Rational Numbers

In Maths, rational numbers are represented in p/q form where q is not equal to zero. It is one of the most important Maths topics. Any fraction with non-zero denominators is a rational number. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational. Also, learn irrational numbers here.

We will learn here the properties of rational numbers along with with its types, the difference between rational and irrational numbers. Solved examples help to understand the concepts in a better way. Also, learn the various rational numbers examples and learn how to find the rational numbers in a better way. To represent rational numbers on a number line, we need to simplify and write in decimal form first.

Let us see what topics we are going to cover here in this article.

Table of contents:

Rational Number Definition

A rational number, in Mathematics, can be defined as any number which can be represented in the form of p/q where q is greater than 0. Also, we can say that any fraction fit under the category of rational numbers, where denominator and numerator are integers and the denominator is not equal to zero.

How to identify rational numbers?

To identify if a number is rational or not, check the below conditions.

  • It is represented in the form of p/q, where q≠0.
  • The ratio p/q can be further simplified and represented in decimal form.

The set of rational numerals are:

  1. Include positive, negative numbers, and zero
  2. Can be expressed as a fraction

Examples of Rational Numbers: 

p

q p/q

Rational/Irrational Number

10

2 10/2 =5

Rational 

1

1000 1/1000 = 0.001

Rational 

50

10 50/10 = 5

Rational 

7

0 7/0

Irrational (because q=0)

Types of Rational Numbers

A number is rational if we can write it as a fraction, where both denominator and numerator are integers.

Below diagram helps us to understand more about the number sets.

Rational Numbers Definition

  • Real numbers (R) include all the rational numbers (Q).
  • Real numbers include the integers (Z).
  • Integers involves the natural numbers(N).
  • Every whole number is a rational number because every whole number can be expressed as a fraction.

Standard Form of Rational Numbers

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.

For Example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number ⅓ is in standard form.

Positive and Negative Rational Numbers

Positive Rational Numbers Negative Rational Numbers
If both the numerator and denominator are of same signs. If numerator and denominator are of opposite signs.
All are greater than 0 All are less than 0
Example: 12/17, 9/11 and 3/5 are positive rational numbers Example: -2/17, 9/-11 and -1/5 are negative rational numbers

Arithmetic Operations on Rational Numbers

In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how can we perform these operations on rational numbers, say p/q and s/t.

Addition: When we add p/q and s/t, we need to make the denominator same. Hence, we get (pt+qs)/qt.

Example: 1/2 + 3/4 = (2+3)/4 = 5/4

Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.

Example: 1/2 – 3/4 = (2-3)/4 = -1/4

Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).

Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8

Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/qs

Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3

Rational Numbers Properties

  • The result of two rationals is always a rational number is we multiply, add or subtract them.
  • A rational number remains the same is we divide or multiply both numerator and denominator with the same number.
  • Sum of zero and a rational number revert the same number itself.

Learn more properties of rational numbers here.

Rational Numbers and Irrational Numbers

There is a difference between rational and Irrational Numbers. A fraction with non-zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by the integer 2. All the numbers that are not rational are called irrational.

Rational Numbers and Irrational Numbers

Rationals can be either positive, negative or zero. While specifying a negative rational number, the negative sign either in front or with the numerator and that is the standard mathematical notation. For example, we denote negative of 5/2 as -5/2.

An irrational number cannot be written as a fraction but can be written as a decimal. It has endless non-repeating digits after the decimal point. Some of the irrational numbers are mentioned below.

π = 3.142857…

√2 = 1.414213…

Rational Numbers Examples

Example 1:

Identify each of the following as irrational or rational: ¾ , 90/12007, 12 and √5.

Solution:

Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both denominator and numerator are whole numbers.

  • ¾ is a rational number as it can be expressed as a fraction.
  • Fraction 90/12007 is rational.
  • 12, also be written as 12/1. Again a rational number.
  • Value of  √5 = 2.2360…. , does not end. Cannot be written as a fraction. It is an irrational number.

Example 2:  

Identify whether mixed fraction, 1½ is a rational number.

Solution: 

The Simplest form of 1½ is 3/2

Numerator = 3, which is an integer

Denominator = 2, is an integer and not equal to zero.

So, yes, 3/2 is a rational number.

Example 3:

Determine whether the given numbers are rational or irrational.

(a) 1.75 (b) 0.01   (c) 0.5  (d) 0.09   (d) √ 3

Solution:

The given numbers are in decimal format. To find whether the given number is decimal or not, we have to convert it into the fraction form (i.e.,) p/q

If the denominator of the fraction is not equal to zero, then the number is rational, or else, it is irrational.

Decimal Number Fraction Rational Number

1.75 

7/4

yes

0.01

1/100

yes

0.5

1/2

yes

0.09

1/11

yes

√ 3

?

No 

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Frequently Asked Questions on Rational Numbers

How to identify the rational numbers?

If a number that can be expressed in the form of p/q is called rational numbers.
Here p and q are integers, and q is not equal to 0. A rational number should have a numerator (p) and denominator (q). Examples: 10/2, 30/3, 100/5.

What is the difference between rational and irrational numbers?

A rational number is a number that is expressed as the ratio of two integers, where the denominator value should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Example of the rational number is 10/2, and for an irrational number is a famous mathematical value Pi(π) which is equal to 3.14.

Is 0 a rational number?

Yes, 0 is a rational number because it is an integer, that can be written in any form such as 0/1, 0/2, where b is a non zero integer. It can be written in the form: p/q = 0/1. Hence, we conclude that 0 is a rational number.

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