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Home , Babylonian mathematics, Ishango bone, Lebombo Bone, Numeral system, Positional notation, Radix

Before the c Ken W. Smith, 2012

Last modified on February 15, 2012

Contents

1 Mathematics Before the Greeks 2 1.1 Basic systems ...... 2 1.1.1 (primitive counting) ...... 2 1.1.2 Simple grouping ...... 2 1.1.3 Complex grouping and multiplicative systems ...... 3 1.1.4 Ciphered systems ...... 3 1.1.5 Positional ...... 4 1.1.6 notation ...... 6 1.1.7 References for early counting systems ...... 9 1.2 ...... 10 1.2.1 Babylonian computations ...... 10 1.2.2 Compound interest ...... 11 1.2.3 Quadratic ...... 11 1.2.4 Rhetorical ...... 14 1.2.5 Plimpton 322 and Pythagorean triples ...... 14 1.2.6 Summary and references for ancient Babylonian mathematics ...... 16 1.3 Egyptian Mathematics ...... 17 1.3.1 The Egyptian dynasties ...... 17 1.3.2 Egyptian unit ...... 17 1.3.3 by adding – Egyptian doubling ...... 19 1.3.4 The Method of False Position ...... 20 1.3.5 The circumference and diameter of a circle ...... 21 1.3.6 References for ancient Egyptian mathematics ...... 22

1 1 Mathematics Before the Greeks 1.1 Basic counting systems 1.1.1 Tally marks (primitive counting) The need to count objects – possessions, sheep, friends, enemies, days – has existed from the dawn of human history. Deep in the past, humans used simple tally marks to enumerate their possessions – and for simple problems, we still do this today, often counting on fingers or making marks on paper. Primitive societies had various levels of complexity in counting. At the most primitive level there is simple counting, with stones, rocks, fingers/toes, in a one-to-one fashion. Each rock, each mark, was related to a single object. In Paleolithic marks or notches were made on bones to record . Presumably these were part of a contract or a recording of property (animals, for example.) See the Wikipedia article on tally sticks. Examples of these “tally sticks” include the and the , probably twenty to thirty thousand years old. (The textbook by Burton has a photo of an ancient wolfbone on page 5.)

1.1.2 Simple grouping After simple tally marks, as the numbers grow larger, we begin to introduce symbols for certain groupings of the marks. In typical elementary school counting, we four tally marks (||||) with a slash (6 ||||) to indicate the five. A different symbol (a larger rock?) might mean five or ten of the smaller ones. This is “simple grouping”, with a few different symbols meaning different sizes or units. The Egyptian hieroglyphic system was a grouping system base ten; there were symbols for ten, for a hundred, for a thousand and so on. The Roman is a grouping system, with V, X, L, C, D and M standing for five, ten, fifty, one hundred, five hundred and a thousand. It is a subtractive system in that one could simplify the system by subtracting instead of adding; IV meant V less I, 5-1=4. The simple grouping concepts persists today in our cash system. We have one, five, ten and twenty dollar bills in our billfold. The ordering of the bills in our billfold is not important....

The advantage of simple grouping is that it is very easy to interpret and it (generally) requires no ordering of the symbols. In the most simple counting of the Egyptian system, ||| ∩ %% stood for 213, as did %% ∩ |||. There were customs in different cultures, however, as to how one tended to write the symbols. The Egyptians, for example, tended to put the most significant symbols on the right. For example, they would have written 213 as ||| ∩ %%.

The Egyptian symbols and system The Egyptian notation from 3200 B.C.E. (?) is a nice example of a simple grouping system. Like many civilizations, the Egyptians used a simple mark for “one” and multiple copies of the same mark for the smaller numbers. But they also used a hobble or heel bone for ten and a coil of for a hundred. (I will attempt to imitate these symbols with the set intersection ∩ and Greek “rho”, %.)

The Egyptians used a lotus flower to represent a thousand and had additional symbols for higher powers of ten. (I will simplify the lotus flower with the symbol ↓. I have copied the symbols, above, with permission, from the webpage http://www.eyelid.co.uk/.) The Egyptians used a curved finger for ten thousand, a fish (or frog?) symbol for one hundred thousand and the image of a astonished man (as in “I just won the lottery!”) for one million. I will use the symbol h for the curved finger and α for the fish/frog. When I need to use the excited man symbol, I’ll give up and just write “excitedman” ( ). See the following webpages for more on, Egyptian mathematics:

2 1. Don Allen’s webpages on Egyptian mathematics,

2. the St. Andrews page on Egyptian mathematics 3. another UK webpage, “eyelid”, on Egyptian mathematics, 4. a math history webpage at the University of Wichita

The Roman numeral system The Romans, like most societies, used a simple dash, I, for “one.” They used the letter V to represent five and the letter X to represent ten. Other groupings were L, C, D, M for fifty, a hundred, five hundred and a thousand. They put a bar over a letter to represent multiplication by one thousand so XII meant 12, 000. Since a simple grouping system does not rely on position, the Romans took advantage of that by writing the most significant letters on the left and then use a deviation from that standard to represent subtraction. This was done only in a few, very limited cases. The notation VI meant ”five + one” while IV meant ”five − one = 4.” This allowed simpler for 4 (IV), 9 (IX), 40, (XL), 90 (XC) and larger numbers like 900 (CM) and 1900 (MCM). Medieval used a version of until the fifteenth century when the needs of commerce (business accounting) forced a transition to the Hindu- positional system.

