The Universe of Discourse

Wed, 21 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.

  1. 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.
  2. 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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