1. Real Numbers and Complex Numbers1
Real Numbers and Complex Numbers
Case Study
1.1 Real Number System
1.2 Surds
1.3 Complex Number System
Chapter Summary

2. Case Study
I think I can do it by drawing a square
of side 1 first.
If you are given a pair of compasses and a ruler only, do you know how to represent the irrational number
on a number line?
In junior forms, we learnt from Pythagoras’ theorem that the diagonal of a square of side 1 is
As shown in the figure, after drawing the diagonal of the square, we can use a pair of compasses to draw an arc with radius OB and O as the centre.
The point of intersection of the arc and the number line is the position of (i.e., point C).

3. 1.1 Real Number System
We often encounter different numbers in our calculations,
For example,
0.16 means … .
1, 2, 4, 7, 0, , 2.5, 0.16, p, , … .
These numbers can be classified into different groups.

4. 7, 4, 0, 1, 2 1.1 Real Number System A. Integers
1, 2, 4, 7 and 0 are all integers.
Positive integers (natural numbers) are integers that are greater than zero.
Negative integers are integers that are less than zero.
Zero is neither positive nor negative.
Integers
7, 4, , , 2
Negative Integers
Positive Integers
(Natural Numbers)

5. 1.1 Real Number System B. Rational Numbers
is a fraction, 2.5 is a terminating decimal and
0.16 is a recurring decimal. .
Recurring decimals are also called repeating decimals.
All of them are rational numbers.
A rational number is a number that can be written in
the form , where p and q are integers and q  0.
Recurring decimals can be converted into fractions, as shown in the next page.
Note that
and
Any integer n can be written as Therefore, integers are also
rational numbers.

6. 1.1 Real Number System B. Rational Numbers Express 0.16 as a fraction:
Let n  0.16 .
 … (1)
10n  … (2)
(2)  (1): 10n  n  1.5
9n  1.5
n  
In other recurring decimals, such as a  0.83 and b  0.803,
. .
. .
consider 100a  and 1000b  , then we obtain
99a  83 and 999b  803.
. .
. .

7. 1.1 Real Number System C. Irrational Numbers
Numbers that cannot be written in the form are irrational
numbers.
Examples: p, , and sin 45
Irrational numbers can only be written as non-terminating and non-recurring decimals:
is just an approximation
of p.

8. 1.1 Real Number System D. Real Numbers
If we group all the rational numbers and irrational numbers together, we have the real number system.
That is, a real number is either a rational number or an irrational number.
Real numbers
1, 2, 4, 7, 0, , 3.5, 0.16, p, .
Rational numbers
1, 2, 4, 7, 0, , 3.5, 0.16
Fractions
Terminating
decimals
Recurring
Integers
Irrational numbers p,
Negative integers
Zero
Positive integers

9. a2  0 for all real numbers a.
1.1 Real Number System
D. Real Numbers
We can represent any real number on a straight line called the real number line. . . . 1 . .
2.5 . p
Real numbers have the following property:
a2  0 for all real numbers a.
For example:
 is a real number since
 is not a real number since

10. 1.2 Surds
In junior forms, we learnt the following properties for surds:
For any real numbers a and b, we have
In general, 1. 2.

11. 1.2 Surds A. Simplification of Surds
For any surds, when we reduce the integer inside the square root sign to the smallest possible integer, such as:
then the surd is said to be in its simplest form.

12. 1.2 Surds B. Operations of Surds
Like surds are surds with the same integer inside the square root sign, such as and
When two surds are like surds, we can add them or subtract them:

13. 1.2 Surds
B. Operations of Surds
Example 1.1T
Simplify
Solution:

14. 1.2 Surds
B. Operations of Surds
Example 1.2T
Simplify
Solution:

15. 1.2 Surds
B. Operations of Surds
Example 1.3T
Simplify
Solution:

16. 1.2 Surds C. Rationalization of the Denominator
Rationalization of the denominator is the process of changing an irrational number in the denominator into a rational number, such as:

17. Example 1.4T 1.2 Surds Solution: C. Rationalization of the Denominator
Simplify
Solution:

18. a2  0 for all real numbers a.
1.3 Complex Number System
A. Introduction to Complex Numbers
In Section 1.1, we learnt that
a2  0 for all real numbers a.
For example:
 is a real number since
 is not a real number since
Therefore, in a real number system, equations such as x2  1 and (x  1)2  4 have no real solution:
Define
 i
Then
Complex numbers
 1  2i

19. 1.3 Complex Number System A. Introduction to Complex Numbers
Properties of complex numbers:
1. The complex number system contains an imaginary unit, denoted by i, such that
i2  1.
2. The standard form of a complex number is
a  bi,
where a and b are real numbers.
3. All real numbers belong to the complex number system.