1.1.3 Complex grouping and multiplicative systems A multiplicative group system is a system where some type of marker or symbol would be used to change the meaning of a mark, multiplying it by a power of ten or some other larger number. For example, while |||| by itself might mean the number 4, the notation ||||ten or ||||t might mean 40. In a another version of a multiplicative system, a single symbol, say ‘4‘, might mean four but a multiplier like ten or hundred would change it into 40 or 400. Imagine, in our modern number system, if 4t mean 4 × ten and 4h meant 4 × one hundred. So 4h5t8 would be equivalent to our modern 458. The multipliers, not the positions, are what matters, so 85t4h would also mean the number 458. An English phrase like “four and twenty” is a modern example. Here “twenty” is a contraction of “two-tens” and so “four and two-tens” is the same as “two tens and four.” The traditional Chinese-Japanese numeral system was apparently a multiplicative number system. According to Eves (pages 17-18), a number like 5625 would have been written vertically on a bamboo strip with seven Chinese symbols representing

5 thousand 6 hundred 2 ten 5

1.1.4 Ciphered systems A ciphered numeral system relies on a particular base and then has symbols for each major numeral and step in the base. If the system were base ten, then there would be symbols for 1, 2, 3, 4, ..., 9, 10, 20, 30,40, ..., 90, 100, 200, 300, 400,...

3 The Egyptians, in addition to the simple group hieroglyphic system developed a suitable for writing on papyrus. This system (analogous to our cursive writing) is called the “hieratic” system. During the development of this system, the Egyptians developed a shorter, more compact ciphered system. In that system they had symbols for 2, 3, 4, ..., 9 and then symbols for 10, 20 and so on. (These are described on page 15 in Burton’s textbook.)

The Greeks used their for a ciphered system, α = 1, β = 2, γ = 3, ... and so on, but then ι, κλ (iota, kappa, lambda) meant 10, 20, 30.... The juxtaposition ακ (alpha-kappa) would mean 21. According to Burton (p. 16), the Greek alphabet used for numerals were:

α, β, γ, δ, , ζ (vau or stigma) , ζ, µ, θ which stood for the numbers 1, 2, ..., 9 and

ι, κ, λ, µ, ν, ξ, o, π, φ (koppa) which stoof for 10, 20, ... 90 and ρ, σ, τ, υ, φ, χ, ω, 6 λ (sampi) which stood for 100, 200, ... ,900. (These numerals are also described online as the Greek numeral system in this St. Andrews webpage.) This equating of letters with numerals means that each word has a number associated with it! Just turn each symbol into the associated number and add them up. So each name could also be viewed as number. (For example, if we used a ciphered system then my name, KEN, might stand for the number 20+5+50 = 75.) This probably explains the number 666 in the book of Revelations in the New Testament. It also explains grafitti found in Pompei: “I am in love with her whose number is 545.”

1.1.5 Our modern Hindu-Arabic number system is a positional numeral system. There are ten symbols (0,1,2,3,4,5,6,7,8,9) and a number has a different meaning depending on the position of the number. In our notatin, the rightmost position for an means it is multiplied by 1; move one position to the left and the number is multiplied by 10. And so on. So 458 is very different from 485 or 854. The modern view sees the positional numeral system as the easiest to use and the most natural, most elegant number system. We do not have to use base ten in the positional number system. We could use a different base, say 2 or 8 or 12, so that each shift to the left multiplies by one more power of this base. In base 2, 110 represents 4+2+0, while in base 5, 110 represents 25+5+0. The Babylonians had a positional base sixty system. Not all number systems were base ten. Bases five, twenty and sixty occurred. There were even some combinations which used 360.

The Babylonian symbols The Babylonians wrote with a stylus in clay. Their mark for “one” was simply a sharp triangle ( ) or a triangle on top of stick ( ) created by pressing the tip of the stylus into the clay. They used a sidewaysL wedge, ^ or , for groupings of ten mark (See the St. Andrews site on Babylonian mathematics for details on the Babylonian numerals.) For example, the Babylonians would write the number 8 with a stack of 8 sharp triangles:

and write a number like 29 with a stack of 9 sharp triangles followed by 2 sideways wedges:

4 The Babylonian for the digits 1 through 59 are given below:

(These symbols are from this webpage on Babylonian mathematics.)

But the Babylonians also had a positional system on the top of these clay markings. This allowed them to restrict their numbers to just these two symbols! Their system was a positional system base 60, a “” system. For example, the number

1835087 = 8(603) + 29(602) + 44(60) + 47 would, in sexigesimal positional notation, be written in four digits

8, 29, 44, 47.

The expression for this number would be

The Mayan numeral system The Mayans used dots (pebbles?) for single numbers and then a line for the number 5. But unlike other ancient societies, they had a symbol for zero! Here are the Mayan symbols for the numbers up to 19.

The Mayans also had a positional number system on top of this, one which was base twenty (“vigesi- mal”.) Thus 3 5 2

5 would represent 2 + 5(20) + 3(400) which in our modern notation is 1302. In the Mayan calendar, the Mayans seemed to have made a slight, subtle change in base! The first shift over (or shift up) multiplied by 20. Then the second shift over (up) multiplied by 20(18) = 360 instead of 400. So 3 5 2 would represent 2 + 5(20) + 3(360) = 1182. This second variation is the “calendric” calculation system. (See, for example, this article on the Mayan calendar.) Of course the Mayans did not use our symbols, but dots and dashes. So they would write, instead of 3 / 5 / 2, the following symbols (written vertically, with the most significant at the top)

There is a subtlety to positional notation. One needs a symbol for nothing! Most simple cultures would not have naturally developed a symbol for nothing since there is no need to count nothing! But a positional notation needs a zero just to represent an empty position. There is a natural evolution from simple tallymarks through simple grouping to ciphered systems and positional notation. Each step is represents by some culture or ancient society. The positional notation is the easiest to use and most applicable, allowing representation of large numbers and also easy representation of noninteger .

