By P. F. Hsieh, A. W. J. Stoddart
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Extra info for Analytic theory of differential equations; the proceedings of the conference at Western Michigan University, Kalamazoo, from 30 April to 2 May 1970
In turn, their discoveries found their way west as scholars traveled and translated. Greece gave India the gift of geometry, and India gave the same gift back multiplied in value. A book written by Jyesthadeva (born c. 1500), for example, contains a sum that introduces yet another wonderful interaction between geometry and the infinite. 2 Jyesthadeva’s proof relied on a preliminary result that is itself an elegant blend of geometry and the infinite. He began with the identity 1 D1 1Cx x Â 1 1Cx Ã ; which can be checked by simplifying the right-hand side.
So, for example, we might write ! 6 6Š 6 5 4 3 2 1 D D D 20 3 3Š 3Š 3 2 1 3 2 1 to find the number of ways of choosing 3 of 6 identical objects. (a) List these 20 possibilities, using the illustration of poker chips labeled A, B, C , D, E, F . x C e/6 using the techniques just discussed, rather than expanding by hand. x C e/n for a positive integer n. This equality is called the binomial theorem. ✐ ✐ ✐ ✐ ✐ ✐ “C&IO” — 2012/2/14 — 12:16 — page 43 — #57 ✐ ✐ 4 Indivisibles Every mathematical subject advances thanks to imaginative conjectures.
So if the second process is always lagging behind the first, then how could both processes exactly consume the same original quantity? This sounds paradoxical. 7. This is a geometric representation of a quantity being reduced at two different rates. Oresme countered this argument by challenging the idea that we may only compare the two remainders after the same number of steps. Granted, if we pause at each step and compare what remains of the quantity that is being halved to what remains when n D 5, then the first remainder will always be smaller.
Analytic theory of differential equations; the proceedings of the conference at Western Michigan University, Kalamazoo, from 30 April to 2 May 1970 by P. F. Hsieh, A. W. J. Stoddart