Question

Q. Prove that √5 is irrational.​

Answer 1

Step-by-step explanation:

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

Sp it t 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

Answer 2

Answer:

let us asume √5 rational nbr it's in the form of p/q

p/q=√5....

p and q are coprimes

by squaring on bothside

(p/q)2= (√5)2

square and root will cancelled

P2/q2=√5

p/5-q

5 also divides P2 and also p

p=k2

k2=√5-p/2q

there fore our assumption is wrong........i√5 is rational is wrng .....√5 is irrational