Question

Prove that √5 is
irrational . step by step . (in paper)​

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

Hence proved

Answer 2

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Let us assume that √5 is a rational.

then,we can write,

√5 = a/b , (where a and b are co-primes and b ≠ 0)

Now,

√5 = a/b $\bf\implies$ a² = 5b²..............(i)

$\bf\implies$ a² is a multiple of 5

$\bf\implies$ a is a multiple of 5

let a = 5c for some positive integers c.

then,

a = 5c $\bf\implies$ a² = 25c²

$\bf\implies$ 5b² = 25c² [Using eq (i)]

$\bf\implies$ b² = 5c²

$\bf\implies$ b² is a multiple of 5

$\bf\implies$ b is a multiple of 5

Thus,

5 is a common multiple of a and b

This is a contradiction,since a and b are co-prime

Since the contradiction arises by assuming that √5 is a rational.

Hence, √5 is irrational.