Real Numbers | Properties

Real Numbers

A number whose square is non-negative, is called a real number.

Example: All rational and irrational numbers are real  numbers.

Addition Properties of real numbers:

  1.    Closure Property : The sum of two real numbers is always a real number.

Example : Two real numbers are 1 and √2, then (1 + √2) is a real number. 

  2.       Commutative Property : a and b are two real numbers, then

                                          a + b = b + a

Example : two real numbers are (2 + √3) and (2 – √3), then

                  (2 + √3) + (2 – √3) = 4

                  (2 – √3) + (2 + √3) = 4

Then                    left hand side = Right hand side

Hence Commutative Property is verified.

  3.       Associative Property : a, b  and c are three real numbers, then

                                  a + ( b + c ) = ( a + b ) + c

Example : Three real numbers are  2 , (5 + √7) and (5 - √7)

                  2 + [(5 + √7) + (5 - √7)] = 2 + 10 = 12

          [2 + (5 + √7) ] + (5 - √7) = (7 + √7) + (5 - √7)   = 12

Then                    left hand side = Right hand side

Hence Associative Property is verified.

  4.       Existence of Additive Identity : For any real number a, then

                                    0 + a = a + 0 = a

0 is called the additive identity for real numbers.

Example : for a real number 5, then

                                    0 + 5 = 5 + 0 = 5

  5.       Existence of Additive Inverse : For each real number a, there exist a real number ( - a) such that a + (- a)  = (-a) + a = 0 then,

a and ( -a ) are called the additive inverse (or negative) of each other.

Example : For a real number (2 – √3), then

                 Additive inverse of 2 – √3 is – (2 – √3 ) = (√3 – 2)  

Multiplication Properties of Real Numbers :

  1.       Closure Property : The product of two real numbers is always a real numbers.

Example : Two real numbers are 1 and √2, then

                          1 х √2 = √2 is a real number. 

  2.       Commutative Property : a and b are two real numbers, then

                           a х b = b х a

Example : Two real numbers are (2 + √3) and (2 – √3), then

                  (2 + √3)(2 – √3) = 2² – ( √3 )² = 4 – 3 = 1

                  (2 – √3)(2 + √3) = 2² – ( √3 )² = 4 – 3 = 1

Then                    left hand side = Right hand side

Hence Commutative Property is verified.

  3.       Associative Property : a, b  and c are three real numbers, then

                                  a х ( b х c ) = ( a х b ) х c

Example : Three real numbers are  2 , (5 + √7) and (5 - √7),

2 х [(5 + √7) х (5 - √7)] = 2 х [5² – (√7)² ] = 2 х [25 – 7 ] 

                                                                    = 2 х 18 = 36

[2 х (5 + √7)] х (5 - √7) = [10  + 2√7 ] х (5 - √7)  

                     =  50 – 10√7 + 10 √7 - 2 х 7 = 50 – 14 = 36

Then                    left hand side = Right hand side

Hence Associative Property is verified.

  4.       Existence of Multiplicative Identity : For any real number, a then,

                                    1 . a = a . 1 = a

1 is the multiplicative identity for any numbers.

Example : for a real number 5, then

                                     1. 5 = 5 . 1 = 5

  5.       Existence of Multiplicative Inverse : For each non-zero real number a, there exist a real number 1/a  such that a . 1/a  = 1/a . a = 1 then,

a and 1/a  are called the multiplicative inverse (or reciprocal ) of each other.

Example : For a real number (2 – √3) then,

Multiplicative inverse of (2 – √3) is 1/ (2 – √3) = (2 + √3)

  6.       Distributive Properties of Multiplication over Addition : a, b and c three real numbers, then

                                a x ( b + c ) = a x b + a x c

Example : Three real numbers are  2 , (5 + √7) and (5 - √7) then,  

                      2 x [(5 + √7) + (5 - √7)] = 2 x 10 = 20

2 x (5 + √7) + 2 x (5 - √7) = 10  + 2√7  + 10 – √7  =  10 + 10 = 20

Then                    left hand side = Right hand side

Hence Distributive Property is verified.