Show That \( \dfrac{1}{\sqrt{2}} \) Is Irrational

Video Explanation

Question

Show that \[ \dfrac{1}{\sqrt{2}} \] is an irrational number.

Solution

Step 1: Assume the Contrary

Assume that \[ \dfrac{1}{\sqrt{2}} \] is a rational number.

Step 2: Use Property of Rational Numbers

If a number is rational, then its reciprocal is also rational (provided it is non-zero).

Hence, the reciprocal of \(\dfrac{1}{\sqrt{2}}\) is \[ \sqrt{2}. \]

This implies that \(\sqrt{2}\) is rational.

Step 3: Reach a Contradiction

But it is known that \(\sqrt{2}\) is irrational.

This contradicts our assumption.

Conclusion

Our assumption is false.

\[ \therefore \quad \dfrac{1}{\sqrt{2}} \text{ is an irrational number.} \]