The square root of 2, denoted sqrt(2), is an irrational number.

Proof: Suppose sqrt(2) is a rational number. Then sqrt(2)=a/b for some integers a and b. Assume that a/b is in lowest terms; that is, assume a and b have no common factors other than 1 or -1.

sqrt(2)=a/b => b*sqrt(2)=a => 2b2=a2

So a is even because 2 is a factor and by Lemma below. Therefore a=2c for some integer c. So a2=(2c)2=4c2 and since a2=2b2, we have 4c2=2b2. It follows that 2c2=b2. Thus b2 is even. By earlier result, since b is an integer and b2 is even, it follows that b is even. Thus b=2d for some integer d. But the fact that 2 is a factor of both a and b is a contradiction. Therefore the assumption that sqrt(2) is rational is incorrect. Thus sqrt(2) is not rational. Q.E.D.

Lemma: Let a be an integer and let a2 be even. Then a is even.

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