Question

Prove that root 5​ is irrational. no.​

Answer 1

Answer:

Given:

√5

We need to prove that √5 is irrational

Proof :

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

:\implies:⟹ √5 = p/q

On squaring both the sides we get,

\pink:\implies:⟹ 5 = p²/q²

\pink:\implies:⟹ 5q² = p² —————–(i)

p²/5 = q²

So 5 divides p

p is a multiple of 5

\pink:\implies:⟹ p = 5m

\pink:\implies:⟹ p² = 25m² ————-(ii)

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

\pink\implies⟹ 5q² = 25m²

\pink\implies⟹ q² = 5m²

\pink\implies⟹ q² is a multiple of 5

\pink\implies⟹ 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