Question

A bit quick with the answer please...

If x = root 2 minus 1 , check whether x-1/x is a rational or irrational number.

Answer 1

$\textbf{ \huge { Hello Mate! }}$

$x = \sqrt{2} - 1 ( Given ) \\ \frac{1}{x} = \frac{1}{ \sqrt{2} - 1} \\ = \frac{1}{ \sqrt{2} - 1 } \times \frac{ \sqrt{2} + 1}{ \sqrt{2} + 1} \\ = \frac{ \sqrt{2} + 1}{ { \sqrt{2} }^{2} - {1}^{2} } = \sqrt{2} + 1 \\ x - \frac{1}{x} = \sqrt{2} - 1 - ( \sqrt{2} + 1) \\ = > \sqrt{2} - 1 - \sqrt{2} - 1 \\ = > - 2$

$\textsf{ \red{ Since, - 2 is rational number }}$

$\textsf{ \blue{ Hence, x - 1 / x is rational number }}$

$\textbf{ Have great future ahead! }$