Classify the following number as rational or irrational
(a) root 15 (b) root 625 (c) 2.662 (d) 0.1501500150001500001……
The Correct Answer and Explanation is:
Let’s classify each of the numbers as either rational or irrational:
(a) ( \sqrt{15} )
The number ( \sqrt{15} ) is irrational. A number is considered irrational if it cannot be expressed as a fraction of two integers. The square root of 15 is a non-repeating, non-terminating decimal. Since 15 is not a perfect square, ( \sqrt{15} ) does not result in a whole number, and its decimal expansion is infinite without repeating. Therefore, ( \sqrt{15} ) is irrational.
(b) ( \sqrt{625} )
The number ( \sqrt{625} ) is rational. ( \sqrt{625} ) equals 25, which is an integer. An integer is always a rational number because it can be expressed as a fraction, with the integer as the numerator and 1 as the denominator (for example, ( 25 = \frac{25}{1} )). Thus, ( \sqrt{625} = 25 ) is a rational number.
(c) 2.662
The number 2.662 is rational. A rational number is any number that can be expressed as a fraction of two integers, including terminating decimals. The decimal 2.662 is a terminating decimal, and it can be written as the fraction ( \frac{2662}{1000} ). Since it is a terminating decimal, 2.662 is a rational number.
(d) ( 0.1501500150001500001\ldots )
The number ( 0.1501500150001500001\ldots ) is rational. This number has a repeating pattern of digits: “150”, which repeats indefinitely. A number with a repeating decimal is always rational, because it can be expressed as a fraction. Specifically, repeating decimals can be converted into fractions using algebraic techniques. In this case, the repeating sequence is “150”, which makes it a rational number.
Summary:
- (a) ( \sqrt{15} ): Irrational
- (b) ( \sqrt{625} ): Rational
- (c) 2.662: Rational
- (d) ( 0.1501500150001500001\ldots ): Rational
Explanation of Rational and Irrational Numbers:
- Rational numbers are numbers that can be written as a fraction of two integers ( \frac{a}{b} ), where ( b \neq 0 ). This includes integers, fractions, and decimal numbers that either terminate or repeat.
- Irrational numbers cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating, such as ( \pi ) or ( \sqrt{2} ). These numbers have an infinite, non-repeating decimal expansion.
Thus, by examining the properties of each number, we can confidently classify them as either rational or irrational.