1. Introduction to Complex Numbers

2. Imaginary Numbers Consider x2 = –1. x2 = –1 or 1 – = x
∵ and
are not real
numbers.
∴ The equation
x2 = –1 has no
real roots.
x2 = –1 or 1 – = x
They are called
imaginary numbers.
The square roots of negative numbers are called imaginary numbers.
e.g , , , 2 - 3

3. (c) For any positive real number p,
(a) is denoted by i. 1 -
 i.e. 1 - = i
(b) 1 - =
 i.e. i 2 = –1
(c) For any positive real number p, 1 - = p
 i.e. i p = -
e.g. 4 - = 1 4 - i 2 = = 9 - 1 9 - 3i =

4. Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers, and 1 -
i =
a + b i
Complex Number:
Real part
Imaginary part

5. Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers, and 1 -
i =
e.g. (i) 2 – i
(ii) –3
(iii) 4i
Real part: 2, imaginary part: –1
Real part: –3, imaginary part: 0
Real part: 0, imaginary part: 4

6. Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers, and 1 -
i =
e.g. (i) 2 – i
(ii) –3
(iii) 4i
Real part: 2, imaginary part: –1
Real part: –3, imaginary part: 0
Real part: 0, imaginary part: 4
For a complex number a + bi, if b = 0, then a + bi is a real number.

7. Complex Numbers
A complex number is a number that can be written in the form a + bi, where a and b are real numbers, and 1 -
i =
e.g. (i) 2 – i
(ii) –3
(iii) 4i
Real part: 2, imaginary part: –1
Real part: –3, imaginary part: 0
Real part: 0, imaginary part: 4
For a complex number a + bi, if a = 0 and b ≠ 0, then a + bi is an imaginary number.

8. Complex Number System Complex numbers Real numbers
e.g. –3, 0
(i) Imaginary numbers
e.g. 4i, –2i
(ii) Sum of a non-zero real number and an imaginary number
e.g. 2 – i, 1 + 5i

9. Follow-up question
It is given that z = (k – 3) + (k + 1)i. If the imaginary part of z is 4,
find the value of k,
is z an imaginary number?
(a)
∵ Imaginary part of z = 4
∴ k + 1 = 4
k = 3

10. Follow-up question
It is given that z = (k – 3) + (k + 1)i. If the imaginary part of z is 4,
find the value of k,
is z an imaginary number?
(b)
∵ Real part = 3 – 3
= 0
Imaginary part ≠ 0
∴ z is an imaginary number.

11. & a = c a + bi = c + di b = d Equality of Complex Numbers
Two complex numbers (a + bi and c + di) are equal when both their
and imaginary
real parts
parts are equal.
a = c
&
a + bi = c + di
Equality
b = d

12. If x – 3i = yi, find the values of the real numbers x and y.
∵ The real parts are equal.
∴ x = 0
∵ The imaginary parts are equal.
∴ y = –3

13. Follow-up question
Find the values of the real numbers x and y if
2x + 4i = –8 + (y + 1)i.
2x + 4i = –8 + (y + 1)i
By comparing the real parts, we have – 8 2 = x 4 – = x
By comparing the imaginary parts, we have 1 4 + = y 3 = y

14. Operations of Complex Numbers

15. Let a + bi and c + di be two complex numbers.
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
e.g. (1 + 2i) + (2 – i) = 1 + 2i + 2 – i
= (1 + 2) + (2 – 1)i
= 3 + i

16. Subtraction
(a + bi) – (c + di) = (a – c) + (b – d)i
e.g. (1 + 2i) – (2 – i) = 1 + 2i – 2 + i
= (1 – 2) + (2 + 1)i
= –1 + 3i

17. Follow-up question
Simplify and express each of the following in the form a + bi.
(a) (4 – 2i) + (3 + i) (b) (–5 + 3i) – (1 + 2i)
(a) (4 – 2i) + (3 + i) = 4 – 2i i
= (4 + 3) + (–2 + 1)i
= 7 – i
(b) (–5 + 3i) – (1 + 2i) = –5 + 3i – 1 – 2i
= (–5 – 1) + (3 – 2)i
= –6 + i

18. Multiplication
(a + bi)(c + di) =
= ac + bci + adi + bdi 2
= ac + bci + adi + bd(–1)
= (ac – bd) + (bc + ad)i
(a + bi)(c)
+ (a + bi)(di)
e.g. (1 + 2i)(2 – i) = (1 + 2i)(2) + (1 + 2i)(–i)
= 2 + 4i – i – 2i2
= 2 + 3i – 2(–1)
= 4 + 3i

19. Division i + - = 2 1 di c bi a + = di c - e.g. i - + = 2 4 ) ( )( di c2 1 di c bi a + = di c -
e.g. i - + = 2 4 2 ) ( )( di c bi a - + = i - + = ) 1 ( 4 2 5 d c i ad bc bd ac 2 ) ( + - = i = 5 i d c ad bc bd ac 2 + - = i =

20. Follow-up question
Simplify and express each of the following in the form a + bi.
(a) (1 + 3i)(–2 + 2i) (b)
(a) (1 + 3i)(–2 + 2i) = (1 + 3i)(–2) + (1 + 3i)(2i)
= –2 – 6i + 2i + 6i2
= –2 – 4i + 6(–1)
= –8 – 4i

21. (b) i - + = 3 1 2 4 i - + = ) 3 ( 1 6 12 2 4 i - + = ) 1 ( 9 6 10 4 i - = 10 i - = 1