Question

prove that √5is not a rational number​

Answer 1

Answer:

5 is one of the factor of q. As, we know p and q are co-prime so it cannot have common factor. But here a contradiction arise that 5 is factor of both p and q. So, by this we can say that √5 is not rational which means √5 is irrational.

Answer 2

Prove that

root 5 is irrational number

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

⇒√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

$\bold \pink{Hence \: proved}$