Decimal Representation of Rational Numbers: Representation and Examples

Decimal Representation of Rational Numbers: A number in the form \(\frac{p}{q}\) or a number that can be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers, and \(q \ne 0\) is called a rational number. There are two types of decimal representation of rational numbers such as terminating and non-terminating repeating.

The non-terminating decimal form of a rational number could be a recurring decimal only. To represent these decimal forms, we need to use the number lines. There are some unique procedures to represent the rational number into its decimal form. In this article, we will see the decimal representations of rational numbers.

Decimal Expansion of Rational Numbers

Definition of Rational Number

The numbers of the form \(\frac{p}{q}\) here \(p\) and \(q\) are integers, and \(q \ne 0\) are called rational numbers.

Examples of Rational Numbers

Each of the numbers \(\frac{3}{7},\,\frac{{ – 4}}{{15}},\,\frac{{ – 8}}{5},\,\frac{{12}}{9}\) are rational numbers, \(0\) is a rational number since we can write \(0 = \frac{0}{1}\). The natural numbers, \(1 = \frac{1}{1},\,2 = \frac{2}{1},\,3 = \frac{3}{1}\) and so on are rational numbers. In the same way, every integer is a rational number. If \(m\) is an integer, then we can write it as \(\frac{m}{1}\), which is a rational number. Every fraction is a rational number. Let \(\frac{a}{b}\) be a fraction. Then, \(a\) and \(b\) are whole numbers and \(b \ne 0\).

Types of Decimal Representation of Rational Numbers

There are two different types of the decimal representation of rational numbers. They are:

1. Terminating decimals
2. Non-terminating recurring decimals

Terminating Decimal Numbers

The decimal numbers having finite numbers of digits after the decimal point are the terminating decimal numbers. Their number of decimal places is finite. These decimal numbers are called exact decimal numbers.
We can represent these decimal numbers in \(\frac{p}{q}\)  form where \(q \ne 0\) or we can represent decimal numbers as rational numbers.

For example, \(2.665\) are the terminating decimal number.

\(2.665\) is represented as \(\frac{{2665}}{{1000}}\), when \(p = 2665\) and \(q = 1000\). So, \(2.665\) is the decimal form of the rational number \(\frac{{2665}}{{1000}}\).

Conversion of Rational Numbers into Terminating Decimals

A rational number is terminated if it can be represented as \(\frac{p}{{{2^n} \times {5^m}}}\). The rational number with no prime factors in its denominator other than \(2\) and \(5\) gives a terminating decimal number. Consider the example \(\frac{{533}}{{200}}\).

Now, the denominator is \(200\) that means \({2^3} \times {5^2}\). To make the denominator \(10′{\rm{s}}\) power, we need to multiply the denominator and the numerator by \(5\).
So,  \(\frac{{533 \times 5}}{{{2^3} \times {5^3}}} = \frac{{2665}}{{1000}} = 2.665\)

Terminating Decimal Representation of a Rational Number Using Number Line

Let us represent this decimal on the number lines.

Now, let us look closely at the portion of the number line between \(2\) and \(3\).

Suppose we divide this into \(10\) equal parts and mark each point of division as the above figure. Then the first mark to the right of \(2\) will represent \(2.1\), the second \(2.2\), and so on. You might find some difficulty in observing these division points between \(2\) and \(3\) in the above figure. To have a clear view, in the same way, you may take a magnifying glass and look at the portion between \(2\) and \(3\).

Now, \(2.665\) lies between \(2.6\) and \(2.7\). So, let us focus on the portion between \(2.6\) and \(2.7\). We imagine dividing this again into \(10\) equal parts and mark each point of division as shown in the figure. The first mark will represent \(2.61\), the next \(2.62\), and so on. To see this clearly, we magnify this.

Again, \(2.665\) lies between \(2.66\) and \(2.67\). So, let us concentrate on this portion of the number line and imagine splitting it again into \(10\) identical parts. We again magnify it to observe more clearly. The first mark represents \(2.661\), the next one representing \(2.662\) and so on. So, \(2.665\) is the \({5^{{\rm{th}}}}\) mark in the subdivision.

