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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 compounding limit for e shows how the same constant arises from repeated small multiplications.

The exponential series at $1$ givesFor every positive integer $q$, multiply by the factorial $q!$ and separate the integer part:Every term in $R_q$ is positive. Each denominator is at least $(q+1{)}^j$, and the inequality is strict when $j\ge 2$. Comparison with the geometric series therefore givesIf $e=p/q$ with integers $p$ and $q\ge 1$, then $q!e=p(q-1)!$ would be an integer. Since $I_q$ is also an integer, $R_q$ would be an integer strictly between zero and one, which is impossible. HenceThis proves the irrationality of e by using its rapidly decreasing series tail.

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: