented as the dif- (2k + 8) / 3.*[23] ference of two positive non-consecutive triangular * 62 * 62 • Either q α > 10 , or p j 2ej > 10 for numbers. There are only three types of non- some j.*[20] trapezoidal numbers: even perfect numbers, powers 2n−1(2n + 1) • N < 2*4*k+1.*[24] of two, and the numbers of the form formed as the product of a Fermat prime 2n +1 with • a power of two in a similar way to the construction of The largest prime factor of N is greater than * 108.*[25] even perfect numbers from Mersenne primes. [35] • • The second largest prime factor is greater than 104, The number√ of perfect numbers less than n is less * and the third largest prime factor is greater than than√c n , where c > 0 is a constant. [36] In fact it * 100.*[26]*[27] is o( n) , using little-o notation. [37] • N has at least 101 prime factors and at least 10 dis- • Every even perfect number ends in 6 or 28, base ten; tinct prime factors.*[20]*[28] If 3 is not one of the and, with the only exception of 6, ends in 1, base * * factors of N, then N has at least 12 distinct prime 9. [38] [39] Therefore in particular the digital root factors.*[29] of every even perfect number other than 6 is 1. • The only square-free perfect number is 6.*[40] In 1888, Sylvester stated:*[30]

...a prolonged on the subject has satisfied me that the existence of any one such 14.6 Related concepts [odd perfect number] —its escape, so to say, from the complex web of conditions which hem The sum of proper divisors gives various other kinds of it in on all sides —would be little short of a numbers. Numbers where the sum is less than the num- miracle. ber itself are called deficient, and where it is greater than the number, abundant. These terms, together with per- Euler stated: “Whether (...) there are any odd perfect fect itself, come from Greek numerology. A pair of num- numbers is a most difficult question”.*[31] bers which are the sum of each other's proper divisors are 14.8. NOTES 99 called amicable, and larger cycles of numbers are called [11] All factors of 2*p − 1 are congruent to 1 mod 2p. For sociable. A positive integer such that every smaller posi- example, 211 − 1 = 2047 = 23 × 89, and both 23 and 89 tive integer is a sum of distinct divisors of it is a practical yield a remainder of 1 when divided by 11. Furthermore, number. whenever p is a Sophie Germain prime—that is, 2p + 1 is also prime—and 2p + 1 is congruent to 1 or 7 mod 8, then By definition, a perfect number is a fixed point of the 2p + 1 will be a factor of 2*p − 1, which is the case for p restricted divisor function s(n) = σ(n) − n, and the aliquot = 11, 23, 83, 131, 179, 191, 239, 251, ... A002515. sequence associated with a perfect number is a constant sequence. All perfect numbers are also S -perfect num- [12] Song Y. Yan (2009). Primality Testing and Integer Fac- bers, or Granville numbers. torization in Public-Key Cryptography. Advances in Infor- mation Security. 11 (2nd ed.). Springer-Verlag. p. 199. A semiperfect number is a natural number that is equal to ISBN 0-387-77268-5. the sum of all or some of its proper divisors. A semiper- fect number that is equal to the sum of all its proper divi- [13] http://www-groups.dcs.st-and.ac.uk/history/HistTopics/ sors is a perfect number. Most abundant numbers are also Perfect_numbers. semiperfect; abundant numbers which are not semiper- fect are called weird numbers. [14] O'Connor, John J.; Robertson, Edmund F., “Abu Ali al- Hasan ibn al-Haytham”, MacTutor History of Mathemat- ics archive, University of St Andrews.

14.7 See also [15] “GIMPS Home”. Mersenne.org. Retrieved 2013-02-05.

• Jan Brożek [16] GIMPS Milestones Report. Retrieved 2014-02-24 • List of perfect numbers [17] Weisstein, Eric W. “Perfect Number”. MathWorld.

• Multiply perfect number [18] Dickson, L. E. (1919). History of the Theory of Numbers, • Vol. I. Washington: Carnegie Institution of Washington. Superperfect numbers p. 6.

[19] Oddperfect.org. 14.8 Notes [20] Ochem, Pascal; Rao, Michaël (2012). “Odd per- fect numbers are greater than 101500" (PDF). “ [1] Caldwell, Chris, A proof that all even perfect numbers Mathematics of Computation. 81 (279): 1869– ” are a power of two times a Mersenne prime . 1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN [2] Dickson, L. E. (1919). History of the Theory of Numbers, 0025-5718. Zbl pre06051364. Vol. I. Washington: Carnegie Institution of Washington. p. 4. [21] Kühnel, U (1949). “Verschärfung der notwendigen Be- dingungen für die Existenz von ungeraden vollkomme- [3] Commentary on the Gospel of John 28.1.1-4, with further nen Zahlen”. Mathematische Zeitschrift. 52: 201–211. references in the Sources Chrétiennes edition: vol. 385, doi:10.1515/crll.1941.183.98. Retrieved 30 March 2011. 58-61. [22] Roberts, T (2008). “On the Form of an Odd Perfect [4] http://torreys.org/sblpapers2015/S22-05_philonic_ Number” (PDF). Australian Mathematical Gazette. 35 arithmological_exegesis.pdf (4): 244. [5] Roshdi Rashed, The Development of Arabic Mathematics: [23] Grün, O (1952). "Über ungerade vollkommene Zahlen” Between Arithmetic and Algebra (Dordrecht: Kluwer Aca- . Mathematische Zeitschrift. 55 (3): 353–354. demic Publishers, 1994), pp. 328–329. doi:10.1007/BF01181133. Retrieved 30 March 2011. [6] Munich, Bayerische Staatsbibliothek, Clm 14908 [24] Nielsen, PP (2003). “An upper bound for odd perfect [7] Smith, DE (1958). The History of Mathematics: Volume numbers”. Integers. 3: A14–A22. Retrieved 30 March II. New York: Dover. p. 21. ISBN 0-486-20430-8. 2011.

[8] Dickson, L. E. (1919). History of the Theory of Numbers, [25] Goto, T; Ohno, Y (2008). “Odd perfect numbers have Vol. I. Washington: Carnegie Institution of Washington. a prime factor exceeding 108" (PDF). Mathematics of p. 10. Computation. 77 (263): 1859–1868. doi:10.1090/S0025- [9] Pickover, C (2001). Wonders of Numbers: Adventures in 5718-08-02050-9. Retrieved 30 March 2011. Mathematics, Mind, and Meaning. Oxford: Oxford Uni- “ versity Press. p. 360. ISBN 0-19-515799-0. [26] Iannucci, DE (1999). The second largest prime di- visor of an odd perfect number exceeds ten thousand” [10] Peterson, I (2002). Mathematical Treks: From Surreal (PDF). Mathematics of Computation. 68 (228): 1749– Numbers to Magic Circles. Washington: Mathematical 1760. doi:10.1090/S0025-5718-99-01126-6. Retrieved Association of America. p. 132. ISBN 88-8358-537-2. 30 March 2011. 100 CHAPTER 14. PERFECT NUMBER

[27] Iannucci, DE (2000). “The third largest prime divi- • Kanold, H.-J. (1941).“Untersuchungen über unger- sor of an odd perfect number exceeds one hundred” ade vollkommene Zahlen”. Journal für die Reine (PDF). Mathematics of Computation. 69 (230): 867– und Angewandte Mathematik. 183: 98–109. 879. doi:10.1090/S0025-5718-99-01127-8. Retrieved 30 March 2011. • Steuerwald, R. “Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vol- [28] Nielsen, PP (2015). “Odd perfect numbers, Diophan- lkommenen Zahl”. S.-B. Bayer. Akad. Wiss. 1937: ” tine equations, and upper bounds (PDF). Mathematics 69–72. of Computation. 84 (0): 2549–2567. doi:10.1090/S0025- 5718-2015-02941-X. Retrieved 13 August 2015. [29] Nielsen, PP (2007). “Odd perfect numbers have at least 14.10 Further reading nine distinct prime factors”(PDF). Mathematics of Com- putation. 76 (260): 2109–2126. doi:10.1090/S0025- • Nankar, M.L.: “History of perfect numbers,” 5718-07-01990-4. Retrieved 30 March 2011. Ganita Bharati 1, no. 1–2 (1979), 7–8. [30] The Collected Mathematical Papers of James Joseph • Sylvester p. 590, tr. from“Sur les nombres dits de Hamil- Hagis, P. (1973). “A Lower Bound for the set of ton”, Compte Rendu de l'Association Française (Toulouse, odd Perfect Prime Numbers”. Mathematics of Com- 1887), pp. 164–168. putation. 27: 951–953. doi:10.2307/2005530.

[31] http://www.math.harvard.edu/~{}knill/seminars/ • Riele, H.J.J. “Perfect Numbers and Aliquot Se- perfect/handout.pdf quences”in H.W. Lenstra and R. Tijdeman (eds.): Computational Methods in Number Theory, Vol. [32] Makowski, A. (1962). “Remark on perfect numbers”. 154, Amsterdam, 1982, pp. 141–157. Elem. Math. 17 (5): 109. • Riesel, H. Prime Numbers and Computer Methods “ [33] Gallardo, Luis H. (2010). On a remark of Makowski for Factorisation, Birkhauser, 1985. about perfect numbers”. Elem. Math. 65: 121–126.. • Sándor, Jozsef; Crstici, Borislav (2004). Handbook [34] Yan, Song Y. (2012), Computational Number Theory and of number theory II. Dordrecht: Kluwer Academic. Modern Cryptography, John Wiley & Sons, Section 2.3, 2(6), ISBN 9781118188613. pp. 15–98. ISBN 1-4020-2546-7. Zbl 1079.11001.

[35] Jones, Chris; Lord, Nick (1999). “Characterising non-trapezoidal numbers”. The Mathematical Gazette. 14.11 External links The Mathematical Association. 83 (497): 262–263. doi:10.2307/3619053. JSTOR 3619053 • Hazewinkel, Michiel, ed. (2001),“Perfect number” [36] Hornfeck, B (1955). “Zur Dichte der Menge der vol- , Encyclopedia of Mathematics, Springer, ISBN 978- lkommenen zahlen”. Arch. Math. 6 (6): 442–443. 1-55608-010-4 doi:10.1007/BF01901120. • David Moews: Perfect, amicable and sociable num- [37] Kanold, HJ (1956). “Eine Bemerkung ¨uber die Menge bers der vollkommenen zahlen”. Math. Ann. 131 (4): 390– 392. doi:10.1007/BF01350108. • Perfect numbers – History and Theory

[38] H. Novarese. Note sur les nombres parfaits Texeira J. VIII • Weisstein, Eric W.“Perfect Number”. MathWorld. (1886), 11–16. • [39] Dickson, L. E. (1919). History of the Theory of Numbers, “Sloane's A000396 : Perfect numbers”. The Vol. I. Washington: Carnegie Institution of Washington. On-Line Encyclopedia of Integer Sequences. OEIS p. 25. Foundation.

[40] Redmond, Don (1996). Number Theory: An Introduction • OddPerfect.org A projected distributed computing to Pure and Applied Mathematics. Chapman & Hall/CRC project to search for odd perfect numbers. Pure and Applied Mathematics. 201. CRC Press. Prob- lem 7.4.11, p. 428. ISBN 9780824796969.. • Great Internet Mersenne Prime Search • Perfect Numbers, math forum at Drexel. 14.9 References • Grimes, James. “8128: Perfect Numbers”. Num- berphile. Brady Haran. • Euclid, Elements, Book IX, Proposition 36. See D.E. Joyce's website for a translation and discussion of this proposition and its proof. Chapter 15

Digital root

The digital root (also repeated digital sum) of a non- ⌊ ⌋ negative integer is the (single digit) value obtained by an n − 1 iterative process of summing digits, on each iteration us- dr(n) = n − 9 . 9 ing the result from the previous iteration to compute a . The process continues until a single-digit num- ber is reached. 15.2 Abstract multiplication of For example, the digital root of 65,536 is 7, because 6 + digital roots 5 + 5 + 3 + 6 = 25 and 2 + 5 = 7.

Digital roots can be calculated with congruences in The table below shows the digital roots produced by the rather than by adding up all the digits, familiar multiplication table in the decimal system. a procedure that can save time in the case of very large numbers. The table shows a number of interesting patterns and symmetries and is known as the Vedic square. Digital roots can be used as a sort of checksum, to check that a sum has been performed correctly. If it has, then the digital root of the sum of the given numbers will equal 15.3 Formal definition the digital root of the sum of the digital roots of the given numbers. This check, which involves only single-digit calculations, can catch many errors in calculation. Let S(n) denote the sum of the digits of n and let the composition of S(n) be as follows: Digital roots are used in Western numerology, but cer- tain numbers deemed to have occult significance (such as ( ) 11 and 22) are not always completely reduced to a single S1(n) = S(n),Sm(n) = S Sm−1(n) , for m ≥ 2. digit. Eventually the sequence S1(n),S2(n),S3(n), ··· be- The number of times the digits must be summed to reach ∗ the digital sum is called a number's additive persistence; comes a one digit number. Let S (n) (the digital sum in the above example, the additive persistence of 65,536 of n ) represent this one digit number. is 2. 15.3.1 Example

15.1 Significance and formula of Let us find the digital sum of 1853 . the digital root S(1853) = 17 It helps to see the digital root of a positive integer as the S(17) = 8 position it holds with respect to the largest multiple of 9 less than the number itself. For example, the digital root Thus, of 11 is 2, which means that 11 is the second number after 9. Likewise, the digital root of 2035 is 1, which means 2 that 2035 − 1 is a multiple of 9. If a number produces a S (1853) = 8. digital root of exactly 9, then the number is a multiple of For simplicity let us agree simply that 9. With this in mind the digital root of a positive integer n may be defined by using floor function ⌊x⌋ , as S∗(1853) = dr(1853) = 8.

101 102 CHAPTER 15. DIGITAL ROOT

15.3.2 Proof that a constant value exists 15.5 Some properties of digital roots How do we know that the sequence S1(n),S2(n),S3(n), ··· eventually becomes a one digit number? Here's a proof: The digital root of a number is zero if and only if the number is itself zero. m−1 Let n = d1 + 10d2 + ··· + 10 dm , for all i , di is an integer greater than or equal to 0 and less than 10. Then, ··· S(n) = d1 + d2 + + dm . This means that S(n) < n dr(n) = 0 ⇔ n = 0. , unless d2, d3, ··· , dm = 0 , in which case n is a one digit number. Thus, repeatedly using the S(n) function The digital root of a number is a positive integer if and would cause n to decrease by at least 1, until it becomes only if the number is itself a positive integer. a one digit number, at which point it will stay constant, as S(d1) = d1 . dr(n) > 0 ⇔ n > 0.

The digital root of n is n itself if and only if the number 15.4 Congruence formula has exactly one digit.

The formula is: dr(n) = n ⇔ n ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

 The digital root of n is less than n if and only if the num-  0 if n = 0, ber is greater than or equal to 10. dr(n) = 9 if n ≠ 0, n ≡ 0 (mod 9),  n mod 9 if n ̸≡ 0 (mod 9) dr(n) < n ⇔ n ≥ 10. or, The digital root of a + b is digital root of the sum of the digital root of a and the digital root of b .

dr(n) = 1 + ((n − 1) mod 9). dr(a + b) = dr(dr(a) + dr(b)). To generalize the concept of digital roots to other bases b, one can simply change the 9 in the formula to b - 1. The digital root of a - b is congruent with the difference (sequence A010888 in the OEIS) of the digital root of a and the digital root of b modulo 9. The digital root is the value modulo 9 because 10 ≡ k ≡ k ≡ 1 (mod 9), and thus 10 1 1 (mod 9), so regard- dr(a − b) ≡ dr(a) − dr(b)(mod 9). less of position, the value mod 9 is the same – a · 100 ≡ a · 10 ≡ a (mod 9) – which is why digits can be mean- Especially, we can define the digital root of minus n as ingfully added. Concretely, for a three-digit number, follows:

− ≡ − dr(abc) ≡ a·102+b·10+c·1 ≡ a·1+b·1+c·1 ≡ a+b+c (drmod( n9)) dr(n)(mod 9). The digital root of a × b is digital root of the product of To obtain the modular value with respect to other num- the digital root of a and the digital root of b . bers n, one can take weighted sums, where the weight on the kth digit corresponds to the value of 10k modulo n, or analogously for bk for different bases. This is simplest dr(a × b) = dr(dr(a) × dr(b)). for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the • The digital root of a nonzero number is 9 if and only divisibility of a decimal number with respect to 2, 5, and if the number is itself a multiple of 9. 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8). dr(n) = 9 ⇔ n = 9m for m = 1, 2, 3, ··· . Also of note is the modulus 11: since 10 ≡ −1 (mod 11), and thus 102 ≡ (−1)2 ≡ 1 (mod 11), • The digital root of a nonzero number is a multiple taking the alternating sum of digits yields the value mod- of 3 if and only if the number is itself a multiple of ulo 11. 3. 15.6. IN OTHER BASES 103

dr(n) = 3 ⇔ n = 9m + 3 for m = 0, 1, 2, ··· , • The digital root of a centered hexagram, or star dr(n) = 6 ⇔ n = 9m + 6 for m = 0, 1, 2, ··· , number is 1 or 4. Digital roots of star numbers dr(n) = 9 ⇔ n = 9m for m = 1, 2, 3, ··· . progress in the sequence 1, 4, 1.

• The digital root of a factorial ≥ 6! is 9. • The digital root of a centered hexagonal number is 1 or 7, their digital roots progressing in the sequence dr(n!) = 9 ⇔ n ≥ 6. 1, 7, 1.

