Rational and Irrational Numbers

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. 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. 

Rational Numbers Definition

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.

Irrational Numbers Definition

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: \(\sqrt{8}\)=2.828…

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

Classification and Examples

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.

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.
  • \(\sqrt{81}\) 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.
  • \(\pi\) is an irrational number which has value 3.142…and is a never-ending and non-repeating number.
  • \(\sqrt{2}\) is an irrational number, as it cannot be simplified.
  • 0.212112111…is a rational number as it is non-recurring and non terminating.