## 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 $\frac{r}{s}$ is an irrational number.

EXERCISE: Classify the following numbers as rational or irrational:

(i) $2-\sqrt{5}$   (ii) $\left ( 3+\sqrt{23} \right )-\sqrt{23}$   (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$   (iv) $\frac{1}{\sqrt{2}}$   (v) 2π

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

(i)    $3+\sqrt{3}$

(ii)    $\left ( 5+\sqrt{31} \right )-\sqrt{31}$

(iii)    $\frac{5\sqrt{11}}{9\sqrt{11}}$

(iv)    $\frac{1}{\sqrt{25}}$

(v)     $\frac{22}{7}\pi$

PRODUCT OF IRRATIONAL NUMBERS

EXERCISE: Simplify each of the following expressions:

(i) $\left ( 3+\sqrt{3} \right )\left ( 2+\sqrt{2} \right )$   (ii) $\left ( 3+\sqrt{3} \right )\left ( 3-\sqrt{3} \right )$   (iii) $\left ( \sqrt{5}+\sqrt{2} \right )^{2}$
(iv) $\left ( \sqrt{5}-\sqrt{2} \right )\left ( \sqrt{5}+\sqrt{2} \right )$

SELF PRACTICE: Simplify each of the following expressions:

(i) $\left ( 5+\sqrt{5} \right )\left ( 7+\sqrt{7} \right )$
(ii) $\left ( 7+\sqrt{7} \right )\left ( 7-\sqrt{7} \right )$
(iii) $\left ( \sqrt{3}+\sqrt{7} \right )^{2}$
(iv) $\left ( \sqrt{7}-\sqrt{3} \right )\left ( \sqrt{7}+\sqrt{3} \right )$

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, $\pi =\frac{c}{d}$. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

EXERCISE: Represent $\sqrt{9.3}$ on the number line.

SELF PRACTICE: Represent $\sqrt{5.4}$, $\sqrt{7.2}$, and $\sqrt{4.5}$ on the number line.

Discussion