Answer to Question #65250 in Calculus for iqra

Question

why under-root 2 is irrationnal number??

Answer 1

Assume that √(2 ) is a rational number. This means that it can be written as the ratio of two integers m, n:
√(2 )=m/n. (1)

We may assume that m and n have no common factors. (If there are any common factors we cancel them in the numerator and denominator.) Squaring in (1) both sides gives
2=m^2/n^2 ,
which implies
2n^2=m^2.

Thus m^2 is even. The only way this can be true is that m itself is even. But then m^2 is divisible by 4. Hence n^2 must be divisible by 2 (i.e., even) and therefore n must be even. So m and n are both even which is a contradiction to our assumption that they have no common factors. The square root of 2 cannot be rational.
Q.E.D.