Difference Between Rational And Irrational Numbers (With Examples)

What Are Rational Numbers?

Rational numbers are those numbers that can be expressed as a quotient (the result in a regular division equation). Even if you express the resulting number not as a fraction and it repeats infinitely, it can still a rational number. Zero is a rational number.

As per the description, the rational numbers include all integers, fractions and repeating decimals. For every rational number, we can write them in the form of c/q, where c and q are integer values.

For a number to be considered a rational number it must satisfy the following criteria:

• It can be expressed in the form of a simple fraction with numerator (c) divided by a denominator (q).
• Both the numerator and the denominator must be regular integers themselves. An integer can simply be described as whole number like 3, 6, or 15.
• The denominator (q) cannot be zero. The numerator or the denominator can be positive or negative as long as the denominator is not zero.

 Examples of Rational Numbers

• Number 5 can be written as 5/1 where both 5 and 1 are integers.
• 0.5 can be written as ½, 5/10, 25/50 or 10/20 and in the form of all terminating decimals.
• √81 is a rational number, as it can be simplified to 9 and can be expressed as 9/1.
• 0.8888888 is recurring decimals and is a rational number

Facts About Rational Numbers

• The numbers that can be expressed as a ratio of two numbers i.e in the form of c/q are referred to as rational numbers.
• Rational numbers includes numbers which are finite or are recurring in nature.
• Rational numbers consists of numbers that are perfect squares such as 4, 9, 16 25 etc.
• Both numerator and denominator of rational numbers are whole numbers, in which the denominator of rational numbers is not equivalent to zero.
• Example of rational numbers: 5/3= 1.66, 1/7 =0.1428, 8/6=1.33

What Are Irrational Numbers?

An irrational number is a number that cannot be expressed as a ratio of two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.

The irrational numbers can be expressed in the form of non-terminating fractions and in different ways. For example, the square roots which are not perfect squares will always result in an irrational number.

Examples of Irrational Numbers

• 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.

Facts About Irrational Numbers

• Numbers that cannot be expressed as ratio of two numbers i.e in the form of c/q are referred to as Irrational numbers.
• These consist of numbers which are non-terminating and non-repeating in nature.
• The irrational numbers includes surds such as √2, √3, √5, √7 and so on.
• Irrational numbers cannot be represented in fractional form.
• Examples of irrational numbers: √7, √17, √5, √9

Difference Between Rational And Irrational Numbers In Tabular Form

Arithmetic Rules For Rational And Irrational Numbers

• The sum of two irrational numbers may be an irrational number or a rational number, as an example (√2+ 4), (π + 2) are irrational numbers and √2 + (-√2) = 0
• The product of an irrational number to a rational number is an irrational number, as an example, 2√5,2π are irrational number.
• The product of two irrational number may be a rational or irrational number, as an example √2×–√2=-2,√2×√3 = √6
• The product of two identical irrational numbers may be rational or irrational, as an example √2×√2 = 2, 
• The division of two irrational numbers can be rational or irrational, as an example 2√2/3√2= 2/3 , 2√2/√3 etc.