(Sam Houston professor Dustin Jone’s has youtube videos that cover these basic enumeration systems.)

1.1.6 Radix notation Most positional number systems have a fixed base. We will assume for ease of reading that the most significant digits are to the left. (This is merely a cultural choice.) Given a base b (b a positive integer greater than 1), we represent the number

n n−1 n−2 4 3 2 (dn · b ) + (dn−1 · b ) + (dn−2 · b ) + ... + (d4 · b ) + (d3 · b ) + (d2 · b ) + (d1 · b) + d0 (1) with the string of digits dndn−1...d4d3d2d1d0. For example, in base ten, 41975 represents 4 · 104 + 1 · 103 + 9 · 102 + 7 · 10 + 3. In base twelve, 41975 represents 4 · 124 + 1 · 123 + 9 · 122 + 7 · 12 + 3 – which is a very different number. Let us call the form in (1) the explicit form of the number dndn−1...d4d3d2d1d0. The explicit form of the number lets us see the calculation which the positional notation represents; thus the explicit form of 41975 in base ten is 4 · 104 + 1 · 103 + 9 · 102 + 7 · 10 + 3.

We can also represent positive numbers smaller than unity (one) using positional notation. We create a symbol (a point “.” or a mark like a semi-colon “;” or even a “,”) and numbers to the right of that mark involve decreasing powers of the base. In base ten, the explicit form of the number 3.14159 is

3 + (1 · 10−1) + (4 · 10−2) + (1 · 10−3) + (5 · 10−4) + (9 · 10−5).

In a sexagesimal system, we will use a semicolon to set up the negative powers of the base and put between the sexagesimal digits. Thus the number 3; 8, 30 should be interpreted (in explicit form) as 8 30 3 + (8 · 60−1) + (30 · 60−2) = 3 + + . 60 3600

6 and the number 2, 24; 8, 30 is

(2 · 60) + 24 + (8 · 60−1) + (30 · 60−2).

We probably shouldn’t speak of 2, 24; 8, 30 as “” notation since the word “decimal” implies base ten. It is common instead to speak of this as “radix notation.” (The word “radix” is intended to imply a fixed “root” or “base.”)

Radix notation and In any base we can turn a into a number in radix notation using long division.

Example 1. Let us write 2/7 in radix notation base 12. We do long division. (In writing numbers with bases slightly larger than ten, such as base twelve or base sixteen, it is customary to use A for the digit representing ten, B for eleven and so on.) Below is the work one might do by hand:

The explanation for this work follows. Replace 2.0 by the number 20, as if we were sliding the point to the right. Since we are using base 12, 20 is the number 2 dozen, that is, 20 = twenty-four. The number 7 goes into 20 three times with a remainder of 3, since two dozen is 3 copies of 7 (twenty-one) plus 3. Then 7 goes into 30 (three dozen) five times with a remainder of 1. 7 goes into 10 (one dozen) one with a remainder of 5. 7 goes into 50 eight times with a remainder of 4. 7 goes into 40 six times with a remainder of 6. 7 goes into 60 (six dozen) ten times (ten = A) with a remainder of 2. 2 Now the long division begins to repeat. So we can see that the radix notation, base 12, for 7 is 2 = 0.35186A 35186A 35186A 35186A... 7 We may simplify notation for a by placing a bar over the repeating block. Thus

7 the radix notation for 2/7 base twelve is 2 = 0.35186A. 7

Example 2. Let’s do one more example. We will write 2/7 in the sexagesimal notation (base 60.) Since 7 is less than 2, we insert a decimal point (or separator “;”) and divide 7 into 2060. Since 20 in sexagesimal notation represents one hundred and twenty, the number 7 will go in to 2060 seventeen times with a remainder of 1. So our first step yields 0; 17 with a remainder of 1 = 1;0. Move the “decimal” point in 1;0 to the right and divide 7 into 1060. This gives 8 with a remainder of 4 since 1060 = 7 · 8 + 4. Take the remainder 4;0, move the “decimal” point to the right again and divide 7 into 4060. It goes in 34 times with a remainder of 2. At this point our lattice notation for long division our work would look like this:

0; 17, 8, 34 7) 2; 0, 0, 0 1, 59, 1, 0, 0, 56, 4, 0 3, 58 2

Now, if we continue, we will once again divide 7 into 2. So our process begins to repeat. Thus the fraction 2/7 in sexagesimal notation is 2 = 0; 17, 8, 34. 7

Rational numbers N A is a number which can be written in the form where both the numerator N D and the denominator D are . A rational number, when written in radix form, must eventually repeat. (Think about this – our long division shows why that has to happen!) A rational number which, in radix notation, ends with a string of repeating zeroes is said to “termi- nate.” In base ten, 6/25 = 0.240000... while 2/7 = 0.385714 385714 385714 385714 ... so 6/25 terminates while 2/7 does not. Given a base, can we tell which numbers will terminate?

Regular numbers A fraction is regular (in a particular base) if its radix expansion in that base terminates. For example, 2 in base ten, 6/25 is regular since 6/25 = 0.240000... but 2/7 is not regular since 7 = 0.35186A 35186A.... N If a fraction D is regular then we must be able to write it as a finite sum of fractions with denominators which are powers of the base b. For example, 6 2 24 = 0.24 = + . 25 10 102 How can we tell if a fraction is regular? We must be able to write it with a denominator D which is some power of the base b. This means that the denominator (when the fraction is in reduced form) must have its only from the list of primes dividing the base. For example, in base ten, 6 the primes dividing the denominator of a regular fraction can only be 2 and 5. So 25 is regular since the

8 2 primes dividing 25 divide the base 10. The number 7 is not regular in base ten since 7 does not divide 10. There is no way to multiply by powers of ten and get rid of the denominator. 5 The number 18 is regular base 60 since the primes dividing 18 (2 and 3) also divide 60. This means 5 that we can rewrite 18 so that the denominator is a power of sixty. In this case, since 18 = 2 · 3 · 3, let’s multiply 18 by 10 · 20 to get 602. Then 5 5 · 200 1000 = = . 18 18 · 200 602 5 Since one thousand is equal, base sixty, to 16, 40, the fraction 18 is equal to 0; 16, 40.

