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

Answer 2

QUESTION :-

prove that √ 5 is irrational

ANSWER :-

Let us assume that √5 is a rational number.

Let us assume that √5 is a rational number.we know that the rational numbers are in the form of p/q form where p,q are integers.

so, √5 = p/q

so, √5 = p/q p = √5q

we know that 'p' is a rational number. so √5 q must be rational since it equals to p

we know that 'p' is a rational number. so √5 q must be rational since it equals to pbut it doesnt occurs with √5 since its not an integer

therefore, p =/= √5q

this contradicts the fact that √5 is an irrational number

hence our assumption is wrong and √5 is an irrational number.