• The digital root of a square is 1, 4, 7, or 9. Digital • The digital root of a triangular number is 1, 3, 6 or roots of square numbers progress in the sequence 1, 9. Digital roots of triangular numbers progress in 4, 9, 7, 7, 9, 4, 1, 9. the sequence 1, 3, 6, 1, 6, 3, 1, 9, 9, which is palin- dromic after the first eight terms. • The digital root of a perfect cube is 1, 8 or 9, and digital roots of perfect cubes progress in that exact • The digital root of Fibonacci numbers is a repeating sequence. pattern of 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, • The digital root of a prime number (except 3) is 1, 4, 1, 5, 6, 2, 8, 1, 9. 2, 4, 5, 7, or 8. • The digital root of Lucas numbers is a repeating pat- • The digital root of a power of 2 is 1, 2, 4, 5, 7, or tern of 2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 8. Digital roots of the powers of 2 progress in the 7, 9, 7, 7, 5, 3, 8. sequence 1, 2, 4, 8, 7, 5. This even applies to neg- ative powers of 2; for example, 2 to the power of 0 • The digital root of the product of twin primes, other − is 1; 2 to the power of 1 (minus one) is .5, with a than 3 and 5, is 8. The digital root of the product of − digital root of 5; 2 to the power of 2 is .25, with a 3 and 5 (twin primes) is 6. digital root of 7; and so on, ad infinitum in both di- rections. This is because negative powers of 2 share the same digits (after removing leading zeroes) as corresponding positive powers of 5, whose digital 15.6 In other bases roots progress in the sequence 1, 5, 7, 8, 4, 2. • The digital root of a power of 5 is 1, 2, 4, 5, 7 or 8. This article is about the digital root in decimal or base Digital roots of the powers of 5 progress in the se- ten, hence it is the number mod 9. It is nothing different quence 1, 5, 7, 8, 4, 2. This even applies to negative as the number converted to base 9 and then only the last powers of 5; for example, 5 to the power of 0 is 1; 5 digit taken. In other radixes the digital root is number to the power of −1 (minus one) is .2, with a digital mod (base-1) so in base 12 a digital root of a number is root of 2; 5 to the power of −2 is .04, with a digital the number mod 11 (Ɛduod), for example, 1972duod is 1 root of 4; and so on, ad infinitum in both directions. + 9 + 7 + 2 = 19 = 17duod which is 1 + 7 = 8, while in This is because the negative powers of 5 share the decimal the root of the same number (3110) is 5; and in same digits (after removing leading zeroes) as cor- base 16 a digital root of a number is the number mod 15 responding positive powers of 2, whose digital roots (0xF), for example, 0x7DF is 7 + 13 + 15 = 35 = 0x23 progress in sequence 1, 2, 4, 8, 7, 5. which is 2 + 3 = 5, while in decimal the root of the same number (2015) is 8. • The digital roots of powered numbers progress in sequence (only certain for positive powers, although in for some exceptions it also may occur for negative powers), and this is because of one of the previously 15.7 See also shown properties. As the digital root of a b is con- gruent with the multiple of the digital root of a and • Base 9 the digital root of b modulo 9, the digital root of a a will also do it. So, for example, as shown above, • powers of 2 will follows the sequence 1, 2, 4, 8, 7, 5; Powers of 47 (whose digital root is 2) will also fol- • low this sequence. The very sequence follows this Digit sum rule, and is appliable to any othe number. • Hamming weight dr(an) ≡ drn(a)(mod 9). • Multiplicative digital root • The digital root of an even perfect number (except 6) is 1. • Vedic square 104 CHAPTER 15. DIGITAL ROOT

15.8 References

• Averbach, Bonnie; Chein, Orin (27 May 1999), Problem Solving Through , Dover Books on Mathematics (reprinted ed.), Mi- neola, : Courier Dover Publications, pp. 125– 127, ISBN 0-486-40917-1 (online copy, p. 125, at Google Books)

• Ghannam, Talal (4 January 2011), The Mystery of Numbers: Revealed Through Their Digital Root, CreateSpace Publications, pp. 68–73, ISBN 978-1- 4776-7841-1 (online copy, p. 68, at Google Books)

• Hall, F. M. (1980), An Introduction into Abstract Al- gebra, 1 (2nd ed.), Cambridge, U.K.: CUP Archive, p. 101, ISBN 978-0-521-29861-2 (online copy, p. 101, at Google Books)

• O'Beirne, T. H. (13 March 1961), “Puzzles and Paradoxes”, New Scientist, Reed Business Infor- mation, 10 (230): 53–54, ISSN 0262-4079 (online copy, p. 53, at Google Books)

• Rouse Ball, W. W.; Coxeter, H. S. M. (6 May 2010), Mathematical Recreations and Essays, Dover Recre- ational Mathematics (13th ed.), NY: Dover Publi- cations, ISBN 978-0-486-25357-2 (online copy at Google Books)

15.9 External links

• pattern of digital root using MS Excel • Weisstein, Eric W. “Digital Root”. MathWorld. 15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 105

15.10 Text and image sources, contributors, and licenses

15.10.1 Text

• 1 Source: ://en.wikipedia.org/wiki/1?oldid=760602849 Contributors: AxelBoldt, Eloquence, Bryan Derksen, Zundark, The Anome, XJaM, Toby Bartels, William Avery, Valhalla, Heron, Juuitchan, N8chz, Jim McKeeth, Patrick, Infrogmation, Michael Hardy, Tim Starling, Dhum Dhum, GUllman, Rp, Liftarn, MartinHarper, SGBailey, Gbleem, CesarB, Egil, Ihcoyc, Fantasy, Ahoerstemeier, Cyp, Haakon, Docu, Susan Mason, LittleDan, Glenn, Cyan, LouI, Andres, Jeandré du Toit, Evercat, Panoramix, EdH, Vargenau, Schneelocke, Revolver, Dcoet- zee, Wzzrd, Nohat, RickK, Reddi, Random832, Denni, Dysprosia, WhisperToMe, Saltine, Sabbut, Ed g2s, Dmetric, Warofdreams, Dpb- smith, Wolfram~enwiki, Lumos3, RadicalBender, Denelson83, Nufy8, Robbot, Dale Arnett, Astronautics~enwiki, Pigsonthewing, Jaredwf, RedWolf, Moondyne, Altenmann, Rfc1394, Academic Challenger, Texture, Yacht, Radical edward, Hadal, JesseW, Wereon, PrimeFan, Dave Bass, Mattflaschen, Tea2min, Enochlau, Robert Happelberg, BigT27, Giftlite, Dbenbenn, Graeme Bartlett, Oberiko, Sj, ShaneKing, Ringbark, Netoholic, Lupin, Numerao, Anton Mravcek, Bkonrad, Sunny256, Jcobb, Rick Block, Duncharris, Guanaco, Yekrats, Zoney, Ee- quor, Macrakis, Francisu, Sohailstyle, R. fiend, Gdr, Sonjaaa, Noe, Bgbot, Mustafaa, Superborsuk, Mamizou, Kaldari, Ihavenolife, Oneiros, Elroch, MRSC, Arcturus, Jewbacca, Janneok~enwiki, Gerrit, Trevor MacInnis, ELApro, Brianjd, Ta bu shi da yu, The demiurge, Mover- ton, Discospinster, 4pq1injbok, Guanabot, Factitious, Schuetzm, Vsmith, HeikoEvermann, Saintswithin, Guanabot2, Mani1, Paul August, Djordjes, CanisRufus, El C, Rgdboer, Kwamikagami, Quercus, Pablo X, Bobo192, NetBot, AmosWolfe, Smalljim, .:Ajvol:., Bbartlog, Jojit fb, Apostrophe, Haham hanuka, SPUI, Nsaa, Lysdexia, Jumbuck, Nazli, Alansohn, Arthena, CyberSkull, Diego Moya, Verdlanco, ABCD, AzaToth, Thoric, SlimVirgin, Redfarmer, Wtmitchell, SidP, Super-Magician, Crobzub, H2g2bob, Ianblair23, DV8 2XL, LukeSurl, Kitch, Oleg Alexandrov, Velho, Woohookitty, Georgia guy, Etacar11, StradivariusTV, Uncle G, Unixer, Ruud Koot, WadeSimMiser, Bluemoose, Dionyziz, Zzyzx11, Cedrus-Libani, Rocafella, Radiant!, Emerson7, RichardWeiss, Magister Mathematicae, BD2412, Kbdank71, Roger McCoy, Jclemens, Reisio, Mendaliv, Enzo Aquarius, Jshadias, Edison, Josh Parris, Ketiltrout, Nightscream, WingZero, Salix alba, Veg- aswikian, DoubleBlue, Matt Deres, Xsg, PkerUNO, Mathbot, Kolbasz, NavarroJ, Pevernagie, Ahunt, Mongreilf, Chobot, DVdm, Gdrbot, 334a, Gwernol, EamonnPKeane, The Rambling Man, YurikBot, Wavelength, Borgx, Jimp, RussBot, [email protected], Soltras, Epolk, Stephenb, Gaius Cornelius, Bovineone, Guiltyspark343, Circa 1900, NawlinWiki, Thechosenone021, The Thadman, Keka, Yoninah, Gu- rudata, Jimmyre, Mikeblas, Scs, Alex43223, BOT-Superzerocool, Rayc, TheNepnoc, Jkelly, Saric, Loool, Zzuuzz, Denisutku, Arthur Rubin, Reyk, Canley, Pasitbank, GraemeL, Gobexe, ArielGold, Garion96, Tropylium, Auroranorth, Ryūkotsusei, Robertd, Crystallina, MacsBug, SmackBot, Incnis Mrsi, Honza Záruba, Reedy, Herostratus, Melchoir, Jab843, AustinKnight, Kintetsubuffalo, HeartofaDog, GraemeMcRae, Yamaguchi 先生, Peter Isotalo, Andy M. 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DIGITAL ROOT

RandomLittleHelper, Inkbug, Cheese00000, FallingGravity, Boblarazzi, QAZZY!23, FrigidNinja, JPaestpreornJeolhlna, Tobatar, Wamiq, Akhooha, Ginsuloft, Luxure, JDiala, BestPie(DOT)Com, PooManTwelve, Willisius, Kwicbaez, Crow, Jmsoccer24, Adamtt9, Pokllllop, DopeDonut, हेमन्त डबराल, Bilorv, Killlrman, Tigercompanion25, Tommy edwards, SPIDERS 397, Dropescone, Gronk Oz, Lyndsey724, SheriffIsInTown, Haykam1222, Narky Blert, XxX$pace5Xxx, TheEditor1515, TheEditor151515, Matt.w.gardner, AlphaBetaGamma01, SparkyDill128, GeneralizationsAreBad, Happyperson88664422, KasparBot, Nicholas.Horsey, Mmatuz000, Poopad, Adam9007, Jack ,WannaBeEditor, CLCStudent, WhyYouDoDisToMeh, Arjun Wox ,مھتاب احمد ,Mullin, Codcassity13, KaBoom100, CAPTAIN RAJU Layla, the remover, InternetArchiveBot, Eurocus47, Amedeo100, Music1201, Boorider7, Motivação, Owen125864454865, Dofjjjdjfjfjf- fjfjfjfjffjffjffjfjfjjfjfjfrkhfowlenouf, Clagerty, SuperSainGouku, Gogo198763 and Anonymous: 747 • 2 (number) Source: https://en.wikipedia.org/wiki/2_(number)?oldid=759402489 Contributors: AxelBoldt, Eloquence, Bryan Derksen, Zundark, The Anome, Jeronimo, Kowloonese, XJaM, Toby Bartels, William Avery, Ellmist, Heron, Olivier, Michael Hardy, JakeVortex, GUllman, Liftarn, SGBailey, (, Minesweeper, Kosebamse, Egil, Ahoerstemeier, Docu, Samuelsen, Angela, Glenn, Llull, Andres, Cimon Avaro, Jiang, Panoramix, Vargenau, Seth , Schneelocke, Iseeaboar, Dcoetzee, Reddi, Random832, Denni, Dysprosia, WhisperToMe, Haukurth, Judzillah, Maximus Rex, Hyacinth, Sabbut, Ed g2s, Dmetric, Warofdreams, Dpbsmith, Jusjih, Wolfram~enwiki, RadicalBen- der, Denelson83, Robbot, Dale Arnett, Jaredwf, Moriori, Fredrik, Altenmann, Gandalf61, Rfc1394, Academic Challenger, Flauto Dolce, Texture, Yacht, JesseW, PrimeFan, Mattflaschen, Dave6, Snobot, Dawoodmajoka, Giftlite, Dbenbenn, Graeme Bartlett, Sj, Numerao, Bradeos Graphon, Anton Mravcek, Rick Block, Duncharris, Guanaco, Eequor, Chameleon, Andycjp, Alexf, Gdr, Noe, Mustafaa, Su- perborsuk, Mamizou, Secfan, Starx, Eric B. and Rakim, Urhixidur, Rgrg, Xrchz, Astronouth7303, Heegoop, Venu62, GaidinBDJ, Jiy, Indosauros, Discospinster, 4pq1injbok, Rich Farmbrough, Guanabot, HeikoEvermann, VT hawkeye, ReallyNiceGuy, RJHall, CanisRu- fus, Kwamikagami, Jpgordon, Bobo192, NetBot, Reinyday, BrokenSegue, Jojit fb, Minghong, Chicago god, Pogo747, Ranveig, Jumbuck, Stephen G. 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Antares5245, Dcshop8384, GT5162, FrescoBot, MHR65, Pepper, MichealH, Xiaosunmath, Robo37, Pinethicket, I dream of horses, Adamgold33, HRoestBot, Edderso, Hamtechperson, Number , Crusoe704, FoxBot, Double sharp, ArtistScientist, TobeBot, Royal Blue Jersey, Aslanio, Lotje, Vrenator, Epic Penguin123, Duoduoduo, Davish Krail, Gold Five, Bellamusic2, Nicola.tesla.rowe, Diannaa, Stephen MUFC, Tbhotch, Mean as custard, NerdyScienceDude, Graeme 2, EmausBot, Ajraddatz, RA0808, Wikipelli, Dcirovic, As- ceticRose, DeadShrimpBlues, Interwebninja, AvicBot, ZéroBot, Fæ, Maxviwe, Poop12345555, Traxs7, Pliming, Mar4d, Killernerd24, Rockclaw1030, Pickuptha'Musket, TheodoreNg, Люди поезда ежедневные, Neozuki, Vunio, L Kensington, Donner60, Twh5125, Chuis- pastonBot, Venugopalreddykonda, Senator2029, Forever Dusk, Rocketrod1960, ClueBot NG, Ezekiel63745, Akuindo, LutherVinci, Yp- nypn, MelbourneStar, Frietjes, Cntras, Newsuperdude102, Imyourfoot, Widr, Pluma, Crosstemplejay, Whatisthisdudes, Wbm1058, Jcon- siglimerrill, BG19bot, Shrohanoldtownbbsr, Bmusician, Arnavchaudhary, Vagobot, Kangaroopower, Janbur95, MusikAnimal, Chamalani, Googlesalot2, NJHartley, Rawrrawr132, Justincheng12345-bot, Miszatomic, Serchia, 22dragon22burn, SuperHero2111, MadGuy7023, Lugia2453, Mmitchell10, Hillbillyholiday, Carinameqwerty3, Xshadow creeper225, HenryAffleck, Wamiq, My name is ,عربستاني not dave, Spraguebrody12345, JaconaFrere, हेमन्त डबराल, Flash5lk, Colombo Eagle, Mcmyrick13, Wjxb, Philip.holy, Thrtih, Haim10, Megan elyse, Matt.w.gardner, AlphaBetaGamma01, Othtvd, RedPanda25, John132456789, Jack Mullin, EnigmaLord515, Michaelr- tookhismouse, Lingo210, WhyYouDoDisToMeh, Qzd, Doogle707, Vijender Rana, Gulumeemee, Bignasty123456789, Jakejake1234, Jcobblwe100, Joe1w, Dop90890, Here2help, AppleBottomJeans and Anonymous: 491 • 3 (number) Source: https://en.wikipedia.org/wiki/3_(number)?oldid=759420735 Contributors: AxelBoldt, Kowloonese, XJaM, Toby Bar- tels, William Avery, Heron, Karl Palmen, Sfdan, Olivier, Michael Hardy, GUllman, SGBailey, Dcljr, Egil, Ihcoyc, Ellywa, Ahoerstemeier, Cyp, DavidWBrooks, Docu, Muriel Gottrop~enwiki, Snoyes, JWSchmidt, Glenn, UserGoogol, Poor Yorick, Netsnipe, Marknew, Lee M, Schneelocke, Alex S, Timwi, Dcoetzee, RickK, Reddi, Denni, Dysprosia, Fuzheado, WhisperToMe, Hyacinth, Sabbut, Ed g2s, Dmetric, Joy, Warofdreams, Dpbsmith, AnonMoos, Pakaran, Wolfram~enwiki, Jeffq, Wst~enwiki, RadicalBender, Denelson83, PuzzletChung, John 15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 107

