Question

√5 is irration prove.

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Answer 1

Answer:

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

Hence,√5 is an irrational number.

Answer 2

$\huge\rm{\underline{\overline{\red{Answer}}}}$

★ 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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