# Converting Decimals and Fractions

A **rational number** is any number than can be written as a ratio of two integers, such as \(\frac{4}{5}\) or \(\frac{14}{658}.\) Rational numbers can be written in decimal form or fraction form.

## Introduction

Integers, terminating decimals, and repeating decimals can be written in fractional form. Likewise, all fractions can be written in decimal form.

## Given \( \sqrt{4}, 3, \pi , \sqrt{5}, 0.333......\), which of the numbers can be written in fractional form?

\(\sqrt{4}=2\) is an integer and can be written in fractional form as \(\frac{2}{1}.\)

3 is also an integer which can be written in fractional form as \(\frac{3}{1}.\)

\(\pi=3.1415926......\) goes on forever without a certain number or numbers repeating, so it can't be written in fractional form. You might have seen \(\pi\) written as \(\frac{22}{7}\), but that is only an approximation and thus accurate only to the tenth decimal place.

\(\sqrt{5}=2.236......\) goes on forever without certain numbers repeating, so it can't be written in fractional form.

Even though \(0.333......\) is non-terminating, it is a repeating decimal in \(3\), whose fractional form is \(\frac{1}{3}\).

## Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert \(\frac{1}{4}\) to a decimal, we divide 1 by 4 to get the quotient 0.25.

\[\frac{1}{4}=0.25\]

## What is the decimal representation of \(\frac{1}{6}?\)

Performing long division when \(1\) is divided by \(6\), the quotient doesn't have a terminating point in that the 6 goes on forever. So, to represent a repeating number, we use a hyphen on the top: \[\frac{1}{6}=0.1\overline{6}. \]

## What is the decimal representation of \(\frac{1}{5}?\)

Performing long division when 1 is divided by 5, we get the quotient to be 0.2. Thus,\[\frac{1}{5}=0.2.\]

## Converting Decimals to Fractions

To write an **integer** as a fraction, make the denominator one. For example, the number \(9\) can be written in fractional form as \(\frac{9}{1}\).

To write a **terminating decimal** as a fraction, write the decimal number into an equivalent fraction by making the numerator the number itself and the denominator 1. For example, if the number is \(0.6\), it becomes \(\frac{0.6}{1}\). After that, multiply both the numerator and the denominator with a number that takes the decimal place of the numerator into the last place. So multiply by 10 \(\left(\frac{0.6\times 10}{1 \times 10}\right)\) to get \(\frac{6}{10}\). Simplify to get \(\frac{3}{5}\).

## What is the fractional representation of \(0.623?\)

We have

\[0.623=\frac{0.623}{1}.\]

Multiply both the numerator and the denominator by 1000, to move the decimal point of the numerator to the zero's place:

\[\frac{0.623\times1000}{1\times 1000}=\frac{623}{1000},\]

which can't be simplified any further. \(_\square\)

To write a **repeating decimal,** such as 0.343434..., as a fraction, we can begin by setting it equal to a variable:
\[0.\overline{34} = x.\]

Our decimal repeats after the hundredths place, so we can write another equation, multiplying both sides of our original equation by 100: \[34.\overline{34}... = 100x.\]

Subtracting the first equation from the second, we get \(34 = 99x, \) or \(x = \frac{34}{99}.\)

## What is the fractional representation of \(0.\overline{8}?\)

We have

\[0.\overline{8}=x.\]

Multiply both sides of the equation by 10:

\[8\overline{8}=10x.\]

Subtracting the first equation from the second, we have \(8 = 9x\) and \(x=\frac{8}{9}.\)

You can read about other methods for converting repeating decimals to fractions here.

**Cite as:**Converting Decimals and Fractions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/decimals-fractions/