Trapp, AlexPlank, Robbot, Dale Arnett, Jaredwf, Moriori, Fredrik, Altenmann, Lowellian, Merovingian, Ashdurbat, Yarvin, Academic Challenger, Texture, Yacht, Anthony, PrimeFan, Mattflaschen, Carnildo, Snobot, Exploding Boy, Giftlite, Dbenbenn, Graeme Bartlett, Wizzy, Andy, Sj, Aphaia, Zigger, Herbee, Numerao, Anton Mravcek, Capitalistroadster, WHEELER, Rick Block, Frencheigh, Dunchar- ris, Eequor, Andycjp, Sonjaaa, Noe, Aulis Eskola, Ihavenolife, Secfan, Karl-Henner, Urhixidur, Andy Christ, Trevor MacInnis, Grstain, Talkstosocks, Mike Rosoft, CALR, Indosauros, Discospinster, 4pq1injbok, Cacycle, Dbachmann, Edgarde, ZeroOne, CanisRufus, Mr. , El C, Pjrich, Kwamikagami, RoyBoy, Thu, Bobo192, Whosyourjudas, Chrisvls, Giraffedata, Jojit fb, Hesperian, Haham hanuka, Chicago god, Geschichte, Alansohn, Dachannien, Verdlanco, Batmanand, Snowolf, Wtmitchell, Velella, Archagon, LukeSurl, Bookandcof- fee, Jakes18, Mmxbass, Mahanga, Zntrip, WilliamKF, Angr, Woohookitty, Georgia guy, Guy M, Uncle G, Nameneko, Jeff3000, Wikiklrsc, Macaddct1984, Waldir, Ajshm, Radiant!, Dysepsion, Jobnikon, BD2412, Roger McCoy, Mendaliv, Jshadias, Josh Parris, Viswaprabha, Rjwilmsi, Koavf, Urbane Legend, Notorious4life, Xsg, Ian Pitchford, Mathbot, Nihiltres, Ewlyahoocom, Gurch, Valermos, Thecurran, Haonhien, CJLL Wright, Chobot, Sbrools, DVdm, Mhking, Gdrbot, 334a, Gwernol, Tone, Algebraist, YurikBot, Wavelength, Borgx, Tex- asAndroid, Spacepotato, Jimp, Mukkakukaku, JarrahTree, RussBot, Rsrikanth05, Cunado19, Dysmorodrepanis~enwiki, Bachrach44, Xun- flash, Ostione, BrainyBroad, Ptcamn, Deskana, Trovatore, Keka, Yoninah, Malcolma, RUL3R, Ospalh, Antoshi, Olympic god, Nick123, Wknight94, Closedmouth, Arthur Rubin, Spinboy 11, BorgQueen, Gobexe, HoratioVitero, Allens, Katieh5584, Amor fati, Andrew73, Paul Erik, SmackBot, Elonka, [email protected], Prodego, Unyoyega, Speight, Jagged 85, Davewild, J.J.Sagnella, Kintetsub- uffalo, GraemeMcRae, Ohnoitsjamie, -wolves, Madmedea~enwiki, PrimeHunter, Apeloverage, Akanemoto, Baa, DHN-bot~enwiki, Colonies Chris, Camtin, Audriusa, Can't sleep, clown will eat me, Shalom Yechiel, Metallurgist, Sangajin, Jennica, Lantrix, Yidish- eryid, DaNASCAT, TonySt, Greenshed, Pax85, COMPFUNK2, Kingdon, Smartkid90, Aelffin, A.J.A., Tanyakh, DMacks, Treelzebub, SashatoBot, Rockvee, Lambiam, U-571, Kuru, Khazar, Rigadoun, Bando26, Bampowthoom, Cielomobile, Cstella23, Waggers, Amir- man, Benatkin, Galadh, Cat's Tuxedo, Iridescent, Colonel Warden, Skalman, Brady8, Newyorkbrad, Scarlet Lioness, Zerakith, RSido, Hetzer, CmdrObot, Sir Vicious, Anubis3, Trav1085, CBM, DSachan, Dgw, NickW557, Shandris, WeggeBot, Asztal, Bill Sayre, An- drewHowse, Doctormatt, Cydebot, Mientkiewicz5508, Gogo Dodo, Andre666, Lugnuts, Porsche997SBS, Eu.stefan, Odie5533, Dumb- BOT, Bookgrrl, Thijs!bot, Epbr123, Barticus88, Calvinballing, Timo3, Mojo Hand, Berria, Sobreira, Missvain, Bobblehead, JustAGal, Nick Number, Tylop~enwiki, RoboServien, Escarbot, Science watch, Stannered, AntiVandalBot, WinBot, ForgottenSteeds, Paul from Michigan, Oodelsofned, Silver2195, Ozgod, Dhrm77, Sluzzelin, JAnDbot, Deflective, Leuko, Barek, Instinct, Awilley, Andonic, Michael- bud, Leolaursen, PhilKnight, Bhall87, Geniac, Magioladitis, Bongwarrior, VoABot II, Jacce, TVfanatic2K, AtticusX, Kajasudhakarababu, Redaktor, ***Ria777, Emilng, Twsx, Trinitymix, WODUP, 7008, Peed O'Phile, Catgut, 28421u2232nfenfcenc, Baba farouq, David Epp- stein, Edward321, Georbe, Xtifr, Welshleprechaun, Bramc~enwiki, Mjbyars1, Motley Crue Rocks, Drewmutt, Anaxial, AlexiusHoratius, RockMFR, J.delanoy, Doughmuffins, Vishwanathgangal, Maurice Carbonaro, Brest, Dave Dial, Cpiral, Katalaveno, DarkFalls, Gman124, McSly, Nemo bis, Shannahanbury, Ramses30, STBotD, Spiesr, Jamesontai, DH85868993, SBKT, Bonadea, Ja 62, Squids and Chips, Steel1943, Zakuragi, Sam Blacketer, VolkovBot, Kelapstick, Jeff G., JohnBlackburne, AlnoktaBOT, Jacroe, Philip Trueman, TXiKi- BoT, Anonymous Dissident, 811nd51d3d, JayC, Djkrajnik, Warrush, Steven J. 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Mathar, Killernerd24, Brastworld, Pickuptha'Musket, A930913, TheodoreNg, Wayne Slam, JoeSperrazza, LWG, Alborzagros, Mayur, Donner60, Ego White Tray, ChuispastonBot, Wan- dersonza, 28bot, Rocketrod1960, Reingelt, OwnedU2Fast, ClueBot NG, Ezekiel63745, Lorrielorrelson5, RS12118, LutherVinci, Yp- nypn, MelbourneStar, Vikasjain2, Thoroughgoodness, Bped1985, Baseball Watcher, Vacation9, TruPepitoM, Frietjes, Hapticfever, Widr, Jaffaroll, Jayman17460, Helpful Pixie Bot, Xxanniexx123, Sdfdfs, Electriccatfish2, Snajafi, S196635, Wbm1058, Lowercase sigmabot, Zoester122333, BG19bot, EgoSumAbbas, Vagobot, Chrysalifourfour, Wheatsing, Flsehop, Researchprojectsorry, MusikAnimal, Amp71, Mark Arsten, 6561jayson, Wgr3, Sandakelum, LVTDUDE, XxxFilipks, Nutsackandasscrack, Jfhutson, Anbu121, RichardMills65, Con- norsmac, Cyberbot II, The Illusive Man, Serchia, Calu2000, 22dragon22burn, Fugicara, FifthCrow, SiBr4, Michael Anon, MrTechno kitty, Jo-Jo Eumerus, Telfordbuck, Mr pwnzy, Ekips39, Rainbow Dash loves girls, Clueless9898, P0kemaztr, Epicgenius, Vanamonde93, Deli- cacysmegma, JPaestpreornJeolhlna, ProtossPylon, SirInik, Jakec, Faceoffbob, Jeffreysv97, Gaben3, LCS check, Wamiq, Bolasbolasbolas, Monochrome Monitor, LieutenantLatvia, Kennethaw88, MagicatthemovieS, Naoise Celtic, JaconaFrere, हेमन्त डबराल, Zombiex100, Jay- dreams, 3primetime3, Shae1405, Julietdeltalima, TaqPol, AlphaBetaGamma01, Quivico, Bajabberwabbers, MahuCG, Wwwasserar, The user; square root of 19, Alban15398, Fitindia, KGirlTrucker81, ContentContentConten, Gulumeemee, Jiklo1569, Imminent77, Cookie Monster 123455, 420 Sonorian 420 and Anonymous: 853

• 4 (number) Source: https://en.wikipedia.org/wiki/4_(number)?oldid=759424323 Contributors: AxelBoldt, Rjstott, Jagged, Kowloonese, XJaM, Rmhermen, Christian List, Deb, William Avery, Heron, Olivier, Robertolyra, AntonioMartin, RTC, Michael Hardy, Matchboy, Llywrch, GUllman, SGBailey, Egil, Bdonlan, Ahoerstemeier, Docu, Samuelsen, Glenn, Llull, Cimon Avaro, Panoramix, Schneelocke, Hashar, Dcoetzee, Denni, Dysprosia, WhisperToMe, Wik, Grendelkhan, Sabbut, Ed g2s, Topbanana, Dmetric, Fvw, Warofdreams, Wol- fram~enwiki, Wst~enwiki, RadicalBender, Denelson83, AlexPlank, Robbot, Dale Arnett, Jaredwf, Iam, Moriori, RedWolf, , Lowellian, Academic Challenger, Texture, Yacht, Dehumanizer, JesseW, PrimeFan, Mattflaschen, Giftlite, Dbenbenn, Graeme Bartlett, Mikez, Sj, Numerao, Anton Mravcek, Rick Block, Duncharris, Guanaco, Dmmaus, Eequor, Tagishsimon, Andycjp, Lst27, Dvavasour, Jorm, Beland, Aulis Eskola, Mzajac, Maximaximax, Mozzerati, Spottedowl, Urhixidur, Andy Christ, Picapica, J Bytheway, 朝彦, Zondor, 108 CHAPTER 15. DIGITAL ROOT

Reflex Reaction, Kate, Mike Rosoft, Brianjd, Liquideagle, Davidbod, Bornintheguz, Diagonalfish, 4pq1injbok, Rich Farmbrough, Guan- abot, Cacycle, Rspeer, Silence, Xezbeth, Martpol, ESkog, CanisRufus, Kwamikagami, Bobo192, Flying Hamster, Reinyday, Flxmghvgvk, Jguk 2, Giraffedata, Jojit fb, Chicago god, Alansohn, Hektor, Njaard, Wiki-uk, Verdlanco, Sligocki, Mailer diablo, Cdc, Bart133, Jakek101, Dirac1933, Skatebiker, RickDeNatale, Kitch, Mmxbass, Oleg Alexandrov, WilliamKF, Imaginatorium, Woohookitty, Georgia guy, Uncle G, Dodiad, Mandarax, Jobnikon, BD2412, Terryn3, Kbdank71, FreplySpang, Roger McCoy, Mendaliv, Jshadias, Josh Parris, Viswaprabha, Odysseyandoracle, Omnieiunium, Salix alba, Jb-adder, NeonMerlin, Notorious4life, Nandesuka, Xsg, Anskas, Mathbot, Arasaka, Alex Sims, Meeve, Joedeshon, WouterBot, Chobot, Benlisquare, DVdm, Gdrbot, 334a, Gwernol, Mercury McKinnon, YurikBot, Wavelength, Borgx, Samuel Wiki, Jimp, RussBot, Vlad4599, Muchness, Aree, Ori Livneh, Wimt, NawlinWiki, BlackWidower, Xunflash, Yoninah, Cholmes75, My winkey has a key, GoAround, Zwobot, BOT-Superzerocool, Bota47, Gokugohan382, Matrixfusion, Komuta, Allchopin, Mrbluesky, Nlu, The Joker, Theodolite, Closedmouth, Arthur Rubin, Pb30, Pasitbank, Gobexe, Curpsbot-unicodify, SmackBot, Cubs Fan, Prodego, TestPilot, KocjoBot~enwiki, Jab843, Kintetsubuffalo, Edgar181, GraemeMcRae, Primaryspace, Commander Keane bot, Yamaguchi 先生, Gilliam, Quidam65, Betacommand, Richfife, MikeRM, PrimeHunter, Silly rabbit, Akanemoto, DHN-bot~enwiki, The owner of all, Audriusa, Yidisheryid, Parent5446, Jjjsixsix, Pax85, Ken S., Jwy, Nakon, Aelffin, Weemspm, Bws2002, DMacks, Hen- rydms, Bdiscoe, FlyHigh, SashatoBot, Lambiam, BrownHairedGirl, Cesium 133, Aspirex, Majorclanger, Hvn0413, Midnightblueowl, Novangelis, Outofline, BranStark, Iridescent, Blehfu, The Haunted Angel, HDCase, JForget, CBM, DSachan, Gegorg, Gregbard, An- drewHowse, Doctormatt, Cydebot, The Chaotician, Andre666, TicketMan, Khatru2, Porsche997SBS, Eu.stefan, DumbBOT, Narcisso, Rider kabuto, Barticus88, Dasani, Daniel, Mojo Hand, Berria, Bobblehead, Graemec2, JustAGal, Aericanwizard, Ideogram, Chandler, RichardVeryard, Nick Number, Tylop~enwiki, Nixon11d7, RoboServien, Escarbot, Stannered, AntiVandalBot, WinBot, Paul from Michi- gan, Shinsuk, Alan Smithee 87, Storkk, Bjenks, Dhrm77, JAnDbot, Winndm31, Rearete, Magioladitis, Adias, Bongwarrior, VoABot II, Jacce, Lemon martini, Oiuytre, Kajasudhakarababu, Nyttend, Rich257, JJ Harrison, Adrian J. Hunter, Estan williams, Eswaran, Cpl Syx, JMyrleFuller, Georbe, Illwauk, Nevit, Welshleprechaun, Oroso, STBot, R'n'B, Pbroks13, Verdatum, Nono64, RockMFR, J.delanoy, Love Krittaya, Bogey97, Tikiwont, Brest, Jreferee, Skumarlabot, Josh Biderman, Notyou7, Gman124, Nemo bis, HoldenV8, Shannahan- bury, 97198, Hennessey, Patrick, KylieTastic, STBotD, Vanished user 39948282, Idioma-bot, Zakuragi, Geeky Randy, VolkovBot, Safe- mariner, JohnBlackburne, Philip Trueman, TXiKiBoT, The Tetrast, Mmbabies, Broadbot, Tean Butcher, Luuva, Hatkirby, Madhero88, AmberAlert1713, Falcon8765, Draconx, Alcmaeonid, Ceranthor, AlleborgoBot, FlyingLeopard2014, Rider of the Storm, GoonerDP, SieBot, TJRC, WereSpielChequers, BotMultichill, Triwbe, Yintan, Snickers92, Arda Xi, Bentogoa, Flyer22 Reborn, Oxymoron83, Byri- albot, Fratrep, GodShiru, Rjfost, Knodeltheory, Tesi1700, Felizdenovo, Mx. Granger, ClueBot, PipepBot, Gazza-Xtreme, SerduchkaFan, Arakunem, Drmies, Wildbird30, BoyInTheTree, JJIG, Estevoaei, The 888th Avatar, Etalli, Snowdreamer2000, Robert Skyhawk, Ex- cirial, Johndeerboy247, Alexbot, Domestic Correction, Chickiedude1, Bremerenator, Dekisugi, Teutonic Tamer, ChrisHodgesUK, La Pi- anista, Thingg, Jonverve, 7, Versus22, Qwfp, SoxBot III, Tyguy997, WikHead, Nukes4Tots, Sha721, D.M. from Ukraine, Addbot, Some jerk on the Internet, Freakmighty, Halosean, CarsracBot, Chzz, Favonian, SpBot, Tassedethe, Numbo3-bot, Tide rolls, Romanskolduns, Bradboss283, Cashelini, CountryBot, Sneaky Oviraptor18, Luckas-bot, Yobot, 2D, Numbuh1cubfan, Fraggle81, Melanippa, Kamikaze- Bot, Daniel 1992, Daft oparin, AnomieBOT, Madgey7, DemocraticLuntz, Blah42b10, JackieBot, Ulric1313, Bluerasberry, Mahmud- masri, Materialscientist, BlurTento, GB fan, ArthurBot, LilHelpa, Xqbot, Hanberke, Jeffwang, The Biting Pear Of Salamanca, Gap9551, Nasa-verve, GrouchoBot, Cruz-iglesia, Ute in DC, RibotBOT, Intruductionetrondocapriccioso, Cartque, Nevermiand., N419BH, Shadow- jams, Aaron Kauppi, Berenluck, FrescoBot, Tangent747, MHR65, Mitizhi, ThiagoRuiz, Maverick9711, Oashi, DivineAlpha, Biker Biker, Pinethicket, HRoestBot, Mahaodeh, Dac04, Double sharp, TobeBot, Royal Blue Jersey, TheStrayCat, EvilEGrammeler69, Dinamik-bot, Vrenator, I am my own Master, Villainus, Jeffrd10, Onel5969, Aircorn, EmausBot, John of Reading, Immunize, RA0808, Minimac's Clone, Klbrain, Moswento, Wikipelli, K6ka, DeadShrimpBlues, ZéroBot, Josve05a, Érico, Pliming, Kellrissa, Mar4d, Derekleungtszhei, Hereforhomework2, Killernerd24, Tolly4bolly, JoeSperrazza, TyA, Davanzo56, Exfiveare, Screwyouwiki, Donner60, Ego White Tray, Or- ange Suede Sofa, ChuispastonBot, Wandersonza, Venugopalreddykonda, Senator2029, Lguipontes, DASHBotAV, Thezaphod42, ClueBot NG, Ezekiel63745, Akuindo, Ypnypn, MelbourneStar, Oxandco, Baseball Watcher, Jake Golle, Frietjes, Widr, Jaffaroll, Jorgenev, Help- ful Pixie Bot, Plantdrew, Youfuckingsuck69, BG19bot, Snaevar-bot, StoopEditor, Vagobot, MusikAnimal, Mark Arsten, Jazzmaster59, Wikih101, Jonathan63096, Randomking333, Artemis710, Serchia, 22dragon22burn, Najta, Awu1996, Cpt.a.haddock, Mogism, TwoT- woHello, Lugia2453, Frosty, Kanjilearner358, Cheese00000, Sarkar89, Wiki4Blog, Wamiq, Noisavni, Ï¿½, Kharkiv07, AddWittyName- Here, FrB.TG, Vampirelazarus, हेमन्त डबराल, Wgml98, Wjxb, 3primetime3, JonathanHopeThisIsUnique, Tedjam, Monumenteer2014, Ferretsrock, Snow Lion Fenian, Zppix, Onedilovthemsjd, Andreaaaaaa23, AlphaBetaGamma01, SoSivr, Joseph2302, JackNewswanger69, BrundelFly999, Jdcomix, Plmkkjhjhg, Meemees, CAPTAIN RAJU, Check the name leon, CeleryKnightAP, InternetArchiveBot, GreenC bot, Gulumeemee, Inimesh and Anonymous: 544

