Is square root of 1/5 a rational number
The Correct Answer and Explanation is :
The square root of ( \frac{1}{5} ) is not a rational number.
Explanation:
A rational number is defined as any number that can be expressed as the ratio of two integers, i.e., in the form ( \frac{p}{q} ), where ( p ) and ( q ) are integers, and ( q \neq 0 ). The key characteristic of rational numbers is that they either terminate or repeat when expressed as a decimal.
Step-by-Step Analysis:
- Expression of ( \sqrt{\frac{1}{5}} ):
The square root of a fraction can be written as the square root of the numerator over the square root of the denominator. So:
[
\sqrt{\frac{1}{5}} = \frac{\sqrt{1}}{\sqrt{5}} = \frac{1}{\sqrt{5}}
] - Nature of ( \sqrt{5} ):
- The number ( \sqrt{5} ) is irrational. This can be shown by the fact that 5 is not a perfect square, meaning there is no integer whose square equals 5. Therefore, ( \sqrt{5} ) cannot be expressed as a ratio of two integers, and its decimal expansion is non-terminating and non-repeating (approximately 2.236067977…).
- Dividing by an irrational number:
Since ( \sqrt{5} ) is irrational, the expression ( \frac{1}{\sqrt{5}} ) is also irrational. Dividing a rational number (1) by an irrational number ( ( \sqrt{5} ) ) results in an irrational number. - Decimal Expansion:
The decimal expansion of ( \sqrt{\frac{1}{5}} ), which is approximately 0.447213595…, is non-terminating and non-repeating, further confirming that it is irrational.
Conclusion:
Because ( \sqrt{\frac{1}{5}} ) cannot be written as a simple fraction of two integers and has a non-terminating, non-repeating decimal expansion, it is irrational.