Question

Prove that 5 is an irrational number. ​

Answer 1

Answer:

Prove that √5 is irrational

If √5 is rational, that means it can be written in the form of a/b, where a and b integers that have no common factor other than 1 and b ≠ 0. This means 5 divides a². ... This means b² is divisible by 5 and so b is also divisible by 5. Therefore, a and b have 5 as a common factor.

Answer 2

Answer:

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

√5 is an irrational number.

Hence proved