Rational and Irrational Numbers

Rational and irrational numbers are the complex form of representation of number in Mathematics. The rational numbers have properties different from irrational numbers. A number which is written in the form of a ratio of two integers is a rational number whereas an irrational number has endless non-repeating digits. 

The example of a rational number is 1/2 and of irrational is π = 3.141.  In this article, we are going to explain about rational and irrational numbers in detail and difference between them with the help of some examples. 

Table of contents:

Rational and Irrational Numbers Definition

What is Rational number?

The 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.

Rational word 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 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 the 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 below 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.

Rational and Irrational Numbers

Difference Between Rational and Irrational Numbers

  • It is expressed in the ratio, where both numerator and denominator is the whole number
  • It is impossible to express irrational numbers in 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-finite and non-recurring decimals are executed

Also, read: Difference Between Rational Numbers And Irrational Numbers


The list of examples of rational and irrational numbers are given here.

List 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

List 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 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 non-repeating 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 above-given examples, which differentiate rational numbers and irrational numbers.

Rational and Irrational Numbers Rules

Here are some rules based on arithmetic operations such as addition and multiplication performed on 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)

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