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Why Are Repeating Decimals Rational Numbers?

are repeating decimals rational: DECIMAL spelled using wooden blocks

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:

are repeating decimals rational: fractional form

In infinite decimal expansion, decimal digits repeat on forever with no end. Numbers with repeating decimals can have an overline above the last number:

are repeating decimals rational: line above the 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:

Rational number in fractional 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:

Rational numbers with terminating decimal

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:

Example of 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.

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