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College Math Resources - Topics in Precalculus

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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**.