Find the value
\[ a = \sqrt{2} + 1 \]
Solution:
\[ \frac{1}{a} = \frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} \]
\[ = \frac{\sqrt{2} – 1}{2 – 1} = \sqrt{2} – 1 \]
\[ a – \frac{1}{a} = (\sqrt{2} + 1) – (\sqrt{2} – 1) \]
\[ = 2 \]
\[ a = \sqrt{2} + 1 \]
\[ \frac{1}{a} = \frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} – 1}{\sqrt{2} – 1} \]
\[ = \frac{\sqrt{2} – 1}{2 – 1} = \sqrt{2} – 1 \]
\[ a – \frac{1}{a} = (\sqrt{2} + 1) – (\sqrt{2} – 1) \]
\[ = 2 \]