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Prove that √3 is irrational ​

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Answer:

for proving √3 as irrational, let it be a rational number.

so,

\sqrt{3} = \frac{p}{q}

where, p and q are co-prime INTEGERS, q≠0

on squaring both SIDES;

3 = \frac{ {p}^{2} }{ {q}^{2} }

or

3q²=p²....(1)

So, we can say that 3 divides p² and thus divides p also.

From the above OBSERVATION, we can conclude that;

p=3m for any INTEGER 'm'

Substituting (1) in the above eq.ⁿ

Now,

(3m)²=3q²

or 9m²=3q²

or 3m²=q²

So, we can say that 3 divides q² and thus divides q also.

Also, it divides q² and q (proved above)

But, we know that these numbers are co prime in nature. This contradiction arises because of our wrong assumption.

Therefore, √3 is irrational.

Hence, proved


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