20. 1.3 Complex Number System A. Introduction to Complex Numbers
Notes:
1. For a complex number a  bi, a is called the real part and b is called the imaginary part.
Complex numbers do not have order. So we cannot compare which of the complex numbers 2  3i and 4  2i is greater.
2. When a  0, a  bi  0  bi  bi, which is a purely imaginary number.
3. When b  0, a  bi  a  0i  a, so any real number can be considered as a complex number.
4. When a  b  0, a  bi  0  0i  0.
Two complex numbers are said to be equal if and only if both of their real parts and imaginary parts are equal.
If a, b, c and d are real numbers, then
a  bi  c  di
if and only if a  c and b  d.

21. 1.3 Complex Number System B. Operations of Complex Numbers
The addition, subtraction, multiplication and division of complex numbers are similar to the operations of algebraic expressions.
In the operation of algebraic expressions, only like terms can be added or subtracted.
We classify the real part and the imaginary part of the complex number as unlike terms in algebraic expressions.
For complex numbers z1  a  bi and z2  c  di, where a, b, c and d are real numbers, we have:
(1) Addition
z1  z2  (a  bi)  (c  di)
e.g. (3  6i)  (5  8i)
 a  bi  c  di
 (3  5)  [6  (8)]i
 (a  c)  (b  d)i
 8  2i

22. 1.3 Complex Number System B. Operations of Complex Numbers
(2) Subtraction
z1  z2  (a  bi)  (c  di)
e.g. (9  7i)  (2  3i)
 a  bi  c  di
 (9  2)  [7  (3)]i
 (a  c)  (b  d)i
 7  4i
(3) Multiplication
This term belongs to the real part because i2  1.
z1z2  (a  bi)(c  di)
 ac  adi  bci  bdi2
 (ac  bd)  (ad  bc)i

23. Example 1.5T 1.3 Complex Number System Solution:
B. Operations of Complex Numbers
Example 1.5T
Simplify (7  2i)(5  3i)  4i(3  i).
Solution:
(7  2i)(5  3i)  4i(3  i)  (35  21i  10i  6i2)  (12i  4i2)
 35  21i  10i  6  12i  4
 33  19i

24. 1.3 Complex Number System B. Operations of Complex Numbers
(4) Division
The denominator contains
The process of division is similar to the rationalization of the denominator in surd.
(p  q)(p  q)  p2  q2

25. Example 1.6T 1.3 Complex Number System Solution:
B. Operations of Complex Numbers
Example 1.6T
Simplify and express the answer in standard form.
Solution:

26. Chapter Summary 1.1 Real Number System Real numbers Rational numbers
Fractions
Terminating
decimals
Recurring
Integers
Irrational numbers
Negative integers
Zero
Positive integers

27. Chapter Summary 1.2 Surds 1. For any positive real numbers a and b:

28. Chapter Summary 1.3 Complex Number System
1. Every complex number can be written in the form a  bi, where a and b are real numbers.
2. The operations of complex numbers obey the same rules as those of real numbers.

29. Follow-up 1.1 1.2 Surds Solution: B. Operations of Surds
Simplify the following expressions.
(a) (b)
Solution:
(a)
(b)

30. Follow-up 1.2 1.2 Surds Solution: B. Operations of Surds
Simplify the following expressions.
(a) (b)
Solution:
(a)
(b)

31. Follow-up 1.3 1.2 Surds Solution: B. Operations of Surds
Simplify the following expressions.
(a) (b)
Solution:
(a)
a2  b2  (a  b)(a  b)
(b)

32. Follow-up 1.4 1.2 Surds Solution:
C. Rationalization of the Denominator
Follow-up 1.4
Simplify the following expressions.
(a) (b)
Solution:
(a)
(b)

33. Follow-up 1.5 1.3 Complex Number System Solution:
B. Operations of Complex Numbers
Follow-up 1.5
Simplify (5  2i)(7  3i)  3(2  5i).
Solution:
(5  2i)(7  3i)  3(2  5i)  (35  15i  14i  6i2)  (6  15i)
 35  15i  14i  6  6  15i
 23  44i

34. Follow-up 1.6 1.3 Complex Number System Solution:
B. Operations of Complex Numbers
Follow-up 1.6
Simplify and express the answer in standard form.
Solution: