The mathematical constant
(the ratio of circumference to the diameter of the circle) is an irrational number.
In other words, it cannot be expressed as a ratio between two integers.
Let us assume that
is rational, so there exist
such that
.
For all
let us define a polynomial

and so we get
![{\displaystyle {\begin{aligned}f^{(k)}\!(x)=(-1)^{k}f^{(k)}\!(\pi -x)&={\begin{cases}\displaystyle \sum _{m\,=\,n}^{2n}{\frac {k!}{n!}}{\binom {m}{k}}c_{m}x^{m-k}&:0\leq k\leq n-1\\[5pt]\displaystyle \sum _{m\,=\,k}^{2n}{\frac {k!}{n!}}{\binom {m}{k}}c_{m}x^{m-k}&:n\leq k\leq 2n\end{cases}}\\[5pt]f^{(k)}\!(0)=(-1)^{k}f^{(k)}\!(\pi )&={\begin{cases}0&:0\leq k\leq n-1\\[5pt]\displaystyle {\frac {k!}{n!}}c_{k}&:n\leq k\leq 2n\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da1b463db667809860751de4d137c15098f4821c)
Now let us define
. The integrand is positive for all
and so
.
Repeated integration by parts gives:

The remaining integral equals zero since
is the zero-polynomial.
For all
the functions
take integer values at
, hence
is a positive integer.
Nevertheless, for all
we get

hence
. But for sufficiently large
we get
. A contradiction.
Therefore,
is an irrational number.