• 5 (number) Source: https://en.wikipedia.org/wiki/5_(number)?oldid=760609509 Contributors: AxelBoldt, The Anome, Kowloonese, XJaM, Christian List, William Avery, Heron, Olivier, Patrick, RTC, Norm, Dominus, GUllman, SGBailey, Minesweeper, Tregoweth, Egil, Ahoerstemeier, Docu, Angela, Darkwind, Glenn, Llull, Cimon Avaro, Panoramix, Schneelocke, Nikola Smolenski, Norwikian, Hashar, Dcoetzee, JonMoore, Denni, Dysprosia, WhisperToMe, Zoicon5, Radiojon, Paul-L~enwiki, Sabbut, Ed g2s, Dmetric, Wi- waxia, Warofdreams, Dbabbitt, Wolfram~enwiki, Carlossuarez46, RadicalBender, Denelson83, AlexPlank, Robbot, Dale Arnett, RedWolf, Christie~enwiki, Yarvin, Academic Challenger, Texture, Yacht, JesseW, JackofOz, Miles, Anthony, PrimeFan, Asparagus, Mattflaschen, Enochlau, Snobot, Giftlite, Dbenbenn, Graeme Bartlett, Andy, Sj, Zigger, Herbee, Numerao, Anton Mravcek, Daysra, Rick Block, Nite- owlneils, Duncharris, Beta m, Prosfilaes, Eequor, Chameleon, Andycjp, Sohailstyle, Noe, Kusunose, Aulis Eskola, Jorgengb, Urhixidur, Joyous!, Andy Christ, Corti, Mike Rosoft, Discospinster, 4pq1injbok, Guanabot, Colin Angus Mackay, Martpol, ESkog, Neko-chan, Can- isRufus, Kwamikagami, Warpozio, Bobo192, Reinyday, Giraffedata, Jojit fb, RazorChicken, Chicago god, QuantumEleven, Stephen Bain, Martin S Taylor, Conny, Alansohn, Csabo, Lokicarbis, Atlant, Verdlanco, Cjthellama, ABCD, Samohyl Jan, Wtmitchell, Kdau, Evil Mon- key, Skatebiker, Fountainofignorance, Oleg Alexandrov, Anonymous4367, WilliamKF, Imaginatorium, Woohookitty, Swamp Ig, Uncle G, BillC, Jacobolus, GregorB, Macaddct1984, Gniw, Liface, Radiant!, Peligroso, Jobnikon, BD2412, Kbdank71, Roger McCoy, Mendaliv, Josh Parris, Viswaprabha, Dimitrii, Koavf, Mike s, Tomtheman5, Notorious4life, Matt Deres, Xsg, John Baez, Mathbot, Wars, Brash, Kanthoney, Alphachimp, Thecurran, Spencerk, Haonhien, CJLL Wright, Chobot, Sherool, DVdm, Gdrbot, 334a, Gwernol, Algebraist, EamonnPKeane, YurikBot, Borgx, JWB, Jimp, Icarus3, Pi Delport, RadioFan2 (usurped), Stephenb, Rsrikanth05, Keka, Yoninah, Hy- perqube, Pietras1988, DryaUnda, BOT-Superzerocool, Cardsplayer4life, Bnitin, Nlu, Deville, Doktorbuk, Arthur Rubin, Jogers, Canley, Pasitbank, Gobexe, Leeannedy, Katieh5584, Meegs, Umeditor, GrinBot~enwiki, Daya.sharma, SmackBot, Prodego, C.Fred, Jrockley, KelleyCook, Kintetsubuffalo, GraemeMcRae, Yamaguchi 先生, Chinamanjoe, Gilliam, Quidam65, Betacommand, Psiphiorg, Quinsareth, Musikfabrik, PrimeHunter, MalafayaBot, Timneu22, SchfiftyThree, Akanemoto, DHN-bot~enwiki, Colonies Chris, Audriusa, Can't sleep, clown will eat me, All in, Neo139, OrphanBot, Yidisheryid, DaNASCAT, Rrburke, El edgar, TheLimbicOne, Aelffin, Lcarscad, BryanG, DMacks, Mehnazhyder, Lambiam, Nathanael Bar-Aur L., Accurizer, Hulmem, Mr. Vernon, Midnightblueowl, Naumz, Iridescent, Neuril- lon, Tawkerbot2, InvisibleK, Geo8rge, Mattbr, DSachan, Gegorg, Neelix, Lordmaster913, BT14, Wild Wizard, AndrewHowse, Doctor- matt, Shaulbehr, Mientkiewicz5508, Offspring9, Clayoquot, Andre666, Tfgbd, Chasingsol, Porsche997SBS, Rtiztik, Bmitchelf, Forbidden- 15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 109

Word, Daniel Olsen, Thijs!bot, Epbr123, Barticus88, TonyTheTiger, Bigwyrm, Berria, Bobblehead, Graemec2, JustAGal, Sorbus, Natalie Erin, Northumbrian, Escarbot, Stannered, WinBot, Paul from Michigan, SummerPhD, Fireice, Dhrm77, JAnDbot, Tigga, DuncanHill, Sko- morokh, Epeefleche, Zero132132, Ecrvich, PhilKnight, Adias, Bongwarrior, VoABot II, Twsx, Froid, David Eppstein, DerHexer, Georbe, Illwauk, Esanchez7587, War-Mage, Tulip19, Patstuart, Glp~enwiki, Hdt83, Mjbyars1, Nono64, Bucs10, RockMFR, J.delanoy, Pharaoh of the Wizards, Trusilver, Uncle Dick, SteveLamacq43, Maurice Carbonaro, Headinthedoor, KILNA, TomS TDotO, Katalaveno, Notyou7, WoWaddict, Gman124, Starnestommy, Richard D. LeCour, Userme, DH85868993, BernardZ, TBF Bri 10, CardinalDan, Idioma-bot, Funandtrvl, Zakuragi, Jeff G., Aesopos, Dougie monty, Philip Trueman, DoorsAjar, TXiKiBoT, Oshwah, Sean D Martin, Djkrajnik, Mmbabies, Jackfork, Tean Butcher, Ilyushka88, Luuva, AmberAlert1713, Boom dizzle, Equine.Scientist, Insanity Incarnate, Why Not A Duck, AlleborgoBot, Wolfenator, SieBot, StAnselm, BotMultichill, Swaq, Merotoker1, Arda Xi, Keilana, Thesavagenorwegian, Toddst1, Oda Mari, Byrialbot, Nuttycoconut, JackSchmidt, OKBot, Dravecky, Knodeltheory, 1zackman, Li4kata, Felizdenovo, Denisarona, Steve, WakingLili, Martarius, Rich2809, ClueBot, Bob1960evens, PipepBot, Fyyer, The Thing That Should Not Be, Cliff, IceUnshattered, SerduchkaFan, MIDI, Drmies, Wildbird30, JPCBR, Eka1007, DragonBot, Excirial, Alexbot, Jusdafax, Kanguole, Joshram, Tj82472, Morel, SounderBruce, Ewsraven, Warof7, C628, Thingg, Aitias, Jonverve, 7, Qwfp, Egmontaz, DumZiBoT, Darkicebot, XLinkBot, Dark Mage, King Willan Bot~enwiki, WikHead, Halosean, Jarrraddamiani, CL, Zarcadia, Download, CarsracBot, Favonian, SpBot, Tassedethe, Numbo3-bot, Tide rolls, Romanskolduns, Luckas-bot, Ptbotgourou, Numbuh1cubfan, TaBOT-zerem, N80r, Tempodivalse, Daft oparin, NerdsLimited, Drinkaboutit, AnomieBOT, Madgey7, Blah42b10, Jim1138, Emperor1992, Galoubet, Piano non troppo, Rpetrov~enwiki, Nazrani, Mahmudmasri, Materialscientist, GB fan, Apollo, ArthurBot, Xqbot, Lily20, TinucherianBot II, Franklin67, Jeffrey Mall, Han- berke, Herr Mlinka, Ched, Gap9551, Nasa-verve, Nevermiand., Shadowjams, Shariant, A.amitkumar, Rtrimm, FrescoBot, LucienBOT, Lagelspeil, Gapress34, Rachel993, Ilostbec, Maverick9711, Tlork Thunderhead, Aleister Wilson, Vhann, CHUMOROCKS, Elockid, HRoestBot, Mewto55555, Karlrulez77, Hellzies, John123521, FoxBot, Royal Blue Jersey, Duoduoduo, Candrias, Reaper Eternal, 4, Fastilysock, Brambleclawx, Five-O-Klock, DARTH SIDIOUS 2, Balletblonde, Mean as custard, Bento00, Galloping Moses, DRAGON BOOSTER, DASHBot, EmausBot, John of Reading, Orphan Wiki, Hamster25, GoingBatty, RenamedUser01302013, Varia4blackfire, Wikipelli, Dcirovic, DeadShrimpBlues, Savh, ZéroBot, Traxs7, Pliming, R. J. Mathar, Killernerd24, Potatosiris, Access Denied, Frigotoni, Ocaasi, Gee-Unit65, ChuispastonBot, Wandersonza, Venugopalreddykonda, Senator2029, Ha897, ClueBot NG, Ezekiel63745, Smtcha- hal, Giggett, Akuindo, Ypnypn, Pieninepa, Vacation9, Proscribe, Frietjes, Diesel fuell, Billiam1185, Widr, Philip K Glass, Helpful Pixie Bot, Novusuna, Wbm1058, BG19bot, Krenair, Snaevar-bot, Vagobot, Celloplayer115, Ugncreative Usergname, Noambz, Gallosexybeast, Anbu121, Mdann52, Panzertruth, Dj232222, Serchia, 22dragon22burn, Angiewangie!, Poopeepootagger, Osnwa, Thisbigman, Sijmon- flippa, Lugia2453, Frosty, Cakeawesome, Tomneal93, Epicgenius, B14709, DavidLeighEllis, Wamiq, Ugog Nizdast, Glaisher, Newman- srus, हेमन्त डबराल, Flash5lk, Prof. Mc, Ihuntpanda, Gokarting54, TranquilHope, Loraof, Snow Lion Fenian, Orduin, TaqPol, Righ- teousbison, AlphaBetaGamma01, SparkyDill128, SoSivr, Macaronman, GeorgeHarold1, CAPTAIN RAJU, Kermosk, Qzd, Deliciously sweet, Barravince, Rocko409, Gulumeemee, John“Hannibal”Smith, Abodh1986, Ydgjjhv, Giovanni Resta, Culverkwan, Cynulliad3 and Anonymous: 608

• 6 (number) Source: https://en.wikipedia.org/wiki/6_(number)?oldid=758101889 Contributors: AxelBoldt, Tarquin, Carpentis, XJaM, Christian List, William Avery, Heron, Karl Palmen, Olivier, Patrick, JohnOwens, GUllman, SGBailey, Dcljr, Pcb21, Egil, Docu, Samuelsen, 5ko, Glenn, Llull, Panoramix, Schneelocke, Dcoetzee, Denni, Dysprosia, WhisperToMe, Zoicon5, Timc, Grendelkhan, Sabbut, Ed g2s, Dmetric, Wiwaxia, Fvw, Lumos3, RadicalBender, Denelson83, AlexPlank, Robbot, Dale Arnett, Iam, RedWolf, Academic Challenger, Texture, Yacht, PrimeFan, Mattflaschen, Fabiform, Giftlite, Gwalla, Dbenbenn, Graeme Bartlett, Sj, Aphaia, MSGJ, Numerao, Anton Mravcek, Jorend, Darkhunger, Rick Block, Duncharris, Dmmaus, Eequor, Andycjp, Sigfpe, Lst27, Aulis Eskola, OwenBlacker, Pmander- son, Urhixidur, Askewchan, Canterbury Tail, Kate, Mike Rosoft, Discospinster, 4pq1injbok, Vsmith, CanisRufus, Kwamikagami, Triona, Bobo192, Arlewis, Reinyday, Jguk 2, Chrisvls, Giraffedata, Jojit fb, Chicago god, Jumbuck, Verdlanco, Shadowlord, Flata, Saga City, Kdau, Skatebiker, Stemonitis, Georgia guy, Uncle G, GregorB, Rad Racer, Graham87, Jobnikon, BD2412, Kbdank71, Roger McCoy, Mendaliv, Notorious4life, Matt Deres, Xsg, RobertG, Mathbot, GünniX, Kolbasz, Chobot, Sherool, DVdm, Gdrbot, 334a, YurikBot, Borgx, Jimp, Kafziel, RussBot, Cate, Alex Bakharev, Carps, NawlinWiki, Joth, Gwaihir, Introgressive, Yoninah, Misza13, BOT-Superzerocool, D-Day, Wknight94, Ageekgal, Arthur Rubin, Pb30, Dspradau, Canley, Pasitbank, JuJube, Vicarious, GrinBot~enwiki, SmackBot, Classicfilms, Deleted14857, Od Mishehu, TomStike, Jab843, Kintetsubuffalo, GraemeMcRae, Quidam65, Betacommand, Skizzik, Chris the speller, Father McKenzie, DStoykov, PrimeHunter, Necronudist, Mattythewhite, Akanemoto, DHN-bot~enwiki, Audriusa, Maguirer, N.MacInnes, Yidisheryid, DaNASCAT, VMS , Addshore, Pax85, Aelffin, MartinRe, Polymath69, DMacks, Filpaul, Mlpkr, Lambiam, Michael J, Waggers, Drtjumper, Cbuckley, Sjsharksrs, Wwagner, Thefuguestate, BranStark, Skalman, LadyofShalott, Tawkerbot2, CRGreathouse, CBM, DSachan, THF, Dgw, FlyingToaster, WeggeBot, INVERTED, Chris83, AndrewHowse, Doctormatt, Cydebot, TylerJG, Road Wizard, Achangeisasgoodasa, Andre666, Porsche997SBS, Rtiztik, Verdy p, DumbBOT, Legis, Epbr123, Barticus88, Qwyrxian, Berria, Headbomb, Aericanwizard, Noclevername, CompositeFan, Whisperednumber, RoboServien, Escarbot, Stannered, WinBot, Skynet1216, Salgueiro~enwiki, Dhrm77, JAnDbot, Magiczoe, Fuzzybyte, Huphelmeyer, Magioladitis, Adias, Bongwarrior, VoABot II, AtticusX, James- BWatson, Kajasudhakarababu, Arthur B, Singularity, Froid, JaGa, Georbe, Smartings, Pax:Vobiscum, Charitwo, Welshleprechaun, STBot, R'n'B, CommonsDelinker, Nono64, LedgendGamer, RockMFR, J.delanoy, Maurice Carbonaro, MrBell, Jreferee, BBrucker2, DubmoB, Nemo bis, Shannahanbury, Million Moments, Zakuragi, Malik Shabazz, Deor, VolkovBot, JohnBlackburne, Dougie monty, Philip True- man, TXiKiBoT, Oshwah, Vipinhari, Rei-bot, Vttale, Record307, Martin451, JhsBot, Tean Butcher, Gibson Flying V, Luuva, Wenli, Dirkbb, Synthebot, Sue Rangell, AlleborgoBot, Angelastic, Steven Weston, Murkee, SieBot, PlanetStar, BotMultichill, Oldag07, Angle- hence, Arda Xi, Tiptoety, Brian R Hunter, Byrialbot, Tomdaartichoke, Fratrep, Knodeltheory, Felizdenovo, Denisarona, QWIES, Escape Orbit, ClueBot, PipepBot, The Thing That Should Not Be, Scartboy, Meekywiki, Wildbird30, Alexbot, Abrech, Bercant, Jjforau, Jonverve, Urakanine, Qwfp, SoxBot III, Darkicebot, XLinkBot, Spitfire, Sakura Cartelet, Little Mountain 5, Avoided, Penale52, Tyguy997, King Willan Bot~enwiki, SilvonenBot, Good Olfactory, Doug butler, Slowmotion7, Drlight11, Proofreader77, RandySavageFTW, Some jerk on the Internet, Halosean, Tcncv, Ronhjones, Cst17, Download, Favonian, SpBot, Bloodrunsblack, Numbo3-bot, Issyl0, Tide rolls, Bfigura's puppy, Lightbot, Romanskolduns, Teles, Bartledan, Luckas-bot, Yobot, TaBOT-zerem, II MusLiM HyBRiD II, Oricato, AnomieBOT, Madgey7, Blah42b10, Piano non troppo, Ulric1313, Mahmudmasri, Materialscientist, GB fan, Knowledge Incarnate, ArthurBot, Xqbot, Kinewma, C3pblow, Hanberke, Herr Mlinka, NASCAR Nathan, Tyrol5, Gap9551, XjWoRk17, Nasa-verve, RibotBOT, 78.26, Small- man12q, Bearnfæder, Us441, MHR65, LucienBOT, Lagelspeil, Michael93555, MichealH, BenzolBot, Ehsan baghaki, Plexity42, HRoest- Bot, Mahaodeh, Arctic Night, CHawc, Karlrulez77, Piandcompany, Ambarsande, TobeBot, Vrenator, Barroxmaximus, Ronandoyler, Dian- naa, Fastilysock, Suffusion of Yellow, TylerGood, Reach Out to the Truth, Reubenator18, Onel5969, MMS2013, Cheesedude3, DASHBot, EmausBot, Ibbn, Swayback Maru, GoingBatty, PoeticVerse, NotAnonymous0, Wikipelli, DeadShrimpBlues, AvicBot, ZéroBot, Pliming, Wackywace, Robd75, Killernerd24, Pickuptha'Musket, TheodoreNg, Aceofclubs1414, TyA, Num Ref, L Kensington, ChuispastonBot, Wandersonza, Venugopalreddykonda, Forever Dusk, DASHBotAV, ClueBot NG, Ezekiel63745, Ypnypn, This lousy T-shirt, Movses-bot, Floatjon, Frietjes, half-moon bubba, Cntras, Widr, Names are hard to think of, In actu, Strings And Verses, Mjpam, Mahali syarifud- din, Ramaksoud2000, Vagobot, MusikAnimal, Celloplayer115, Mark Arsten, Suricate9600, Ugncreative Usergname, 220 of Borg, Zack- 110 CHAPTER 15. DIGITAL ROOT

mann08, Justincheng12345-bot, Laodah, Serchia, 22dragon22burn, JYBot, Mogism, Tentinator, Dustin V. S., Flat Out, Wamiq, NottNott, Ben373119, Noyster, Fiveandfive, TheEpTic, ColRad85, हेमन्त डबराल, Wabbbash, DJ is a DJ, Musamir Azad, KH-1, Lumenf, Velvel2, Luigi9001, Eliseo 3.14, AlphaBetaGamma01, SparkyDill128, Rioluloveraaa, Cnbr15, Aleuuhhmsc, Boehm, Rico Raccoon, Acid420b, Ks-M9, CLCStudent, InternetArchiveBot, Jwegryn, Alban15398, GreenC bot, Gulumeemee, Subrat Hellstorm, Nishant Kharel, TheTruth- Warrior and Anonymous: 422

