Euler's number ($e$)

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.

If Euler's number were $a/b$, choose $n\ge \max (b,2)$ and multiply its factorial-series tail by $n!$. This produces an integer, but the tail is positive and smaller than ${\sum }_{j\ge 1}(n+1{)}^{-j}=1/n<1$. The contradiction proves that the number is an irrational number. The argument illustrates how a rapidly convergent rational series can force a hypothetical rational remainder into an impossible integer interval.