Rational Numbers: Rational Numbers are the numbers that can be expressed in the form of p/q or in between two integers where q is not equal to zero (q ≠ 0). The set of rational numbers also contains the set of integers, fractions, decimals, and more. All the numbers that can be expressed in the form of a ratio where the denominator is not one are referred to as rational numbers. For example, 9 is a rational number since it can be expressed as 9/1. Even 0 is a rational number. Some of the other examples are 2/3, 5/2, 5.907, and more.

Students will learn the different types of rational numbers and their related information and calculation in this article. This article also discusses the concept of Rational Numbers in detail, including their definition, classifications, and solved examples. Read on to know more about this topic.

What is a Rational Number?

Every rational number is defined as a number that can be expressed in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\)

Numerator and Denominator: In the given form \(\frac{{\rm{p}}}{{\rm{q}}},\) the integer \(p\) is the numerator and the integer \({\rm{q\;}}\left( { \ne 0} \right)\) is the denominator. So, in \(\frac{{ – 3}}{7}\) the numerator is \( – 3\) and the denominator is \(7.\)

Rational Numbers Examples

Some Examples of Rational Numbers are:

\(p\)	\(q\)	\(\frac{p}{q}\)	Rational
\(1\)	\(2\)	\(\frac{1}{2}\)	Yes
\( – 3\)	\(4\)	\(\frac{{ – 3}}{4}\)	Yes
\(\;0.3\)	\(1\)	\(\frac{3}{{10}}\)	Yes
\( – 0.7\)	\(1\)	\(\frac{{ – 7}}{{10}}\)	Yes
\(0.141414 \ldots .\)	\(1\)	\(\frac{{14}}{{99}}\)	Yes

How to Identify a Rational Number?

We know that a rational number can be expressed as a fraction or an integer. Every integer is a rational number. Each of these numbers is considered a rational number. Now, to identify whether the given number is a rational number or not we need to check with the following conditions:

1. We can represent the number as a fraction of integers like \(\frac{p}{q},\) where \(q \ne 0.\)
2. The ratio \(\frac{p}{q}\) can be simplified and represented in the decimal form which is either terminating or non-terminating recurring.

Types of Rational Numbers

1. Positive rational numbers: \(\frac{2}{5},\;0.2,\;6,\) are some examples of positive rational numbers. Here \(0.2\) can be written as \(\frac{1}{5}\) and \(6\) can be written as \(\frac{6}{1}.\)
2. Negative rational numbers: \( – \frac{2}{7},\; – 0.5,\; – 8,\) are some examples of negative rational numbers. Here \( – 0.5\) can be written as \(\frac{1}{2}\) and \( – 8\) can be written as \( – \frac{8}{1}.\)
3. Integer form of rational number: As we know that all integers are rational numbers because we can write them in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\;\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\) Example, \(6\) can be written as \(\frac{6}{1}.\)
4. Decimal form of rational number: Terminating and non-terminating recurring decimal numbers are rational numbers. Example, \(0.3\) is terminating decimal number which can be written as \(\frac{3}{{10}}\) and \(0.33333 \ldots \) is a non-terminating recurring decimal number which can be written as \(\frac{1}{3}.\)

Standard Form of Rational Numbers

Let us observe the rational numbers: \(\frac{3}{5},{\rm{\;}}\frac{{ – 5}}{8},{\rm{\;}}\frac{2}{7},{\rm{\;}}\frac{{ – 7}}{{11}}.\)

The denominators of those rational numbers are positive integers and \(1\) is the only common factor between the numerators and denominators. Further, the negative sign occurs only in the numerator. These rational numbers are said to be in standard form or simplest form or lowest form.

Definition: A rational number is claimed to be in its standard form if its denominator is a positive integer and therefore, the numerator and denominator do not have any common factor other than \(1.\)

If you remember the method of reducing the fractions to their lowest forms, we divide the numerator and the denominator of the fraction by the same nonzero positive integer. We shall use an equivalent method for reducing the rational numbers to their standard form.

