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It is much easier to prove that e (the base of natural logarithms) is an irrational number than that pi is irrational. The key is the famous series for e :
$e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots$
So, suppose that $e=m/n.$
Multiply both sides by $n!$ .
The lefthand side becomes an integer.
The first $n+1$ terms of the righthand side become integers.
The rest of the righthand side is $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots$ which is positive but smaller than 1 and therefore not an integer.
So, integer = integer + notinteger; contradiction.
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