Pi is an irrational number
Date: Around 1995
What is the proof that pi is an irrational number?
The proof is not quite a one-liner. See P. Neckmann,"A History of Pi",
Chapter 16, who refers the reader to E.W. Hobson's 1913 book, "Squaring the
Circle," (reprinted by Chelsea Publishing Co., NY).
The previous response gives the right reference. Basically the proof that
pi is irrational comes from the theory of continued fractions and was not
discovered until the late 1700's. Pi is not only irrational but also
transcendental: that was not proved until the late 1800's and is basically
just Euler's formula: e^(i*pi) = -1
the proof that sqrt(2) is irrational is much easier: assume it is rational
and equals p/q. p and q are reduced so they cannot both be even. Rearrange
and square and we have p^2 = 2*q^2. Therefore p is even, and equal to 2*r.
Now we have 4*r^2 = 2*q^2, and q turns out to have to be even, which is a
contradiction. Therefore, the premise that sqrt(2) is rational is false.
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Update: June 2012