Question

prove thar √2 is irrational

Answer 1

Let us assume that √2 is a rational number
therefore, there exists p and q such that ,√2=p/q
where, p and q are co-prime to each other.
therefore, √2×q=p
squaring on both sides ,2q²=p²
it implies, 2 divides p²
Therefore, 2 divides p ,it implies, p is even
Let, p=2k where k is an integer
Now, 2q²=(2k)² ,2q²=4k²
q²=2k² it implies q is even
therefore, both p and q are even.
It implies p and q have a common factor 2.
This is contradictory to our assumption that p and q are co-prime.
it implies, our assumption that √2 is a rational number is wrong.
Therefore, √2 is an irrational number
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