Babylon/Mesopotamia. Mesopotamia = between two rivers, namely the Tigris and Euphrates.

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1 Babylon/Mesopotamia Mesopotamia = between two rivers, namely the Tigris and Euphrates. Civilization dates from before 3000 BCE covering several empires with varying borders: Sumerians, Akkadians, Babylonians, etc. Made cuneiform tablets (wedge-shaped writing on clay tablets). Most tablets date either from the time of Hammurabi (c.100 BCE) and the Seleucid dynasty (c.300 BCE) which ruled after Babylon was conquered by Alexander the Great. Thousands of tablets have been recovered. Two main types: tables of values (multiplication, reciprocals, measures) and worked problems copied out by scribes. Enumeration The Babylonians mainly used two symbols: a vee for 1 and a wedge for 10. In practice these would have been made by the same stylus rotated 90. Any number up to 59 was written with combinations of these: e.g. 53 = Numbers larger than 60 were written positionally using powers of 60: e.g. 3 = = In this example, the vee s on the left each correspond to 60. Think about how we read decimal numbers: in 3, the first represents 00 and the second 0. This is what we mean by a positional system of enumeration: the same symbol means different things depending on its position. There was no symbol for zero (to space out vees and wedges) until very late in Babylonian history, nor any sexagesimal point, so determining position can be very difficult. For instance, the above might have represented or = or = depending on context. The Babylonians were aware of the Egyptian method of unit fractions, but they preferred to use powers of 60 in the same way as we use decimals.

2 To make things easier to read, we will write a Babylonian number in hybrid fashion, using commas to separate terms and a semicolon to denote the sexagesimal point. Thus 3, 1, 0; 15 = = Why base 60? There are many theories, but no-one is sure precisely why. For instance: 60 has many divisors, making it easy to divide quantities in many different ways. Indeed many sexagesimal fractions have a nicer representation than as decimals: for example, 1 3 =; 0 is a terminating sexagesimal which much simpler than the decimal A year has approximately 360 days which is divisible by 60. A degree could therefore be used to measure approximately how far the sun travels round the ecliptic each day. Units of Babylonian measure often used 60 when moving up and down the magnitude scale similarly to how we use multiples of 1000 to move from, say, joules to kilojoules to megajoules. Calculations Addition and subtraction are done just as we would with decimals. For instance, to calculate 1309 = summed with 17 = we might write 1,49; + 3,37; + 1 4,6; to yield = 156. Notice how we carry over 60 s just like we are used to doing with 10 s in decimal arithmetic: = 86 = 1, 6. Multiplication is a pain as you need many more times tables at your disposal to do it quickly. Here is a suggestion of how to multiply two small numbers in a hybrid fashion. 49; 37; 5,43; + 4,30; 30,13; Our first calculation reads 49 7 = 343 = 5, 43. The second reads = These are then summed to obtain = 1813 = Fractions/Division The Babylonians didn t typically use fractions, instead relying on sexagesimal calculation. They produced tables of reciprocals, for instance 1 = =; 5, 7, 16, 1, 49, 5,

3 which could then be used to divide: Similarly, 5 11 = 5 (; 5, 7, 16,...) = ; 16, 1,... 1 = =; 8, 34, 17, 8, 34, 17,... 7 yields 10 7 = 10 (; 8, 34, 17,...) = 1; 5, 4,... In practice, the approximations usually only had one to three sexagesimal places. Often, when required to compute a fraction such as 1 11 a scribe would write 11 does not divide, perhaps indicating that the following solution would be an approximation. It is not known whether the Babylonians understood that sexagesimal expansions of rational numbers would be (eventually) periodic, as they are for decimals. It seems unlikely, as the periods are typically very long, and the accuracy given by even the second sexagesimal place (< ) would have been enough for even the most demanding application. Many tables have been found to assist with multiplication and division. For example, a table listing all the ways in which an integer < 10 might be multiplied to get 10 might be given as follows: We omit the symbols for separation and the sexagesimal point, as they did not exist. This actually means that the table has greater application: for instance, that 10 6 = 1; = 1, 6; 40 (the location of the sexagesimal point is decided at the end) The Yale Tablet YBC 789 One of the most famous mathematical tablets concerns an approximation to. YBC stands for the Yale Babylonian Collection which contains over 45,000 objects. The tablet is shown below along with an enhanced representation of the numerals.

