Site map 

 Contact Graeme 

 Skip Navigation LinksMath Help > First Principles > Logic and Proof > Irrationality Proofs

Irrationality Proofs

Contents of this section:

Skip Navigation Links.

In general, irrationality proofs work by assuming a number is rational, multiplying it by a multiple of its supposed denominator, and then showing the result is not an integer.

Proof that sqrt(n) is irrational, if n is an integer that is not a perfect square.

Proof that e is irrational -- defining e as the limit of 1/0! + 1/1! + 1/2! + ..., and also e=p/q, then we quickly arrive at the fact that q!e is not an integer, a contradiction.

Proof that π is irrational -- defining some functions in terms of π, sin(x) and cos(x), and using the fact that the second derivative of sin(x) is -sin(x), we eventually arrive at a contradiction.

Internet references 

Related Pages in this website

Perfect Squares -- proof that sqrt(n) is irrational, as long as n isn't a perfect square.

The webmaster and author of this Math Help site is Graeme McRae.