The Irrationality of $e$ and $\pi$
Unlike an ordinary fraction with an infinite numerator and denominator, an infinite continued fraction must be irrational. Euler was the first to show that the base of the natural logarithm $e$ is irrational, by numerically estimating its continued fraction, and showing the infinite sequence of convergents $p_n/q_n$ so obtained converged to $e$ using the Ricatti equation. Hermite showed that the recurrence relations for these convergents correspond to the recurrence relations between certain improper integrals, so proving the continued fraction tends to $e$ in the limit of infinite $n$. Here we provide a motivation for the integrals involved and obtain closed form integral representations for $p_n$ and $q_n$ for all $n$. An interesting feature, is that the above results provide a short cut to the standard proof of the irrationality of $\pi$.