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:
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 nonrepeating digits after the decimal point. Below is an example of the irrational number:
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






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 neverending and nonrepeating number.
 √2 is an irrational number, as it cannot be simplified.
 0.212112111…is a rational number as it is nonrecurring and nonterminating.
There are a lot more examples apart from the abovegiven 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)
Class Related Links
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Frequently Asked Questions – FAQs
What are rational and irrational numbers?
What is the difference between rational and irrational numerals?
Give examples of rational and irrational numbers?
Examples of irrational numbers are √2, √3, pi(π), etc.