Question

prove that√5 is irrational​

Answer 1

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

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Answer 2

To Prove:-

√5 is irrational​

Solution:-

☆Let us assume that $\sqrt{5}$ is rational

So it t can be expressed in the form p/q where p and q are co-prime integers and q≠0.

$\sqrt{5}$ = $\frac{p}{q}$

On squaring both the sides we get,

⇴ 5 = $\frac{p²}{q²}$

⇴ 5q² = p²

⇴ $\frac{p²}{5}$ = q²

⇴ So 5 divides p  

⇴ p is a multiple of 5

⇴ p = 5m

⇴ p²=25m²

⇴ 5q²=25m²

⇴ q²=5m²

⇴ q² is a multiple of 5

⇴ q is a multiple of 5.

p and q have same common factor 5.

This contradiction is arisen because of our incorrect  assumption.

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