In Alexandria mathematicians first began to develop algebra independent from geometry.

Transcription

1 The Rise of Algebra

2 In response to social unrest caused by the Roman occupation of Greek territories, the ancient Greek mathematical tradition consolidated in Egypt, home of the Library of Alexandria. It was in Alexandria that Ptolemy developed his table of chords to aid the study of astronomy. Other notable mathematicians of this era include Heron, Diophantus, Pappus, Theon, and Hypatia.

3 In Alexandria mathematicians first began to develop algebra independent from geometry. We ve seen how algebra can be understood geometrically, but we ve also experienced the limitations of this approach. (a x)bx = c a x x bx a

4 In around 200 AD Heron translated geometric procedures into algebraic ones, and formulated and solved algebraic problems without appeal to geometry either for motivation or justification. Perhaps the most famous such procedure is the one our book labels Problem 7.1. What is Heron s formula for the area of a triangle with sides a, b, c? a b c

5 The climax of Alexandrian Greek algebra can be found in the work of Diophantus. We don t know exactly where he originally came from, or when exactly he lived (probably around 250 AD), but we do know how long he lived thanks to an algebra problem composed in his honor. How old was he when he died? For 1/6 of his life Diophantus was a boy. A 1/12 part later he grew a beard, and married after 1/7. 5 years later he had a son. The son lived only 1/2 the father s life, and Diophantus died 4 years later. Diophantus influence on the history of mathematics began long after his death. He wrote his major work, the thirteen books of the Arithmetic, as the Dark Ages descended upon Europe, where its contents were ignored for centuries.

6 Diophantus wrote the Arithmetic in the style of the Rhind papyrus. It is a collection of different problems, each solved in a unique way, with no attempt made to collect together problems with similar solutions, and with no deductive proofs. Diophantus claimed to have written it as a series of exercises for his students. In the Arithmetic we see the first use of symbols denoting unknown quantities to aid in problem solving. (Because the author of our textbook wants to present the history of mathematics in its original form, yet doesn t want us to get hung up on the unfamiliar Greek letters, he suggests the use of familiar Latin letters that correspond to the Greek ones.)

7 The appearance of algebraic symbolism was a major advancement in the history of mathematics. Over many centuries it has become increasingly revised and refined. A surprising feature of Diophantus symbolism is the consideration of powers greater than three. Until this point in the history of mathematics, the Greeks had no use for such powers since they had no geometric meaning. When number is separated from magnitude, however, powers higher than three do make sense, and Diophantus used them without hesitation. Δ Δ Δ, ΔK, K K

8 Seven out of the thirteen books of the Arithmetic have been lost. Of the six that remain, the first consists mainly of problems leading to determinate (having a unique solution) first degree equations in one or two unknowns. The remaining books consider indeterminate second degree equations.

9 Of all Diophantus problems, the one about dividing a given square number into two squares (Proposition 7.5) is the most famous. Why? This is a Latin translation of Diophantus Arithmetic which includes comments by Pierre de Fermat. It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

10 Fermat was one of the two most prominent mathematicians of the 17 th century (the other was Descartes). His work paved the way for differential calculus and he is considered the founder of modern number theory. He refused to publish his work, but his friends and relatives preserved it. Fermat had written notes in his copy of Diophantus Arithmetic, and his son published a version of this book complete with his father s notes. For over three centuries the world s best mathematicians failed to discover the marvelous proof of what became known as Fermat s Last Theorem (the last to be proved, that is). None were successful until 1994 when Andrew Wiles finally announced a proof over one hundred pages long.

11 The range of equations that Diophantus could reduce and solve is impressive. We don t know how he discovered his methods, but we do know they are considerably different from anything already known at the time. The equations that arise from the problems included in this section are today referred to as Diophantine equations. The exercises for section 7.4 involve such equations.

12 Although Diophantus did consider fractions to be numbers, as opposed to strictly ratios, he did not recognize complex, irrational, or even negative numbers. In the solution to Problem 7.9, for example, Diophantus dismisses a negative solution to a problem as absurd.

13 Problem 7.9: Find three numbers in geometrical progression, so that each of them, added to a given number, gives a square. Given number 20. Solution: Take a square that, added to 20, gives a square: say 16. Let this be one of the extremes; let the other be 1S, so the mean is 4x (remember that if three terms are in geometric progression, the product of the extremes is the square of the mean). This it is required that both 1S 20U and 4x 20U be squares. Their difference is 1S M 4x, or 1x times 1x M 4U. The usual method gives 4x 20U equal to 4, which is absurd, because 4x 20U ought to be greater than 20.

14 Diophantus did know how to expand products of the form (x a)(x b), which means he knew a negative times a negative should be a positive, and a negative times a positive should be a negative. This realization complicated conceptual understanding of imaginary numbers. If a negative times a negative is a positive, then 1 can t be negative, since 1 1 = 1. On the other hand, it can t be positive either, since any positive times a positive is a positive.

15 Diophantus was not the only mathematician to avoid irrational, imaginary, and negative numbers. Reluctance to fully accept these quantities continued until the 19 th century! Why? Numbers counted things. Four sheep, four children, and four jars all have something in common. This common property is described by the number four. Counting was refined with the introduction of fractions, but still numbers described quantity. In this sense even zero is a difficult concept. If numbers represent quantity, then zero must not be a number. In Chapter 8 we will see how zero began life as a place holder, a separate symbol used in positional number systems to indicate the absence of number in a particular position.

16 In the 9 th century, Indian mathematicians made a conceptual leap that marked a major advancement in the history of mathematics. They began to treat the place holding symbol as an quantity on par with the counting numbers. By creating rules for adding, subtracting, multiplying, and dividing by zero, they showed how it could be treated as a number even though it didn t count anything. This revolutionary idea took a long time to gain acceptance, and even in the 16 th and 17 th centuries some prominent mathematicians were still reluctant to accept zero as a root of an equation.

17 Given the reluctance to accept zero as a number, it should come as no surprise that the use of negative, irrational, and complex numbers was firmly resisted. Negative numbers arise naturally as solutions to equations. For example, what equation is suggested by the following problem? I am 7 years old and my sister is 2. When will I be exactly twice as old as my sister? The equation is (x + 7) = 2(2 + x) and the solution is x = 3. What happens if we replace the ages 7 and 2 by 18 and 11? The equation is (x + 18) = 2(11 + x) and the solution is x = 4.

18 The thread of mathematical history we trace backwards from modern times moved out of the Mediterranean altogether as the influence of Christianity grew and its leaders became suspicious of knowledge from pagan (i.e., non-christian) sources. During this period Pappus tried to revitalize interest in theoretical mathematics, but was ultimately defeated by the growing interest in, and authority of, questions addressing religious subjects.

19 Theon was an important commentator (person who expands and explains the work of others) during this turbulent time, as was his daughter Hypatia. It is believed that the oldest existing copies of Diophantus Arithmetic came from a commentary by Hypatia. She met a dramatic and tragic end.

20 As Europe entered the Dark Ages, mathematics continued its growth elsewhere. Important contributions come from China, India, and the Islamic world.