# Decimal Representation of Rational Numbers: Definition, Types, Facts

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## What Is the Decimal Representation of Rational Numbers?

Decimal representation of rational numbers is a method of finding decimal expansion of the given rational number or converting a rational number into an equivalent decimal number using long division.

Decimal form of a rational number example: $\frac{5}{4} = 5\div4= 1.25$

Rational numbers are the numbers of the form $\frac{p}{q}$, where p and q are integers and $q\neq0$. Decimals are the numbers that have a whole number part and a fractional part separated by a decimal point. Every rational number is either a terminating or repeating decimal.

## Writing Rational Numbers in the Decimal Form

To convert the rational number $\frac{p}{q}$ into a decimal, we divide the number p by the number q using the long division process. Writing rational numbers in decimal form involves two cases. The decimal form of a rational number we get can be of 2 types:

1. When Remainder $= 0$, the decimal expansion is terminating.

Terminating decimal is the decimal number in which there is an end-digit. There are a finite number of digits after the decimal point.

For example, if we divide 5 by 1, we get 0.2. The number terminates and doesn’t continue after 2. So, it’s a terminating number.

1. When Remainder $\neq0$, the decimal expansion is non-terminating and repeating.

In a non-terminating and repeating decimal, a single digit or a block of digits repeat themselves infinitely after the decimal point.

For example, we get $0.\overline{09}=0.09090909$ … on dividing 1 by 11. Here, the group of digits, 09, keep on repeating.

So, how are rational numbers written as decimals? Let’s discuss both the cases.

## Terminating Decimal Representation of Rational Numbers

When the decimal expansion of $\frac{p}{q}$, $q\neq 0$ comes to an end after a finite number of digits, the decimal expansion is called terminating. In this case, when we divide p by q using the long division method, we get remainder 0.

Example 1: Find the decimal representation of the rational number $\frac{4}{5}$ .

Thus, $\frac{4}{5} = 0.8$ … terminating decimal expansion

Example 2: Find the decimal representation of the rational number $\frac{3}{4}$.

Here, $\frac{3}{4} = 0.75$ … terminating decimal expansion

## Non-terminating and Repeating Decimal Representation of Rational Numbers

In some cases, when we divide p by q to find the decimal expansion of $\frac{p}{q}$, $q\neq 0$, the remainder never becomes 0. Also, the remainder repeats after some steps, which results in a non-terminating and recurring decimal expansion.

Example 1: $\frac{10}{3} = 3.333$…

Example 2: $\frac{7}{11} = 0.636363$ …

## Facts about Decimal Representation of Rational Numbers

• A rational number has either a terminating decimal expansion or a non-terminating and recurring (repeating) decimal expansion.
• The converse of the above statement is also true. If the decimal expansion of a number is terminating or non-terminating and recurring (repeating), then the number is a rational number.
• The decimal expansion of an irrational number is non-terminating and non-repeating.

## Conclusion

In this article, we learned about the two types of decimal expansions of rational numbers. Let’s solve a few examples and practice problems for revision.

## Solved Examples on Decimal Representation of Rational Numbers

1. Express $\frac{3}{8}$ in the form of a decimal.

Solution:

Divide 3 by 8 using the long division method.

The rational number $\frac{3}{8}$ has a terminating decimal expansion.

$\frac{3}{8} = 0.375$

1. What is the decimal expansion of the rational number $\frac{20}{11}$?

Solution:

Divide 20 by 11.

The rational number $\frac{20}{11}$ has a non-terminating and recurring decimal expansion.

$\frac{20}{11} = 0.181818$…

1. Write the rational numbers in the decimal form?