• 7 (number) Source: https://en.wikipedia.org/wiki/7_(number)?oldid=758575753 Contributors: AxelBoldt, Wesley, Amillar, XJaM, Chris- tian List, William Avery, Heron, Olivier, Patrick, Michael Hardy, SGBailey, Ixfd64, Karada, Paul A, Alfio, Tregoweth, Egil, Bdonlan, DavidWBrooks, Docu, Darkwind, LittleDan, Glenn, Poor Yorick, Cimon Avaro, Panoramix, Schneelocke, Hashar, Timwi, Dcoetzee, Wikiborg, RickK, Denni, Dysprosia, Tedius Zanarukando, WhisperToMe, Zoicon5, Peregrine981, Jakenelson, Furrykef, Hyacinth, An- drewKepert, Sabbut, Dfarmer, Dmetric, Wiwaxia, Stormie, Wolfram~enwiki, Lumos3, RadicalBender, Denelson83, Phil Boswell, Donar- reiskoffer, Bearcat, AlexPlank, Robbot, Dale Arnett, Pbrooks, RedWolf, Academic Challenger, Texture, Yacht, Wereon, PrimeFan, Mattflaschen, Dina, Enochlau, BigT27, Giftlite, Dbenbenn, Graeme Bartlett, Sj, Numerao, Anton Mravcek, Moyogo, Capitalistroadster, Michael Devore, Rick Block, Duncharris, Gilgamesh~enwiki, Jason Quinn, Eequor, Crystalphoenix, Christopherlin, Wmahan, Utcursch, Andycjp, Sigfpe, Telso, R. fiend, Noe, Antandrus, Mustafaa, Fra74~enwiki, Aulis Eskola, Andux, JimWae, Icairns, Urhixidur, Joyous!, Andy Christ, Zondor, Mike Rosoft, Jayjg, Freakofnurture, AliveFreeHappy, Poccil, Imroy, Stepp-Wulf, Lubaf, Discospinster, 4pq1injbok, Rich Farmbrough, Luvcraft, Cacycle, Rspeer, Smyth, Moochocoogle, Mikkel, Dbachmann, Martpol, Bender235, ZeroOne, TerraFrost, Scumbag, PlasmaDragon, RJHall, CanisRufus, Aecis, Bennylin, Rgdboer, Kwamikagami, Art LaPella, Thu, Adambro, Bobo192, Whosy- ourjudas, Robotje, Alcidebava, StoatBringer, Jojit fb, ACW, RenesisX, Apostrophe, Chicago god, Martin S Taylor, Alansohn, Eric Kvaalen, Mlabar, Wiki-uk, Philip Cross, Verdlanco, That, ABCD, Goldom, Adkins, Ramsquire, Snowolf, Wtmitchell, Kdau, ReyBrujo, Cburnett, Evil Monkey, Ianblair23, DV8 2XL, Drbreznjev, Bookandcoffee, Oleg Alexandrov, Mahanga, Tournesol, George Hernandez, Bacteria, Imaginatorium, Will-h, Woohookitty, Georgia guy, GVOLTT, Kenkam, Uncle G, Chaosmotic, NeoChaosX, Jeff3000, Xhin, Tabletop, Schzmo, Wikiklrsc, Damicatz, Mangojuice, Eaolson, SDC, Waldir, Pictureuploader, Dedalus, Wikipedian231, Radiant!, Dysepsion, Mr- Someone, RichardWeiss, Jobnikon, BD2412, Imersion, RingbearerNZ, Roger McCoy, DePiep, Mendaliv, Josh Parris, Sjö, Rjwilmsi, Sei- denstud, Wahoofive, SMC, Badhotra, Lairor, Concordia, The wub, DoubleBlue, Notorious4life, Ptdecker, Platypus222, Xsg, Ianthegecko, Tufflaw, Mathbot, Alhutch, Hottentot, Rune.welsh, RexNL, Gurch, IlGreven, NavarroJ, Thecurran, Mechasheherezada, Haonhien, Chobot, Bornhj, DVdm, Gdrbot, 334a, Bgwhite, Algebraist, Wack'd, Banaticus, YurikBot, Wavelength, Borgx, Quentin X, Rt66lt, Jimp, Wolf- mankurd, RussBot, Rapomon, GusF, Christy747, RadioFan2 (usurped), Akamad, Stephenb, C777, Gaius Cornelius, Theelf29, Tavilis, Ugur Basak, Royalbroil, K.C. Tang, Odysses, NawlinWiki, Xunflash, Grafen, Ptcamn, Mike Halterman, Welsh, Phoenix79, Keka, Darker Dreams, Yoninah, Brandon, The Filmaker, Froth, Mgcsinc, Ospalh, Dbfirs, Cawas, BOT-Superzerocool, Mysid, DeadEyeArrow, Tirerim, Nick123, Wknight94, Rwxrwxrwx, Zzuuzz, C i d, Wotnarg, Denisutku, Arthur Rubin, LiquidFire, Donald Albury, KGasso, Pasitbank, Petri Krohn, Stormymax, CWenger, Ybbor, Allens, Katieh5584, Cmglee, Minnesota1, MaeseLeon, Kicking222, SmackBot, GarfieldP, Ralphbk, Elonka, Zazaban, Chandarsubramanian, Stux, McGeddon, Bigbluefish, Od Mishehu, Renesis, Frymaster, Spireguy, Kintetsubuf- falo, GraemeMcRae, HalfShadow, SmartGuy Old, Commander Keane bot, Gilliam, Quidam65, Ohnoitsjamie, Betacommand, Skizzik, Quadratic, Jzimbert, Andy M. Wang, Psiphiorg, Hraefen, Chris the speller, Bluebot, Kurykh, Pietaster, MikeRM, Persian Poet Gal, Blindsuperhero, PrimeHunter, Anchoress, MalafayaBot, Akanemoto, Neo-Jay, Selwinbriggs, Outlando, DHN-bot~enwiki, Riphamilton, Colonies Chris, Antonrojo, Darth Panda, Zachorious, Audriusa, Modest Genius, Houdin654jeff, Mooncow, Can't sleep, clown will eat me, Kristbg, JustUser, Msr69er, Onorem, Vanished User 0001, Michael.Pohoreski, Yidisheryid, Reycount, Dmeastis, DaDoc540, Nakon, TGoodman, VegaDark, Blake-, Quixada, Aelffin, Mtmelendez, Jjmcspooh, PPBlais, DWM, Clean Copy, Andrew c, Das Baz, Rothesay, Mwtoews, DMacks, Mlpkr, Kukini, John Reid, Lambiam, Eliyak, Derek farn, Axem Titanium, Soap, Khazar, Kurounmo~enwiki, Golden- Tie, Jidanni, DEmerson3, T.s.khwaja, Carroller, Don't give an Ameriflag, Goodnightmush, Matt Pizzashack, A.Z., Ki11meshining, 16@r, Drumlineramos, Allilinin~enwiki, Shimmera, Stupid Corn, SliceNYC, Stwalkerster, NJMauthor, InedibleHulk, Vonn, Wsw70, Sarkie, No- vangelis, Saxbryn, Aino2000, BranStark, Sonic3KMaster, HisSpaceResearch, Iridescent, Sjrush, Maestlin, J Di, DJ HEAVEN, Twas Now, Facedogg, Majora4, Courcelles, Gorilla Knight, Tawkerbot2, Ioannes Pragensis, Tommywommy117, Kanjilearner, J Milburn, JForget, Will314159, Ric36, CBM, DSachan, WATP, Dgw, Eibwen~enwiki, El aprendelenguas, Gegorg, Raptor007, WeggeBot, Moreschi, Dandart, Karenjc, LaFoiblesse, Revolus, AndrewHowse, Doctormatt, ChristTrekker, Mientkiewicz5508, Gunluva, MC10, Gogo Dodo, Andre666, Corpx, Llort, Tfaman, Porsche997SBS, Arrowned, Rtiztik, RC Master, Doug Weller, Christian75, Dan Lawrence, Xantharius, Omicronper- sei8, [email protected], Badgerbadger, Thijs!bot, Jessemonroy650, Epbr123, Barticus88, Jpark3909, Billy Bishop, Keraunos, Andyjsmith, Timo3, Oerjan, Gamer007, Miguelfms, Mojo Hand, Berria, Gapgirl9, Marek69, Bobblehead, James086, Graemec2, JustAGal, Sorbus, Dfrg.msc, Ludde23, Douglas.kastle, TangentCube, Tylop~enwiki, X96lee15, Dawnseeker2000, CompositeFan, RoboServien, Northum- brian, Escarbot, Justyn, Stannered, AntiVandalBot, WinBot, Xmzx, Luna Santin, Seaphoto, Paul from Michigan, Prolog, MC7777777, Dan- ger, Aspensti, Salgueiro~enwiki, ThejaswiUdupa, MaXiMiUS, GuyRGifford, DMP3G, Signeco, Dhrm77, JAnDbot, Leuko, GromXXVII, Arch dude, PhilKnight, MegX, Helge Skjeveland, Evening Breeze, Penubag, Adias, Bongwarrior, VoABot II, Dekimasu, AtticusX, Ka- jasudhakarababu, Swpb, Flame0430, King Mir, Froid, Maxfieldwh, 28421u2232nfenfcenc, Absolon, Ludvikus, DerHexer, Garik 11, Jesse Jones~enwiki, Anewgeneration, Welshleprechaun, Len W, Erpbridge, Gwern, FisherQueen, Mjbyars1, MartinBot, MarkRoberts, Dogma100, Timothy Titus, Tolukra, R'n'B, CommonsDelinker, Alexnevzorov, Nono64, YassirLaCama, PrestonH, Bucs10, Tgeairn, RockMFR, J.delanoy, Trusilver, Bogey97, Uncle Dick, Interplanet Janet, Thaurisil, Bluesquareapple, Greymane, Delphiki27, Katalaveno, Sevennn, Notyou7, Dahliarose, Itohacs, Doctor Sunshine, Nemo bis, Wofo17, Shannahanbury, M-le-mot-dit, Belovedfreak, NewEng- landYankee, Amber Firecat, Hughtcool, Chenhsi, Celiahelena~enwiki, STBotD, WJBscribe, Nikki311, Mercredi-Soir, Diego420, Great- estrowerever, Remember the dot, Rolfenstein, U.S.A.U.S.A.U.S.A., Pdcook, TheNewPhobia, Sabre Knight, Davecrosby uk, Greene36, MelkorDCLXVI, Manu rulz psk, Smallfixer, ACSE, Zakuragi, Vranak, X!, Deor, VolkovBot, CWii, Thedjatclubrock, Flyingidiot, Enders shadow89, Jeff G., JohnBlackburne, Katydidit, Hendybhoy, Aesopos, Dougie monty, Philip Trueman, Snedden, TXiKiBoT, Oshwah, Ann Stouter, Porkrind, Taliesyn1, Lincolnshep, Anna Lincoln, Ferengi, Dajaggman, Bloodset777, Luuva, Retor, Jjmckool, Rifleman jay, Am- berAlert1713, Pious7, Billinghurst, Lerdthenerd, Meters, Zanethers, Trolleytimes, Synthebot, Joseph A. Spadaro, Snoborder93, Tamorlan, DMurf, Prieto21, Falcon8765, Claredelune, Brianga, Drr797, AlleborgoBot, HansHermans, Pediaknowledge, Shrikant soman, Nijumania, Blaximus The Great, Byobob, Delusional 1, SieBot, StAnselm, Coffee, Hornymanatee100, Scarian, BotMultichill, Thzith~enwiki, Geraki- bot, Josh the Nerd, Caltas, Matthew Yeager, Bielle, Smsarmad, Hdpercar, Spread your legs for my virus, Arda Xi, Keilana, Jjjjjjjjjjtttttttttt, Happysailor, Flyer22 Reborn, Casper1027, Twentythreethousand, Hatkkk, Furgle, GaryColemanFan, Cpenarsky, JackSchmidt, Hobar- timus, Taggard, Thirteen squared, Quesoman87, Macy, Hellstrm, Ravanacker, OKBot, Werldwayd, Calatayudboy, TheTaternumber13, StaticGull, Knodeltheory, Mr hyperchee, Gambitthf, Felizdenovo, Moe1234567890000, Jons63, Escape Orbit, Resistgravity, Spice of life, Martarius, ClueBot, Qqq14, Phoenix-wiki, PipepBot, KingofPi, Fyyer, The Thing That Should Not Be, Scartboy, Swissbernie, Mapleaf1, Witchwooder, Cptmurdok, Wutsje, Nobody 343, Drmies, Wildbird30, Regibox, AirdishStraus, Fuzzbeans, Leodmacleod, Niceguyedc, LizardJr8, John J. Bulten, Faerybox, Asher13, Excirial, Alexbot, Jusdafax, TheDudez, Erebus Morgaine, Clark, Roserobin, Juancen- tro, Winston365, BirgerH, Rhododendrites, NuclearWarfare, Huntergrl, Antodav2007, PirateMonkeyAlpha, Razorflame, Ebolamunkee, 15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 111

Gazooki, Aitias, Jonverve, Ted4d7, Versus22, Jacquelineleo, Qwfp, Antstamper, ClanCC, Vanished User 1004, Mdeby, DumZiBoT, Green rockgirl, Bishoy.atef, Chimchar monferno, CBMIBM, Darkicebot, Alchemist Jack, Londonsista, XLinkBot, AmeliaElizabeth, Gonzonoir, PeruAlonso, Not Something and Not Someone, King Willan Bot~enwiki, WikHead, Deantalley1, NellieBly, Frood, WikiDao, That- guyflint, Wyatt915, Addbot, Iemaster77, Willking1979, Sajieman, B2üücking, Kongr43gpen, Ronhjones, ʝ, Lets Enjoy Life, Levasseur, CanadianLinuxUser, Fluffernutter, Rejectwater, CarsracBot, Buster7, LAAFan, Glane23, Chzz, Favonian, SpBot, Terrillja, Tassedethe, Numbo3-bot, Tide rolls, Lightbot, OlEnglish, Romanskolduns, Vazde, Auos11, YersiniaP, Jbug321, Ben Ben, Luckas-bot, ZX81, Yobot, Japlha, Themfromspace, Systemizer, Ptbotgourou, Numbuh1cubfan, Sarahmint, TaBOT-zerem, Legobot II, Leoneris, Dimitrissss, En- DaLeCoMpLeX, Iroony, Daniel 1992, Wiki Roxor, AnomieBOT, Madgey7, Rubinbot, IRP, Galoubet, 90, Chuckiesdad, Orangeville29, Flopsy Mopsy and Cottonmouth, Kaltdrache89, Ccmint, Ulric1313, Bluerasberry, Gt4kill3r, Mahmudmasri, Materialscientist, Senio eil- liw, 90 Auto, The High Fin Sperm Whale, Big papi001, Kevcotra, GB fan, ArthurBot, DirlBot, Senitary, Xqbot, Over9000726, JimVC3, Capricorn42, Drilnoth, Randy Seltzer, Hanberke, Crmsonknight, Herr Mlinka, NFD9001, Nanataktuk, Gap9551, Srich32977, Hi878, Nasa-verve, GrouchoBot, Riotrocket8676, Vandalism destroyer, UnconstructiveEditorsRule, RibotBOT, Cartque, Lover 1213, Xenocidic, Shadowjams, Chexandy89, Aaron Kauppi, Roger-is-picante, Aerosapien, Barnibusthecat7, FrescoBot, LucienBOT, C.pazoki, Applewind, Dalyup!, Nightlion, Wildboy7, Robo37, HamburgerRadio, Habibicb, 04brennanr, Maher27777, Pinethicket, Elockid, HRoest- Bot, Actorglada, Nopuns, LittleWink, El estremeñu, Angstorm, Calmer , TakeThatWiki, Tyler shayde, Karlrulez77, Serols, Space- Flight89, Dmfeastis, Meaghan, RandomStringOfCharacters, ISR10h, HenryAyoola, December21st2012Freak, Solferino7, Purplesoab, SamanthaIsGay, Caseless, Chris5858, TobeBot, Vrenator, MOOGLEGAZJA, Clarkcj12, Bvdrunner, January, Cowlibob, Reaper Eternal, Jimmyjons3639, Minimac, Woogee, TjBot, InbalabnI, Togoman, NerdyScienceDude, Slon02, Mouadino, EmausBot, Bobjoejrthethird, John of Reading, Cymonkey, Ndkl, Racerx11, Swayback Maru, GoingBatty, RA0808, Heart reaper, RenamedUser01302013, NotAnony- mous0, ZxxZxxZ, Tommy2010, Wikipelli, Mdickey99, PatrickNiedzielski, DeadShrimpBlues, Savh, Mz7, ZéroBot, Gulsparv, Fæ, Ti- tanzed0, Marthavi, Davykamanzi, Imperial Monarch, Pliming, Hellothere567, Kingcae199, R. J. Mathar, J0mama666, KuduIO, Killern- erd24, Pickuptha'Musket, TheodoreNg, Navidsohrab, Cbg22, Aceofclubs1414, Frigotoni, Tolly4bolly, JoeSperrazza, Bullmoosebell, TyA, L Kensington, Donner60, Rizikipedia, Ego White Tray, Hickslagan, Xcalizorz, ChuispastonBot, Sol Lanitis, NTox, Venugopalreddykonda, Rocketrod1960, Ham!?no, ClueBot NG, Ezekiel63745, Gareth Griffith-Jones, Akuindo, Robee990, Takuboku, Ypnypn, Sevenburgers, Frietjes, Cntras, Dreth, Widr, Allymillar4, Karl 334, IgnorantArmies, Iste Praetor, Ninjastriker13, Toby Patten, HMSSolent, Night- enbelle, Calabe1992, Howtheocritushadsung, Xyzm146, Silverlands, Jeraphine Gryphon, BG19bot, Ivory to, Walrus068, Snaevar-bot, ,יהודה שמחה ולדמן ,Vagobot, Adam.bell567, Suffolkdialect, Peawormsworth, MusikAnimal, TROLLU4FUN, Vin09, Reynarttt, ITWT Jawadreventon, Sixty-sixBULLETS, CaptainCommanderBenjamin, Justin Clin d'Œil, Magooyi, David.moreno72, Toasted Onion, Ser- chia, Mysterion6318, 22dragon22burn, Saung Tadashi, Webclient101, Lugia2453, Frosty, Juzumaru, Psychetube, Davidavid1997, Hel- loworld32123, SimplicityDown, JPaestpreornJeolhlna, Werr077, Mysterious Island, EvergreenFir, ElHef, Wamiq, Haminoon, Teddywil- son33, FOREAL7, Ieatpie7, Djmoo, TheMeaningof7, Thoseguys34, Szqecs, JAYMERICA56, Ginsuloft, Sam Sailor, MagicatthemovieS, Hollolalaane, Stamptrader, Olorin Aiwendil, Ericosity, हेमन्त डबराल, Justin15w, Flash5lk, Chase987654, Mkrgolf, Fireball24fire, The- HoundOfYes, Seven7seven7seven, Snowycats, HMSLavender, Wikieditor180, AdrianGamer, Fsfg33, Jaytt7, Danroc1, Captain big dick, CLMW97, Snow Lion Fenian, Jorshua7102, Jackkwernerr, Derrib, Jstar700, Whitle the amazing, AlphaBetaGamma01, SparkyDill128, Robert.stefan.m, Bennyboy317, MusikBot, Graphicsindiaonline, CAPTAIN RAJU, Wc.1233.chan, CLCStudent, Gavriil Khipés, Scoutyy, RastafariMorgan23, DatGuy, Arjun Wox, Mahoney 30, Fuortu, Gulumeemee, Gidman2021, PearlCrystalGem, Eyesgreennn, Dankmeme- and Anonymous: 1503 יקותיאל סודרי ,saremylife, Spock012, ABAY2255