We call this process of visualising the representation of numbers on the number line as successive magnification through a magnifying glass. So, we have noticed that it is possible by sufficient consecutive magnifications to think about the position of a rational number with a terminating decimal representation on the number line.

Now, let us try and imagine the position of rational numbers with a non-terminating recurring decimal representation on the number line. We can see at appropriate intervals with a magnifying glass and visualise the position of the number on the number line by successive or continuous magnifications.

Non-Terminating Recurring Decimal Numbers

The decimal numbers having infinite numbers of digits after the decimal point are known as non-terminating. The decimal numbers have infinite numbers of digits after the decimal point. The digits are repetitive at the same intervals after the decimal point are known as the recurring decimal numbers.

For example, \({\rm{0}}{\rm{.111}}…{\rm{,}}\,{\rm{4}}{\rm{.444444}}…{\rm{,}}\,{\rm{5}}{\rm{.232323}}…{\rm{,}}\,{\rm{21}}{\rm{.123123}}…\) etc., are the recurring decimals. We can represent recurring decimal numbers in \(\frac{P}{q}\)  form where \(q \ne 0\) or we can represent these decimal numbers as rational numbers.

Conversion of Rational Numbers into Recurring Decimals

Let us take an example and understand. Method of finding the decimal expansion of  \(\frac{1}{3}\) is:

Divide the numerator by the denominator.

The decimal expansion of \(\frac{1}{3}\) is \(0.333333\). Here, the remainder is \(1\) in every step, and the divisor is \(3\).

Recurring Decimal Representation of a Rational Number Using Numberline

Let us represent these decimals on the number lines. Now, we continue with successive magnification and successively decrease the lengths of the sections of the number line in which \(5.3\overline 7 \) is situated. \(5.3\overline 7 \) is the decimal form of the rational number. First, we see that \(5.3\overline 7 \) is located between \(5\) and \(6\). In the next step, we locate \(5.3\overline 7 \) between \(5.3\) and \(5.4\).

To get a clearer exact visualisation of the representation, we split up this portion of the number line into \(10\) equal parts and use a magnifying glass to observe that \(5.3\overline 7 \) lies between \(5.377\) and \(5.378\). Now to observe \(5.3\overline 7 \)  more accurately, we divide the section between \(5.377\) and \(5.378\) into \(10\) identical parts and observe the representation of \(5.3\overline 7 \) in the given figure. Observe that \(5.3\overline 7 \) is situated nearer to \(5.3778\) than to \(5.3777\).

Note: We can continue endlessly in this way, successively viewing through a magnifying glass and simultaneously imagining the decrease in the length of the portion of the number line in which \(5.3\overline 7 \) is located. The size of the portion of the line we specify depends on the degree of accuracy we would like for the visualisation of the position of the number on the number line.

Solved Examples – Decimal Representation of Rational Numbers

Q.1. Represent \(\frac{{28}}{{10}}\) on a number line.
Ans: Given, a rational number is \(\frac{{28}}{{10}}\).
The decimal form of  \(\frac{{28}}{{10}}\) is \(2.8\)
 Now, let us look closely at the portion of the number line between \(2\) and \(3\).

Suppose we divide this into \(10\) equal parts and mark each point of division as the above figure. Then the first mark to the right of \(2\) will represent \(2.1\), the second \(2.2\), and so on. You might find some difficulty in observing these division points between \(2\) and \(3\) in the above figure. To have a clear view, in the same way, you may take a magnifying glass and look at the portion between \(2\) and \(3\).
Thus, we can mark \(2.8\) on the number line.

