MthT 430 Apostol's Irrationality

The following proof of the irrationality of Ö2 is credited to Thomas Apostol. It relies on geometry and the principal of mathematical induction.

Suppose that there are natural numbers m and n such that m² = 2 n² and that n is the smallest natural number for which m² = 2 n². Draw a circle centered at A so that AB = BC = n and AC = m.

Let DE be tangent to the circle at D and BC be tangent to the circle at B. Then DE = EB = DC = m - n. Then DDCE is a right triangle with sides m - n < n and hypotenuse n - (m- n) = 2n - m and n is not the smallest natural number satisfying m² = 2 n².

Tom M. Apostol, Irrationality of The Square Root of Two - A Geometric Proof, American Mathematical Monthly 107, No. 9 (Nov., 2000), pp. 841-842.

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On 12 Sep 2007, 09:55.