Pythagoras's theorem states that the diagonal of a square with sides of integral
length
cannot be rational. Assume is rational and equal to where and are integers with no common factors.
Then
so
and ,
so
is even. But if
is even, then is even. Since is defined to be expressed in lowest terms, must be odd; otherwise and would have the common factor 2. Since is even, we can let , then . Therefore, , and , so must be even. But cannot be both even and odd, so there are no and such that is rational, and must be irrational.
In particular, Pythagoras's constant is irrational.
Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar
proofs for
(the golden ratio) and using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality
of
is given by Wiedijk (2006).
of a square with sides of integral
length 
cannot be rational. Assume 
is rational and equal to 
where 
and 
are integers with no common factors.
Then
,
so 
is even. But if 
is even, then 
is even. Since 
is defined to be expressed in lowest terms, 
must be odd; otherwise 
and 
would have the common factor 2. Since 
is even, we can let 
, then 
. Therefore, 
, and 
, so 
must be even. But 
cannot be both even and odd, so there are no 
and 
such that 
is rational, and 
must be irrational.
is irrational.
Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar
proofs for 
(the golden ratio) and 
using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality
of 
is given by Wiedijk (2006).
