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
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and so we get
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Now let us define . The integrand is positive for all and so .
Repeated integration by parts gives:
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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
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hence . But for sufficiently large we get . A contradiction.
Therefore, is an irrational number.