Example: Reduce to standard form \(\frac{{36}}{{ – 24}}.\)

Solution: So, the HCF of the numbers \(36\) and \(24\) is \(12.\). So, the standard form can be obtained by dividing the given fraction by \( – 12.\)

\(\frac{{36}}{{ – 24}} = \frac{{36 \div \left( { – 12} \right)}}{{ – 24 \div \left( { – 12} \right)}} = \frac{{ – 3}}{2}\)

Positive and Negative Rational Numbers

Rational numbers can be differentiated as positive and negative rational numbers. When the numerator and the denominator are both positive or negative, it’s referred to as a positive rational number. When one of the numerators or the denominator is a positive integer, and the other is a negative integer, it is called a negative rational number.

Positive Rational Numbers	Negative Rational Numbers
When both the numerators and the denominators are of the same sign then it is called a positive rational number. Example: \(\frac{3}{8}\) is a positive rational number.	When both the numerator and the denominator are of different signs then they are known as negative rational numbers. Example: \(\frac{{ – 8}}{9}\) is a negative rational number.
All the numbers are greater than zero.	All the numbers are less than zero.

Properties of Rational Numbers

A rational number is the subset of the real number which will obey all the properties of the real number system. A few of the important properties are as follows:

1. Whenever we multiply, add, subtract, or divide any two rational numbers the result is in always a rational number.
2. Rational number remains the same when we divide or multiply the numerator and the denominator with same number.
3. When we add zero to any rational number, we get the same number as the result.
4. Rational numbers are closed under the subtraction, addition, and multiplication.

Are Integers Also Rational Numbers?

What are Equivalent Rational Numbers

A rational number can be written using different numerators and denominators. For example: we will take the rational number \(\frac{{ – 2}}{3}.\)
\(\frac{{ – 2}}{3} = \frac{{ – 2 \times 2}}{{3 \times 2}} = \frac{{ – 4}}{6}.\) We can see that \(\frac{{ – 2}}{3}\) is same as \(\frac{{ – 4}}{6}.\)

Also, \(\frac{{ – 2}}{3} = \frac{{\left( { – 2} \right) \times \left( { – 5} \right)}}{{3 \times \left( { – 5} \right)}} = \frac{{10}}{{ – 15}}.\) So, \(\frac{{ – 2}}{3}\) is also same as \(\frac{{10}}{{ – 15}}.\)
Thus, \(\frac{{ – 2}}{3} = \frac{{ – 4}}{6} = \frac{{10}}{{ – 15}},\)
So such rational numbers that are equal are known as equivalent rational numbers.

Arithmetic Operations on Rational Numbers

As you already know how to add, subtract, multiply or divide the integers as well as fractions. Now, let us see these basic operations on rational numbers.

Addition: When adding the rational numbers with the same denominator, add only the numerators by keeping the denominator the same. Two rational numbers with the different denominators are added by taking out the LCM of the two denominators and then convert both the rational numbers to their equivalent forms to have the LCM as the denominator.

Example: \(\frac{{ – 2}}{3} + \frac{3}{8} = \frac{{ – 16}}{{24}} + \frac{9}{{24}} = \frac{{ – 16 + 9}}{{24}} = \frac{{ – 7}}{{24}}.\) So, the LCM of the numbers \(3\) and \(8\) is \(24.\)

Subtraction: While subtracting two rational numbers, we add the additive inverse of the rational number that’s being subtracted from the other rational number.

Example: \(\frac{7}{8} – \frac{2}{3} = \frac{7}{8} + \) additive inverse of \(\frac{2}{3} = \frac{7}{8} + \frac{{\left( { – 2} \right)}}{3} = \frac{{21 + \left( { – 16} \right)}}{{24}} = \frac{5}{{24}}.\)

Multiplication: To multiply the two rational numbers, multiply their numerators and denominators separately and write the product as \(\frac{{{\rm{product\;of\;numerators}}}}{{{\rm{product\;of\;denominators}}}}.\)

When you multiply the rational number with a positive integer, then multiply the numerator by that integer, by keeping the denominator unchanged.

Here, multiply a rational number by the negative integer:

\(\frac{{ – 2}}{9} \times \left( { – 5} \right) = \frac{{ – 2 \times \left( { – 5} \right)}}{9} = \frac{{10}}{9}\)

Division: To divide one rational number by the other non-zero rational number, we multiply the rational number by the reciprocal of the other rational number.