Q.2. Find the decimal form of a rational number \(\frac{3}{4}.\)
Ans: A rational number is terminating if it can be represented as \(\frac{p}{{{2^n} \times {5^m}}}\). The rational number whose denominator has no factors other than \(2\) and \(5\) gives a terminating decimal number. Now, in \(\frac{3}{4}\) the denominator is \(4\) that means \({2^2}\). To make the denominator \(10′{\rm{s}}\) power, we need to multiply the denominator and the numerator by \({5^2}\).
So,  \(\frac{{3 \times {5^2}}}{{{2^2} \times {5^2}}} = \frac{{75}}{{100}} = 0.75\)
Hence, the decimal expansion form is \(0.75\).

Q.3. Find the decimal form of a rational number \(\frac{{10}}{3}.\)
Ans: A rational number is a terminating decimal if it can be represented as \(\frac{p}{{{2^n} \times {5^m}}}\). The rational number whose denominator has prime factors other than \(2\) and \(5\) gives a non-terminating recurring decimal number. Now, in \(\frac{{10}}{3}\) the denominator is \(3\) that is not the factor of \(2\) and \(5\).
Therefore, to find its decimal form, we need to apply the long division method.
For example, consider \(\frac{{10}}{3}\).

We got \(\frac{{10}}{3} = 3.333 \ldots \) 
In \(3.333 \ldots \), the period is \(3\) and the periodicity is \(1.\)
Hence, the decimal expansion form is \(3.\overline 3 .\)

Q.4. Represent \(\frac{3}{2}\) into the decimal form on a number line.
Ans:

Suppose we divide this into \(10\) equal parts and mark each point of division as the above figure. Then the first mark to the right of \(1\) will represent \(1.1\), the second \(1.2\), and so on. You might find some difficulty in observing these division points between \(1\) and \(2\) in the above figure. To have a clear view, in the same way, you may take a magnifying glass and look at the portion between \(1\) and \(2\).
Thus, we can mark \(1.5\) on the number line.

Q.5. Find the decimal form of a rational number \(\frac{2}{{25}}.\)
Ans: A rational number is terminated if it can be represented as \(\frac{p}{{{2^n} \times {5^m}}}\). The rational number whose denominator has no factors other than \(2\) and \(5\) gives a terminating decimal number. Now, in \(\frac{2}{{25}}\) the denominator is \(25\) that means \({{5^2}}\). To make the denominator \(10′{\rm{s}}\) power, we need to multiply the denominator and the numerator by \({{2^2}}\).
So,  \(\frac{{2 \times {2^2}}}{{{5^2} \times {2^2}}} = \frac{8}{{100}} = 0.08\)
Hence, the decimal expansion form is \(0.08\).

Summary

A rational number can be represented as a decimal number. There are two types of decimal representation of rational numbers as terminating and recurring decimal numbers. To represent the decimal forms of rational numbers, we should use the number lines. There are some specific rules to represent the rational number into its decimal form. This article explained the decimal representations of rational numbers and showed some examples of decimal representations of rational numbers.

Learn All the Concepts on Rational Numbers

Frequently Asked Questions (FAQs)

Q.1. How do you find the decimal of a rational number?
Ans: Using the long division method, we can find the decimal form of any rational number. There is a special method to find the terminating decimal expansion of a rational number whose denominator has no prime factors other than \(2\) and \(5\) gives a terminating decimal number. The rational number whose denominator has prime factors other than \(2\) and \(5\) gives a non-terminating recurring decimal.

Q.2. What is the decimal representation of an irrational number?
Ans: The decimal representation of an irrational number is always non-terminating and non-recurring decimal.

Q.3. What are the types of decimal expansion?
Ans: There are two different types of decimal expansion. They are,
1. Terminating decimals
2. Non-terminating decimals
(a) Non-terminating recurring decimal
(b) Non-terminating non-recurring decimals

Q.4. What are rational numbers?
Ans: The numbers of the form \(\frac{p}{q},\,w\) here \(p\) and \(q\) are integers, and \(q \ne 0\) are called rational numbers.

Q.5. What type of decimal representation does a rational number have?
Ans: There ate two types of decimal representations of rational numbers: they are terminating decimal representations and non-terminating recurring decimals.