# 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/