Demenstrate as directed — task. Maths CBSE Live product, Class 10 (2026-27).

Prove that \(\frac{1}{1- \sqrt{2}}\) is an irrational number.

Proof:

Let us assume that \(\frac{1}{1- \sqrt{2}}\) is .

By definition, \(\frac{1}{1- \sqrt{2}}\) \(=\)

\frac{i}{i}

where \(p, q\) are integers and \(q \neq 0\).

Consider the LHS.

Let us rationalise \(\frac{1}{1- \sqrt{2}}\).

\begin{array}{l} \frac{1}{1 - \sqrt{2}} = \frac{1}{1 - \sqrt{2}} \times \frac{1 i \sqrt{2}}{1 i \sqrt{2}} \\ = \frac{i + \sqrt{i}}{i - i} \\ = \frac{i + \sqrt{i}}{i} \end{array}

Therefore, it can be written as \(- 1 - \sqrt{2}\) \(=\)

\frac{i}{i}

where \(p, q\) are integers and \(q \neq 0\).

\(\Rightarrow\)

\frac{- i q - i}{i} = \sqrt{i}

Here, the LHS is in the above equation.

This implies that \(\sqrt{2}\) is , which is a contradiction.

Therefore, \(\frac{1}{1- \sqrt{2}}\) is an irrational number.

Hence, proved.

Did you find an error?

8. Demenstrate as directed