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Mathematical Amazements and Surprises on Pi-Day [Mathematik - - -mag man eben!] o. Univ. Prof. Dr. Alfred S. Posamentier Executive Director for Internationalization and Sponsored Programs Long Island University, New York
Former Dean, School of Education Professor Emeritus of Mathematics Education The City College of The City University of New York
This Is the ultimate special -Day
3.14-15 9:26:53
3.1415926535897932384626433832... George Washington was born on February 22, 1732.
The string 02221732 occurs at position 9,039,149.
This string occurs 3 times in the first 200M digits of , counting from the first digit after the decimal point. (The 3. is not counted. )
The string and surrounding digits: 3.14159265... 478149013493242974180222173249756451826 61284136 Where the symbol came from: • In 1706 William Jones (1675 – 1749) published his book, Synopsis Palmariorum Matheseos, where he used to represent the ratio of the circumference of a circle to its diameter. • In 1736, Leonhard Euler began using to represent the ratio of the circumference of a circle to its diameter. But not until he used the symbol in 1748 in his famous book Introductio in analysin infinitorum did the use of to represent the ratio of the circumference of a circle to its diameter become widespread. Diameter of circumcircle = 1 A A
B C
E B
C B A D D C
A F A B
A B a L C 1/2 K
x K D F C E B O
J E
E D
I F
1 360 180 H G x D C 2 nn a The perimeter of the n-sided regular polygon sinxa 2 1 is then n times the side length, which makes 2 this perimeter 180 180 180 One side length =sin 2a nnsin , lim sin n nnn n Perimeter of inscribed polygon of n sides n Perimeter of circumscribed polygon of 3 2.5980762113533159402911695122588… n sides 3 5.1961524227066318805823390245176… 4 2.8284271247461900976033774484194… 4 4.0000000000000000000000000000000… 5 2.9389262614623656458435297731954… 5 3.6327126400268044294773337874031… 6 3.0000000000000000000000000000000… 6 3.4641016151377545870548926830117… 7 3.0371861738229068433303783299385… 7 3.3710223316527005103251364713988… 8 3.0614674589207181738276798722432… 8 3.3137084989847603904135097936776… 9 3.0781812899310185973968965321403… 9 3.2757321083958212521594309449915… 10 3.0901699437494742410229341718282… 10 3.2491969623290632615587141221513… 11 3.0990581252557266748255970688128… 11 3.2298914223220338542066829685944… 12 3.1058285412302491481867860514886… 12 3.2153903091734724776706439019295… 13 3.1111036357382509729337984413828… 13 3.2042122194157076473003149216291… 14 3.1152930753884016600446359029551… 14 3.1954086414620991330865590688542… 15 3.1186753622663900565261342660769… 15 3.1883484250503318788938749085512… 24 3.1326286132812381971617494694917… 24 3.1596599420975004833166349778332… 36 3.137606738915694248090313750149… 36 3.1495918869332641879926720996586… 54 3.1398207611656947410923929097419… 54 3.1451418433791039391493421086004… 72 3.140595890304191984286221559116… 72 3.1435878894128684595626030399174… 90 3.14095470322508744813956634628… 90 3.1428692542572957450362363196353… 120 3.1412337969447783132734022664935… 96 3.1427145996453682981688590937721… 180 3.1414331587110323074954161329369… 120 3.1423105883024314667236592753428… 250 3.1415099708381519785686472871987… 180 3.141911687079165437723201139551… 500 3.1415719827794756248676550789799… 250 3.1417580308448944353707690613384… 1,000 3.1415874858795633519332270354959… 500 3.1416339959448860645952957694732… 10,000 3.141592601912665692979346479289… 1,000 3.1416029890561561260413432901054…
10,000 3.1415927569440529197246707719118… 3.141592653589793238462643383279502…. To justify the formula for the area of a circle
C. Since , the base length is, therefore, r.
C ;Cr 2 ; d Base of "Parallelogram" r ; Area r r r 2 Compare the area of the circle, radius r, r to the area of the square, side 2r.
2r 22 Areasquare (2rr ) 4 4 22 1.273239 Areacircle r r 4 Therefore, 3.141594 1.273239 Legislating the value of • On 18th January 1897, a monograph by • Edward J. Goodwin was entered into the Indiana legislature as House Bill No. 246 –for the first time. • “A bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted by the official action of the legislature of 1897.” • His values could have been: = 4, 3.160494, 3.232488, 3.265306, 3.2, 3.333333, 3.265986, 2.56, and 3.555556. • Fortunately, public ridicule killed this effort! The value of in the Bible
The description of a pool (or fountain) in King Solomon’s temple is referred to in the passages: 1 Kings 7:23 and 2 Chronicles 4:2, and reads as follows:
“And he made the molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line of thirty cubits did compass it round about.”
