Prove that 2+3 is an irrational number. - Vidyadip

Question

Prove that $\sqrt{2}+\sqrt{3}$ is an irrational number.

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Answer

Let us suppose that $\sqrt{2}+\sqrt{3}$ is rational.
Let $\sqrt{2}+\sqrt{3}=\text{a},$ where a is rational.
Therefore, $\sqrt{2}=\text{a}-\sqrt{3}$
Squaring on both sides, we get
$2=\text{a}^2+3-2\text{a}\sqrt{3}$
Therefore,
$\sqrt{3}=\frac{\text{a}^2+1}{2\text{a}}$
which is a contradiction as the right hand side is a rational number while $\sqrt{3}$ is irrational.
Hence, $\sqrt{2}+\sqrt{3}$ is irrational.

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