# Converting Decimals and Fractions

## Introduction

When converting between decimals and fractions, we have to realize that fractions are closed under rational numbers. That is, any number that can be written in fractional form is a **rational number**. This includes integers, terminating decimals, and non-terminating or repeating decimals.

An **integer** can simply be written as fraction by making the numerator the number itself and the denominator one. For example, the number \(9\) can be written in fractional form as \(\frac{9}{1}\).

A **terminating decimal** can be written as fraction by writing it the way you say it. For example, the decimal \(1.5\) is **one and one half** or \(1\frac{1}{2}\). Adding the terms gives \(1+\frac{1}{2}=\frac{2}{2}+\frac{1}{2}=\frac{3}{2}\).

A **repeating decimal** can also be written as fraction using algebraic method.

If a number doesn't have the above property, then it is not a rational number. So it can't be written in fractional form. For example, \(\sqrt{2}=1.414213......\), in which the decimal representation is non-terminating and non-repeating, and thus can't be converted into fractional 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 thus 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 a certain number or 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 \(\frac{1}{3}\). \(_\square\)

## Converting Fractions to Decimal

When converting fractions to decimals, simply divide the numerator by the denominator, and the quotient after performing the long division will be the decimal form of the fraction. Let's look at an example.

What is the decimal representation of \(\frac{1}{4}?\)

Performing long division when \(1\) is divided by \(4\), we get the quotient to be \(0.25\), which is the decimal representation of \(\frac{1}{4}\):

\[\frac{1}{4}=0.25. \ _\square\]

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

Performing long division when \(1\) is divided by \(6\), we find that, unlike the previous example, 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:

\[\frac{1}{6}=0.1\overline{6}. \ _\square\]

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, which is the decimal representation of \(\frac{1}{5}.\) Thus,

\[\frac{1}{5}=0.2. \ _\square\]

What is the decimal representation of \(\frac{1}{3}?\)

Performing long division when 1 is divided by 3, the quotient doesn't have a terminating point in that the 3 goes on forever. So, to represent a repeating number, we use a hyphen on the top as follows:

\[\frac{1}{3}=0.\overline{3}. \ _\square\]

## Converting Decimals to Fractions

When switching from a terminating decimal to 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 \(0.25?\)

We have

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

Multiply both the numerator and the denominator by 100, to move the decimal point of the numerator to the zero's place:

\[\frac{0.25\times100}{1\times 100}=\frac{25}{100}.\]

Simplify to get \(\frac{1}{4}.\ _\square\)

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. \(_\square\)

What is the fractional representation of 0.35?

We have

\[0.35=\frac{0.35}{1}.\]

Multiply both the numerator and the denominator by 100, to move the decimal point of the numerator to the zero's place, then

\[\frac{0.35\times100}{1\times 100}=\frac{35}{100}.\]

Simplify to get \(\frac{7}{20}. \ _\square\)

What is the fractional representation of 1.45?

We have

\[1.45=\frac{1.45}{1}.\]

Multiply both the numerator and the denominator by 100, to move the decimal point of the numerator to the zero's place, then

\[\frac{1.45\times100}{1\times 100}=\frac{145}{100}.\]

Simplify to get \(\frac{29}{20}. \ _\square\)

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

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