From this the Bible has 30 3 10 The Hidden value of in the Bible
• A late 18th century Rabbi, Elijah of Vilna (1720 – 1797), was one of the great modern biblical scholars, who earned the title “Gaon of Vilna” (meaning brilliance of Vilna). • Today Vilnius is the capital of Lithuania.
• He used a technique called Gematria. Discovered in the Bible was: • Letter values: =100, = 6, and = 5. The spelling for “line measure” • In 1 Kings 7:23 = 5 + 6 + 100 = 111, • In 2 Chronicles 4:2 = 6 + 100 = 106 • Using gematria the ratio of these two values:
111 • Then: 1.0472 106 • Multiply by the “estimate” of 3: 3 1.0472 3.1416 • Correct to 4 decimal places!!! -Mnemonics
First 14 decimal places of (3.14159265358979) from the sentence: “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.” What Fibonacci contributed to our knowledge of
• In 1223 he wrote Practica geometriae, where, making use of a regular polygon of 96 sides, he computed the value of to be:
1440 3.1418181818204667768595057940195 1 458 3 A 16th Century Artist’s approximation of
• The famous German artist and mathematician, Albrecht Dürer (1471 - 1528) used an approximation for of
1 3 3.125 8 Who calculated When Number of Value found decimal place accuracy Babylonians 2000? BCE 1 3.125 = 3 + 1/8
2 Egyptians 2000? BCE 1 8 3.16045 4 9
Bible (1 Kings 7:23) 550? BCE 1 (4) 3 (3.1416)
Archimedes 250? BCE 3 3.1418
Vitruvius 15 BCE 1 3.125
Ptolemy 150 3 3.14166
Liu Hui 263 5 3.14159
Tsu Ch'ung Chi 480? 7 355 3.1415926 113
Brahmagupta 640? 1 3.162277 10
Al-Khowarizmi 800 4 3.1416
Fibonacci 1220 3 3.1418181818204667768595…
Viete 1593 9 3.1415926536
Romanus 1593 15 3.141592653589793
Van Ceulen 1615 35 3.1415926535897932384626433832795029
Newton 1665 16 3.1415926535897932 Year Mathematician Number of place accuracy Time for of calculation
1954 S. C. Nicholson & J. Jeenel 3,092 13 minutes
1954 G. E. Felton 7,480 33 hours (Generated 10,021 places but only 7,480 were correct due to machine error.)
1958 François Genuys 10,000 100 minutes
1959 Jean Guilloud 16,167 4 hours, 20 minutes
1961 Daniel Shanks[1] & John W. Wrench, Jr. 100,265 8 hours, 43 minutes
1966 M. Jean Goilloud & J. Filliatre 250,000 41 hours, 55 minutes
1967 M. Jean Goilloud & Michele Dichampt 500,000 44 hours, 45 minutes
1973 M. Jean Goilloud & Martine Bouyer 1,001,250 23 hours, 18 minutes
1981 Kazunori Miyoshi & Kazuhika Nakayama 2,000,036 137 hours, 20 minutes
1982 Yoshiaki Tamura & Yasumasa Kanada 8,388,576 6 hours, 48 minutes
1982 Yoshiaki Tamura & Yasumasa Kanada 16,777,206 Less than 30 hours
1988 Yoshiaki Tamura & Yasumasa Kanada 201,326,551 About 6 hours
1989 Gregory V. & David V. Chudnovsky 1,011,196,691 Not known
1992 Gregory V. & David V. Chudnovsky 2,260,321,336 Not known
1994 Gregory V. & David V. Chudnovsky 4,044,000,000 Not known
1995 Takahashi & Yasumasa Kanada 6,442,450,938 Not known
1997 Takahashi & Yasumasa Kanada 51,539,600,000 About 29 hours
1999 Takahashi & Yasumasa Kanada 206,158,430,000 Not known
2002 Yasumasa Kanada 1,241,100,000,000 About 600 hours
2013 Shigeru Kondo 12,100,000,000,000 90 days
2014 houkounchi 13,300,000,000,000 Formulas used for computation
For the main computation, the following formula was used: Chudnovsky Formula:
Verification was done using the following two BBP formulas: Plouffe's Formula:
Bellard's Formula: Euler - the most prolific mathematician • Euler’sand approximation “namer” –ofcorrect to 126 decimal places (and the fastest to date):
2 1 1 1 1 1 6 23452 2 2 2
1 1 1 1 1 6 1 6 2 2 2 2 2 2345 i 1 i Probability of •If your wonderment has not been stretched till now, here is one that will surely get you to “wonder.” Probability of
• French naturalist, Georges Louis Leclerc, Comte de Buffon (1707-1788) espoused the following: • Take a sheet of paper with equally spaced ruled lines, an a needle of length equal to the distance between the lines. Probability that the needle will intersect Numberthe line of line-touchingis: tosses P Number of all tosses To calculate : Toss the needle (equal in length to the space between the lines) and tally the line-touching tosses and the total number of tosses. Then put them into this formula: 2 number of all tosses number of intersection tosses In 1901, the Italian mathematician Lazzerini tried with 3,408 tosses of the 355 needle.113 He got = = 3.1415929. Why are sewer covers round?