Thus \(\frac{{ – 7}}{2} \div \frac{4}{3} = \frac{{ – 7}}{2} \times \left( {{\rm{reciprocal\;of}}\;\frac{4}{3}} \right) = \frac{{ – 7}}{2} \times \frac{3}{4} = \frac{{ – 21}}{8}.\)

Rational Numbers on a Number Line

Here, we will learn how to represent numbers on the number line, so let us draw a number line.

So, here the points to the right of the \(0\) are denoted by \( + \) sign and are positive numbers. The point to the left of \(0\) are denoted by \( – \) sign and are negative numbers.

So, represent the number \( – \frac{1}{2}\) on the number line.

As rational number \( – \frac{1}{2}\) is negative it will be marked on the left side of the \(0.\)

So, while marking the integers on the number line, successive integers are marked at equal intervals. Also, from \(0\) the pair \(1\) and \( – 1\) is at same distance. So, are the pairs \(2\) and \( – 2,3\) and \( – 3\) and so on.

Similarly, the rational numbers \(\frac{1}{2}\) and \( – \frac{1}{2}\) would be at equal distance from \(0.\) Now, you know how to mark the rational number \(\frac{1}{2}.\) This is marked at a point which is half the distance between \(0\) and \(1\) So, \( – \frac{1}{2}\) is marked at a point half the distance between \(0\) and \( – 1.\)

Now, try to mark \( – \frac{3}{2}\) on the number line. It lies on the left side of \(0\) and is the same distance as \(\frac{3}{2}\) from \(0.\) In decreasing order, we have \(\frac{{ – 1}}{2},{\rm{\;\;}}\frac{{ – 2}}{2}\left( { = – 1} \right),{\rm{\;\;}}\frac{{ – 3}}{2},{\rm{\;\;}}\frac{{ – 4}}{2}\left( { = – 2} \right),\) which shows that \(\frac{{ – 3}}{2}\) lies between \( – 1\) and \( – 2.\) Thus, \(\frac{{ – 3}}{2}\) lies halfway between \( – 1\) and \( – 2.\)

All the other rational numbers which are with different denominators can be represented in the same way.

Relation Between Rational Number and Irrational Number

We know that the numbers which are not rational are known as irrational numbers. The comparison between the rational and the irrational number has been given below:

Rational Numbers	Irrational Numbers
Rational numbers are the numbers that can be expressed as fractions of integers. Examples: \(0.75,{\rm{\;\;}}\frac{{ – 31}}{5}\)	Irrational numbers are numbers that cannot be expressed as fractions of integers. Example: \(\surd 2,\pi .\)
These numbers can be terminating decimals.	These numbers can never be terminating decimals.
Rational numbers can be non-terminating decimals with repetitive patterns of decimals.	Irrational numbers always have non-terminating decimal expansions with no repetitive patterns of decimals.
The set of rational numbers contains natural numbers, whole numbers, and integers.	The set of irrational numbers is a separate set that does not contain any of the other sets of numbers.

Look at the given diagram for a better understanding.

Multiplicative Inverse of Rational Number

A rational number \(\frac{q}{p}\) is known as multiplicative inverse or reciprocal of \(\frac{p}{q}\) and is denoted by \({\left( {\frac{p}{q}} \right)^{ – 1}}….\) The numbers \(1\) and \( – 1\) are the only rational numbers that are their own reciprocals. No other rational number is its own reciprocal. Rational number \(0\) has no multiplicative inverse.
Example: Number is \(\frac{2}{8}\) its multiplicative inverse is \(\frac{8}{2}.\)

How to Identify the Rational Number Between Two Rational Numbers?

The number of integers between two integers is limited (finite). Will the same happen in the case of rational numbers too? We will see that below.

