7 Difference Between Rational And Irrational Numbers (With Examples)

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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

BASIS OF COMPARISON RATIONAL NUMBERS IRRATIONAL NUMBERS
Description 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.   Numbers that cannot be expressed as ratio of two numbers i.e in the form of c/q are referred to as Irrational numbers.  
Nature Of Numbers Rational numbers includes numbers which are finite or are recurring in nature.   These consist of numbers which are non-terminating and non-repeating in nature.  
Consist Of Rational numbers consists of numbers that are perfect squares such as 4, 9, 16 25 etc.   The irrational numbers includes surds such as √2, √3, √5, √7 and so on.  
Representation Both numerator and denominator of rational numbers are whole numbers, in which the denominator of rational numbers is not equivalent to zero.   Irrational numbers cannot be represented in fractional form.  
Examples 5/3= 1.66, 1/7 =0.1428, 8/6=1.33   √7, √17, √5, √9  

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.