Question

prove that √5 is irrational​

Answer 1

Step-by-step explanation:

Prove root 5 is an 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

Answer 2

Answer:

let's prove it by the method of contradiction -

say root 5 is a rational . it can be expressed in the form p/q where p,q are co-prime integer

root 5 = p/q

5 =p^2/q^2

5q^2=p^2 (1)

p^2 is multiple of 5

p is also a multiple of 5

p=5m

p^2=25m^2 (2)

from equation 1&2 we get,

5q^2=25m^2

q^2=5m^2

q^2 is a multiple of 5

q is a multiple of 5

hence p,q has common factor 5 . this contradicts that they are co-prime

therefore p/q not a rational no. this proves that root 5 is a irrational no.