Prove that (2+ 3 ) is irrational. - Vidyadip

Question

Prove that $\big(2+\sqrt3\big)$ is irrational.

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Answer

If possible, let $\big(2+\sqrt3\big)$ be rational.
Then 2 and $\sqrt3$ are rational.
$\Rightarrow2+\sqrt3-2$ is rational $\dots(\because$ difference of two rationals is rational$)$
$\Rightarrow\sqrt3$ is rational
This contradicts the fact that $\sqrt3$ is irrational.
The contradiction arises by asuming that $\big(2+\sqrt3\big)$ is rational.
Hence, $\big(2+\sqrt3\big)$ is irrational.

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