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 left-hand side becomes an integer.

The first $n+1$ terms of the right-hand side become integers.

The rest of the right-hand 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 + not-integer; contradiction.

Enrichment task

Complete the proof by showing that $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots~<~1$

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E Is Irrational

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