• 8 (number) Source: https://en.wikipedia.org/wiki/8_(number)?oldid=754627926 Contributors: AxelBoldt, Kowloonese, William Avery, SimonP, Heron, Hephaestos, Olivier, Patrick, D, Michael Hardy, Mahjongg, SGBailey, Zanimum, Minesweeper, Egil, Ahoerstemeier, Docu, Samuelsen, Nveitch, Glenn, Kimiko, Cimon Avaro, Schneelocke, Hike395, Timwi, Dcoetzee, Rob.derosa, Denni, Dysprosia, Whis- perToMe, Sabbut, Dmetric, Wiwaxia, Wolfram~enwiki, Jeffq, Wst~enwiki, RadicalBender, Denelson83, Robbot, Dale Arnett, Fredrik, RedWolf, Gandalf61, Sverdrup, Yarvin, Academic Challenger, Texture, Yacht, Auric, JesseW, Wikibot, PrimeFan, Pifactorial, Mat- tflaschen, Dina, Enochlau, Giftlite, Dbenbenn, Graeme Bartlett, Sj, Numerao, Anton Mravcek, No Guru, Rick Block, Duncharris, Beardo, Guanaco, Skagedal, Eequor, Andycjp, Sigfpe, Lst27, R. fiend, Noe, Zantolak, Kevin B12, Karl-Henner, Mike Rosoft, Ssmith619, Pyrop, UrmasU, Discospinster, 4pq1injbok, Rich Farmbrough, Guanabot, Vague Rant, Dbachmann, Night Gyr, ESkog, Syp, RJHall, CanisRufus, Kwamikagami, Bobo192, AmosWolfe, Reinyday, Coconino, Jojit fb, Apostrophe, Chicago god, Nsaa, Jumbuck, Alansohn, Borisblue, Verdlanco, ABCD, Sade, Kotasik, Magetoo, Wtmitchell, ProhibitOnions, Kdau, Evil Monkey, Woohookitty, Linas, Anilocra, Etacar11, Jftsang, Uncle G, Polyparadigm, Dodiad, Graham87, TAKASUGI Shinji, Keeves, Jobnikon, BD2412, Kbdank71, Roger McCoy, Men- daliv, Search4Lancer, Sjö, Rjwilmsi, Ian Page, Hack-Man, Harro5, Omnieiunium, MarnetteD, Yamamoto Ichiro, Xsg, John Baez, Mathbot, Nihiltres, Joedeshon, Thecurran, Chobot, DVdm, Gdrbot, 334a, Bgwhite, Siddhant, YurikBot, Wavelength, Borgx, Retodon8, Stormerne, RussBot, Vlad4599, Red Slash, Stephenb, Sacre, Varnav, Ugur Basak, BlackWidower, Grafen, Keka, Stenun, Yoninah, Zwobot, BOT- Superzerocool, Wknight94, Phgao, N-Bot, Nikkimaria, Closedmouth, Arthur Rubin, Pasitbank, Wikipeditor, Pouchkidium, SmackBot, F, McGeddon, Unyoyega, Bmearns, Spireguy, Kintetsubuffalo, GraemeMcRae, Gilliam, Quidam65, Betacommand, Psiphiorg, Hraefen, Jeffro77, Simon123, PrimeHunter, Maxwellgoodwin, Akanemoto, DHN-bot~enwiki, Killingthedream, Audriusa, MyNameIsVlad, Jus- tUser, Yidisheryid, Matthewrpeterson, Pax85, Dharmabum420, Hateless, Decltype, Aelffin, Dreadstar, Andrew c, DMacks, Ohconfucius, Lambiam, Nathanael Bar-Aur L., Khazar, J 1982, RandomCritic, Mets501, Michael Greiner, Midnightblueowl, PhatePunk, Colonel War- den, Antonio Prates, Amakuru, Switchercat, CRGreathouse, CmdrObot, CBM, JohnCD, DSachan, Dgw, ShelfSkewed, Eccentrix inc, WeggeBot, Mikapell, Misof, Doctormatt, Warhorus, Mientkiewicz5508, Achangeisasgoodasa, Andre666, Porsche997SBS, NorthernThun- der, Calumnewton, Thijs!bot, Barticus88, Berria, RevolverOcelotX, Marek69, DmitTrix, WillMak050389, A3RO, Graemec2, QuasyBoy, Tylop~enwiki, Dawnseeker2000, RoboServien, Escarbot, Stannered, WinBot, Hyderziaee, Paul from Michigan, Mackan79, Modernist, Salgueiro~enwiki, Alphachimpbot, Once in a Blue Moon, Altamel, Dhrm77, JAnDbot, Leuko, Barek, UtDicitur, Ecphora, Guy0307, Stephen's black friend, Magioladitis, Adias, VoABot II, Lemon martini, Kajasudhakarababu, Animum, VegKilla, David Eppstein, JaGa, Georbe, Hbent, Xtifr, Welshleprechaun, Mjbyars1, MartinBot, Arjun01, Dogma100, Pbroks13, Nono64, EdBever, RockMFR, Pharaoh of the Wizards, Numbo3, Alantex, Nemo bis, Floaterfluss, Junafani, Calebfultz, Railwayfan2005, Juliancolton, Equazcion, Popndopalis8, Sabre Knight, Idioma-bot, Funandtrvl, Montchav, Zakuragi, Deor, VolkovBot, Ytenin, JohnBlackburne, AlnoktaBOT, Philip Trueman, TXiKiBoT, Oshwah, A4bot, And1987, Anonymous Dissident, CoJaBo, Otaku-jin, Monkey Bounce, Djkrajnik, Fitheach~enwiki, Mmba- bies, JhsBot, Gonzo 405, Tean Butcher, Themat21III, Synthebot, Squalk25, AlleborgoBot, PGWG, MrChupon, HybridBoy, JohnWunder, Hmwith, Rypcord, SieBot, Djkizza, WereSpielChequers, BotMultichill, Oldag07, Dawn Bard, RJaguar3, Merotoker1, Arda Xi, Momo san, Taemyr, Fbarw, Halcionne, MookieZ, OKBot, Knodeltheory, Anchor Link Bot, Mygerardromance, Felizdenovo, Denisarona, Es- cape Orbit, Maxschmelling, Wboyce39, Mx. Granger, Dancingwombatsrule, Martarius, ClueBot, PipepBot, Snigbrook, Fyyer, Bleed- ingshoes, The Thing That Should Not Be, Sandelb, Drmies, Mild Bill Hiccup, Wildbird30, PolarYukon, Chstdu, Passargea, Alexbot, QWerk.fi, Winston365, Souryoku, SchreiberBike, Sacksodon, Aitias, Jonverve, 7, Shashijeevan, Darkicebot, Heironymous Rowe, Roxy the dog, OptimisticFool, BjkliBucho, Doug butler, Iemaster77, ConCompS, Goddard123, Some jerk on the Internet, B2üücking, Jojhut- 112 CHAPTER 15. DIGITAL ROOT

ton, ʝ, Diablokrom, Stewartdc8, Glane23, Favonian, Tsange, David Hale Clark, Numbo3-bot, Ehrenkater, Tide rolls, Romanskolduns, Jarble, HerculeBot, Thespyofcharles, Luckas-bot, Yobot, Tohd8BohaithuGh1, Fraggle81, TaBOT-zerem, KamikazeBot, Dhiru.patel, Fun- nyfaces, AnomieBOT, Madgey7, Blah42b10, Jim1138, Tucoxn, Kingpin13, RandomAct, Mahmudmasri, Materialscientist, Intractable, GB fan, Knowledge Incarnate, Santa Claus of the Future, DirlBot, Thebirdistheword, Xqbot, Hanberke, Cyndaquazy, Herr Mlinka, Gap9551, Nasa-verve, GrouchoBot, SassoBot, Cartque, Shadowjams, Simbrex, FrescoBot, LucienBOT, Ryryrules100, Oldlaptop321, Maverick9711, Robo37, HamburgerRadio, Citation bot 1, Tintenfischlein, Ehsan baghaki, Pinethicket, I dream of horses, HRoestBot, RedBot, TobeBot, DixonDBot, Davish Krail, Gold Five, Ricardodiazpr, 4, Louislee95, Stroppolo, MegaSloth, RjwilmsiBot, Aircorn, EmausBot, Acather96, Bigtony123, Immunize, GoingBatty, Nicwazh3re, DeadShrimpBlues, ZéroBot, Jesse231, Jlowther91, Killern- erd24, Cbg22, Ilikebooks123, JoeSperrazza, Coasterlover1994, Bulwersator, Puffin, ChuispastonBot, Sdkb, Venugopalreddykonda, Clue- Bot NG, Ezekiel63745, Akuindo, 1234567loversyou, Ypnypn, MelbourneStar, Movses-bot, Jake Golle, Bmoch, Frietjes, Widr, Flu- orineGas, Helpful Pixie Bot, Plantdrew, BG19bot, Vagobot, ISTB351, NickyCastle, Carnold30, Gorthian, Dainomite, Wheres Dan, Sourced much, The Light Burns, Cyberbot II, Serchia, ChrisGualtieri, 22dragon22burn, Mogism, Saehry, DavidPKendal, Passengerpi- geon, Dschslava, Gagejo23, Thatguywhoeverybodylikes, PWNGWN, Tentinator, Wamiq, 0alx0, Jackace101, MagicatthemovieS, हेमन्त डबराल, Molecularmachine, Fireball24fire, Gronk Oz, PErdos, Tophet, CyanLights, TheCoffeeAddict, Wynstol, AlphaBetaGamma01, Thelimiter, Azealia911, Rukyruky, KSFT, Liamrosco, InternetArchiveBot, AirshipJungleman29, Gulumeemee, BrettWheeler, Bender the Bot and Anonymous: 527 • 9 (number) Source: https://en.wikipedia.org/wiki/9_(number)?oldid=760482022 Contributors: Damian Yerrick, AxelBoldt, Zundark, The Anome, Tarquin, -- April, Setok, Kowloonese, XJaM, Christian List, William Avery, Heron, Karl Palmen, Olivier, Patrick, Michael Hardy, Dmd3e, JakeVortex, Liftarn, SGBailey, Eric119, Egil, Ahoerstemeier, Cyp, Docu, Samuelsen, Suisui, Darkwind, Andrewa, Glenn, Nikai, Cimon Avaro, Panoramix, Schneelocke, Hashar, Charles Matthews, Adam Bishop, Dcoetzee, Denni, Dysprosia, WhisperToMe, Zoicon5, AndrewKepert, Sabbut, Dmetric, Wiwaxia, Wolfram~enwiki, Wst~enwiki, RadicalBender, Denelson83, Robbot, Dale Arnett, Cgranade, Korath, Tomchiukc, RedWolf, Altenmann, Psychonaut, Henrygb, Academic Challenger, Texture, Hadal, Scooter~enwiki, PrimeFan, Mat- tflaschen, Enochlau, Giftlite, Dbenbenn, Graeme Bartlett, Sj, Herbee, Numerao, Anton Mravcek, Emrys2, Rick Block, Duncharris, Gua- naco, Eequor, SMWhat, Lst27, Antandrus, Dnas, JnB987, Pacian, Tothebarricades.tk, Trevor MacInnis, Mr Bound, Eyrian, Jiy, Discospin- ster, 4pq1injbok, Rich Farmbrough, Pjacobi, Legalizeit, Dbachmann, Si morgan, Bender235, ZeroOne, Violetriga, CanisRufus, Bennylin, Kiand, Kwamikagami, Shanes, Susvolans, Thu, Bobo192, NetBot, Smalljim, Defrosted, Reinyday, Coconino, Azure Haights, Jojit fb, Acjelen, Chicago god, Bladeswin, Jumbuck, Alansohn, Wiki-uk, Verdlanco, ABCD, Kotasik, Zerofoks, David de Cooman, Wtmitchell, Velella, SidP, Twisp, Kdau, Evil Monkey, Skatebiker, Arthur Warrington Thomas, DV8 2XL, Djsasso, Kazvorpal, WilliamKF, GVOLTT, Uncle G, Barrylb, MattGiuca, JFG, Duncan.france, UnholyDiver, Mandarax, Jack Cox, Jobnikon, Kbdank71, Roger McCoy, Mendaliv, Sjakkalle, Jake Wartenberg, Vegaswikian, Bhadani, Ed Nieuwenhuys~enwiki, Matt Deres, Sango123, Xsg, Red Deadeye, Mathbot, Ni- hiltres, Mitsukai, Mordien, Thecurran, Haonhien, King of Hearts, Chobot, DVdm, Gdrbot, 334a, YurikBot, Borgx, Alpha Prime, Deep- trivia, Rt66lt, Jimp, RussBot, Bergsten, Cunado19, NawlinWiki, Den68cube, Grafen, Keka, JJ4sad6, Yoninah, Fshepinc, DryaUnda, Dead- EyeArrow, Sandstein, Closedmouth, Arthur Rubin, BorgQueen, 2fort5r, Leeannedy, RenamedUser jaskldjslak904, Ilmari Karonen, King- boyk, Veinor, SmackBot, Nihonjoe, Incnis Mrsi, Kintetsubuffalo, Flannel, Gilliam, Quidam65, Ohnoitsjamie, Psiphiorg, Chris the speller, PrimeHunter, Astaroth5, Akanemoto, DHN-bot~enwiki, Darth Panda, Killingthedream, Audriusa, Can't sleep, clown will eat me, JustUser, Saberlotus, DaNASCAT, Pax85, COMPFUNK2, Fuhghettaboutit, Cybercobra, Kevlar67, Blake stapler, Andrew c, Rothesay, DMacks, Ck lostsword, Lambiam, ArglebargleIV, Interfusor, Goodnightmush, Jim.belk, Aspirex, Dicklyon, Optakeover, VdSV9, E-Kartoffel, Dr.K., Jggouvea, JoeBot, Igoldste, Laserbream, Tawkerbot2, Mapsax, CRGreathouse, CmdrObot, Legaia, DSachan, Gegorg, WeggeBot, Edc- shah, Neelix, Azure, AndrewHowse, Mientkiewicz5508, Stukorn, Andre666, Porsche997SBS, Soetermans, Omicronpersei8, Gimmetrow, [email protected], Thijs!bot, Barticus88, Brokenbeaker, Nalvage, Berria, JustAGal, Tylop~enwiki, RoboServien, Escarbot, C4jones2, Stannered, WinBot, Luna Santin, Paul from Michigan, Mackan79, Pichote, Salgueiro~enwiki, Dhrm77, JAnDbot, Arch dude, Skorpio-88, Fuzzybyte, Adias, Bongwarrior, VoABot II, Kajasudhakarababu, Jéské Couriano, Froid, Ch0s3n0n3, Chris Ssk, Darkage7, David Epp- stein, Jay Kana, JMyrleFuller, DerHexer, Xtifr, Connor Behan, Mjbyars1, Juansidious, Agroanoftedium, Nono64, RockMFR, J.delanoy, Numbo3, Uncle Dick, Nemo bis, Stratman, NewEnglandYankee, DadaNeem, Kenneth M Burke, Yeatesy, Ja 62, Sabre Knight, Idioma- bot, Zakuragi, VolkovBot, Gajbireland, BAKAJANAI, JohnBlackburne, Katydidit, Zhou yi777, Dougie monty, TXiKiBoT, Oshwah, A4bot, Rich Janis, JhsBot, Tean Butcher, Master Bigode, Gibson Flying V, Luuva, Saturn star, AmberAlert1713, Staka, Jaredstein, Zephyrus19, Synthebot, Falcon8765, AlleborgoBot, SieBot, Interselector, Teh nubkilr, BotMultichill, Jordi9~enwiki, Winchelsea, Ben- togoa, Dbomp, Hobartimus, Atosecond, Diorcouture0369, Lordmagnusen, Felizdenovo, LarRan, Nine ix, Skyward, ClueBot, PipepBot, The Thing That Should Not Be, P0mbal, Sandelb, Icchan1107, Wildbird30, PolarYukon, Estevoaei, TheSmuel, Bbb2007, Uswhps7, Jm- rburgard, Alexbot, Jusdafax, ECRidah8991, Antodav2007, CMW275, SchreiberBike, Thingg, Jonverve, 7, Certes, Smarkflea, RMFan1, Darkicebot, Ladsgroup, Skarebo, Renthead101, HexaChord, Wacokunkel, Belong2cc, Willking1979, SpBot, Guffydrawers, Numbo3-bot, Tide rolls, OlEnglish, Romanskolduns, MuZemike, Luckas-bot, Yobot, Bunnyhop11, Numbuh1cubfan, KamikazeBot, Coypu, Eric-Wester, AnomieBOT, Blah42b10, Flewis, Mahmudmasri, ImperatorExercitus, Iamtheealmighty, Jtamad, GB fan, ArthurBot, Acebamcole, Xqbot, Jayarathina, Hanberke, Nineboyd, Herr Mlinka, The Evil IP address, Gap9551, Nasa-verve, GrouchoBot, RibotBOT, Rabby16, Winterwa- ter, AlasdairEdits, MHR65, LucienBOT, Kmg952, Sky Attacker, Robo37, Shogami, Pinethicket, HRoestBot, Fkush, Gkgemini, Abhijit- sengupta ias, RedBot, Salvidrim!, Orenburg1, FoxBot, TobeBot, Royal Blue Jersey, Bigez620, Minimac, DARTH SIDIOUS 2, Jawadfaisal, Ninukk, Kid with ideas, EmausBot, John of Reading, WikitanvirBot, Katherine, Racerx11, Faolin42, GoingBatty, Tommy2010, Wikipelli, DeadShrimpBlues, Captaingimpy, AvicBot, Dolovis, ShadowKN, Jenks24, Kingcae199, Mar4d, Dondervogel 2, SPARTAN T-82, Ilike- books123, Wayne Slam, L Kensington, Donner60, ChuispastonBot, King of numbers with 9s, Venugopalreddykonda, AgentSniff, DASH- BotAV, 1007D, ClueBot NG, Ezekiel63745, Peter James, Gareth Griffith-Jones, Akuindo, Ypnypn, Alza08, Starchild1884, Frietjes, Dru of Id, Racecar123456, Widr, Whywatsup, North Atlanticist Usonian, Helpful Pixie Bot, AlternoBreak, Titodutta, Raishvar, BG19bot, Gurt Posh, J991, John M. Johnson, M1CHAEL9Official, Altaïr, CitationCleanerBot, Nikkywikky321, Zzzto, Lolasmark, David.moreno72, ,Frosty, Graphium, Ogray43, Jiawhein, Glyphexpert ,عربستاني ,Serchia, Jig309, 22dragon22burn, Total-MAdMaN, Avengingbandit Daalfabet, Wamiq, Ntwild97, Pho Ther, M briglia05, JaconaFrere, हेमन्त डबराल, Tophet, R3kt, Loraof, Snow Lion Fenian, Funbfunbi77, Iamonlycreatingthisaccountbconething, 1c.jr12, AlphaBetaGamma01, Sundayclose, Zoepaige, Jpchahine, Meowmaster534, JJMC89, BU Rob13, Nslookup, Sirchchris11, Eurocus47, Yet Another User 2, Jason Kruger72, Gulumeemee, Tonasky, Bender the Bot, Angry Bakers, Johnnymanziel2222, Cyrus noto3at bulaga, Joe1w and Anonymous: 498 • 10 (number) Source: https://en.wikipedia.org/wiki/10_(number)?oldid=759154832 Contributors: AxelBoldt, Bryan Derksen, XJaM, William Avery, Heron, Olivier, BrianHansen~enwiki, MartinHarper, SGBailey, SebastianHelm, Egil, Docu, Samuelsen, TUF-KAT, Suisui, Glenn, Schneelocke, Hashar, Dcoetzee, Reddi, Denni, Dysprosia, WhisperToMe, Zoicon5, Prumpf, Markhurd, Saltine, AndrewKepert, Sabbut, Taoster, Wiwaxia, Warofdreams, Wolfram~enwiki, Mordomo, Denelson83, Robbot, Dale Arnett, Jaredwf, RedWolf, Altenmann, Academic Challenger, Rholton, Yacht, PrimeFan, Enochlau, Psb777, Fabiform, Giftlite, Sj, Robin Patterson, Geeoharee, Tom harrison, Numerao, Anton Mravcek, Rick Block, Duncharris, Eequor, Andycjp, Sigfpe, Lst27, Andjb, Noe, Beland, Dnas, Kusunose, Lumidek, Crei- 15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 113

