Properties of Irrational Numbers

1. Irrational numbers satisfy the commutative, associative and distributive laws for addition and multiplication.

2. (i) Sum of two irrationals need not be an irrational.

Example: Each one of (2 + √3) and (3 - √3) is irrational.

                  But,  (2 + √3) + (3 - √3) = 5, which is rational.

   

   (ii). Difference of two irrationals need not be an irrational.

Example: Each one of ( 4 + √5) and ( 2 + √5 ) is irrational.

                   But,   ( 4 +√5) - ( 2 + √5 ) = 2, which is rational.

      

   (iii). Product of two irrationals need not be an irrational.

Example: √5 is irrational.

                    But, √5 x √5 = 5, which is rational.

      

    (iv). Quotient of two irrationals need not be an irrational.

Example: Each one of 3√5 and √5 is irrational.

                     But, 3√5/√5 = 3, which is rational.

  3 (i). Sum of a rational and an irrational is irrational.

Example: 2 is rational and √3 is irrational then,

                  (2 + √3) is a irrational number.

    (ii). Difference of a rational and an irrational is irrational.

Example: 5 is rational and √7 is irrational then,

                  (5 - √7) is a irrational number.

   (iii). Product of a rational and an irrational is irrational.

Example: 2 is rational and √3 is irrational then,

                  (2√3) is a irrational number.

   (iv). Quotient of a rational and an irrational is irrational.

Example: 2 is rational and √3 is irrational then,

                  (2/√3) is a irrational number.