College Math Resources - Topics in Precalculus

The Irrationality of Root 2

Any ratio of integers can be reduced to "lowest terms" by cancelling out common factors.
So if root 2 is rational, then it can be expressed as a ratio m/n where m and n have no common factors.
Now if m/n is root 2, then (m^2)/(n^2)=2, so m^2 = 2*(n^2) is even.
But the prime factorisation of  m^2 includes just the same factors as in m (each repeated twice as often). So if m^2 is even, then the factor 2 must occur in m. So m^2 is actually a multiple of 4.
But then 2*(n^2)=(m^2)=4*something, so (n^2)=2*something, and so it too is even.
So n must be even also.
But then m and n both have a factor of two and we already cancelled out all common factors!
Thus if we assume that root 2 is a ratio of whole numbers then we get a contradiction,
and so it cannot be true.