Decimal Representation of Rational Numbers: Definition, Types, Facts

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.

2. 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: $\frac{7}{11} = 0.636363$ …

Decimal Form of a Rational Number Chart

Rational Number	Decimal form
$\frac{1}{2}$	0.5
$\frac{99}{2}$	49.5
$\frac{1345}{100}$	13.45
$\frac{1}{3}$	0.333…
$\frac{5}{9}$	0.5555…
$\frac{6}{5}$	1.2
$\frac{8}{100}$	0.08

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.  

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

2. 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. 

$\frac{20}{11} = 0.181818$…

3. Write the rational numbers in the decimal form?

i) $\frac{2}{10}$ ii) $\frac{174}{100}$ ii) $\frac{56}{1000}

Solution:

i) $\frac{2}{10} = 0.2$

ii) $\frac{174}{100} = 1.74$

iii) $\frac{56}{1000} = 0.056$

Practice Problems on Decimal Representation of Rational Numbers

Frequently Asked Questions on Decimal Representation of Rational Numbers