# Decimal to Fraction

Welcome to decimal to fraction, our website about converting decimals to fractions. Here you can find all you want to know about turning decimals into fractions, including the step by step instructions with examples. Make sure to also check out our decimals to fractions calculator towards the end of this article. We start with a few necessary definitions, but then come quickly to the point of converting decimals into fractions.

## What is a Decimal

A decimal number is a number in the base 10 positional notation system which contains a decimal point between the integer part and the fractional part.

For example, the decimal number 29.31 has 29 as integer part and 0.31 as a fractional part, separated by the decimal separator.

There are two kind of decimal numbers, irrational numbers and rational numbers:

An irrational number, e.g. pi (π), cannot be expressed as a fraction because it is a non-repeating and non-terminating decimal: 3.1415926535897932384626433832795…

In contrast, every rational number is terminating or repeating, and can be written as a quotient p/q of two integers, $\hspace{3px} p, q \hspace{3px}\epsilon\hspace{3px}\mathbb{Z},\hspace{3px} q\neq 0$.

P is called the nominator and q is called the denominator. The set of all rational numbers is written as $\hspace{3px}\mathbb{Q},\hspace{3px}$ and the set of all irrational (non-fractional) numbers is usually denoted $\hspace{3px}\mathbb{R-Q}\hspace{3px}$.

All numbers in $\hspace{3px}\mathbb{Q}\hspace{3px}$ are either terminating decimals like 0.125, or repeating decimals with an infinitely repeated pattern in the fractional part.

For example, 0.08333… is a repeating decimal which is usually written as 0.083; the overlined number 3 indicates that the sequence of 3s never ends.

The infinitely-repeated sequence of such a periodic decimal is called repetend or reptend. The repetend can be indicated as ellipsis 0.08333…, by a vinculum 0.083, as dots $0.08\dot{3}\hspace{3px}$ or with parentheses 0.08(3).

## Convert Decimal to Fraction

To convert decimals to fractions the decimals must be rational. Depending on the decimal fraction under consideration, proceed by using the corresponding method below:

### Convert a Terminating Decimal to Fraction

All terminating decimals can be changed to fractions by

1. Counting the number of decimal places
2. Putting the integer part, if any, aside, or using n + p/q = (nq + p)/q
2. Putting the decimal’s digits over 1 followed by the number of zeroes equal to the number of digits after the decimal point and finally adding the integer part, if any, back

For instance, 0.68 has two decimal places so we write 68/100. Using the greatest common divisor aka greatest common factor of 68 and 100, gcf(68/100) = 2, we can bring it to lowest form:

0.68 = $\frac{68}{100}$ = $\frac{\frac{68}{2}}{\frac{100}{2}}$ = $\frac{29}{50}$.

Along the same lines: 1.68 = 1 + $\frac{68}{100}$ = 1 + $\frac{\frac{68}{2}}{\frac{100}{2}}$ = $\frac{50}{50}$ + $\frac{29}{50}$ = $\frac{79}{50}$ .

Second example 2.5:

2.5 = 2 + 0.5 = 25/10 = $\frac{\frac{25}{gcf(25/10)}}{\frac{10}{gcf(25/10)}}$ = $\frac{\frac{25}{5}}{\frac{10}{5}}$ = $\frac{5}{2}$.

Make sure to understand what the gcf stands for by following the link above. If the gcf(p,q) of any fraction equals one, then the fraction is already in simplest form.

Our frequent, detailed conversions might as well serve as examples:

Now it’s your turn: Convert 1.55 to a fraction in lowest form and check your result using our calculator.

### Convert a Repeating Decimal to Fraction

All repeating decimals, which are non-terminating, can be changed to fractions, we a bit of algebra:

1. Multiply the number x so that the repeating sequence appears ones before the decimal point
2. Multiply the number x such that the repeating sequence starts right after the decimal point
3. Build the difference and divide it by the factor of x

Say we have a number x = 0.91666…

To achieve our first objective we have to multiply x by 1000: 1000x = 916.6666666…
To achieve the second objective we have to multiply x by 10: 10x = 91.6666666…

1000x = 916.66666…
-10x = 91.66666…
990x = 825

⇒ x = 825/900 = $\frac{\frac{825}{gcf(825/900)}}{\frac{900}{gcf(825/900)}}$ = $\frac{\frac{825}{75}}{\frac{900}{75}}$ = $\frac{11}{12}$.

In this second example we assume x = 0.83333… = 0.83:

100x = 83.33333…
-10x = 8.33333…
90x = 75

⇒ x = 75/90 = $\frac{\frac{75}{gcf(75/90)}}{\frac{90}{gcf(75/90)}}$ = $\frac{\frac{75}{15}}{\frac{90}{15}}$ = $\frac{5}{6}$.

Now, try to convert 0.07142857142857142857142857142857… = 0.0714285 using the algorithm above. Then compare you result using our calculator.

## Decimal to Fraction Calculator

If you like you can use our decimal to fraction calculator. It will convert any decimal to fraction as you type, be it either a negative or positive number.

Note that our calculator cannot only handle all rational numbers, repeating as well as terminating, but it will also present you an approximation if your decimal number is irrational.

Don’t forget to bookmark our tool now as decimal to fraction calculator.

You can also look for many decimals to fractions conversion by using the search form in the sidebar. And here you can find the inverse operation, fraction to decimal.

This brings us to the end of our article decimal to fraction. We have tried to explain you all on how to convert a decimal to a fraction, writing decimals as fractions and convert decimals to fractions in an straightforward way.

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