MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000
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Not much is known about the life of Diophantus. He was probably Greek. From a puzzle put forth in his honor, it appears that he married at 33 and had a son who died at 42 when Diophantus was 80, four years before he died. The only other information we have is a report that Anatolius, the bishop of Laodicea in A.D.270, made a dedication to Diophantus on an Egyption computation tract written by Anatolius. [Calinger]
Diophantus used fractions. Also, Diophantus treated numbers as entities independent of measurement, following laws and principles even when their physical meaning was unclear. For example, he would represent an unknown number as a variable, say x, and then proceed to manipulate this "number" like any other, even though he could not know at the outset if it even existed. This faith in principles rather than physical intuition allowed Diophantus the freedom to create new principles that worked even though the "number" he was searching for had an uncertain existance. Another person who put faith in physical principles ahead of intuition was Cardano. He created several new methods, such as completing the cube, which worked even though some steps seemed to his contemporaries to be arbitrarily defined, see Cardano. Here is a sample of Diophantus’ work [Arithmetica,Book I]: "3. To divide a given number into two numbers such that one is a given ratio of the other plus a given difference.
Look familiar? Good old algebra. The algebraic steps that Diophantus understood but left out of this explanation would lead to x = (80-4)/4, which of course is 19. Now try this one. "7. From the same (required) number to subtract two given numbers so as to make the remainders have to one another a given ratio.
Here is another example: "10. Given two numbers, to add to the lesser and to subtract from the greater the same (required) number so as to make the sum in the first case have to the difference in the second case a given ratio.
It can easily be seen from the examples above that Diophantus used reverse-engineering (algebra). That is, he assumed an answer in the form of x, such as (20+x) = 4(100-x), and then cleared away the clutter until an equation, x=76, was left with x on one side and a single number on the other. The methods of Diophantus and Euclid are by no means incompatible. As an example, take problem 1 in The Arithmetica. Euclid could just as easily solved it. I will first show how Diophantus solved it, and then I will show how Euclid might have solved it.
In conclusion, while Euclid may have understood some rudimentary techniques of reverse-engineering numerical problems, Diophantus was the first to develop it into a rigorous, systematic approach, which we call algebra. Select a link to continue, or scroll to the top to find a link to the main page.
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Euclid could have solved the
same problem in the following way:
