Famous Theorems of Mathematics/π is irrational

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. Proof Let us assume that is rational, so there exist such that . For all let us define a polynomial and so we get 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 . Nevertheless, for all we get By the triangle inequality for integrals we get But , hence for sufficiently large we get . A contradiction. Conclusion: is an irrational number.