Prove that root 3 + root 5 is irrational - Maths - Real Numbers

Question

Prove that root 3 + root 5 is irrational

Harleen Birdi · Accepted Answer

Let us assume on contrary that $\sqrt{3} + \sqrt{5}$ is a rational number.

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

$\sqrt{3} + \sqrt{5} = \frac{p}{q} \Rightarrow \frac{p}{q} - \sqrt{3} = \sqrt{5} \Rightarrow (\frac{p}{q} - \sqrt{3})^{2} = (\sqrt{5})^{2} \Rightarrow \frac{p^{2}}{q^{2}} - \frac{2 \sqrt{3}}{q} p + 3 = 5 \Rightarrow \frac{p^{2}}{q^{2}} - 2 = \frac{2 \sqrt{3}}{q} p \Rightarrow \frac{p^{2} - 2 q^{2}}{q^{2}} = \frac{2 p}{q} \sqrt{3} \Rightarrow \frac{p^{2} - 2 q^{2}}{2 p q} = \sqrt{3}$

$\Rightarrow \sqrt{3}$ is a rational number $[∵ p, q are integers ∴ \frac{p^{2} - 2 q^{2}}{2 p q} is rational]$

But this contradicts the fact that $\sqrt{3}$ is irrational.

So, our assumption is wrong.

Hence, $\sqrt{3} + \sqrt{5}$ is irrational