Because they can then never fall in the hole. Which other geometric shape can also be used?
The German engineer Franz Reuleaux (1829 – 1905) developed a shape that has properties similar to a circle.
The shape that is now called a Reuleaux triangle. All opposite points are everywhere equidistant. Therefore, it can be used on a fire hydrant.
• …and it can also be used as a sewer cover! The Austrian mathematician Wilhelm Blaschke (1885-1962) proved that given any number of such figures of equal breadth, the Reuleaux triangle will always possess the smallest area, and the circle will have the greatest area. Here is another figure of constant breadth. An amazing zero sum!
• 1237892 + 5619452 + 6428642 – 2428682 – 7619432 – 3237872 = 0
• We will delete the 100-thousands place (the left-most digit) from each number: • 237892 + 619452 + 428642 – 428682 – 619432 – 237872 and the sum remains 0.
• We repeat this process by deleting the left-most digit of each number from each of the following:
• 37892 + 19452 + 28642 – 28682 – 19432 – 37872 = 0
• 7892 + 9452 + 8642 – 8682 – 9432 – 7872 = 0
• 892 + 452 + 642 – 682 – 432 – 872 = 0
• 92 + 52 + 42 – 82 – 32 – 72 = 0 This time delete the units digit from each of the numbers. Again the resulting sum each time will be zero:
1237892 + 5619452 + 6428642 – 2428682 – 7619432 – 3237872 = 0
123782 + 561942 + 642862 – 242862 – 761942 – 323782 = 0
12372+ 56192 + 64282 – 24282 – 76192 – 32372 = 0
1232+ 5612 + 6422 – 2422 – 7612 – 3232 = 0
122+ 562 + 642 – 242 – 762 – 322 = 0
12 + 52 + 62 – 22 – 72 – 32 = 0 Now combine the two types of deletions into one. Removing the right and left digit from each number.
Again we will retain zero sums!
1237892 + 5619452 + 6428642 – 2428682 – 7619432 – 3237872 = 0 23782 + 61942 + 42862 – 42862 – 61942 – 23782 = 0 372 + 192 + 282 – 282 – 192 – 372 = 0 A Most Unusual Number 666 • In Roman numerals 666 = DCLXVI
• Where all the numerals less than 1,000 are used and in descending order!
------
• D = 500, C = 100, L = 50, X = 10, V = 5, I = 1 666 is equal to the sum of the first 36 natural numbers:
1 2 3 4 5 6 7 8 9 10 11 12 30 31 32 33 34 35 36 666 . More about 666!
• 666 is the sum of the squares of the first 7 prime numbers: 2, 3, 5, 7, 11, 13, 17.
22 3 2 5 2 7 2 11 2 13 2 17 2 666
• 4 + 9 + 25 + 49 + 121 + 169 + 289 = 666 The sum of the digits of 666 (6 + 6 + 6) is equal to the sum of the digits of its prime factors.
Prime factoring: 666 = 2 · 3 · 3 · 37
Sum of the digits of the prime factors: 2 + 3 + 3 + 3 + 7 = 6 + 6 + 6.