4 The tablet clearly depicts a square of side 30 (or possibly 1 = 0; 30) and labels the diagonal in two ways: 1; 4, 51, 10 as an approximation to. This is accurate to within %! 4; 5, 35 as an approximation to the diagonal when the side is 30. More usually the Babylonians used the simpler approximation 1; 5 = which is still very close. Given the insane accuracy of this approximation, it is reasonable to ask how it was obtained. No-one knows for sure, but two methods are theorized since both were used by the Babylonians to solve other problems. 1: Square root approximation a ± b a ± b a. This is essentially the linear approximation from elementary calculus. The idea is to choose a rational number for a whose square is very close to, then the error should be very small. For instance: = = 1; 30 ( = 4 ) /9 8/3 = = 17 1 = 1; 5 ( = 7 ) /5 14/5 = = 1; 4, 51, 5, 4, 51, 5, 4, 51,... : Method of the Mean This is an iterative method. Choose a starting value a 1 and define a sequence using the recurrence relation a n+1 = a n+/a n. Recalling the AM-GM inequality a + b ab with equality iff a = b we see that, unless some a 1 =, the sequence a n will always be greater than for n. Indeed it is easy to check that a n+1 = 1 (a n an ) + < 1 an (a n ) Let us apply this to the sequence starting with a n = 1. a 1 = 1, a = 3 = 1; 30, a 3 = 17 = 1; 5 1 a 4 = = = 1; 4, 51, 10, 35, 17, a 5 = = 1; 4, 51, 10, 7, It seems incredible that they d have bothered to go so far with these calculations to obtain the observed accuracy.

5 The same analysis can be used to approximate other roots. For example, we could start with with a = 3 to approximate 11 via a n+1 = 1 (a n + 11 a n ), whence we obtain a = 10 3 = 3; 0, a 3 = = 3; 19, a 4 = 7901 = 3; 18, 59, 50, 57, 17, Quadratic Equations The Bablylonians could apply the above methods for extracting square-roots to general quadratic equations. Questions might be phrased as follows: I added twice the side to the square; the result is, 51, 40. What is the side? We are essentially asking for the solution to x + x = = Questions such as these were solved using templates. In the above example, the template is for solving x(x + p) = q where p, q > 0. Other templates were required for the other types of quadratic equation (x = px + q, etc.), since the Babylonians did not recognize negative numbers. Here is their algorithm applied to a simpler equation x + 4x = : Set y = x + p (y = x + 4) then the equation can be decoupled: { xy = q y x = p Now use this to solve for x + y: { xy = y x = 4 4xy + (y x) = p + 4q 4xy + (y x) = (y + x) = p + 4q (y + x) = 4 x + y = p + 4q x + y = 4 ; 54 x + p = p + 4q x + 4 4; 54 p + 4q p x = x ; 7 The square-root was was approximated 4 = = 4; Everything done without the benefit of any of our modern notation! Only found the positive solution. x = px + q and x + q = px had similar templates. Students just determined form and applied template. This method (quadratic formula/completing the square) is at least 4000 years old! Restricted cubic equations could be managed similarly.

6 Linear systems These were solved by a mixture of the false position method (guess and modify as done by the Egyptians) and an approach modelled on homogeneous equations. Example Solve the system of equations { 3x + y = 11 x + y = 7 1. Choose an equation, say the second, and set x = y. Now solve, for instance using false position to obtain x = 7 3 = ; 0.. All solutions to the second equation have the form x = x + d and y = y d, since (d, d) is the general solution to the homogeneous equation x + y = Substitute into the first equation: ( ) ( ) = d + 3 d = d = d = 3 4. Now solve x = 3, y = 1. Pythagorean Triples Among the many tables of values created by the Babylonians are lists of Pythagorean triples. The Plimpton 3 tablet (also at Yale) has a large number of these (albeit with some mistakes). Because of the strange way in which the triples were encoded, it took a long time before scholars realized what the list was. The table is also broken on the left so some columns are probably missing. Usually written v = 1 + u, probably found by solving (v u)(v + u) = 1. Simply choose v + u and find v u in the table fo reciprocals.

7 Example Line 15 of the table describes the triple 53 = as follows. The last two entries indicate line number 15. The third entry is 53. The second entry is 8. The first entry is ( ) = 1; 3, 13, 46, 40 (exact) It is theorized that these entries were produced by finding rational solutions to the equation v = 1 + u (equivalently (v + u)(v u) = 1). The scribe probably started with v + u and used a table of reciprocals to calculate v u. In our example, let v + u = 9 5 = 1; 48 and compute v u = 5 9 =; 33, 0 to get v = 1; 10, 40 = and u =; 37, 0 = 45. It is not known how the table was completed, although the first column exhibits a descending pattern that provides clues to its construction. The Plimpton tablet has been the source of enormous scholarship, so for more details... Geometry We will not discuss particular examples, but many geometric problems have also been found in cuneiform tablets. For instance, they used both π 3 and π to approximate areas of circles. They had correct and incorrect formulæ for the volume of a frustrum (truncated pyramid). They also knew that the altitude of an isosceles triangle bisects its base and that the angle in semicircle is a right angle. None of these statements were presented as theorems in a modern sense; the fact that computations using these facts exist indicates that at least some Babylonians believed these facts. Summary Sexagesimal positional enumeration. (Essentially) No zero or fractions. More advanced than Egypt but still very utilitarian. Perhaps only appears more advanced because we have much more evidence (1000 s of tablets versus a handful of papyri). Mostly examples without abstraction, like Egypt. Limited distinction (e.g. does not divide ) between approximate and exact results. Limited geometry compared to algorithmic/numerical methods.