Enable contrast version

# TutorMe Blog

## Why Are Repeating Decimals Rational Numbers?

Inactive
Jana Russick
May 11, 2021 Are repeating decimals rational? The answer is yes. But before we talk about why, let's review rational numbers. A rational number is a fraction in its lowest term. It's written in form a/b, where both a and b are integers, and b is a non-zero denominator.

Now, let’s talk about why repeating decimals are considered rational numbers.

## Are Repeating Decimals Rational?

Repeating or recurring decimals are decimal representations of numbers with infinitely repeating digits. Numbers with a repeating pattern of decimals are rational because when you put them into fractional form, both the numerator a and denominator b become non-fractional whole numbers.

For example, when you use long division to divide 1 by 3, the resultant quotient is 0.33333…. However, when put it into fractional form, it's made of positive integers that don’t have decimal points: In infinite decimal expansion, decimal digits repeat on forever with no end. Numbers with repeating decimals can have an overline above the last number: The overline is an easier, shorter way to indicate infinite decimal expansion without having to write a bunch of repeating numbers. Though repeating digits like this don't seem like rational numbers, they can take the form of a rational expressions when converted to their fraction form: This is because the repeating part of this decimal no longer appears as a decimal in rational number form. Instead, it’s represented by non-repeating, natural numbers 4 and 9. Remember — irrational numbers cannot be written as fractions.

## What Is a Terminating Decimal?

When converted to decimal form, some rational numbers have a terminating decimal. This means that there's a finite number of digits after the decimal point: The terminating decimal of this rational number is 5 since no decimal numbers follow it.

### What Is a Non-Terminating Decimal?

You can never count the decimal places of a number with repeating decimals. Because of this, repeating decimals are called non-terminating decimals.

Here's an example of a non-terminating decimal: In this sequence of digits, the same number combination of 09 will be infinitely repeated.

## Repeating Decimals Are Rational

Because rational numbers are used at all levels of math, it's important to know what makes a number rational. It might not seem like numbers with repeating decimals are rational numbers. But when you use your algebraic skills to convert repeating decimals into the fractional form a/b, you prove that repeating decimals are indeed rational.

### More Math Homework Help

BEST IN CLASS SINCE 2015
TutorMe homepage
Made in California by Zovio
© 2013 - 2021 TutorMe, LLC
High Contrast Mode
On
Off