complex numbers

1. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 63
COMPLEX NUMBERS
Summary of the theory
Washngton pp 334 - 361
THEORY
 Argand diagram: A representation of a complex number in a complex plane
o The horizontal axis represents the real part of the complex numbers, while the vertical axis represents the imaginary part of the
complex number
o When we multiply a complex number by 1
j= − , its representation in an Argand diagram rotates anti-clock wise through 900
o A complex number and its conjugate is symmetrical about the real axis
o A complex number is often represented by a directed line segment (an arrow), similar to a vector
 Argument/amplitude/phase angle: If z r θ
= ∠ , then the argument, denoted by arg( )
z , is given by arg( )
z θ
=
 Cartesian form: Rectangular form
 Complex conjugate
o Rectangular form: If z x jy
= + , then the complex conjugate of z, denoted by z , is given by z x jy
= −
o Polar form: If (cos sin )
z r r j
θ θ θ
= ∠ = + , then ( ) (cos sin )
z r r j
θ θ θ
= ∠ − = −

2. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 64
o Exponential form: If j
z re θ
= , then j
z re θ
−
=
 Complex number: Any number z x jy
= + where ,
x y ∈ and 1
j= −
o Pure real numbers and pure imaginary numbers are sub sets of the set of complex numbers
 Complex plane: The x-y plane where the x-axis represents the real numbers and the y-axis represents the imaginary numbers
 Conversions: ( )
2 2 1
cos ; sin ; ; tan y
x
x r y r r x y
θ θ θ −
= = = + =
 De Moivre's theorems: If 1 1 1
z r θ
= ∠ and 2 2 2
z r θ
= ∠ , then
o 1 2 1 2 1 2
( )
z z rr θ θ
= ∠ +
o 1 1
2 2 1 2
( )
z r
z r θ θ
= ∠ −
o 1 1 1
( ) ( )
n n
z r nθ
= ∠
o ( ) ( )
1 1
360 2
1 1 1 , 0,1,2, ,( 1)
k k
n n n
n n
z r r k n
θ θ π
+ ° +
=
∠ =
∠ = −

 Equality: Two complex numbers 1
z a jb
= + and are equal if and only if a c
= and b d
=
 Euler's formula: cos sin
j
e j
θ
θ θ
±
= ±
 Exponential form: A complex number in the form j
z re θ
=
o θ MUST be in radians!
o De Moivre: 1/ 1/ ( 2 )/
n n j k n
z r e θ π
+
=
o Logs: ln( ) ln( ) ln
j
z re r j
θ
θ
= = +
 Fact: Remember, ,
a b ab a b
= ∀ ∈ but if ,
a b ab a b
≠ ∈
 Imaginary number: Any number aj where a∈ and 1
j= −
 Imaginary part: If z x jy
= + , then Im( )
z y
=
 Imaginary unit: 1
i j
= = −
 Modulus/magnitude/absolute value: If z r θ
= ∠ , then the modulus, denoted by z , is given by z r
=

3. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 65
 Phasor: The representation of a wave by a rotating arm in an Argand plane, called a phasor plane
 Polar form: A complex number in the form ( ) ( )
cos sin cis
z r r j r r
θ θ θ θ
= ∠ = + =
o Also called the modulus-argument or trigonometric form
o θ MAY be in radians or degrees!
o ( 180 ;180 ]
θ ∈ − ° °
 Powers of j: 2 3 4 1
1; ; 1; j
j j j j j
=
− =
− ==
−
 Principal argument: ( 180 ;180 ]
θ ∈ − ° °
o Calculators give the principal argument when converting from rectangular to polar coordinates
 Real part: If z x jy
= + , then Re( )
z x
=
 Rectangular form: A complex number in the form x jy
+ where ,
x y ∈ and 1
j= −

4. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 66
CALCULATIONS
 By hand:
Operation Rectangular Polar Exponential
1 2
,
z z ∈; n∈ 1 1 1
z x jy
= +
2 2 2
z x jy
= +
1 1 1
z r θ
= ∠
2 2 2
z r θ
= ∠
θ in either radians or degrees
1
1 1
j
z re θ
=
2
2 2
j
z r e θ
=
θ in radians
Conjugate 1 1 1
z x jy z x jy
= + ⇔ = − 1 1 1 1 1 1
( )
z r z r
θ θ
= ∠ ⇔ = ∠ − 1 1
1 1 1 1
j j
z re z re
θ θ
−
= ⇔ =
Addition/subtraction 1 2 1 2 1 2
( ) ( )
z z x x j y y
± = ± + ± Covert to rectangular Covert to rectangular
Multiplication 1 2 1 1 2 2
( )( )
z z x jy x jy
=+ + ( )
1 2 1 2 1 2
z z rr θ θ
= ∠ + ( )
1 2
1 2 1 2
j
z z rr e
θ θ
+
=
Division 1 1 1 2 2
2 2 2 2 2
z x jy x jy
z x jy x jy
+ −
+ −
= × ( )
1 1
2 2 1 2
z r
z r θ θ
= ∠ − ( )
1 2
1 1
2 2
j
z r
z r e θ θ
−
=
Powers 1 1 1
( ) ( )
n n
z x jy
= + or convert to polar ( ) ( ) ( )
1 1 1
n n
z r nθ
= ∠ ( ) ( ) ( )
1
1 1
n n j n
z r e
θ
=
Roots Convert to polar ( ) ( ) ( )
( ) ( )
1
1
1/ 1/ 360
1 1 1
1/ 2
1
n n k
n
n
n k
n
z z r
r
θ
θ π
+ °
+
= = ∠
= ∠
( ) ( ) ( )
1
1/ 1/ 2 /
1 1
n n j k n
z r e θ π
+
=
Logs Convert to exponential Convert to exponential ( ) ( )
1 1 1
ln ln
z r jθ
= +
 Practice using your calculator!

5. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 67
APPLICATIONS
 Electrical
o Reactance – effective resistance of a part in a circuit
o Impedance – total effective resistance to the flow of a current
o For a circuit with a resistor with resistance R, a capacitor with reactance XC and an inductor with reactance XL the impedance is
( )
L C
Z R j X X
=
+ +
 Magnitude: 2 2
( )
L C
z R X X
= + −
 Phase angle: ( )
1
tan L C
X X
R
θ −
−
=
SUPPLEMENTARY EXERCISE 4 (JJ Verlinde)
1. Simplify the complex numbers below. Do the operations in the rectangular form and leave your answer in the simplest rectangular form.
1.1 35
j 1.2 41
j 1.3 74
j 1.4 (2 )(3 4)
j j
− +
1.5 (1 3)(1 3)
j j
− + 1.6 ( 2)( 2)
j j
+ − (j+2) 1.7
3
2
1
−
−
j
j
1.8
j
+
−1
2
1.9
j
j
−
1
2
2. Convert the following complex numbers to the polar form, with the argument in radians, accurately to 2 decimal places where necessary.
2.1 1 j
+ 2.2 3 j
− 2.3 2 2
j
− + 2.4 4 3
j
− −
2.5 5
j
− 2.6 8
−

6. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 68
3. Convert the following complex numbers to the rectangular form:
3.1
3
3
4
π
3.2 2 π 3.3 2120
3.4 2 240
− 
3.5 4 0 3.6 5
2
π
−
4. Simplify the following complex numbers. Do not convert to another form. Leave your answers in the simplest polar form.
4.1
4
2 3
4 3
4
2
π π
π
 
 
 
 
    4.2
6
2
8
π
−
 
 
 
4.3
( )( )
4 30
2150 3 60

 
5. Simplify the following complex numbers by using De Moivre’s theorem. Leave your answer in the polar form with the angles in radians.
5.1 4
)
1
(
4
j
−
5.2 2 2
j
− + 5.3 4
2
( 1 )
j
j
− +
5.4 8
)
1
( j
−
6. Use De Moivre’s theorem to solve for z and leave your answer in the polar form with the angles in radians.
6.1 2
4
z j
= 6.2 8
3
−
=
z 6.3 3
2 2
z j
= − 6.4 3
1
2
j
z +
−
=
6.5 16
4
−
=
z
7. Write the following complex numbers in the simplest rectangular (Cartesian) form, accurately to 3 decimal places.
7.1 2
3 π
j
e −
7.2 2
ln3 j
e
π
−
7.3 3
2
j
e
π
8. Write the following complex numbers in the exponential form.
8.1 1 j
+ 8.2 3 j
− 8.3 2 2
j
− + 8.4 4 3
j
− −
8.5 5
j
− 8.6 8
−
9. Solve for x and y, both real, if:
9.1 )
3
5
(
)
3
2
(
2 j
y
j
xj
j +
−
−
= 9.2 )
2
3
)(
2
(
)
1
( j
j
x
y
j
xj −
+
=
+
−

7. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 69
ANSWERS 4
If your answers are different from those given here, first do some further calculations – your answers may still be the same! If they are not,
please discuss it with your lecturer.
1.1 -j 1.2 j 1.3 -1 1.4 10 5
j
+ 1.5 4
1.6 10 5
j
− − 1.7
1 1
2 2
j
− + 1.8 1 j
− − 1.9 1 j
− +
2.1 2 1.410.79
4
π
≈ 2.2 2 2 0.52
6
π
− ≈ − 2.3
3
8 2.83 2.36
4
π
≈ 2.4 5 3.79 5 2.50
= − 2.5 5 5 1.57
2
π
− ≈ −
2.6 8 8 3.14
π ≈ 3.1
3 3
2 2
j
− + 3.2 -2 3.3 1 3
j
− + 3.4 1 3
j
− +
3.5 4 3.6 5
j 4.1
5
12
6
π
4.2
1 3
8 4
π
− 4.3
2
180
3
− 
5.1 π 5.2 4 3
8
8
π
5.3
1
2 2
π
− 5.4 16 0
6.1 2
4
z
π
= or
5
2
4
z
π
= 6.2 2
3
z
π
= or 2
z π
= or
5
2
3
z
π
=
6.3 6
8
12
z
π
= − or 6 7
8
12
z
π
= or 6 5
8
4
z
π
= 6.4 2
3
z
π
= or
4
2
3
z
π
=
6.5 2
4
z
π
= or
3
2
4
z
π
= or
5
2
4
z
π
= or
7
2
4
z
π
= 7.1 3
je
− 7.2 3
j
− 7.3 1 3
j
+
8.1 4
2
j
e
π
8.2 6
2
j
e
π
−
8.3
3
4
8
j
e
π
8.4 2.50
5 j
e−
8.5 2
5
j
e
π
−
8.6 8 j
e π
9.1 ( ; ) (10;6)
x y = 9.2 ( ; ) (1;9)
x y =