666 is equal to the sum of two consecutive palindromic prime numbers, 313 + 353 = 666 A strange occurrence of the number 666 is when we insert + signs into the sequence of numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9
Here are two ways to do it:
1 + 2 + 3 + 4 + 567 + 89 = 666
or 123 + 456 + 78 + 9 = 666 Now consider the reverse sequence: 9, 8, 7, 6, 5, 4, 3, 2, 1
We can create 666 as follows:
9 + 87 + 6 + 543 + 21 = 666 The sum of the first 144 decimal places of equals 666.
3.14159265358979323846264338327950288419716 9399375105820974944592307816406286208998 6280348253421170679821480865132823066470938 44609550582231725359 The Fabulous Fibonacci Numbers
Leonardo Pisano Short Biography
• Latin “filius Bonacci,” or a son of Bonacci, but it may more likely derive from “de filiis Bonacci,” or family of Bonacci. • Born: Pisa 1175 – 1250 • Wrote several books. Most noteworthy: “Liber Abaci” or “Book of Calculations” • First use of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 • Chapter 12: “The Rabbit Problem” “The nine Indian figures are: Fibonacci used the term Indian figures for the Hindu numerals. 9 8 7 6 5 4 3 2 1. (It is assumed that Fibonacci wrote the numerals in order from right to left, since he took them from the Arabs who write in this direction.) With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written, as demonstrated below. A number is a sum of units, and through the addition of them the number increases by steps without end. First one composes those numbers, which are from one to ten. Second, from the tens are made those numbers, which are from ten up to one hundred. Third, from the hundreds are made those numbers, which are from one hundred up to one thousand. … and thus by an unending sequence of steps, any number whatsoever is constructed by joining the preceding numbers. The first place in the writing of the numbers is at the right. The second follows the first to the left.” Regeneration of Rabbits from Fibonacci’s posed problem
Red=Baby Rabbits The Fibonacci numbers as a recursive sequence
1 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13 8 + 13 = 21 13 + 21 = 34 21 + 34 = 55 34 + 55 = 89 55 + 89 = 144 89 + 144 = 233 144 + 233 = 377 233 + 377 = 610 377 + 610 = 987 610 + 987 = 1597 ... The Fibonacci numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,… The Bracts of a Pineapple
Notice the Fibonacci numbers 5, 8, and 13. Spiral arrangement of the bracts of a pine cone – also exhibit the Fibonacci numbers What is the sum of the (initial) consecutive Fibonacci numbers? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
• The sum of the first 4 numbers = 7 = 8-1 • The sum of the first 5 numbers = 12 = 13-1 • The sum of the first 6 numbers = 20 = 21-1 • The sum of the first 7 numbers = 33 = 34-1 • The sum of the first 8 numbers = 54 = 55-1 • The sum of the first 9 numbers = 88 = 89-1 • The sum of the first 10 numbers =143 =144-1
What might we now conclude? The sum of the first n consecutive Fibonacci numbers equals the (n+2)nd – 1
Since FFFFFFn n1 n 2 , we get: n n 2 n 1 By substituting increasing values for n , we get:
FFF1 3 2
FFF2 4 3
FFF3 5 4
FFF4 6 5
FFFn11 n n
FFFn n21 n
Sum of the first n Fibonacci numbers Fnn2 F 2 F 2 1.
Therefore, FFFFFF1 2 3 4 nn 2 1 Take any four consecutive numbers in the Fibonacci sequence: 3, 5, 8, 13
Find the difference of the squares of the middle two numbers: 822 5 64 25 39
Then find the product of the outer two numbers: 3 13 39 Notice: they are equal!
In general, we then have: 22 FFFFn1 n n 1 n 2 The number 666 is also related to the Fibonacci Numbers.
th Fn is the n Fibonacci number:
FFFF1 9 11 15 1 34 89 610 666 and when you inspect the subscripts, you get: 191115 18 6 6 6
Similarly, for the cubes of some Fibonacci numbers:
33333 FFFFF1 2 4 5 6 1 1 27 125 512 666 and now the subscripts give us the following sum: 1245618 666. Pythagorean Triples
a2 +b2 = c2
32 4 2 5 2 52 12 2 13 2 112 60 2 61 2 392 80 2 89 2 A Novel Method for Generating Pythagorean Triples 1 2 3 4 5 6 7 n 1,2,3,4,5,6,7 ,, n 3 5 7 9 11 13 15 21n Convert each of the fractions in this sequence to an improper fraction to get the following sequence: 4 12 24 40 60 84 112n (2 n 1) n 2 n ( n 1) ,,,,,,,, 35 7 9111315 21nn 21
The fractions (when reduced to lowest terms) will produce the first two members of a Pythagorean triple: 3,4,5 – 5,12,13 – 7,24,25, -- 9,40,41 – 11,60,61 – 13,84,85 –15,112,113.
