RATIONAL AND IRRATIONAL NUMBERS
Rational and Irrational numbers both are real numbers but different with respect to their
properties. A rational number is the one which can be represented in the form of P/Q where P
and Q are integers and Q ≠ 0. But an irrational number cannot be written in the form of simple
fractions. ⅔ is an example of rational numbers whereas √2 is an irrational number.
Let us learn more here with examples and the difference between them.
Table of contents:

Definition

How to Classify

Difference

Examples


List of Rational Numbers

List of Irrational Numbers
Rules
Definitions
What is a Rational number?
Rational numbers are numbers which can be expressed as a fraction and also as positive numbers,
negative numbers and zero. It can be written as p/q, where q is not equal to zero.
The word rational is derived from the word ‘ratio’, which actually means a comparison of two or
more values or integer numbers and is known as a fraction. In simple words, it is the ratio of
two integers.
Example: 3/2 is a rational number. It means integer 3 is divided by another integer 2.
What is Irrational Number?
The numbers which are not a rational number are called irrational numbers. Now, let us
elaborate, irrational numbers could be written in decimals but not in the form of fractions, which
means it cannot be written as the ratio of two integers.
Irrational numbers have endless non-repeating digits after the decimal point. Below is an
example of the irrational number:
Example: √8=2.828…
How to Classify Rational and Irrational Numbers?
Let us see how to identify rational and irrational numbers based on the given set of examples.
As per the definition, the rational numbers include all integers, fractions and repeating decimals.
For every rational number, we can write them in the form of p/q, where p and q are integers
value.
Venn Diagram
The below image shows the Venn diagram of rational and irrational numbers which come under
real numbers.
Difference Between Rational and Irrational Numbers

It is expressed in the ratio, where both
numerator and denominator are the whole
numbers

It is impossible to express irrational
numbers as fractions or in a ratio of two
integers

It includes perfect squares

It includes surds

The decimal expansion for rational number
executes finite or recurring decimals

Here, non-terminating and non-recurring
decimals are executed
Also, read: Difference Between Rational Numbers And Irrational Numbers
Examples
A list of examples of rational and irrational numbers are given here.
Examples of Rational Numbers

Number 9 can be written as 9/1 where 9 and 1 both are integers.

0.5 can be written as ½, 5/10 or 10/20 and in the form of all termination decimals.

√81 is a rational number, as it can be simplified to 9 and can be expressed as 9/1.

0.7777777 is recurring decimals and is a rational number
Examples of Irrational Numbers
Similarly, as we have already defined that irrational numbers cannot be expressed in fraction or
ratio form, let us understand the concepts with a few examples.

5/0 is an irrational number, with the denominator as zero.

π is an irrational number which has value 3.142…and is a never-ending and nonrepeating number.

√2 is an irrational number, as it cannot be simplified.

0.212112111…is a rational number as it is non-recurring and non-terminating.
There are a lot more examples apart from the above-given examples, which differentiate rational
numbers and irrational numbers.
Properties of Rational and Irrational Numbers
Here are some rules based on arithmetic operations such as addition and multiplication
performed on the rational number and irrational number.
#Rule 1: The sum of two rational numbers is also rational.
Example: 1/2 + 1/3 = (3+2)/6 = 5/6
#Rule 2: The product of two rational number is rational.
Example: 1/2 x 1/3 = 1/6
#Rule 3: The sum of two irrational numbers is not always irrational.
Example: √2+√2 = 2√2 is irrational
2+2√5+(-2√5) = 2 is rational
#Rule 4: The product of two irrational numbers is not always irrational.
Example: √2 x √3 = √6 (Irrational)
√2 x √2 = √4 = 2 (Rational)