The square root
of 2, denoted sqrt(2)
, is an irrational number
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
So a is even
because 2 is a factor and by Lemma
below. Therefore a=2c for some integer c. So a2
and since a2
, we have 4c2
. It follows that 2c2
. 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.
: Let a be an integer and let a2
be even. Then a is even.