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 => 2b
2=a
2
So a is
even because 2 is a factor and by
Lemma below. Therefore a=2c for some integer c. So a
2=(2c)
2=4c
2 and since a
2=2b
2, we have 4c
2=2b
2. It follows that 2c
2=b
2. Thus b
2 is even. By
earlier result, since b is an integer and b
2 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 a
2 be even. Then a is even.