In each case, we use the Pythagorean theorem to get the hypotenuse TO MAKE A PYTHAGOREAN TRIPLE FROM THE FIBONACCI NUMBERS
Take any four consecutive Fibonacci numbers: 3, 5, 8, and 13. Follow these rules: 1. Multiply the middle two numbers and double the result: The product of 5 and 8 is 40 then we double this to get 80 2. Multiply the two outer numbers: Here the product of 3 and 13 is 39
3. Add the squares of the inner two numbers:
22 Here 5 8 25 64 89
We created a Pythagorean triple: 39, 80, 89.
2 2 2 39 80 1521 6400 7921 89 . Therefore 392 + 802 = 892 A Famous Numerical Loop
• You are asked to follow two rules as you select any natural number: •If the number is odd then multiply by 3 and add 1. •If the number is even then divide by 2.
• Regardless of the number you select, after continued repetition of the process, you will always end up with the number 1. Let’s try it for the arbitrarily selected number 7.
7 is odd, therefore, multiply by 3 and add 1 to get: 7 3 + 1 = 22 22 is even, so we simply divide by 2 to get 11 11 is odd, so we multiply by 3 and add 1 to get 34. 34 is even, so we divide by 2 to get 17. 17 is odd, so we multiply by 3 and add 1 to get 52. 52 is even, so we divide by 2 to get 26. 26 is even, so we divide by 2 to get 13. 13 is odd, so we multiply by 3 and add 1 to get 40. 40 is even, so we divide by 2 to get 20. 20 is even, therefore, divide by 2 to get 10. 10 is even, therefore, divide by 2 to get 5. 5 is odd, so we multiply by 3 and add 1 to get 16. 16 is even, so we divide by 2 to get 8. 8 is even, so we divide by 2 to get 4. 4 is even, so we divide by 2 to get 2. 2 is also even, so we again divide by 2 to get 1. A word of caution!
• When you begin with 9, •it will take 19 steps
• When you begin with 41, •it will take 109 Steps Getting into an Endless Loop
• Choose a 4-digit number (not one with all four digits the same). • Rearrange the digits to make the biggest and smallest number. • Subtract the two numbers. • With this new number, continue this process. • Soon you will get 6,174. • But keep going! • What do you notice? We will (randomly) select the number 3,203 • The largest number formed with these digits is: 3320. • The smallest number formed with these digits is: 0233. • The difference is: 3087. • The largest number formed with these digits is: 8730. • The smallest number formed with these digits is: 0378. • The difference is: 8352. • The largest number formed with these digits is: 8532. • The smallest number formed with these digits is: 2358. • The difference is: 6174. • The largest number formed with these digits is: 7641. • The smallest number formed with these digits is: 1467. • The difference is: 6174.
• And so the loop is formed, since you keep on getting 6174 if you continue Another loop!
• Get the sum of the squares of the digits of any number:
• Let us use the number 5. • Keep on taking the sum of the squares of the digits.
52 = 25, 22 + 52 = 29, 22 + 92 = 85, 82 + 52 = 89,
82 + 92 = 145, 12 + 42 + 52 = 42, 42 + 22 = 20, 22 + 02 = 4, 42 = 16,
12 + 62 = 37, 32 + 72 = 58, 52 + 82 = 89, ... Sum of the cubes of the digits of any number
• Begin with the (arbitrarily selected) number 352 and find the sum of the cubes of the digits: 33 5 3 2 3 27 125 8 160
• Now we use this sum, 160, and repeat the process: • The sum of the cubes of the digits of 160 is: • 1333 6 0 1 216 0 217
• Again repeat the process with 217: 23 1 3 7 3 8 1 343 352 • This is the same number (352) we started with. Mathematics can be entertaining.
• In mathematics, the peculiarities are amazing!
• Most concepts are interconnected; and the fun is to discover these connections.
• There are endlessly many •amazements and surprises awaiting you! Some suggested sources from: www.Prometheus Books.com I hope you enjoyed the entertaining side of mathematics!
•Any questions? •Please ask now!
• Or contact me at: • Dr. Alfred Posamentier • Long Island University, New York • 1 University Plaza • Brooklyn, New York 11201 • Mobile: +1 917-837-1400 • [email protected] • [email protected]