85 For which bases is the fraction regular? Since 108 = (22)(33) then as long as the base b is divisible 108 by both 2 and 3 this fraction will be regular. For example in base twelve, since 12 = (22)(3), this fraction will terminate after three digits (since 108 divides 123.) To find the base 12 radix representation of our fraction we could do long division (as above) or we could just multiply both numerator and denominator by 16 so that the denominator is equal 123 : 85 85 · 16 1360 = = . 108 108 · 16 123 85 Since 1360 = 9(122) + 5(12) + 4 = 954 then the radix notation for ( ) in base 12 is 0.954 . 12 108 10 12 Similarly this fraction is regular in base sixty. But it repeats in base ten since ten is not divisible by the prime 3 dividing the denominator 108. In base ten there is a repeating block 703 703 703.... 85 = 0.78703. 108

???

Now that we understand bases and positional notation, we are ready for the t-shirt humor worn by mathematicians and scientists: “There are 10 types of people in the world – those who can count in binary and those who can‘t!”

1.1.7 References for early counting systems This material follows 1. Chapter 1 of Eves, 2. Sections 1.1 to 1.3 of Kline,

3. Chapter 1 of Burton. (It is also included in sections 2.1 to 2.3, pp. 19-25 of The Rainbow of Mathematics by Ivor Grattan- Guiness and Part 2 (chapters 7 to 14) and Part 3, chapters 15 & 16 of From Five Fingers to Infinity edited by Frank Swetz.)

Exercises & Worksheets. Do Worksheets 1 and 2 on Primitive Counting.

9 1.2 Babylonian Mathematics The Babylonian civilization reached its height around 2000 BC, some four thousand years ago. (They played an important role throughout the Hebrew Old Testament: Abraham left the Sumerian region around 1900 BC and the Jewish people are later exiled to around 587 BC.) The Babylonian civilization had a significant impact on later Greek civilizations. The nation of Iraq roughly approximates the ancient Babylonian empire situated along the Tigris and rivers.

1.2.1 Babylonian computations The Babylonians wrote on wet clay using a stick with a wedge shape at the end. (The word “cuneiform” describes this wedge shape.) Because of the durability of the dried clay records, we have a fair number of ancient tablets from Babylon and so we know quite a about their mathematics. Many of their mathematical tablets were translated by the German mathematician Otto Neubauer early in the twentieth century. The Babylonian mathematical tables often contained a of problems with solutions. The work appeared to involve practice with certain . There is very little mathematical theory which survives but they did have methods for solving quadratic equations, for doing compound interest, for fractions and computing areas and volumes. The Babylonians knew the area of some simple polygons and had approximations for the volumes of objects like the . There is some evidence that they may have had some rudimentary knowledge of geometric series.

As discussed earlier, the Babylonians used a sophisticated base sixty (sexagesimal) positional notation. Their astrologers divided the circle of the heavens into 360 parts, corresponding to 360 days in the year; we still have this measurement preserved in our measurement for . They subdivided the degree into 60 and divided minutes into 60 .

The Greeks later continued the use of the Babylonian sexagesimals and we still use their mathematics with both our angles and our time measurements. The subdivision of the into minutes and seconds dates to the Babylonians 4000 years ago!

The Babylonians could compute the “decimal” (sexagesimal) form of fractions, not just integers. They were comfortable with multiplication and division and computed a table of reciprocals of integers for ease in division. They apparently knew how to compute the length of the hypotenuse of a right triangle and had a good approximation for the of two. Here,√ below, is a photgraph of a clay√ tablet where a Babylonian has constructed a hypotenuse of length 2 and approximated the value of 2 in sexagesimal notation.

(The original tablet is in the Yale Babylonian Collection; this photograph was taken by Bill Casselman at the University of British Columbia. It is part of the Wikipedia site on Babylonian mathematics.)

10 1.2.2 Compound interest The Babylonians approximated compound interest using a method we now call linear interpolation. As an example, we will do a problem (Eves, p. 57, problem 2.2) from the Louvre tablet: find the doubling time for interest which is compounded annually at 20%. After n years, the amount of money in the account will be f(n) = (1+20%)n = (1.2)n. Computing (by hand) we see that f(3) = (1.2)3 = 1.728 while f(4) = (1.2)4 = 2.0736. So the doubling time is a little less than four years. To get a better value for the correct answer, we apply linear interpolation between those two values, that is, we pretend that the function y = f(n) is a line (it is not!) and find the point (n, 2) on the line connection (3, 1.728) with (4, 2.0736). Since the slope of this line is 2.0736 − 1.728 = 0.3456 we see that 2.0736 − 2 = 0.3456 4 − n and so 0.0736 = 0.3456(4 − n) 0.0736 23 4 − n = = 0.3456 108 and so 23 85 n = 4 − = 3 ≈ 3.78703. 108 108 The Babylonians did this all in sexagesimal notation obtaining the sexagesimal form for this number:

3; 47, 13, 20. ln 2 (For comparison, in modern notation, the true doubling time for 20 percent interest is ≈ 3.802 ln 1.2 years.)