He converted them into rational numbers with the same denominators.
So, \(\frac{{ – 3}}{5} = \frac{{ – 9}}{{15}}\) and \(\frac{{ – 1}}{3} = \frac{{ – 5}}{{15}}\)

 

We have \(\frac{{ – 9}}{{15}} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}} < \frac{{ – 5}}{{15}}\) or \(\frac{{ – 3}}{5} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}} < \frac{{ – 1}}{3}\)

 

He could find rational numbers \(\frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}}\) between \(\frac{{ – 3}}{5}\) and \(\frac{{ – 1}}{3}.\)

 

Are the numbers \(\frac{{ – 8}}{{15}},{\rm{\;}}\frac{{ – 7}}{{15}},{\rm{\;}}\frac{{ – 6}}{{15}}\) the only rational numbers between \( – \frac{3}{5}\) and \( – \frac{1}{3}?\)

 

We have \(\frac{{ – 3}}{5} = \frac{{ – 18}}{{30}}\) and \(\frac{{ – 8}}{{15}} = \frac{{ – 16}}{{30}}\)

 

And \(\frac{{ – 18}}{{30}} < \frac{{ – 17}}{{30}} < \frac{{ – 16}}{{30}}.\) i.e., \(\frac{{ – 3}}{5} < \frac{{ – 17}}{{30}} < \frac{{ – 8}}{{15}}\)

 

Hence, \(\frac{{ – 3}}{5} < \frac{{ – 17}}{{30}} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}} < \frac{{ – 1}}{3}\)

 

So, he could find one more rational number between \(\frac{{ – 3}}{5}\) and \(\frac{{ – 1}}{3}.\)

 

By using the above method, we can insert as many rational numbers as we want between two different rational numbers. We can find an unlimited number of rational numbers between any two rational numbers.

Solved Examples

1. Find four rational numbers equivalent to the given rational number: \(\frac{3}{4}\)
Solution: We have\(\frac{3}{4} = \frac{{3 \times 2}}{{4 \times 2}} = \frac{{3 \times 3}}{{4 \times 3}} = \frac{{3 \times 4}}{{4 \times 4}} = \frac{{3 \times 5}}{{4 \times 5}}\)
Therefore, \(\frac{3}{4} = \frac{6}{8} = \frac{9}{{12}} = \frac{{12}}{{16}} = \frac{{15}}{{20}}\)
Thus, the four rational numbers equivalent to \(\frac{3}{4}\) are \(\frac{6}{8},\;\frac{9}{{12}},\;\frac{{12}}{{16}}\) and \(\frac{{15}}{{20}}.\)

2. Write each of the following rational numbers with positive denominators:
\(\frac{3}{{ – 8}},\;\frac{7}{{ – 12}},\;\frac{{ – 5}}{{ – 2}},\;\frac{{ – 13}}{{ – 8}}\)
Solution: We have \(\frac{3}{{ – 8}} = \frac{{3 \times \left( { – 1} \right)}}{{\left( { – 8} \right) \times \left( { – 1} \right)}} = \frac{{ – 3}}{8};\),
\(\frac{7}{{ – 12}} = \frac{{7 \times \left( { – 1} \right)}}{{\left( { – 12} \right) \times \left( { – 1} \right)}} = \frac{{ – 7}}{{12}};\)
\(\frac{{ – 5}}{{ – 2}} = \frac{{\left( { – 5} \right) \times \left( { – 1} \right)}}{{\left( { – 2} \right) \times \left( { – 1} \right)}} = \frac{5}{2};\)
\(\frac{{ – 13}}{{ – 8}} = \frac{{\left( { – 13} \right) \times \left( { – 1} \right)}}{{\left( { – 8} \right) \times \left( { – 1} \right)}} = \frac{{13}}{8}\)
3. Express \(\frac{{ – 5}}{{13}}\) as a rational number with positive numerator.
Solution: We have:
\(\frac{{ – 5}}{{13}} = \frac{{\left( { – 5} \right) \times \left( { – 1} \right)}}{{13 \times \left( { – 1} \right)}} = \frac{5}{{ – 13}}\)
4. Express \(\frac{{ – 4}}{7}\) as a rational number with (i) numerator \(\; = – 12\) and (ii) numerator \( = 20.\)
Solution: (i) Numerator of \(\frac{{ – 4}}{7}\) is \( – 4\)
By what number should we multiply \(\left( { – 4} \right)\) to get \(\left( { – 12} \right)\)?
Clearly such number is \(\left( { – 12} \right) \div \left( { – 4} \right) = 3\)
So, we multiply its numerator and denominator by \(3.\)
Therefore, \(\frac{{ – 4}}{7} = \frac{{\left( { – 4} \right) \times 3}}{{7 \times 3}} = \frac{{ – 12}}{{21}}.\)
Hence, \(\frac{{ – 4}}{7} = \frac{{ – 12}}{{21}}\)
(ii) Numerator of \(\frac{{ – 4}}{7}\) is \( – 4.\)
By what number should we multiply \(\left( { – 4} \right)\) to get \(20\)?
Clearly, such number is \(\left( {20} \right) \div \left( { – 4} \right) = – 5.\)
Therefore, \(\frac{{ – 4}}{7} = \frac{{\left( { – 4} \right) \times \left( { – 5} \right)}}{{7 \times \left( { – 5} \right)}} = \frac{{20}}{{ – 35}}.\)
Hence, \(\frac{{ – 4}}{7} = \frac{{20}}{{ – 35}}.\)
5. Find \(x\) such that \(\frac{{ – 3}}{8}\) and \(\frac{x}{{ – 24}}\) are equivalent rational numbers.
Solution: It is given that \(\frac{{ – 3}}{8} = \frac{x}{{ – 24}}\)
Therefore, \(\frac{{ – 3}}{8} = \frac{x}{{ – 24}}\)
So, \(8 \times x = \left( { – 3} \right) \times \left( { – 24} \right)\)
So, \(8 \times x = 72\)
So, \(x = \frac{{72}}{8} = 9\)
Hence, \(x = 9\)

