Operations on Real Numbers

Operations on Real Numbers

You have learnt, in earlier classes that rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication. Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number (that is, rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division). It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always

Let us look at what happens when we add and multiply a rational number with an irrational number. For example,  is irrational. What about 2 +  and  since  has a non-recurring decimal expansion, the same is true for 2 +  and . Therefore, both 2 +  and  are also irrational numbers.