1.2.3 Quadratic equations The modern public view of mathematics focuses on mechanical manipulations of symbols, what we now call algebra. But our algebra is a relatively modern concept. Ancient societies (and even European soci- eties in the middle ages and Renaissance times) reasoned from a geometric, visual viewpoint. Arguments about squaring a term really involved a literal square! A cubic involved cubes; the concept of a was generally unknown since it had no physical meaning! This visual geometric viewpoint to mathematics is a major theme is this course! Modern students and teachers generally miss this concept and greatly underestimate the role of geometric reasoning in mathematics up until the seventeenth century!

To compare the ancient and modern view, let us solve a . Suppose we are given constants a and b and need to solve the equation

x2 + bx = a. (2)

b2 The modern algebraic approach is to “complete the square.” We add the quantity to both sides and 4 then write the left-hand side as a perfect square:

b2 b2 x2 + bx + = a + 4 4 b b2 (x + )2 = a + 2 4

11 Now taking square roots of both sides, we obtain r b b2 x + = ± a + 2 4 and so r b b2 x = − ± a + (3) 2 4 This is often rewritten slightly and encoded as the “quadratic formula” for algebra students to memorize; the quadratic formula is merely the final result of completing the square.

But prior to the algebra of the seventeenth century, one was more likely to truly complete a (literal) square. One might assume that both a and b are positive values, so that a is the area of some figure and b b was a certain length. Thus we might consider the square with sides of length x + 2 :

The rest of the work involves the following steps.

12 (This image is part of the Wikimedia Commons and available under the GNU Free Documentation License. It was copied from the Wikipedia page on completing the square.) Note that the final answer would not involve the ± ; it is assumed that x is positive since x is a physical length. Furthermore, the work would be quite different if a or b was negative. In general, negative values were seen as meaningless, so that if, in modern terms, a was negative, one would not consider the equation x2 + bx = a but would instead consider the equation x2 + bx + (−a) = 0 so that the equation uses the positive term (−a).

The figure

suggests a way to approximate square roots. The Babylonians approximated the square root of x2 + bx b by x + 2 . After a change of symbols, we see that this is equivalent to approximating the x x 2 x2 a + x by a + 2a . This is reasonable, algebraically, since the square of a + 2a is a + x + 4a2 . If x is small and a large then the last term is very small and so can be ignored in an approximation. All we lose by this approximation is the area of the very small square – the blue one – in the figure above. (This approximation is equivalent, in modern mathematics, to the first two terms in the binomial theorem’s 2 1/2 expansion of (a + x) or in the first two terms√ of the Taylor series for the square root√ function.) 2 1 Thus the Babylonians would approximate 5 by choosing a = 4 and so estimate 5 ≈ 2 + 4 .

An ancient Babylonian problem (from Burton, p. 72, problem 9) says, essentially, “7 times the side of a square plus 11 times the area is equal to 6; 15. Find the side of the square.” In modern notation we seek x in the quadratic equation 25 11x2 + 7x = . 4 The Babylonians would have recognized(geometrically) that 7 7 7 (x + )2 = x2 + x + ( )2 22 11 22 and so after multiplying by 112, we have

7 72 (11x + )2 = 11(11x2 + 7x) + ( ) 2 4 25 Substituting for 11x2 + 7x implies that 4 7 72 25 49 324 (11x + )2 = 11(11x2 + 7x) + ( ) = 11( ) + = = 81. 2 4 4 4 4 7 Therefore (11x + )2 is the square root of 81. From this we see that 2 7 11x + = 9 2 which implies that 18 − 7 11 11x = = 2 2

13 and so 1 x = . 2 The answer they gave was, of course, in sexagesimals:

0; 30.

1.2.4 Rhetorical algebra The Babylonians did have some tools. All of their algebraic manipulation was written out; there were no symbols. This is an early example of rhetorical algebra. In this course we will follow the evolution of algebra through three stages. The first stage, rhetorical algebra, is simply algebra written in words. Today we solve 3x2 + 7 = 19 by writing 1 3x2 + 7 = 19 =⇒ 3x2 = 19 − 7 = 12 =⇒ x2 = ( )12 = 4 =⇒ x = ±2. 3 Using rhetorical algebra this would instead be written out:

Let us solve for an unknown quantity if 7 more than 3 times the square of the quantity is 19. Here is the solution: To solve for the unknown quantity, subtract 7 from the 19 and divide that answer by 3 to obtain 4.

Since the square of 2 is 4, the unknown quantity is 2.

Later, in India, the middle east and Europe, people would develop syncopated algebra. Syncopated algebra is similar to rhetorical algebra but it has abbreviations for the words and operations. Using the example, above, syncopated algebra might look like this:

Solve for quan. if 7 mo. 3 tim. sq. quan. equ. 19.

Here is the solution: 7 mo. 3 tim. sq. quan. equal. 19 implies 3 tim sq. quan. equ. 19 less 7 equ 12.

3 sq. quan. equ. 12 implies sq. quan. equ. 4 so quan. equ. 2.

Our modern algebra, with symbols for unknowns and for operations and for is synthetic algebra.

1.2.5 Plimpton 322 and Pythagorean triples The Babylonians knew the relationship between the legs and hypotenuse of a right triangle. We now call this the “” but the Babylonians knew this over a thousand years before the life of ! Information about Babylonian mathematics has come from numerous clay tablets. The most mathe- matically significant tablet is the Plimpton 322 tablet. The Plimpton 322 tablet (Eves, p. 46, see also Wikipedia) supposedly provides Pythagorean triples which measure angles one degree at a time. One might think of the Plimpton 322 tablet as a trigonometric table, showing how one might draw angles with degrees between 30 degrees and 45 degrees. (The tablet, and possible interpretations of it, is discussed in depth in Math Monthly article by Robson, 2002, available online.)

