Arithmetic Fundamental Groups and Noncommutative Algebra by Fried M., Ihara Y. (eds.)

By Fried M., Ihara Y. (eds.)

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Diophantus always worked with a single unknown quantity ζ. In order to solve this specific problem, he assumed as given certain values that allowed him a smooth solution: a = 30, r = 2, b = 50, s = 3. Now the two numbers sought were ζ + 30 (for y) and 2ζ − 30 (for x), so that the first ratio was an identity, 2ζ/ζ = 2, that was fulfilled for any nonzero value of ζ. For the modern reader, substituting 30 7 Algebra 7 these values in the second ratio would result in (ζ + 80) (2ζ − 80) = 3. By applying his solution techniques, Diophantus was led to z = 64.

7 Algebra 7 the status of 1 was ambiguous in certain texts, since it did not really constitute a collection as stipulated by Euclid. Such a numerical limitation, coupled with the strong geometric orientation of Greek mathematics, slowed the development and full acceptance of more elaborate and flexible ideas of number in the West. Diophantus A somewhat different, and idiosyncratic, orientation to solving mathematical problems can be found in the work of a later Greek, Diophantus of Alexandria (fl.

In order to do so, he introduced a unit length that served as a reference for all other lengths and for all operations among them. For example, suppose that Descartes was given a segment AB and was asked to find its square root. He would draw the straight 43 7 The Britannica Guide to Algebra and Trigonometry 7 line DB, where DA was defined as the unit length. Then, he would bisect DB at C, draw the semicircle on the diameter DB with centre C, and finally draw the perpendicular from A to E on the semicircle.

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