On the Proof That e Is Irrational

关于e是无理数的证明 苏剑林 · 2009-10-17 · Mathematics proof irrational number

In mathematics, we denote the limit

$$\lim_{x->\infty}(1+1/x)^x$$

by e, and we can compute that e = 2.7182818284590452353602..., a number just as important as π (some books even consider it more important than π). Like π, this number is also an "irrational number." Let's now prove this. more

Suppose $e=p/q$ is rational, where p, q are positive integers. From the Maclaurin expansion of e, we have:
$$p/q=e=1+1/2+1/{3!}+...+1/{q!}+\varphi^{q+1}/{(q+1)!}\tag{1}$$$$p/q-(1+1/2+1/{3!}+...+1/{q!})=\varphi^{q+1}/{(q+1)!}\tag{2}$$
where $\varphi \in(0,1)$
Now, multiplying both sides of (2) by q!, the left-hand side becomes an integer, while the right-hand side equals $\varphi^{q+1}/{(q+1)}$, which is not an integer. So the two sides cannot be equal. This contradicts the assumed equality! Hence our assumption is false, and e is irrational.

This proof via the Maclaurin expansion is a relatively simple one — provided, of course, that one is familiar with the Maclaurin expansion of e. Compared with the proof given on Wikipedia, I find this one considerably more concise and intuitive!