14 (This photo is taken from the Wikipedia article on the Plimpton tablet and part of the public domain.)

A translation of this table (again from Wikipedia) is as follows:

Table 1: Plimpton 322, from Wikipedia

(1:)59:00:15 1:59 2:49 1 (1:)56:56:58:14:50:06:15 56:07 1:20:25 2 (1:)55:07:41:15:33:45 1:16:41 1:50:49 3 (1:)53:10:29:32:52:16 3:31:49 5:09:01 4 (1:)48:54:01:40 1:05 1:37 5 (1:)47:06:41:40 5:19 8:01 6 (1:)43:11:56:28:26:40 38:11 59:01 7 (1:)41:33:45:14:03:45 13:19 20:49 8 (1:)38:33:36:36 8:01 12:49 9 (1:)35:10:02:28:27:24:26 1:22:41 2:16:01 10 (1:)33:45 45 1:15 11 (1:)29:21:54:02:15 27:59 48:49 12 (1:)27:00:03:45 2:41 4:49 13 (1:)25:48:51:35:06:40 29:31 53:49 14 (1:)23:13:46:40 56 1:46 15

15 We might interprete row 1 as the four numbers (1; 59, 00, 15 | 1; 59 | 2; 49 | 1) which in modern notation would be: (28561/14400) | s = 119/60 | d = 169/60 | 1). An alternative interpretation views row 1 as (0; 59, 00; 15 | 1; 59 | 2; 49 | 1) which in modern notation would be: (14161/14400 | s = 119/60 | d = 169/60 | 1). There is a (120, 119, 169) and so if we divide everything by 60 we see that there is a right triangle (` = 2, s = 119/60, d = 169/60). Here s represent the short side of the right triangle, ` the long side and d represents the length of the diagonal (hypotenuse.) The first entry in the table would either be the the square of s/` or the square of d/`. The eleventh row of the table might be interpreted as (1; 33, 45 | 0; 45 | 1; 15 | 11) which in modern notation would be: (25/16 | s = 3/4 | d = 5/4 | 11) An alternative interpretation views row 1 as (0; 33, 45 | 0; 45 | 1, 15 | 11) which in modern notation would be: (9/16 | s = 3/4 | d = 5/4 | 11) In either view, the second and third entries represent parts of the Pythagorean triple (3/4, 1, 5/4) which is a multiple of the standard (3, 4, 5) Pythagorean triple.

1.2.6 Summary and references for ancient Babylonian mathematics Babylonian mathematics, dating back to almost 2000 BC, was quite sophisticated, with a very workable positional notation, rhetorical algebra (which recognized quadratic and cubic equations), and that used the Pythagorean Theorem. As far as we know, the Babylonians did not have the concept of proof. The concept of careful reasoning through basic mathematical principles would have to wait for the Greek age. But they were capable of fairly significant computations.

The material in this section follows 1. Sections 2.1 to 2.6 of Eves, 2. Sections 2.5 and 2.6 of Burton, 3. Sections 1.5 to 1.8 of Kline, (Other resources are sections 2.4 and 2.5 of The Rainbow of Mathematics by Grattan-Guiness and Part 3, chapters 17-20 of From Five Fingers to Infinity edited by Swetz.) More more on the Babylonian mathematics see the St. Andrews website and the Wikipedia article on Babyonian mathematics.

Exercises & Worksheets Do Worksheet 3 on Primitive Counting.

16 1.3 Egyptian Mathematics Simple counting will suffice for shepherds tracking their sheep, but more sophisticated societies keep records, invest money, collect taxes and survey land. The mathematics they use must allow for the computation of interest (including compound interest) and allow one to measure area. The must be convenient so that the result can be put into a record. Computations need to be relatively easy and readable to other officials in other communities. Linear and quadratic equations needed solving. Geometry was sufficient to compute areas of fields, so in these societies we see the first examples of the Pythagorean theorem. Much of the mathematics we have recovered from this period involves a series of computations. There is no evidence of a theory behind the computations; we have no record of the derivations of these results. The ancient records were often written on material that decayed rapidly. Paper was nonexistent. The ancient languages and computations have been difficulty to decipher. This is why objects such as the Rosetta stone have played a significant role. The main civilizations involved in these early developments were the major early civilizations, the Babylonians and the Egyptians. Paper was nonexistent; papyrus was rare, so most old documents are clay tablets.

1.3.1 The Egyptian dynasties Information about the Egyptians, including mathematics, was recorded on tablets and papyrus. An ancient papyrus, called the Moscow papyrus, dates to about 1850 BC, during the Middle Kingdom of Egypt. (This is probably 400 to 500 years before Moses lived in Egypt.) That papyrus, written in hieratic script, consists of a number of mathematics problems and computations. The Moscow papyrus has computational problems and geometric problems including areas and volumes. Several problems computed “a quantity” which is unknown, a primitive form of algebra. The Rhind papyrus is a larger work, with numerous math problems, most likely written 200 years later, by a scribe named Ahmes. It is a scroll about a foot in height and about 16 feet long. The papyrus scroll describes a series of mathematical procedures including computations with Egyptian unit fractions (see below.) The mathematical procedures appearing in the Moscow and Rhind papyri appear to be done by rote; they are prescriptive, without defense or conceptual philosophy. Much of what we know about Egyptian hieroglyphics can be traced to the Rosetta stone. The Rosetta stone is an ancient rock, four feet high, inscribed in three languages. It was probably created about 200 BC and provided a first break in our understanding of Egyptian hieroglyphics because it contained, carved into the rock, a king’s decree, written in three different languages. One of the languages was the Egyptian hieroglyphics but another language was ancient Greek. When the stone was found in 1799 by members of Napolean’s army, scholars could translate the ancient Greek and use that translation has a key to unlocking the Egyptian writing.

1.3.2 Egyptian unit fractions (See Burton section 2.3 or Swetz’s anthology, p. 135-144, for the article by Gillings on Problems 1-6 of the Rhind Mathematical Papyrus. Also see Wikipedia’s article on unit fractions.) According to the Rhind papyrus (written by the scribe Ahmes), the Egyptians wrote every fraction (with the exception of 2/3) as a sum of reciprocals of integers. A sum of reciprocals of integers is called a “unit fraction since the only numerators are units (ones.) Lets place a big dot (•) before a number to represent its reciprocal. (The Egyptians really placed a elliptical symbol above the number.) Using my altered notation, we write •4 + •28 in place of 2/7 since 2/7 = 1/4 + 1/28 and we write •7 + •13 + •91 in place of 3/13 since 3/13 = 1/7 + 1/13 + 1/91. The last fraction would have been achieved by pulling off 1/13 from 3/13 and then writing 2/13 as a unit fraction. They then had tables for doubling a unit fraction such as 13 and so would have recognized 2/13 as •7 + •91.

17 In order to make the unit fractions work, the Egyptians had (in the Rhind papyrus) a table of the doubles of unit fractions. It is not clear how they created their table, but here is one for doubling a single unit fraction: To compute 2/n, where n is odd, first find the smallest integer s such that 2/n > 1/s. In modern terms, this would mean choosing s = (n + 1)/2. Then find a larger integer t such that 2/n = 1/s + 1/t. In modern terms, this would mean computing 2 1 2s − n 1 − = = . n s ns ns Then 2 1 1 = + . n s ns For example, 2 1 1 = + 7 4 28 and 2 1 1 = + . 13 7 91

Exercise. Use this algorithm to write the double of •91.

2 2 1 1 Solution. Since is slightly larger than = we subtract and obtain 91 92 46 46 2 1 1 − = . 91 46 91 · 46 So 2 1 1 = + = •46 + •4186. 91 46 91 · 46

It is not clear if the Egyptians had this algorithm; some of the entries in the Rhind papyrus do not 2 follow this procedure. The Rhind papyrus has a table for for odd integers n, 5 ≤ n ≤ 97. n Thus one could look up (7) •4 , •28 2 when one wanted to write out . 7 5 So how would the Egyptians work out ? 7 Since 5 = (2)(2) + 1 then 5 2 1 = (2) + . 7 7 7 2 1 1 2 2 2 1 1 Now 7 = 4 + 28 so (2) 7 = 4 + 28 = 2 + 14 and so 5 1 1 1 = + + . 7 2 14 7 Thus 5 would be written •2 + •7 + •14. 7

18 7 Let’s work out . 13 7 4 + 2 + 1 4 2 1 = = + + . 13 13 13 13 13 Now 2 1 1 = + . 13 7 91 So 4 2 2 = + . 13 7 91 2 1 1 2 1 1 We go to our doubles table and write as + and write as + . 7 4 28 91 46 4186 Thus 7 4 + 2 + 1 4 2 1 1 1 1 1 1 = = + + = ( + ) + ( + ) + . 13 13 13 13 13 46 4186 7 91 13 So in place of 7/13 we would write •7 + •13 + •46 + •91+•4186.

1.3.3 Multiplication by adding – Egyptian doubling The papyrus with Egyptian mathematics implies that multiplication of fractions by integers was viewed as a combination of doubling and adding. For example, if one were to multiply an entity (x) by six, one could double it (to get 2x) and then double again (to get 4x) and then add those two pieces. If we wish to multiply by the integer 7, we write 7 in binary, in powers of two. 7 = 111 in binary, that is, 7 = 4 + 2 + 1. Then, if one want to multiply a number x by 7, one simply doubles x and then doubles that and then adds up the terms 4x, 2x and x. More generally, if one wants to multiply by the integer m, then multiplication can be replaced by doubling and adding, guided by by the binary (base two) expansion of m. Multiplication by 2j is the k k−1 2 same as doubling j times so if m = dk(2 ) + dk−1(2 ) + ...d2(2 ) + d1(2) + d0 (where the digits dj are either 0 or 1) then multiplication by m involves doubling j times anytime dj = 1. This is Egyptian doubling (or just binary multiplication.)

Another example 1 Let’s do one more example of this. Suppose we wanted to multiply •5 = by 7. We do this by first 5 1 1 2 4 writing 7 in binary, that is 7 = 1112 = 4 + 2 + 1. Thus 5 times 7 is 5 + 5 + 5 . By our tables for unit 2 fractions, we write 5 = •3+•15. 2 1 1 2 We double 5 by doubling 3 and 15 . By our table of unit fractions 15 = •8 + •120. (The Babylonians 2 had a special symbol fo 3 so we won’t write that as unit fractions. The final answer:

2 (•5) × 7 = •5 + •3 + •15 + •8 + •120 + 3 .

The Egyptians used this concept with their fractions and the concept is used today in modern computer software where we use squaring and multiplication instead of . Suppose we want to raise some large number to the . Since 7 = 4 + 2 + 1 then

x7 = (x4) · (x2) · (x) and so we can successively square x, getting x2 and x4 and multipliply them together at the end.

A worked xample

19 Here is a more practical and realistic example: What are the last (least significant) three digits of 233328? Try computing 233328 on a calculator and looking for the least significant digits. The number 233328 has 4532 digits! Your calculator can’t handle that. Instead, let us first write 3328 in binary. It is 11 10 8 3328 = 11010000000002 = 2 + 2 + 2 . So

11 10 9 233328 = (232 ) · (232 ) · (232 ).

9 We create 232 by squaring 23 nine times. We are only interested in the last three digits so here are our results:

232 = 529. 234 = (232)2 = (5292) = 279841 ≡ 841 mod 1000. 238 = (234)2 ≡ (8412) ≡ 281 mod 1000. 2316 = (238)2 ≡ (2812) ≡ 961 mod 1000. 2332 = (2316)2 ≡ (9612) ≡ 521 mod 1000. 2364 = (2332)2 ≡ (5212) ≡ 441 mod 1000. 23128 = (2364)2 ≡ (4412) ≡ 481 mod 1000. 23256 = (23128)2 ≡ (4812) ≡ 361 mod 1000. 23512 = (23256)2 ≡ (3612) ≡ 321 mod 1000. 231024 = (23512)2 ≡ (3212) ≡ 41 mod 1000. 232048 = (231024)2 ≡ (412) ≡ 681 mod 1000. Therefore

233328 = (232048) · (231024) · (23256) ≡ (681)(41)(361) = 10079481 ≡ 481 mod 1000.

So the three least significant digits of 233328 are 481.

As of 2011, one can use Wolfram-Alpha and really compute 233328. If we do that we get

676453995 ...thousands of digits ... 2564075284481.

Modern secure communication (including internet purchasing) is based on the RSA public-key en- cryption scheme and typically computes XE mod N where X,E and N are integers with hundreds of digits. These computations are done rapidly by this square-and-multiply process. Although the Egyptians avoid multiplication by doubling-and-adding while modern avoid exponentiation by squaring-and-multiplying, the principle is the same.

1.3.4 The Method of False Position The “method of false position” is a natural way of solving a problem for an unknown without the explicit use of algebra. Instead of using a (like x) one makes an intelligent guess, tests it in the problem, computing the answer, and then adjusts the guess at the end. For example, suppose we are told that one-third of a number and one-half of a number add to twenty. What is the number?

20 1 1 The modern approach is to write 3 x + 2 x = 20, finds a common denominator (or multiplies by a common multiple of the denominators), and solves for x. Here are typical steps: 1 1 x + x = 20 3 2 2 3 ( + )x = 20 6 6 5 ( )x = 20 6 5x = 20 · 6 20 x = · 6 = 24. 5 The Egyptians and Babylonians would have made a guess of something like “make the unknown equal to six.” (This is a good guess because it is easy to take both one-third of six and one-half of six.) They would then compute one-third of six added to one-half of six and get 2 + 3 = 5. But we really seek an unknown that gives an answer of 20, not 5. That is easy to fix – multiply everything by 4! So the unknown must really be 6 · 4 = 24. This method of guess-and-fix, in place of an explicit variable, is often called the method of false position. (The first guess is a false one which we correct later. The “false guess” takes the position of the more modern algebra “variable”.) We will return to the method of false position in our study of the “quiet millenium” after the Greek age.

1.3.5 The circumference and diameter of a circle The ancient Babylonians viewed the circle as a curved line and apparently wrote in terms of the circum- ference. (See Robson’s article “Words and Pictures...”, section 3.) At one point they approximated the area of a circle (in square units) as about one-twelfth of the square of the circumference. Today we tend to focus on the ratio of the circumference of a circle to its diameter, even giving it a C (Greek) name, (π.) We use, today, the symbol π to stand for the ratio of d where C is the circumference of a circle and d is its diameter. But before we even begin discussing the ancient approximations to π, one has to ask, “Is the ratio of the circumference C of a circle to its diameter d a constant? Couldn’t it vary with the size of the circle? Apparently π is a constant. (Why???) Ancient societies had a variety of estimates for the ratio of the circumference and diameter of a circle. (It is clearly bigger than 3!) The Egyptians and Babylonians had approximations for the area of a circle based on the circumference. One estimate by the Egyptians for the area of a circle was A = (8/9d)2. This puts π at 256/81. The Greeks got serious about evaluating π – we will look at the work of on that when we get to the Greek age. When I was in high school (1970) I heard about some California high school students using a mainframe computer to calculate π to 100,265 digits. I went to the local library and found the article (by Shanks and Wrench) in which the 100,000 digits were printed (10,000 to a page!!) and I photocopied the ten pages of 100,000 digits. I was awed by the fact that the number could run on and on forever! I kept those pages for many years. But now one can get a computer or a webpage to give you thousands of digits of pi. For example, if one goes to WolframAlpha and types in N[pi, 100000] and waits half-a- second, the Mathematica program at the WolframAlpha site will compute these 100,000 digits. The site may not display all 100,000 digits but by clicking on the link “Download as: pdf” one can download the digits to a pdf file. (I still think that’s pretty cool!)

21 Nowadays 100,000 digits is pretty small. For forrty years new mainframe computers were taken for a “test drive” by computing more and more digits of pi. In 2011 the record was somewhere around five trillion digits – but this time the work was all done on a desktop computer! For more reading on this interesting constant, one might check out Wikipedia’s article (of course) and Jonathan Borwein’s article on pi. All of this computing doesn’t excuse us from returning to the original question: “Is the ratio of the circumference of a circle to its diameter a constant? If so, why??”

1.3.6 References for ancient Egyptian mathematics This material follows 1. Sections 2.7 to 2.10 of Eves, 2. Sections 2.1 to 2.4 of Burton,

3. Chapter 2 of Kline, (Other resources are sections 2.6 to 2.9 of The Rainbow of Mathematics by Grattan-Guiness and Part 3, chapters 21 & 22 of From Five Fingers to Infinity edited by Swetz; see also chapter 23 on False Position.)

Exercises & Worksheets Do Worksheets 4 and 5 on Primitive Counting.

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