Question

Prove that
$\sqrt{5}$
is an irrational number​

Answer 1

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Let us assume√5 is rational.

√5=$\frac{p}{q}$ [p and q are co-prime]

p=√5q ...(1)

p²=5q [Squaring both the sides]

$\frac{p²}{5}=q²$...(2)

p² divides 5, p also divides 5.

p=5m [m is any integer]

From equation 1,

√5q=5m

q=$\frac{5m}{√5}$

q=√5m

q²=5m² [Squaring both the sides]

5 divides both p and q.

But p and q are co-primes.

It means our assumption is wrong.

√5 is irrational

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