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

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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   is a positive integer.

Nevertheless, for all   we get

 

hence  . But for sufficiently large   we get  . A contradiction.

Therefore,   is an irrational number.