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Euler's number is the value $\exp (1)={\sum }_{n=0}^{\infty }1/n!$ of the exponential function, approximately $2.71828$. It is the base of the natural logarithm. Its factorial-series tail gives an elementary proof of the irrationality of e.

The exponential function series gives Euler's number $e={\sum }_{k=0}^{\infty }1/k!$. Suppose $e=a/b$ with integers $a,b$ and $b>0$. Choose $n\ge \max (b,2)$. Since $b$ divides $n!$, the numberis an integer. But its positive tail satisfiesThe inequality is strict because the factors after the first exceed $n+1$ when $j\ge 2$. No integer lies strictly between zero and one. This proves the irrationality of e: