Question

PROVE That ✓ 5 is irrational​

Answer 1

Step-by-step explanation:

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.

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

Hence proved

Answer 2

#then it must be in form of qp where, q=0 ( p and q are co-prime)

#p2 is divisible by 5.

#So, p is divisible by 5.

#So, q is divisible by 5.

#Thus p and q have a common factor of 5.

#We have assumed p and q are co-prime but here they a common factor of 5.