Prove that is an irrational number. It is given that is an irrational number.

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Solution

Q.
Prove that $5 - 2 \sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.

Ans.
Let $5 - 2 \sqrt{3}$ be a rational number such that

$5 - 2 \sqrt{3} = a,$ where $a$ is a non-zero rational number.

$\Rightarrow - 2 \sqrt{3} = a - 5 \Rightarrow \sqrt{3} = \frac{a - 5}{- 2}$

Since 5 and 2 are integers and a is a rational number, therefore $\frac{a - 5}{- 2}$ is a rational number

$\Rightarrow \sqrt{3}$ is a rational number which contradicts the fact that $\sqrt{3}$ is an irrational number.

Therefore, our assumption is wrong.

Hence $5 - 2 \sqrt{3}$ is an irrational number

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