Question

Prove that `sqrt2` is an irrational number.

Answer 1

Let `sqrt2` is rational.
∴ `sqrt2 = "p"/"q"` where p and q are co-prime integers and q ≠ 0.

⇒ `sqrt2"q" = "p"`
⇒ 2q² = p²  ....(1)
⇒ 2 divides p²
⇒ 2 divides p  ......(A)

Let p = 2c where c is an integer
⇒ p² = 4c²
⇒ 2q² = 4c²  ...[from (1)]
⇒ q² = 2c²
⇒ 2 divides q²
⇒ 2 divides q  .....(B)

From statements (A) and (B), 2 divides p and q both that means p and q are not co-prime which contradicts our assumption.
So, our assumption is wrong.

Hence `sqrt2` is irrational.
Proved.