Question

prove that root 2 + root 3 is an irrational number​

Answer 1

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Let us assume on the contrary that $2 + \sqrt{3}$ is rational. Then there exists co-prime positive integers a and b such that :-

$2 + \sqrt{3} = \frac{a}{b} \\ = > \sqrt{3 } = \frac{a}{b} - 2 \\ = > \sqrt{3} = \frac{a - 2b}{b} \\ = > \sqrt{3} = rational$

(Since, a and b are integers)

But this contradicts the fact that $2 + \sqrt{3}$ is irrational. So, our assumption is not correct.

Therefore, it can be concluded that $2 + \sqrt{3}$ is an irrational number.

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