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
(2ba)/(ab), which a little manipulation shows is also
equal to √2 but has smaller numerator and denominator,
contradiction.
 According to Cuttheknot,
the proof was anticipated in 1892 by A. P. Kiselev and appeared on
page 121 of his book Geometry.
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