Question

Prove that √5 is irrational.​

Answer 1

Let us assume that √5 is a rational number.

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒ √5 = p/q

On squaring both the sides we get,

⇒5 = p²/q²

⇒5q² = p² —————–(i)

p²/5 = q²

So 5 divides p

p is a multiple of 5

⇒ p = 5m

⇒ p² = 25m² ————-(ii)

From equations (i) and (ii), we get,

5q² = 25m²

⇒ q² = 5m²

⇒ q² is a multiple of 5

⇒ q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

$\large\green{√5 \: is \: a \: rational \: number}$

Answer 2

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0. ... Therefore, p/q is not a rational number. This contradicts our given assumption. So, root 5 is an irrational number.23-Aug-2020