lecture3 - HISTORY OF MATHEMATICS – LECTURE 3 – TUESDAY...

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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 3 – TUESDAY 30TH AUGUST Over half a million tablets from Mesopotamia have been excavated, many of which contain evidence of Babylonian mathematics. These were largely unstudied until the 1920s, at which point the Austrian mathematician Otto Neugebauer began to look at the tablets dating from between 1800 and 1600 B.C. Neugebauer discovered that the texts were of two kinds: “table texts” which listed the values of perfect squares, perfect cubes, and Pythagorean triples, as well as “problem texts” which demonstrated the extent to which the Babylonians used and understood algebra. While nothing was written using modern notation, Neugebauer found solutions for problems involving quadratic and some cubic equations. Ex. Here is a translated problem from a four thousand year old tablet: “I have added the area and two‐thirds of the side of my square and it is 0;35. What is the side of my square?” If we translate this using modern notation we get the quadratic equation: 2 3 35 60 The given solution reads: “Two‐thirds of 1 is 0;40. Half of this, 0;20, you multiply by 0;20, and to the resulting 0;6,40 you add 0;35 to get 0;41,40, which has 0;50 as its square root. From this you subtract 0;20, and 0;30 is the side of the square.” When converted to modern notation this reads: This indicates that the Babylonians knew that the general solution to the formula is which can easily be derived by completing the square. Note: While we know that there are actually two solutions to a quadratic equation, the concept of a negative number was not accepted by Babylonian mathematicians, likely because their problems were based on physical dimensions which had to be positive. Indeed, while there is evidence that negative numbers were used by Chinese mathematicians two thousand years ago, they were not accepted in Europe until the 17th century. Plimpton 322 A Babylonian tablet of particular interest is known as Plimpton 322, and was deciphered in 1945 by Neugebauer and Abraham Sachs (see the photograph on page 75). Although only a fragment it contains four columns of numbers which led many to believe that each row represented a Pythagorean triple, i.e. three numbers that satisfy the result of Pythagoras’ Theorem for right traingles. This is remarkable not only because the tablets pre‐date Pythagoras by over 1000 years, but also indicates that the Babylonians were aware of the formulas for generating triples. Below is a table containing all the values listed on Plimpton 322, written in base 60 form, with the errors corrected: I. z2/x2 1, 59, 0, 15 1, 56, 56, 58, 14, 50, 6, 15 1, 55, 7, 41, 15, 33, 45 1, 53, 10, 29, 32, 52, 16 1, 48, 54, 1, 40 1, 47, 6, 41, 40 1, 43, 11, 56, 28, 26, 40 1, 41, 33, 59, 3, 45 1, 38, 33, 36, 36 1, 35, 10, 2, 28, 27, 24, 26, 40 1, 33, 45 1, 29, 21, 54, 2, 15 1, 27, 0, 3, 45 1, 25, 48, 51, 35, 6, 40 1, 23, 13, 46, 40 II. Length of the width y 1, 59 56, 7 1, 16, 41 3, 31, 49 1, 5 5, 19 38, 11 13, 19 9, 1 1, 22, 41 45 27, 59 7, 21, 1 29, 31 56 III. Length of the diagonal z 2, 49 3, 12, 1 1, 50, 49 5, 9, 1 1, 37 8, 1 59, 1 20, 49 12, 49 2, 16, 1 1, 15 48, 49 4, 49 53, 49 53 IV. Reference number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 So, for example, in row 11, the width is 45, the diagonal is 75, and hence the corresponding Pythagorean triple is (45, 60, 75). Note: It is plausible to assume that the values for the length, x, were contained in the missing part of the tablet, since it is clear that they were explicitly calculated in order to obtain the numbers in the first column. The fact that the scribe tabulated the values in the first column indicates that the Babylonians were aware of the proof given below. Theorem: The formulas x = 2mn, y = m2 – n2, and z = m2 + n2, where all the variables represent positive integers, and m > n, will generate Pythagorean triples (x, y, z) that will solve the equation x2 + y2 = z2. Proof: Note: If we also assume that m and n are relatively prime (have no common divisors) then the formulas will generate all the primitive Pythagorean triples (those which don’t have a common divisor). Note: The first three rows of the table (respectively) correspond to m = 12 and n = 5, m = 64 and n = 27, m = 75 and n = 32. Diophantus While we will discuss Greek mathematics during the next three lectures, it is appropriate to consider Diophantus now, given his role as the “father of algebra”, though it must be noted that his work was not markedly more advanced than that of the Babylonians, leading us to conclude both that the mathematics of Mesopotamia was ahead of its time, and that relatively little development occurred for the next 1500 years. However Diophantus was the first person to use symbols rather than words to describe equations (see pages 221‐222). Although likely of Greek descent, Diophantus lived in Alexandria, Egypt, during the 3rd century A.D.. While little is known of his life, his work was documented in Arithmetica, a 13 volume work, of which six books survived, eventually being translated into many other languages, and inspiring future mathematicians, in particular Pierre de Fermat, whose famous Last Theorem was written in the margin of a Latin copy of the Arithmetica, next to the problem below. Ex. (Book II, Problem 8) Divide a given number, say 16, into the sum of two squares. Ex. (Book II, Problem 13) Find a number such that if both 6 and 7 are subtracted from it, both remainders are perfect squares. Ex. (Book II, Problem 20) Find two numbers such that the square of either added to the other yields a perfect square. ...
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