Question

prove that √3 is irrational ​

Answer 1

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

Answer 2

Answer:

Answer: Let us assume to the contrary that √3 is a rational number. where p and q are co-primes and q≠ 0. ... This demonstrates that √3 is an irrational number.