Rational and Irrational Numbers

All the numbers used in Mathematical computations are broadly classified into two kinds. They are real numbers and imaginary numbers. Real numbers are those numbers that exist in reality and are used in most of the Mathematical computations. Imaginary numbers are the numbers that do not exist in reality. However, they are assumed to be existing to ease a few Mathematical and Scientific computations. Real numbers comprise the entire list of rational and irrational numbers. The chart below describes the difference between rational and irrational numbers. 

Rational Numbers Definition:

Rational numbers are the numbers that can be written in the form of a fraction where numerator and denominator are integers. In the case of rational numbers, numerator and denominator should be coprime and denominator should not be equal to zero. Mathematically, rational numbers definition is given as the number a/b if a and b are coprimes, and b is not equal to zero. 


Examples for Rational Numbers:

  • 5 is a rational number because ‘5’ can be written as \[\frac{5}{1}\]. Here 5 and 1 are coprimes and 1 is not equal to zero.

  • 2.343 is a rational number because it can be written as 2343/1000 

  • The square root of perfect square numbers

Irrational Numbers Definition:

Any real number is said to be an irrational number if the number cannot be expressed in the form of a fraction where the denominator is not equal to zero. 

Mathematically, the definition of the irrational number is given as a number that cannot be expressed in the form of a/b where a and b are coprime and b is not equal to zero.

Examples for irrational numbers:

  • The square root of a prime number is an irrational number. \[\left( {\sqrt 2 ,\sqrt 3 ,\sqrt 5 ,\sqrt 7 ,\sqrt {13} ,{\text{ }}etc} \right)\]

  • Mathematical constant π is an irrational number because it is a non-terminating recurring decimal number.

From the above explanations, the difference between rational and irrational numbers is evident.

Properties of Rational Numbers:

  • Rational numbers are closed under addition, subtraction, multiplication, and division. This means that 

  • The sum of rational numbers is rational.

  • The difference between the two rational numbers is rational.

  • The product of two rational numbers is rational

  • The quotient of two rational numbers is also rational. However, rational numbers are not closed under division if the divisor is zero.


  • Rational numbers are commutative for addition and multiplication. However, they are not commutative for subtraction and division. If ‘c’ and ‘d’ are two rational numbers, then

  • \[c{\text{ }} + {\text{ }}d{\text{ }} = {\text{ }}d{\text{ }} + {\text{ }}c\]

  • \[c{\text{ }} \times {\text{ }}d{\text{ }} = {\text{ }}d{\text{ }} \times {\text{ }}c\]

  • \[c{\text{ }} - {\text{ }}d{\text{ }} \ne {\text{ }}d{\text{ }} - {\text{ }}c\]

  • \[c{\text{ }} \div {\text{ }}d{\text{ }} \ne {\text{ }}d{\text{ }} \div {\text{ }}c\]


  • Addition and multiplication are associative for rational numbers whereas subtraction and division are not associative. If‘ ’j’, ‘k’ and ‘l’ are three rational numbers, then

  • \[\left( {j{\text{ }} + {\text{ }}k} \right){\text{ }} + {\text{ }}l{\text{ }} = {\text{ }}j{\text{ }} + {\text{ }}\left( {k{\text{ }} + {\text{ }}l} \right)\]

  • \[\;\left( {j{\text{ }} \times {\text{ }}k} \right){\text{ }} \times {\text{ }}l{\text{ }} = {\text{ }}j{\text{ }} \times {\text{ }}\left( {k{\text{ }} \times {\text{ }}l} \right)\]

  • \[\;\left( {j{\text{ }} - {\text{ }}k} \right){\text{ }} - {\text{ }}l{\text{ }} \ne {\text{ }}j{\text{ }} - {\text{ }}\left( {k{\text{ }} - {\text{ }}l} \right)\]

  • \[\;\left( {j{\text{ }} \div {\text{ }}k} \right){\text{ }} \div {\text{ }}l{\text{ }} \ne {\text{ }}j{\text{ }} \div {\text{ }}\left( {k{\text{ }} \div {\text{ }}l} \right)\]

 

  • Rational numbers obey the distribution of multiplication over addition. If ‘j’, ‘k’ and ‘l’ are rational numbers, then

\[j{\text{ }}\left( {k{\text{ }} + {\text{ }}l} \right){\text{ }} = {\text{ }}\left( {j{\text{ }} \times {\text{ }}k} \right){\text{ }} + {\text{ }}\left( {j{\text{ }} \times {\text{ }}l} \right)\]

Rational and Irrational Numbers examples:

Categorize the following into the list of rational and irrational numbers. Justify your answer.

\[\left( {0.99,{\text{ }}2.12341234 \ldots  \ldots ,{\text{ }}57,{\text{ }}\frac{{16}}{{26}},{\text{ }}2\sqrt 5 ,{\text{ }}\frac{{10}}{0}} \right)\]

Solution:

  1. 0.99 is a rational number because, by rational numbers definition, it can be expressed as \[\frac{{99}}{{100}}\]

  2. 2.12341234…. is an irrational number because it is a non-terminating decimal that cannot be expressed in the form of a fraction.

  3. 57 is a rational number because it can be written in the form of a fraction as 57/1.

  4. \[\frac{{16}}{{26}} = \frac{8}{{13}}\] which is a rational number.

  5. \[2\sqrt 5 \] is irrational because the denominator is not an integer and the number satisfies the irrational numbers definition.

  6. \[\frac{{10}}{0}\] is equal to infinity which is undefined. So the number does not fall under rational and irrational numbers examples.

Fun facts:

  • For a number to be called rational, only denominators should not be equal to zero. However, 0 may occur in the place of the numerator. ‘0’ is a rational number because 0 can be written as \[\frac{0}{1}\]. Here denominator is not equal to zero.

  • Coprimes are the numbers that have only 1 as a common factor. For example, 3 and 10 are coprimes because the only common factor between them is 1 whereas 3 and 6 are not co primes because they have a common factor 3 along with 1.

  • The entire list of rational and irrational numbers can be called as real numbers. However, all real numbers cannot be uniquely rational or irrational.

  • All non-repeating and repeating terminating decimals are rational numbers and all non-terminating repeating and non-repeating decimals are irrational numbers.

  • The best rational and irrational numbers examples are square roots of perfect squares and non-perfect square numbers respectively.

FAQ (Frequently Asked Questions)

1. Do irrational numbers obey closure property?

  • Irrational numbers do not obey closure property. 

  • When two irrational numbers are added, the sum need not be irrational. The sum of 2 + √3 and 4 - √3 is equal to 6 which is not irrational. 

  • When two irrational numbers are subtracted, the difference may not be irrational. The difference between 5√2 and 5√2 is 0 which is a rational number. 

  • When two irrational numbers are multiplied, the product need not be irrational. The product of √7 and √7 is equal to √49  = 7. This is not irrational. 

  • If two irrational numbers are divided, the quotient need not be irrational. If √6 and 7√6 is divided, the quotient is 1/7 which is a rational number.

What happens when basic Mathematical operations are performed between a rational number and an irrational number?

  • If any of the mathematical operations are performed between a rational and an irrational number, the result obtained is irrational.

  • When an irrational number is added to a rational number, the sum obtained is an irrational number.

  • The difference between rational and irrational numbers is an irrational number.

  • The product of two numbers is irrational if one and only one of the numbers is irrational. 

  • The quotient obtained by dividing a rational number by an irrational number or vice versa is an irrational number.