A proof begins by assuming that 2=ab\displaystyle \sqrt{2}=\frac{a}{b}2=ba, where a a\,a and b b\,b are positive integers with no common factor, and hence that a2=2b2a^2=2b^2a2=2b2.
Complete the proof by contradiction that 2 \sqrt{2}\,2 is irrational.