We assume that sqrt(3) is a rational number
==> since sqrt(3) is a rational number
we can find a smallest integer k such that k*sqrt(3) is also a integer.
now take m = k*sqrt(3)-k which is also a integer.
And m*sqrt(3) = (k*sqrt(3)-k) * sqrt(3) = 3k-sqrt(3)k is also an integer.
However, m = k*sqrt(3) -k is smaller than k.
And we claimed that k is the smallest integer that will make k*sqrt(3) an integer.
This is the contradiction.
We conclude that our assumption is false and sqrt(3) is irrational.