dieki, Quota, Ouro, Discospinster, 4pq1injbok, Rich Farmbrough, Guanabot, Guanabot2, Hapsiainen, Kwamikagami, Bobo192, Robotje, John Vandenberg, Shenme, R. S. Shaw, Jolomo, Bdamokos, Unknown W. Brackets, Wrs1864, Haham hanuka, Chicago god, Nsaa, Ranveig, Kotasik, Miltonhowe, Fourthords, Knowledge Seeker, Evil Monkey, Bsadowski1, Skatebiker, Djsasso, Ceyockey, Stemonitis, WilliamKF, Will Orrick, Thorpe, Uncle G, JFG, WadeSimMiser, BD2412, Ajnewbold, Kbdank71, Roger McCoy, Mendaliv, Jshadias, Koavf, MZMcBride, Notorious4life, Xsg, Baryonic Being, Mathbot, Nihiltres, ChongDae, Ggb667, Chobot, DVdm, JesseGarrett, Gdr- bot, Korg, 334a, YurikBot, Wavelength, Borgx, Jimp, Pi Delport, Hydrargyrum, Pseudomonas, Iani, Volunteer Sibelius Salesman, Yon- inah, Moe Epsilon, BOT-Superzerocool, ThinkingInBinary, Wknight94, Arthur Rubin, RickReinckens, Pasitbank, Paul Erik, Quadpus, SmackBot, Down10, Kintetsubuffalo, GraemeMcRae, Dsl002, Commander Keane bot, Gilliam, Psiphiorg, PrimeHunter, Jammycakes, Akanemoto, DHN-bot~enwiki, Yidisheryid, DaNASCAT, Azumanga1, Matchups, DaDoc540, B jonas, Aelffin, Dreadstar, Alexandra lb, DMacks, Bkris, Monkey06, Eliyak, Zahid Abdassabur, The--dud, JoeBot, Blehfu, Me lkjhgfdsa, Vindicta, Ytny, Courcelles, Aapold, Pi, Jonathan W, JForget, Sakurambo, DSachan, NickW557, ShelfSkewed, Thepm, Doctormatt, Stukorn, Gogo Dodo, Andre666, Chrislk02, Tortillovsky, Thijs!bot, Epbr123, Barticus88, Jpark3909, Calvinballing, JustAGal, Chandler, Johnwrw, CompositeFan, RoboServien, Es- carbot, AntiVandalBot, WinBot, Seaphoto, Edokter, Joe Schmedley, Tractakid, Farosdaughter, Salgueiro~enwiki, Dhrm77, JAnDbot, Deflective, Hydrostatics, MER-C, Craigantill, Kirrages, Adias, VoABot II, Kenaldinho10, Kajasudhakarababu, Georbe, FredMSloniker, Lunakeet, J.delanoy, Thaurisil, Shawn in Montreal, DarkFalls, Midnight Madness, NewEnglandYankee, Jeterfantwo, Kidlittle, Inspire62, Steel1943, Idioma-bot, VolkovBot, JohnBlackburne, AlnoktaBOT, Philip Trueman, TXiKiBoT, Oshwah, A4bot, Qxz, Someguy1221, Davereject, Mmbabies, JhsBot, Tean Butcher, Raph79, Lampica, AlleborgoBot, SieBot, TJRC, PlanetStar, Tiddly Tom, BotMultichill, Timmyxcore, Flyer22 Reborn, Byrialbot, Steven Crossin, Kumioko (renamed), Denisarona, Sfan00 IMG, ClueBot, Iluvmesodou, Pipep- Bot, Rjd0060, Srussack, Hongthay, Mild Bill Hiccup, Wildbird30, Phuntsok2000, Stepshep, Truegator, Alexbot, ShanRaj 10, BOTarate, Darkicebot, Ladsgroup, Little Mountain 5, NHJG, Thatguyflint, Dranchak2010, UhOhFeeling, ERK, COWS4YOU, Ronhjones, ʝ, Re- jectwater, CarsracBot, SpBot, Numbo3-bot, OffsBlink, Wiki pker, Romanskolduns, HerculeBot, Middayexpress, Luckas-bot, Yobot, Pt- botgourou, Fraggle81, TaBOT-zerem, LittleBuzzard, Nallimbot, KamikazeBot, AnomieBOT, Srobak, Blah42b10, IRP, Emperor1992, Dingold, Grolltech, Ulric1313, Materialscientist, ArthurBot, MauritsBot, Xqbot, Capricorn42, Renaissancee, Hanberke, Prenter~enwiki, Anne Bauval, Gap9551, GrouchoBot, Abce2, RibotBOT, Intruductionetrondocapriccioso, Doulos Christos, FrescoBot, LucienBOT, HJ Mitchell, Robo37, JIK1975, Pinethicket, HRoestBot, RedBot, Onye3, Lacerie, December21st2012Freak, FoxBot, TobeBot, Royal Blue Jersey, Tbhotch, Avedeus, Mean as custard, RobertAlanHarris, MatthewW7, EmausBot, Mrbinky, K6ka, DeadShrimpBlues, Derekleungt- szhei, Puffin, ChuispastonBot, JonRichfield, Petrb, ClueBot NG, Cwmhiraeth, Ezekiel63745, Ypnypn, Frietjes, U.Steele, Kikones34, ,BG19bot, Wiki13, MusikAnimal, Messirulez, Khaledmontoia1234, Citizenconcerned ,ثعلب شبه الجزيره ,Widr, Deep Thought Cyberbot II, Serchia, 22dragon22burn, Dobie80, Numbermaniac, Quinette21, MatthewJ00, Wobblehead, Atlswag69, DavidLeighEllis, Babitaarora, Comp.arch, NottNott, Quenhitran, JWNoctis, MagicatthemovieS, Gronk Oz, Pronum1, Tophet, Blackstockk16, Zppix, Alpha- BetaGamma01, 演歌ビニール, Darkryan306, Vrshaankm, MusikBot, Seminyoon2, Looqmaniac, GreenC bot, Gulumeemee, TJXenon, Tonasky, Bender the Bot, UTHDHSHRFFIFDNEGDCCY, Mukul chilwal, Dr.snellen and Anonymous: 287 • 11 (number) Source: https://en.wikipedia.org/wiki/11_(number)?oldid=760400769 Contributors: The Epopt, XJaM, Christian List, Heron, Edward, Michael Hardy, JakeVortex, SGBailey, Tregoweth, Egil, DavidWBrooks, Docu, Theresa knott, Glenn, Error, Schneelocke, Agtx, Dcoetzee, Denni, Dysprosia, Tedius Zanarukando, WhisperToMe, Wiwaxia, Warofdreams, Wolfram~enwiki, Robbot, Dale Arnett, Celsius1414, Fredrik, Scriptwriter, Hemanshu, Yacht, JB82, Catbar, Wikibot, PrimeFan, Giftlite, Sj, Aphaia, Spencer195, Numerao, An- ton Mravcek, No Guru, Curps, Rick Block, Duncharris, J. 'mach' wust, UgenBot, Lumidek, Sonett72, Latexguy, Kate, D6, Discospinster, Rich Farmbrough, Guanabot, Guanabot2, RJHall, CanisRufus, Bennylin, Kwamikagami, Grue, Reinyday, Lore Sjoberg, Shenme, Chbarts, Jojit fb, Rajah, Minghong, Haham hanuka, Cyrillic, Chicago god, Mareino, Arthena, Borisblue, Andrewpmk, Verdlanco, ABCD, AzaToth, Sligocki, Hu, Malber, Evil Monkey, Djsasso, Bookandcoffee, Oleg Alexandrov, Richard Arthur Norton (1958- ), OwenX, Woohookitty, FeanorStar7, Jersyko, Will Orrick, Oliphaunt, SDC, Crazysunshine, Radiant!, BD2412, SixWingedSeraph, Zzedar, Mendaliv, Jshadias, Josh Parris, Rjwilmsi, Bruce1ee, Crazynas, Notorious4life, DirkvdM, Yamamoto Ichiro, Celestianpower, Chobot, DVdm, Korg, Yurik- Bot, Wavelength, Borgx, Angus Lepper, Jcam, Neitherday, Jimp, RussBot, Cwlq, Chensiyuan, Yoninah, Ospalh, BOT-Superzerocool, Jeh, Nick123, Igiffin, Genjix, EAderhold, Doktorbuk, Arthur Rubin, BorgQueen, GrinBot~enwiki, Quadpus, SmackBot, YellowMon- key, Vidyasagar, Gilliam, Skizzik, Rmosler2100, Hamedog, Vampirewolf3, PrimeHunter, Akanemoto, DHN-bot~enwiki, Yidisheryid, DaNASCAT, Thefanaticgamer, Alexandra lb, Kleuske, DMacks, Nishkid64, U-571, Scientizzle, Gobonobo, Anonymous112, Nong- Bot~enwiki, Jbonneau, Beetstra, Trisbray, InedibleHulk, Mets501, Novangelis, MTSbot~enwiki, PaulGS, T-W, Poolkris, CRGreathouse, Ventifax, Smiloid, DSachan, Laslo piniflex, Dgw, WeggeBot, Neelix, Gogo Dodo, Andre666, Michaelbarnes, [email protected], Hyp- nosadist, Thijs!bot, Epbr123, Barticus88, 91scottduncan, Calvinballing, ByteofKnowledge, Faigl.ladislav, Marek69, SGGH, Cdf333fad3a, JustAGal, Chandler, Philippe, Nick Number, Johnwrw, Azaner, RoboServien, Escarbot, AntiVandalBot, WinBot, Paul from Michigan, AE123, Edokter, Erukto, Darklilac, Oatmealcookiemon, The Dan, Dhrm77, Ioeth, JAnDbot, Asmeurer, Deflective, Barek, Robina Fox, Roleplayer, MegX, Trevordeclercq, Adias, Freedomlinux, Bongwarrior, Lemon martini, Bengha, Drago02129, Mwalimu59, Shocking Blue, Woodenbeam, David Eppstein, DerHexer, MartinBot, STBot, Doogle59, Roastytoast, R'n'B, Trusilver, Qmega, HHP2K, Nemo bis, Spaner, Pyrospirit, Belovedfreak, STBotD, ImMAW, Idioma-bot, Black Kite, 28bytes, Morenooso, JohnBlackburne, Kyle the bot, TXiKiBoT, Oshwah, GcSwRhIc, Qxz, Darcstars, Mmbabies, JhsBot, Justintime516, Broadbot, Yomcat, Jackfork, Tean Butcher, Franchi- seIndustries, Digita, Jyothisv, Ernest lk lam, SieBot, Testing times, BotMultichill, Yintan, Flyer22 Reborn, Willy11009, Ms2150, Baseball Bugs, Sean.hoyland, TypoBot, Ken123BOT, Faithlessthewonderboy, Sfan00 IMG, ClueBot, Venusboysatish, Bucknaked 88, Mild Bill Hic- cup, Laudak, Trivialist, Paulcmnt, Stepshep, Alexbot, Watchduck, Jotterbot, Singhalawap, 11irageb, Thingg, 11rich3k, Certes, SoxBot III, DumZiBoT, Darkicebot, Ladsgroup, Mitch Ames, SilvonenBot, MystBot, Cfcno1fan, Addbot, Basilicofresco, Jkasd, Noozgroop, Lost on Belmont, SpBot, Numbo3-bot, Bungalowbill430, Tide rolls, Romanskolduns, Frehley, Luckas-bot, Ptbotgourou, TaBOT-zerem, Xqt, Jason Recliner, Esq., Nallimbot, KamikazeBot, Mklewis07, AnomieBOT, ThaddeusB, Blah42b10, LlywelynII, BobKilcoyne, Ulric1313, Mate- rialscientist, ImperatorExercitus, ArthurDuhurst, ArthurBot, DirlBot, Xqbot, TinucherianBot II, ChrisTGH, Nasnema, Rtfm1st, FaleBot, Gap9551, Srich32977, RibotBOT, JMK339, Paradise coyote, FrescoBot, Bpmcneilly, Robo37, Safinaskar, Spidey104, MastiBot, Hen- ryAyoola, Pier4r, Double sharp, Royal Blue Jersey, Olejika, Cutelyaware, Mean as custard, Alph Bot, Dkentb3, Skamecrazy123, Smd75jr, EmausBot, WikitanvirBot, Srb149, L235, Solarra, Tommy2010, Wikipelli, Savh, ZéroBot, R. J. Mathar, Pickuptha'Musket, Num Ref, Jesanj, ChuispastonBot, Wandersonza, Bridgetgallagher2011, ClueBot NG, Slapmyspunk, Ypnypn, Sanfazer, Frietjes, O.Koslowski, Widr, Ba11zooka, Calabe1992, BG19bot, Mfgarris, MusikAnimal, Altaïr, Wannabemodel, WebTV3, ChrisGualtieri, EuroCarGT, JojoGirl278, Listroiderbob, தமிழ்க்குரிசில், 069952497a, MatthewHoobin, Comp.arch, MedliAndMakar, Vivaengrey, FrB.TG, JaconaFrere, Anony- moouse...., Mikeforall, INeekster, Gronk Oz, Awze, Jasmineh 11, MRD2014, Lokirulerofearth, AlphaBetaGamma01, 演歌ビニール, MusikBot, 逆襲的天邪鬼, KaBoom100, Sp gaming, Frenchfry444, InternetArchiveBot, Chrissymad, Gulumeemee, TJXenon, Tonasky, Pauciloquence, Islamiccoolguy and Anonymous: 361 • 0 Source: https://en.wikipedia.org/wiki/0?oldid=760310939 Contributors: AxelBoldt, Magnus Manske, The Epopt, Calypso, Sodium, Bryan Derksen, Zundark, The Anome, Manning Bartlett, Ed Poor, Andre Engels, Josh Grosse, Jrincayc, XJaM, Arvindn, Christian 114 CHAPTER 15. DIGITAL ROOT

