1. Number Systems

1.1 Introduction - Learning Objectives

Example 1:

Are the following statements true or false? Give reasons for your answers. (i) Every whole number is a natural number. (ii) Every integer is a rational number. (iii) Every rational number is an integer.

Example 2:

Find five rational numbers between 1 and 2.

Exercise 1.1

Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?
Find six rational numbers between 3 and 4.
Find five rational numbers between 3/5 and 4/5.
State whether the following statements are true or false. Give reasons for your answers.
- (i) Every natural number is a whole number.
- (ii) Every integer is a whole number.
- (iii) Every rational number is a whole number.

1.2 Irrational Numbers

Example 3:

Locate √2 on the number line.

Example 4:

Locate √3 on the number line.

Exercise 1.2

State whether the following statements are true or false. Justify your answers.
Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Show how √5 can be represented on the number line.
Classroom activity (Constructing the ‘square root spiral’).

1.3 Real Numbers and their Decimal Expansions

Example 5:

Find the decimal expansions of 10/3, 7/8 and 1/7.

Example 6:

Show that 3.142678 is a rational number.

Example 7:

Show that 0.333... = 1/3

Example 8:

Show that 1.272727... = 14/11

Example 9:

Show that 0.235353... = 233/990

Example 10:

Find an irrational number between 1/7 and 2/7

Exercise 1.3

Write the following in decimal form and say what kind of decimal expansion each has:
- (i) 36/100
- (ii) 1/11
- (iii) 4/8
- (iv) 3/13
- (v) 2/11
- (vi) 329/400
You know that 1/7 = 0.142857... Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division?
Express the following in the form p/q, where p and q are integers and q ≠ 0.
- (i) 0.6̅
- (ii) 0.47̅
- (iii) 0.001̅
Express 0.99999... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.
Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations. Can you guess what property q must satisfy?
Write three numbers whose decimal expansions are non-terminating non-recurring.
Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Classify the following numbers as rational or irrational:
- (i) √23
- (ii) √225
- (iii) 0.3796
- (iv) 7.478478...
- (v) 1.101001000100001...

1.4 Operations on Real Numbers

Example 11:

Check whether 7√5, 7/√5, 2 + √5, π − 2 are irrational numbers or not.

Example 12:

Add (2√2 + 5√3) and (2√3 − 3√2).

Example 13:

Multiply 6√5 by 2√5.

Example 14:

Divide 8√15 by 2√3.

Example 15:

Simplify the following expressions: (i) (√5 + √7 + 2√5), (ii) (√5 + √5)(√5 − √5), (iii) (√3 + √7)², (iv) (√11 − √7)(√11 + √7)

Example 16:

Rationalise the denominator of 1/√2.

Example 17:

Rationalise the denominator of 1/(√2 + √3).

Example 18:

Rationalise the denominator of 5/(√3 − √5).

Example 19:

Rationalise the denominator of 1/(√7 + √3 + √2).

Exercise 1.4

Classify the following numbers as rational or irrational:
- (i) √2 – √5
- (ii) (3 + √2)(3 – √2)
- (iii) √2 / √7
- (iv) 1/√2
- (v) 2π
Simplify each of the following expressions:
- (i) (√3 + √3 + √2 + √2)
- (ii) (√3 + √3)(√3 – √3)
- (iii) (√5 + √2)²
- (iv) (√5 – √2)(√5 + √2)
Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π = c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Represent √9.3 on the number line.
Rationalise the denominators of the following:
- (i) 1/√7
- (ii) 1/(√7 – √6)
- (iii) 1/(√5 + √2)
- (iv) 1/(√7 – √2)

1.5 Laws of Exponents for Real Numbers

Example 20:

Simplify:

(i) 2^1/3 × 2^2/3
(ii) (3⁵)^1/4
(iii) 7^1/5 ÷ 3^1/3
(iv) 13^1/5 × 17^1/5

Exercise 1.5

Find:
- (i) √64
- (ii) √32
- (iii) √125
Find:
- (i) ³√9
- (ii) √32
- (iii) ⁴√16
- (iv) –³√125
Simplify:
- (i) 2^(1/3) × 2^(2/3)
- (ii) 7^(1/3) ÷ 3^(1/3)
- (iii) 11^(1/2) ÷ 11^(1/4)
- (iv) 7^(1/2) × 8^(1/2)