Prove that (root)2 is irrational

The part of syllabus covered by this question is proof by contradiction. Consequently, in order to achieve a contradiction, you can assume that root2 is rational and thus expressed as a/b where a and b are co-prime (have no common factors). By squaring both sides you get( 2=a^2/b^2). Multiply both sides by b^2 to get 2b^2=a^2. By definition of an even integer, we know a^2 is an even integer as it has a factor of 2. If a^2 is even, we know a is even as (even x even = even). If a is even, we can re-write this integer as 2k. Since a=2k we now know that 2b^2= (2k)^2= 4k^2. Therefore by simplification we deduce that b^2=2k^2. From the same process we applied to a^2, we can deduce that b is also even. If a and b are both even, then our initial statement that a/b has no common factors is a contradiction as a and b both have common factors of 2. Hence concludes our proof that root2 is irrational.