List, Toby Bartels, William Avery, Heron, Camembert, Montrealais, Patrick, Infrogmation, D, Michael Hardy, Wshun, Llywrch, Lexor, GUllman, Liftarn, MartinHarper, Menchi, TakuyaMurata, Delirium, Tregoweth, Egil, Ulrich.fuchs~enwiki, Ellywa, Ahoerstemeier, Docu, Samuelsen, Angela, Alvaro, Kragen, LittleDan, Bogdangiusca, AugPi, Llull, Andres, Jeandré du Toit, Evercat, Mxn, Schneelocke, Asktav, Mulad, Emperorbma, Charles Matthews, Crissov, Dcoetzee, Random832, Dysprosia, Doradus, DJ Clayworth, Natevw, Furrykef, Saltine, LMB, Head, Vaceituno, Jeanmichel~enwiki, Elwoz, Bloodshedder, Jerzy, Bamos, Slawojarek, Owen, Denelson83, Chuunen Baka, Robbot, Fredrik, PBS, RedWolf, Securiger, Rfc1394, Henrygb, Texture, Yacht, Rrjanbiah, Wereon, Paul Murray, PrimeFan, Diberri, Guy Peters, Xanzzibar, Mattflaschen, Mfc, Tea2min, Robert Happelberg, Giftlite, Dbenbenn, Graeme Bartlett, Gene Ward Smith, Sj, Joaopaulo1511, Numerao, Anton Mravcek, Everyking, No Guru, Markus Kuhn, Jcobb, Varlaam, Joe Kress, Rick Block, Duncharris, Mboverload, Eequor, Jackol, AdamJacobMuller, Tagishsimon, Ryanaxp, Doshell, Utcursch, Farside~enwiki, UgenBot, Yath, Noe, Q17, Beland, Pm, Eregli bob, Kaldari, Ihavenolife, Yanamad, Secfan, RetiredUser2, Pmanderson, Indolering, Soman, Ctac, Petershank, Rgrg, Abdull, Trevor MacInnis, Drbalaji md, Lacrimosus, Grstain, Mike Rosoft, Brianjd, Cypa, Jayjg, Poccil, Duja, Flora~enwiki, Discospinster, Steve Farrell, 4pq1injbok, Rich Farmbrough, Guanabot, David Schaich, Dbachmann, Paul August, Wwhitlock, DcoetzeeBot~enwiki, Bender235, Project2501a, Can- isRufus, BenjBot, Livajo, El C, Kwamikagami, Edward Z. 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Spadaro, Jediknil, Enviroboy, Sesshomaru, Insanity Incarnate, Dmcq, Life, Liberty, Property, Arcfrk, Symmm, Angelastic, Portia327, Tvinh, Raflee, PericlesofAthens, Professionalmoron, Wpchen, Radagast3, Billytrousers, TheStarman, S.Örvarr.S, Murkee, Demmy, Obaidz96, Regregex, GoonerDP, SieBot, Alessgrimal, Lanaface34, BotMultichill, Winchelsea, Gerakibot, Dawn Bard, Caltas, TacticalBread, Srushe, Toddst1, BTH, Oxymoron83, Byrialbot, 0rrAvenger, Avnjay, Steven Crossin, Kf- gauss, SimonTrew, Macy, Kudret abi, Thewalkingcrow, Nusumareta, Dodger67, Anchor Link Bot, ChemicalDeathMan, Nn123645, Noth- infukenwerks, Rdhettinger, Frapoz, Francvs, Tuntable, Velvetron, Troy 07, Loren.wilton, Martarius, Toliar, ClueBot, Bob1960evens, Ilu- vmesodou, PipepBot, Laurensmells, The Thing That Should Not Be, Cliff, SerduchkaFan, Techdawg667, Jagun, Wysprgr2005, Garyzx, Drmies, Mild Bill Hiccup, TheOldJacobite, Eclectic hippie, Yixuan05, Rhubbarb, CounterVandalismBot, Niceguyedc, Jalanpalmer, Xinjao, Aua, DragonBot, Wrsh11, Excirial, Alexbot, Jusdafax, CrazyChemGuy, Three-quarter-ten, Monobi, Lartoven, Jotterbot, 7&6=thirteen, Basketball110, IXella007, Real Deuce, Thingg, Nikki Plikki Sikki, Jonverve, DerBorg, Versus22, SoxBot III, Ostinato2, Bücherwürmlein, Theking2, DumZiBoT, Makotoy, ZALASTA111, G20071221, BarretB, AmeliaElizabeth, Tarheel95, Spitfire, Gnowor, Franjosp, Duncan, Kalupinka, Lily W, Denyontheworld, Dthomsen8, Ost316, AndreNatas, SilvonenBot, Coasterror, Dnvrfantj, RyanCross, Kbdankbot, Phan- tom321, Bloodbath 87, Halosean, Jojhutton, Betterusername, DougsTech, Metagraph, Fieldday-sunday, TonyTeacher, WFPM, Download, CarsracBot, Redheylin, Z. Patterson, Glass Sword, Surfcar, SpBot, AtheWeatherman, Quercus solaris, Blaylockjam10, Ledzeppelinfan4, TangLab, Danith123horn, Jasonfitz, Chair5, Bigzteve, Tide rolls, Vikasthakurno1, HerculeBot, Mps, JSR, Mr1744, Luckas-bot, Yobot, Digital fuel, Senator Palpatine, TaBOT-zerem, HIVE MIND!1!1, KamikazeBot, Ali 786 52, South Bay, Synchronism, AnomieBOT, Ar- jun G. Menon, Sagaci, Master of Pies, Royote, JackieBot, Da man times 2, AdjustShift, LlywelynII, Mahmudmasri, The High Fin Sperm Whale, Citation bot, Fleaman5000, Kalamkaar, Empro2, Bob Burkhardt, GB fan, DirlBot, Xqbot, Jayarathina, ManningBartlett, Tin- ucherianBot II, Jeffrey Mall, Timmyshin, Gilo1969, Joey Skywalker, BotPuppet, Wyklety, Nasa-verve, Abce2, Tdjewell, Mario777Zelda, 15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 115

Shirik, RibotBOT, Kyng, SCΛRECROW, Yohannesb, Shadowjams, WebCiteBOT, AJCham, Hueykablooie, Cgersten, Untangledystopia, Thejadefalcon, Captain-n00dle, GliderMaven, FrescoBot, Michael93555, Maverick9711, Tlork Thunderhead, Pxos, Robo37, Citation bot 1, Pshent, LetsPlayMBP, 06jcoxon, Winabet, Kkonnorr, Pinethicket, I dream of horses, Phil1881, LittleWink, 10metreh, Jonesey95, Tinton5, Phearson, Wikiain, Кыс, Meaghan, Fizzle1, Ptarjan, Mikespedia, Jujutacular, Speedy bell, Reconsider the static, Arjrs123, To- -Fallenangel2009, Effi ,علی ویکی ,beBot, Trappist the monk, Superawesome 9001, Zanhe, Carolgregor, Tofutwitch11, Amiodarone ciencyjacky154, Jimmy416, No One of Consequence, Stephen MUFC, Thecoolduke2008, Robixen, H.ehsaan, American Hindu, DARTH SIDIOUS 2, DoctorJeriactrick, Whisky drinker, Flying dustman, Agent Smith (The Matrix), NerdyScienceDude, Salvio giuliano, Gron- demar, EmausBot, Aqwhuzaifa, WikitanvirBot, Gfoley4, Dee Fraser, Shhhnotsoloud, Rarevogel, Ut mehrotra, ZxxZxxZ, Wham Bam Rock II, The Mysterious El Willstro, Tommy2010, Wikipelli, TeleComNasSprVen, ZéroBot, Hudee, PBS-AWB, Akshatme, Fæ, Traxs7, Michealt, Azuris, 1234r00t, TheodoreNg, Kyucasio, Wayne Slam, Ocaasi, SillyLorenzo, Milind 220696, Th3rokkcc3r, Cookiefonster, L Kensington, Vflash10, ChuispastonBot, Sunshine4921, Broderick 0, AMD, Thebobofi, Xanchester, Mikhail Ryazanov, ClueBot NG, Ezekiel63745, Ronakshah1990, LutherVinci, Ypnypn, Winston786, Frietjes, Primergrey, Rurik the Varangian, North Atlanticist Usonian, MerlIwBot, Helpful Pixie Bot, Curb Chain, Wbm1058, Jeraphine Gryphon, Andersw2, Vagobot, Zyxwv99, Solomon7968, Drift chambers, Gorthian, Sbblr geervaanee, Minsbot, Achowat, Pendragon5, WOWIndian, Toploftical, Serchia, Dkyguy1995, Aditya Mahar, Everything Is Numbers, RMCD bot, Wario-Man, Me, Myself, and I are Here, Rupert loup, Donfbreed2, Melonkelon, Ratha K, Mutley1989, Wamiq, Evensteven, Grfr12345, Epitectus, JCMPC, Beren Dersi, हेमन्त डबराल, Tarpuq, Tigercompanion25, Melbinsuresh, SheriffIsInTown, P.O IX, Maths314, BlueworldSpeccie, AlphaBetaGamma01, SoSivr, JulMian22=4, KasparBot, RedPanda25, Equinox, Jack Mullin, Entro3.14, Sdp Knu, *Castle&Gardens*, Arepunjabi, Deyaaanog, GreenC bot, Robot psychiatrist, Bender the Bot, Sephistication and Anonymous: 1030 • Aliquot sum Source: https://en.wikipedia.org/wiki/Aliquot_sum?oldid=723908245 Contributors: Dave.Dunford, Closedmouth, David Eppstein, GeoffreyT2000 and Anonymous: 3 • Perfect number Source: https://en.wikipedia.org/wiki/Perfect_number?oldid=759924781 Contributors: AxelBoldt, LC~enwiki, Bryan Derksen, The Anome, XJaM, Christian List, Toby Bartels, William Avery, Ark~enwiki, Heron, Jim McKeeth, Stevertigo, Michael Hardy, Ixfd64, Seav, Anonymous56789, Jimfbleak, Samuelsen, MyNameIsClare, Ciphergoth, Nikai, Dmoews, Rob Hooft, Schneelocke, Dys- prosia, Jitse Niesen, Hyacinth, Sabbut, Jni, PuzzletChung, Robert2957, Donarreiskoffer, Robbot, Fredrik, Yelyos, Lowellian, Gandalf61, Merovingian, Kesuari, Robinh, PrimeFan, Lzur, Giftlite, Dbenbenn, Smjg, Ævar Arnfjörð Bjarmason, Herbee, Curps, Jackol, Antan- drus, Iceager, Pmanderson, Elroch, Gscshoyru, Everlong, Mormegil, Discospinster, Guanabot, Smyth, Xezbeth, Paul August, Bender235, Kbh3rd, Pt, Bobo192, Pearle, Nsaa, Jumbuck, Logologist, Pouya, Shinjiman, Yogi de, Kdau, Suruena, Bsadowski1, Blaxthos, Oleg Alexan- drov, Nuno Tavares, OwenX, Linas, Jannex, Georgia guy, WadeSimMiser, Chinmin~enwiki, Trevor Andersen, Dysepsion, Graham87, Rjwilmsi, Matt.whitby, Staecker, R.e.b., Bubba73, Bhadani, Sango123, FlaBot, Gordon Stangler, Mathbot, Itinerant1, Intgr, Goudzovski, Glenn L, Chobot, Dylan Thurston, YurikBot, Wavelength, Dmharvey, Xihr, Stephenb, Schoen, Rsrikanth05, Lsdan, Anomalocaris, Nawlin- Wiki, Trovatore, Herve661, Ericswebber, DYLAN LENNON~enwiki, Saberwyn, Aaron Schulz, Gadget850, Ninly, Nekonobaka, Arthur Rubin, Reyk, JoanneB, Alexw~enwiki, Tropylium, Cmglee, SmackBot, RDBury, McGeddon, Jagged 85, Mscuthbert, Eskimbot, Grae- meMcRae, Johnny06man, PrimeHunter, Xaxxon, DHN-bot~enwiki, Hgrosser, Can't sleep, clown will eat me, Laslovarga, Stephlar, Wen D House, Mini-Geek, Richard L. Peterson, Gobonobo, Robofish, JoshuaZ, Jim.belk, Ph89~enwiki, Schildt.a, Waggers, Al11697, Asyndeton, Tsr21, DouglasCalvert, WAREL, Diannuc, Tawkerbot2, Gco, Valoem, Orangutan, MightyWarrior, Shamil, Vaughan Pratt, CRGreathouse, NickW557, WeggeBot, Myasuda, Doctormatt, Cydebot, Ntsimp, SpK, Thijs!bot, Barticus88, John254, Tellyaddict, G Furtado, CZeke, AntiVandalBot, Seaphoto, QuiteUnusual, AstroLynx, Mdotley, Extropian314, .anacondabot, Acroterion, Bongwarrior, JamesBWatson, Xeddy, David Eppstein, Kope, DerHexer, Peter jackson, WATARU, Tholly, Alexei Kopylov, J.delanoy, Nigholith, TomS TDotO, TELL ME that, Icseaturtles, POP JAM, Pet Bottle~enwiki, Lucky Eight, MIYAJ, Policron, Dissimul, Tambora1815, Greatestrowerever, Neil Dodgson, SoCalSuperEagle, Wikieditor06, Chikushi~enwiki, Sugakusha, Kotobakarihakirai, Goodboy Johnny, InterCommunication, Lonely Boyz, Bovineboy2008, TXiKiBoT, Ipi31415, GDonato, Rei-bot, Froggy33, Ocolon, Spinningspark, Bubblylizzie, Newbyguesses, SieBot, Coffee, Randommelon, Nihil novi, Andersmusician, Jvs, Mandsford, Dominik92, PhiEaglesfan712, Lutvokuric, Lain2252, Knodeltheory, Anchor Link Bot, S2000magician, HairyWombat, WickerGuy, Faithlessthewonderboy, ClueBot, Justin W Smith, The Thing That Should Not Be, Jagun, Boing! said Zebedee, Moto neeraj, ChandlerMapBot, Geopol, CrazyChemGuy, Watchduck, Doloco, Ludwigs2, Lartoven, Party, XLinkBot, Katsushi, Sakura Cartelet, Nicoguaro, Dsimic, Addbot, Roentgenium111, Some jerk on the Internet, Tcncv, Bkbyler, Se'taan, Download, PranksterTurtle, Roux, Doniago, 5 albert square, Legobot, Luckas-bot, HuPi, Yobot, AnomieBOT, JackieBot, Lucas Brown, Ul- ric1313, Chesscanoe, Materialscientist, Seghetti, Citation bot, Eumolpo, Nick chang207`, Desi4eva, ArthurBot, Xqbot, Cureden, Drilnoth, Gap9551, Srich32977, Klilikakis, GrouchoBot, Anhydrobiosis, Ernsts, Smallman12q, Drummer2901, Motomuku, Citation bot 1, Jone- sey95, OldManNIck, Paulpro16, Rock-et-86, Radioxid, SnakeHawk, Duoduoduo, Theoryofnumbers, The Perfection, Suffusion of Yellow, Noraft, TjBot, Pyn123, Jurvetson2, Winner 42, Wikipelli, ZéroBot, Quondum, Cymru.lass, Tolly4bolly, Eganfan, Toshio Yamaguchi, Sapphorain, Llightex, Subpopulations, Anita5192, Dweymouth, ClueBot NG, Wcherowi, JUSTINBFAN4EVER&ALWAYS, Seadog- burger, Joel B. Lewis, Widr, Supermint, Helpful Pixie Bot, Mjpam, Tsmc, Strike Eagle, Schuelke~enwiki, BG19bot, Juestr, Chmarkine, Gug01, Hehehe12, Phoenixia1177, Marblestyle, AllenZh, WOLfan112, JordanKyser22, ChrisGualtieri, Perfectdog, TheMrNoa456, Delta- hedron, JurgenNL, Deepuniverse2012, Anshuavi, Pokajanje, CsDix, Beatlesrock4, Blackbombchu, KeithPardue, Poimenlaon, 22merlin, Monkbot, Teddyktchan, XI Ki11JoY IX, GeoffreyT2000, Loraof, Jshatch75, Redditpedia, Crito10, Dollar General 1, Kameronchia1234, 9and50swans, Pascal Ochem, Retardednamingpolicy, GeneralClaudius, Mindotaur, PeterBocan, Kd7tck and Anonymous: 357 • Digital root Source: https://en.wikipedia.org/wiki/Digital_root?oldid=759789398 Contributors: Michael Hardy, Silverfish, Dcoetzee, Bkell, PrimeFan, Giftlite, Ævar Arnfjörð Bjarmason, Anton Mravcek, Sdfisher, Edcolins, Lucky 6.9, Doops, Sam Hocevar, Wroscel, SocratesJedi, ZeroOne, Bennylin, Remuel, Tauwasser, Yogi de, Skatebiker, Oleg Alexandrov, Bkkbrad, Paul Carpenter, Alex Nisnevich, Bubba73, Sango123, FlaBot, Scythe33, Glenn L, Alpt, Larry Fish, Kompik, Arthur Rubin, Messy Thinking, SigmaEpsilon, WeepingElf, SmackBot, Mwazzap, InverseHypercube, Octahedron80, Nbarth, Kcordina, Mets501, JForget, CRGreathouse, Gtalal, RDates, CZeke, JAnDbot, Magioladitis, David Eppstein, Ztobor, MartinBot, David J Wilson, Fruits Monster, The enemies of god, VolkovBot, Sami- mas, Kmhkmh, Voltron, ClueBot, Rumping, Justin W Smith, Luisluz, XLinkBot, Lalvarado3, Dthomsen8, Addbot, Aviados, Andrewrp, Xqbot, FrescoBot, Thinking of England, Petronius2, Losan33, Craphtex, ClueBot NG, Proz, Yourmomblah, Helpful Pixie Bot, JMtB03, David.moreno72, ChrisGualtieri, 1clank2, Starfries, Trilelea, DStanB, Vsrocks101, KoriganStone, Mcdonadt, Aoziwe, Anareth, Demopun, Mindotaur, Limuzi0609 and Anonymous: 60

15.10.2 Images • File:10_playing_cards.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/10_playing_cards.jpg License: CC-BY-SA- 3.0 Contributors: Own work (photo) Original artist: Enoch Lau 116 CHAPTER 15. DIGITAL ROOT

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15.10.3 Content license

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