Is the square root of 29 rational or irrational?
The Correct Answer and Explanation is :
The square root of 29 is irrational.
Explanation:
A rational number can be expressed as the ratio of two integers, where the numerator and the denominator are whole numbers, and the denominator is not zero. Examples of rational numbers include fractions like ( \frac{3}{4} ) or integers like 5 (which can be written as ( \frac{5}{1} )).
On the other hand, an irrational number cannot be expressed as a fraction of two integers. Irrational numbers have non-terminating, non-repeating decimal expansions. Famous examples include numbers like ( \pi ) (approximately 3.14159) and ( \sqrt{2} ) (approximately 1.414).
The square root of a number is rational if the number is a perfect square, meaning it can be expressed as the square of an integer. For example:
- ( \sqrt{4} = 2 ) (since ( 2^2 = 4 ))
- ( \sqrt{9} = 3 ) (since ( 3^2 = 9 ))
However, 29 is not a perfect square. It falls between 25 and 36, the squares of 5 and 6, respectively:
- ( 5^2 = 25 )
- ( 6^2 = 36 )
Thus, ( \sqrt{29} ) is approximately 5.385, but its decimal expansion is non-terminating and non-repeating. Since there is no integer that, when squared, gives exactly 29, and the square root of 29 cannot be expressed as a simple fraction, it is irrational.
In summary, the square root of 29 does not result in a rational number but instead gives a decimal that goes on forever without repeating. This is the hallmark of an irrational number.