FAQs on Rational Numbers

We have answered the most frequently asked questions on Rational Numbers below:

Question 1: Is \(7\) a rational number? Answer: \(7\) is a rational number as it can be written as \(\frac{7}{1}.\)

Question 2: Is \(0\) a rational number? Answer: A rational number is defined as the number that can be expressed in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are integers and \({\rm{q}} \ne 0.\) We can express \(0\) as \(\frac{0}{1}\) which is in the form of \(\frac{p}{q}\) where \(p\) is equal to zero and \(q\) is \(1\) or any integer.

Question 3: What is an irrational number? Answer: An irrational number is a real number which cannot be written in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\) Example of irrational numbers are \(\pi\), ratio of a circle’s circumference to its diameter, Euler’s number e, the golden ratio \({\rm{\varphi }},\) and the square root of two. All square roots of natural numbers are, other than perfect squares, are irrational.

Question 5: Is \(3.14\) a Rational number? Answer: Yes, the number \(3.14 = \frac{{314}}{{100}}\) is a rational number.

Question 6: Is \(\frac{1}{3}\) a rational or irrational number? Answer: \(\frac{1}{3}\) is a rational number.

Question 7: How to identify the rational numbers? Answer: To identify whether the given number is a rational number or not we need to check if we can represent the number in the form of \(\frac{{\rm{p}}}{{\rm{q}}},\) where \(p\) and \(q\) are co-prime integers and \({\rm{q}} \ne 0.\) \(p\) \(q\) \(\frac{p}{q}\) Rational \(1\) \(2\) \(\frac{1}{2}\) Yes \( – 3\) \(4\) \(\frac{{ – 3}}{4}\) Yes \(\;0.3\) \(1\) \(\frac{3}{{10}}\) Yes \( – 0.7\) \(1\) \(\frac{{ – 7}}{{10}}\) Yes \(0.141414 \ldots .\) \(1\) \(\frac{{14}}{{99}}\) Yes

Also Check,

Integers	Even Numbers	Whole Numbers
Composite Numbers	Real Numbers	Natural Numbers
Co Prime Numbers	Odd Numbers	Prime Numbers

Now you have detailed information on Rational Numbers. When you prepare for exams, ensure that you understand all the concepts, topics, and chapters on time. Embibe provides NCERT Solutions For Class 8 Maths Chapter 1 and NCERT Books For Class 8 Maths, both of which will provide ample understanding and practice for Rational Numbers. You can also take Class 8 Maths Mock Test to improve your score in Mathematics.

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