Question

prove that ✓5 is an irrational number.



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Answer 1

Given: √5

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.

Hence proved

Answer 2

we need to prove that root 5 is irrational..

proof :

let assume that root 5 is a rational number.

so it can be expressed in the form p/q where p, q are co - prime integers and q is not equal to 0..

root 5 = p/q

on squaring both of the side we get

$\sqrt{5} = p ^{2} \: by \: q ^{2}$

$5q^{2} = p ^{2}$--------(I)

p square /5 = q square..

so 5 divides p

p is the multiple of 5..

p = 5m

p square = 25m square _______(ii)

from equation i , ii.. we get

$5q^{2} = 25m^{2}$

$q^{2} = 5m^{2}$

q square is the multiple of 5..

q is the multiple of 5..

hence p and q have a common factor 5. this contradicts our assumption that they are co prime.. therefore p/q is not a rational number

root 5 is a irrational number..