True or False: 0.33333 . . . is a rational number.
The Correct Answer and Explanation is:
True.
The number 0.33333…0.33333 \dots0.33333… (with the threes repeating indefinitely) is a rational number. To understand why, let’s break it down:
- Definition of Rational Numbers: A rational number is any number that can be expressed as the fraction of two integers, i.e., in the form pq\frac{p}{q}qp, where ppp and qqq are integers and q≠0q \neq 0q=0. Rational numbers can be written as either terminating or repeating decimals.
- Repeating Decimals: The decimal 0.33333…0.33333 \dots0.33333… is an example of a repeating decimal. Specifically, it is a repeating decimal with the digit “3” repeating indefinitely. Repeating decimals are always rational because they can be written as a fraction.
- Converting to a Fraction: You can convert 0.33333…0.33333 \dots0.33333… to a fraction using a simple algebraic method. Let x=0.33333…x = 0.33333 \dotsx=0.33333…. To eliminate the repeating decimal, multiply both sides of the equation by 10: 10x=3.33333…10x = 3.33333 \dots10x=3.33333… Subtract the original equation x=0.33333…x = 0.33333 \dotsx=0.33333… from this: 10x−x=3.33333⋯−0.33333…10x – x = 3.33333 \dots – 0.33333 \dots10x−x=3.33333⋯−0.33333… 9x=39x = 39x=3 x=39=13x = \frac{3}{9} = \frac{1}{3}x=93=31
- Conclusion: The decimal 0.33333…0.33333 \dots0.33333… is equivalent to the fraction 13\frac{1}{3}31, which is a ratio of two integers. Hence, 0.33333…0.33333 \dots0.33333… is a rational number.
In summary, any repeating decimal can be expressed as a fraction, making it a rational number. Therefore, the statement that 0.33333…0.33333 \dots0.33333… is a rational number is True.
