Tue, 20 Feb 2007
Addenda to Apostol's proof that sqrt(2) is irrational
Yesterday I posted Tom
Apostol's wonderful proof that √2 is irrational. Here are some
additional notes about it.
- Gareth McCaughan observed that:
It's equivalent to the following simple algebraic proof: if a/b is the "simplest" integer ratio equal to √2 then consider (2b-a)/(a-b), which a little manipulation shows is also equal to √2 but has smaller numerator and denominator, contradiction.
- According to Cut-the-knot, the proof was anticipated in 1892 by A. P. Kiselev and appeared on page 121 of his book Geometry.
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