# Question #4b5c8

Oct 27, 2016

$0.33333$ is rational.

#### Explanation:

Any real number whose decimal representation terminates or repeats a certain pattern indefinitely is a rational number.

If $0.33333$ is intended as just that, i.e. the terminating decimal $0.33333000 \ldots$, then we can just multiply and divide by a power of $10$ to find its fractional representation.

$0.33333 = \frac{0.33333 \times {10}^{5}}{10} ^ 5 = \frac{33333}{100000}$

If it is intended as $0.33333 \ldots = 0. \overline{3}$, that is, an unending string of $3 \text{'s}$, then we can divide the repeating portion by ${10}^{k} - 1$, where $k$ is the number of digits the repeating portion. As $3$ is what is repeating, and has a single digit, we have

$0.33333 \ldots = \frac{3}{{10}^{1} - 1} = \frac{3}{9} = \frac{1}{3}$

In either case, we can represent the given number as a ratio of two integers, meaning it is a rational number.