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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 2 – WEDNESDAY 24TH AUGUST Counting is a recursive operation, relying on the previous value, and gives us the natural number system. It then becomes natural to try to combine two or more numbers, using binary operations such as addition and multiplication. This is the notion behind arithmetic (derived from the Greek word arithmos, meaning “number”). Rhind Papyrus The Rhind Papypus is the largest and best preserved document containing the mathematics of ancient Egypt (see the photograph on page 51). It was written by the scribe Ahmes, and discovered during illegal excavations in Thebes, before being purchased in 1858 by the Scotsman Alexander Henry Rhind, after whom it is named. Originally measuring 18 feet by 13 inches, the scroll dates from 1650 B.C., and begins with a list of the value of 2 divided by the odd numbers from 5 to 101 whereby the answer is a sum of fractions whose numerators equal 1 (see page 40). Ex. Ex. Looking at these examples, it is natural to ask how they were derived, and if the decompositions are unique. For the values of the denominator that are multiples of 3 (as in the first example), the scribe used the relationship: 2
2 1 6 However for many other values of the denominator the decompositions are not unique. For example was preferred to and it is not known why. In fact no algorithm has been proposed which will explain all the decompositions on the scroll, though we can always generate a sum of four terms by using the formula: 2 1 1
3 1 6 The fact that the scribe did not always use this formula and some close inspection of all the decompositions indicates that: i) Small denominators were preferred, with none greater than 1000 being used; ii) The fewer the number of terms in the decomposition, the better, as the scribe never used more than four; iii) Even denominators were preferred to odd; iv) Smaller denominators were written first in each decomposition; v) A bigger first denominator was preferred if the subsequent denominators were significantly smaller . For example: was preferred to After listing the values of 2/n, Ahmes filled out the remainder of the Rhind Papyrus with 85 mathematical problems, which give us a indication of how the Egyptians performed basic arithmetic. The Rosetta Stone Before we look at Egyptian arithmetic, it is worth noting that the Rhind Papyrus was written using ancient hieroglyphics, similar to those we looked at in Lecture 1. So it was necessary to decipher the meaning of all the symbols in order to interpret the mathematics. This was made readily possible by the discovery of the Rosetta Stone in 1799 by Napoleon’s army (see the photograph on page 36). The stone contained three translations of a passage of text. Since one version was in classical Greek, and one in Egyptian hieroglyphics, it just required someone with a knowledge of the former to provide a dictionary which could then be used to translate other Egyptian documents. This was duly provided by Jean‐
François Champollion in 1824, and hence the Rhind Papyrus was able to be deciphered quickly upon discovery. Egyptian Arithmetic Due to the 85 problems listed (see Section 2.3), the Rhind Papyrus gave a clear demonstration of how the Egyptians performed arithmetic. It showed a tendency to reduce problems involving multiplication and division to repeated additions, and relies upon the fact that every number has a unique binary representation, i.e. it can be written as the sum of distinct powers of 2. Ex. Ex. Ex. Ex. Babylonian Arithmetic While we have relatively few sources of Egyptian arithmetic, the discovery of 400 clay tablets since 1850 have led to us having a deep understanding of Babylonian arithmetic from 2000‐1600 B.C., and has led to the conclusion that the mathematics of Mesopotamia (today part of Iraq), of which Babylon was a city, was far more advanced than that of the Egyptians. In addition to arithmetic, the Babylonians developed algebra (see Lecture 3) in order to solve the equations arising from practical problems. Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 B.C. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1,4 using the base 60 system from Lecture 1: 82 = 1, 4 = 1
592 = 58, 1 = 58 60 + 4 = 64 60 +1 = 3481 The Babylonians then used the formulas below, along with their knowledge of perfect squares to perform general multiplication: ab = [(a + b)2 a2 b2]/2 ab = [(a + b)2 (a b)2]/4 Ex. Ex. Division is a more difficult process. The Babylonians interpreted division as multiplication by the reciprocal: a/b = a (1/b) so all that was necessary was a table of reciprocals. These tables were written in base 60 form, and started with the following terms: 2 0; 30 3 0; 20 4 0; 15 5 0; 12 6 0; 10 Ex. 9 0; 6, 40 10 0; 6 12 0; 5 15 0; 4 16 0; 3, 45 Ex. 20 0; 3 24 0; 2, 30 25 0; 2, 24 Ex. The table has gaps in it since any denominator whose value is the multiple of a prime 1/7, 1/11, 1/13, etc. are not finite base 60 fractions. In these cases the Babylonians would create an approximation. Ex. ...
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