FARAH MALEK

COMPLEX NUMBERS

Chapter 1

Contents

• Introduction to Complex Numbers

• Basic Operations on Complex Numbers

• Equal Complex Numbers

• Modulus and Argument of Complex Numbers

Introduction to Complex No

x2 = 4

3

x = 2

x2 = 36 x = 6

x2 = -1 x = -1 undefined

Real no

Real no

Imaginary no -1

Denoted by ‘i’

Imaginary no, i

i2 = -1

i3 = i2 i = (-1) i = - i

i4 = i2 i2 = (-1) (-1) = 1

i5 = i2 i2 i = (-1)(-1)i = i

ii 244

33 i

ii 3532575

i1

i1

4

Imaginary no can be added, subtracted,

multiplied or divided

5

6i + 2i = 8i

9i – 4i = 5i

3i 2i = 6i2 = 6(-1) = - 6

8i

2i = 4

Complex No, z

6

z = a + bi

Real no Real no

imaginary no

Eg. 3 + ½ i

- 3 + 5 i

- ½ i

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Basic Operations on Complex No

7

(a) Addition and subtraction

Eg.

(2 + 3i) + (3 – i) = (2 + 3) + (3 – 1)i

= 5 + 2i

(5 + 4i) – (2 + 3i) = (5 – 2) + (4 – 3)i

= 3 + i

(b) Multiplication

Eg.

(2 + 3i)(4 – 2i) = 8 – 4i + 12i – 6i²

= 8 + 8i – 6(-1)

= 14 + 8i

z z* = (a + bi)(a – bi) = a2 – abi + abi + b2

Conjugate numbers (z or z*)

If z = a + bi, z* = a – bi

zz* = a2 + b2 8

(c) Division

Eg. i

i

i

i

i

i

1

1

1

3

1

3

i

i

i

ii

2

2

24

1

232

2

Equal Complex No

9

If a + bi = 6 + 7i,

then a = 6, b = 7

a) (x + yi)(3 – i) = - 3 + 11i

Find the values of x and y in each of the following cases:

b) x – 2i

2 + i +

2 – yi

1 – i = 3 – 2i

Solution:

(x + yi)(3 – i) = - 3 + 11i a)

3x + y + (3y)i – xi = - 3 + 11i

3x + y + (3y – x)i = - 3 + 11i

So 3x + y = - 3 …………(i)

3y – x = 11

x = 3y – 11 …...(ii)

3(3y – 11) + y = - 3 (ii)(i),

y = 3 x = 3(3) – 11 = - 2 10

b) x – 2i

2 + i +

2 – yi

1 – i = 3 – 2i

x – 2i

2 + i

2 – i

2 – i ( ) ( ) +

2 – iy

1 – i ( )

1 + i

1 + i ( ) = 3 – 2i

2x – 2 – 4i – xi

5

2 + y – yi + 2i

2 + = 3 – 2i

4x – 4 – 8i – 2xi + 10 + 5y + 10i – 5yi = 30 – 20i

4x + 6 + 5y + (10 – 5y – 8 – 2x)i = 30 – 20i

4x + 6 + 5y = 30 …………….(i)

10 – 5y – 8 – 2x = - 20

2 – 5y – 2x = - 20

2x + 5y = 22

5y = 22 – 2x ………(ii)

(ii)(i), 4x + 6 + 22 – 2x = 30 x = 1

5y = 22 – 2(1) = 20 y = 4 11

Find the square roots of the complex number 6 + 8i

in the form x+yi, where x and y are real numbers.

Solution:

(6 + 8i)1/2 = x + yi (x, y R) Let

So 6 + 8i = (x + yi)2

6 + 8i = x2 – y2 + 2xyi

x2 – y2 = 6 ………….(i)

2xy = 8

x = 4/y ……….(ii)

(ii)(i), (4/y)2 – y2 = 6

y4 + 6y2 - 16 = 0

(y2 – 2)(y2 + 8) = 0

y2 = 2

y2 = - 8

y = ±√2

is rejected 12

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22

2

4

2when

x

y

22

2

4

2when

x

y

22286 Therefore, ii

13

Argand Diagram

A diagram in which complex numbers

are represented geometrically by a

point P with rectangular Cartesian

coordinates (x, y)

14

y

x

P(a, b)

θ θ

P*(a, -b)

z = a + bi

Eg.

z* = a – bi

y

x

P(a, b)

θ

z = a + bi

r

a

b

Length of OP is called the modulus of z, |z|

|z| = (a2 + b2)1/2

Argument of z, (measured in radians)

- < <

O a

b

baOP

tan

22

,arg z15

Find the modulus and argument of each of the following

complex numbers.

a) 2 + i

b) 5 – 4i

c) – 4 + 2i

d) – 1 - i√3

Solution: y

x

z = 2 + i

|z| = (22 + 12)1/2 = √5

θ = tan-1(1/2) = 26.6o = 0.464 rad

P(2, 1)

θ

a)

z = 5 – 4i

|z| = (52 + (-4)2)1/2 = √41

θ = tan-1(-4/5) = -38.7o = - 0.675 rad

b)

y

x

P(5, -4)

θ

16

y

x

z = - 4 + 2i

|z| = ((-4)2 + 22)1/2 = √20 = 2√5

α = tan-1(2/-4) = - 26.6o

P(-4, 2)

θ

c)

d)

y

x

z = - 1 – i√3

|z| = ((-1)2 + (-√3)2)1/2 = √4 = 2

α = tan-1(-√3/-1) = 60o

P(-√3, -1)

θ

α

θ = 180o – 26.57o = 153.43o = 2.678 rad

α

θ = -(180o – 60o) = -120o = - 2.094 rad

17

If a = 2 – i and b = 1 + 3i, find the modulus and argument

of each of the following.

Solution:

a) a + 2b b) 2a – b

a) a + 2b = 2 – i + 2(1 + 3i)

= 2 – i + 2 + 6i

= 4 + 5i

|a + 2b| = (42 + 52)1/2 = √41

θ = tan-1(5/4) = 51.34o = 0.896 rad

y

x

P(4, 5)

θ

c) ib

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b) 2a – b = 2(2 – i) – (1 + 3i)

= 4 – 2i – 1 – 3i

= 3 – 5i

|2a – b| = (32 + (-5)2)1/2 = √34

θ = tan-1(-5/3) = - 59.04o = - 1.030 rad

y

x

P(3, -5)

θ

c) ib = i(1 + 3i)

= i + 3i2

= - 3 + i

y

x

P(-3, 1)

θ α

α = tan-1(1/-3) = - 18.43o

θ = 180o – 18.43o = 161.57o = 2.820 rad

|ib| = ((-3)2 + (1)2)1/2 = √10

19