Question

Prove that √2 + √3 is irrational.

Answer 1

They both are irrational numbers and the sum of irrational numbers is always irrational.

Answer 2

Answer:

∴ √2 + √3 be a irrational number.

Step-by-step explanation:

Let √2 + √3 be a rational number. Which can be written  in form of p/q.

√2 + √3 = p/q                                 ...................(i)

squaring on both sides

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

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

$2 + 3 + 2\sqrt{6} = \frac{p^2}{q^2}$

$5 + 2\sqrt{6} = \frac{p^2}{q^2}$

$2\sqrt{6} = \frac{p^2}{q^2}\ -\ 5$

$\sqrt{6} = \frac{p^2}{2q^2}\ - \frac{5}{2}$

We know that $\frac{p^2}{2q^2}$is a rational number.

and $-\frac{5}{2}$ is also a rational number.

So,$\frac{p^2}{2q^2}\ - \frac{5}{2}$ will be a rational number. Because we know that sum of rational number is always rational.

∴ √6 must be a rational number, but we know that √6 is a irrational number.

Here, RHS ≠ LHS

It creates a contradict.

∴ √2 + √3 be a irrational number.