# Rational Numbers

In Maths, rational numbers are represented in p/q form where q is not equal to zero. It is also a type of real number. 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, since they give us infinite values. Also, check irrational numbers here and compare with rational numerals.

We will learn here the properties of rational numbers along 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 number examples and learn how to find rational numbers in a better way. To represent rational numbers on a number line, we need to simplify and write in the decimal form first.

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

## 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 ≠ 0. Also, we can say that any fraction fits 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:

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

Examples of Rational Numbers:

 p q p/q Rational 10 2 10/2 =5 Rational 1 1000 1/1000 = 0.001 Rational 50 10 50/10 = 5 Rational

## Types of Rational Numbers

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

The below diagram helps us to understand more about the number sets.

• Real numbers (R) include all the rational numbers (Q).
• Real numbers include the integers (Z).
• Integers involve 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 the 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 we can 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 the 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 results are always a rational number if we multiply, add or subtract any two rational numbers.
• A rational number remains the same if we divide or multiply both numerator and denominator with the same factor.
• If we add zero to a rational number then we will get the same number itself.

## 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. Check the chart below, to differentiate between rational and irrational.

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

An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are:

Pi (π) = 3.142857…

Euler’s Number (e) = 2.7182818284590452…….

√2 = 1.414213…

## Solved 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. 3/4 = 0.75
• Fraction 90/12007 is rational.
• 12, also be written as 12/1. Again a rational number.
• Value of  √5 = 2.2360679775…….. It is a non-terminating value and hence cannot be written as a fraction. It is an irrational number.

Example 2:

Identify whether mixed fraction, 11/2 is a rational number.

Solution:

The Simplest form of 11/2 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 is expressed in the form of p/q then it is a rational number.
Here p and q are integers, and q is not equal to 0. A rational number should have a numerator and denominator. 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 should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Rational numbers are terminating decimals but irrational numbers are non-terminating. Example of the rational number is 10/2, and for an irrational number is a famous mathematical value Pi(π) which is equal to 3.141592653589…….

### 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.

### Is 7 a rational number?

7 is a rational number because it can be written in the form of ratio such as 7/1.

### Is 9 a rational number?

9 is a rational number because it can be written in the form of ratio such as 9/1.

### How to get a rational number between 3 and 4?

Rational number between 3 and 4 = 1/2 (3+4)
= 7/2

1. Shayna

You explain things very well and I know because I’m one of the students following and learning the chapters regularly since long

1. Hfdjei

Please explain me properties of rational numbers