Question

prove that√5 is irrational​

Answer 1

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

So it 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.

Answer 2

Let us assume that √5 is a rational

Where p, q belongs to integers and q ≠ 0

• √5 = $\frac{p}{q}$

Where p, q are co-primes and they do not have any common factor except 1 .

On squaring on both sides

• (√5) ² = ( $\frac{p}{q}$)²
• 5 = $\frac{p²}{q²}$
• 5q² = p² ===> eqn ①

$\frac{p²}{5}$ = q²

5 is divided by p

p is a multiple of 5

Let,

• p = 5m
• p² = (5m)²
• p² = 25m² ===> eqn ②

from eqn ① and ②

we get

• 5q² = 25m²
• q² = 5m²

q² is the multiple of 5

q is also multiple of 5

Here p, q have common factor as 5 .

Hence, our contradiction is wrong and they are co-primes so √5 is an irrational number.

Hence proved