Question

Prove that `sqrt(2)` is an irrational number.

Answer 1

Let `sqrt(2)` be a rational number.

Then, its simplest form = `p/q`

Where p and q are integers having no common factor other than 1, and q ≠ 0.

Now, `sqrt(2) = p/q`

On squaring both sides we get,

`2 = p^2/q^2`

2q² = p²   ...(i)

⇒ 2 divides p²

⇒ 2 divides p   ...(∵ 2 is a prime and divides p² ⇒ 2 divides p)

Let p = 2r for some integer r.

Putting p = 2r in, (i) we get

2q² = 4r²

⇒ q² = 2r²

⇒ 2 divides p²   ...(∵ 2 divides 2r²)

⇒ 2 divides q   ...(∵ 2 is prime and divides q² ⇒ 2 divides q)

Thus, 2 is a common factor of p and q.

But this contradicts the fact that p and q have no common factor other than 1.

Thus, contradiction arises by assuming `sqrt(2)` is rational.

Hence, `sqrt(2)` is irrational.