TITLE: How to prove to be an irrational number? Like $\sqrt{2}$, $\sqrt{3}$, or $\sum\limits_{k=1}^{\infty} \frac{1}{n^2}=\pi^2/6$
QUESTION [7 upvotes]: As we know $\sqrt{2},\sqrt{3}$ are irrational numbers. And I see some proofs on the net.
So I doubt that how $e,\pi$ or already known irrational numbers are proved to be irrational.
In fact, I got interested in Riemann zeta function
$$\zeta(s)=\sum_{n=0}^{\infty} \frac{1}{n^s},$$
we know $\zeta(2)=\pi^2/6$ from Euler, 1737.
One mathematician (sorry to forgot his name) proved $\zeta(3)$ to be also irrational 40 years before.
Can somebody explain how he could do with that? To understand Apéry's theorem, is it very hard?
An question raises that could one real number make up of two different irrationals (for example: $e,\pi$), $e\pi$, or others can be rational?

REPLY [11 votes]: It is known to be very difficult to find the arithmetic nature of most real numbers such as the constants that appear in mathematical analysis, e.g. $\zeta (5)=\sum_{n\geq 1}\frac{1}{n^{5}}$.
Roger Apéry proved directly that $\zeta(2)$ is irrational. And he was the first to prove that $\zeta(3)$ is irrational too. He constructed two sequences $(a_n),(b_n)$ $[1]$
$$\begin{equation*}
a_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2}c_{n,k},
\qquad b_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2},\end{equation*}$$
where 
$$\begin{equation*}
c_{n,k}=\sum_{m=1}^{n}\frac{1}{m^{3}}+\sum_{m=1}^{k}\frac{\left( -1\right)
^{m-1}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}\quad k\leq n.
\end{equation*}$$
The ratio $a_n/b_n\to\zeta(3)$ and has the following properties:

$2(b_{n}\zeta (3)-a_{n})$ satisfies $\lim\sup \left\vert 2(b_{n}\zeta (3)-a_{n})\right\vert^{1/n}\le(\sqrt{2}-1)^4 $.
$b_{n}\in \mathbb{Z},2(\operatorname{lcm}(1,2,\ldots ,n))^{3}a_{n}\in 
\mathbb{Z}$.
$\left\vert b_{n}\zeta (3)-a_{n}\right\vert >0$.

This is enough to prove  the irrationality of $\zeta (3)$ by contradiction. $[2]$.
There is The Tricki entry To prove that a number is irrational, show that it is almost rational that gives two examples and explains the principle of some proofs of the irrationality of numbers. 
References.
$[1]$ Poorten,  Alf., A Proof that Euler Missed…, Apéry’s proof of the irrationality of $\zeta(3)$. An informal report, Math. Intelligencer 1, nº 4, 1978/79, pp. 195-203.
$[2]$ Fischler, Stéfane, Irrationalité de valeurs de zêta (d’ après Apéry, Rivoal, …), Séminaire Bourbaki 2002-2003, exposé nº 910 (nov. 2002), Astérisque 294 (2004), 27-62
$[3]$ Apéry, Roger (1979), Irrationalité de $\zeta2$ et $\zeta3$, Astérisque 61: 11–13