Question

Prove that √2+ √3 is an irrational naumber dont dare to spam ​

Answer 1

To Prove :-

• √2 + √3 is an irrational number

Solution :-

As we know that √2 and √3 both are irrational number .

• Sum of two irrational number may be rational or may be irrational in some cases like if the both irrational number are same but different sign(-,+) (+,-)

So we can say that √2+√3 is also an irrational number

Answer 2

Let as assume that √2 + √3 is a rational number .

Then , there exists co - prime positive integers p and q such that

$\sqrt{2 } + \sqrt{3}$

$= > \frac{p}{q } - \sqrt{3} = \sqrt{2}$

Squaring on both sides,we get.

$= > ( \frac{p}{q} - \sqrt{3} ) {}^{2} = ( \sqrt{2} ) {}^{2}$

$= > \frac{ {p}^{2} }{ {q}^{2} } + 3 - 2 = 2 \sqrt{3} \: \frac{p}{q}$

$= > \frac{p {}^{2} }{q {}^{2} } + 1 = 2 \sqrt{3} \: \frac{p}{q}$

$= > ( \frac{p {} ^ {2} + {q}^{2} } { {q}^{2} } ) \times \frac{q}{2p} = \sqrt{3}$

$= > \frac{p {}^ {2} + {q}^{2} }{2pq} = \sqrt{3}$

$= > \sqrt{3} \: \: is \: a \: rational \: number$

This contradicts the fact that √3 is irrational .

so assumption was incorrect . Here √2 + √3 is irrational.

I HOPE IT HELPS (✿^‿^)