## 1.8 Algebra of Real Numbers [20m]

A chef combines different food ingredients to create new dishes as per well-defined recipes. In a similar manner, a mathematician combines elementary numbers, using certain laws of operations, to create new numbers. How many different numbers can you create by using the numbers 2 and 1/2 and the basic arithmetic operations of addition, subtraction, multiplication, and division? Write down your responses in the discussion board below.

Both 2 and 1/2 are real numbers. Specifically, both of them are rational numbers. What kind of number would their sum, difference, product or quotient be?

If we combine a rational number and an irrational number using different operations, what kind of numbers will we get? In this lesson, we will seek an answer to such questions.

**OPERATIONS on REAL NUMBERS**

1. The SUM/DIFFERENCE/PRODUCT of two real numbers is a real number.

2. The SUM/DIFFERENCE/PRODUCT of two rational numbers is a rational number.

3. The SUM/DIFFERENCE/PRODUCT of two irrational numbers may be a rational or an irrational number.

4. The SUM/DIFFERENCE/PRODUCT of a rational number and an irrational number is an irrational number.

5. If *r* is a rational number and *s* is an irrational number, then is an irrational number.

**EXERCISE**: Classify the following numbers as rational or irrational:

(i) (ii) (iii) (iv) (v) 2π

**SELF PRACTICE**: Classify the following numbers as rational or irrational:

(i)

(ii)

(iii) (iv)(v)

**PRODUCT OF IRRATIONAL NUMBERS**

**EXERCISE**: Simplify each of the following expressions:

(i) (ii) (iii)

(iv)

**SELF PRACTICE**: Simplify each of the following expressions:

(i)

(ii)

(iii)

(iv)

Can you predict which of the products will be a rational number and which will be an irrational number?

**EXERCISE**: Recall, π is defined as the ratio of the circumference (say *c*) of a circle to its diameter (say *d*). That is, . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

**EXERCISE**: Represent on the number line.

**SELF PRACTICE**: Represent , , and on the number line.

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