Question

$prove \: that \: \sqrt{5} \: is \: irrational.$

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Answer 1

Let's assume √5 is a rational number and 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

As, 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

Also, 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.

So, root 5 is an irrational number.

Hence proved.