Show that 5-23 is an irrational number. - Vidyadip

Question

Show that $5-2\sqrt{3}$ is an irrational number.

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Answer

Let us assume that $5-2\sqrt{3}$ is a rational number.
$\therefore\ 5-2\sqrt{3}=\frac{\text{a}}{\text{b}}$ where, a and b are positive co-prime number.
$\Rightarrow\ 5-\frac{\text{a}}{\text{b}}=2\sqrt{3}$
$\Rightarrow\ \frac{5\text{b}-\text{a}}{\text{b}}=2\sqrt{3}$
$\Rightarrow\ \frac{5\text{b}-\text{a}}{2\text{b}}=\sqrt{3}$
We know that $\sqrt{3}$ is an irrational number.
This contradicts our assumption that $5-2\sqrt{3}$ is a rational number.
Hence, $5-2\sqrt{3}$ must be irrational.

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