# Complex Numbers

## Complex numbers are used in alternating current theory and mechanical vector analysis.

There are two main forms of complex numbers

- Cartesian
- Polar

### Complex numbers on the Cartesian form

A complex number consists of a real part and an imaginary part and can be expressed on the Cartesian form as

*Z = a + j b (1)*

*where *

*Z = complex number*

*a = real part*

*j b = imaginary part (it is common to use i instead of j)*

A complex number can be represented in a Cartesian axis diagram with an real and an imaginary axis - also called the **Argand** diagram:

#### Example - Complex numbers on the Cartesian form

The complex numbers

*Z _{A} = 3 + j 2 (2a)*

*Z _{B} = -3 + j 3 (2b)*

*Z _{C} = -2 - j 2 (2c)*

can be represented in the Argand diagram:

#### Addition and Subtraction of Complex numbers

Complex numbers are added/subtracted by adding/subtracting the separately the real parts and the imaginary parts of the number.

##### Example - Adding two Complex numbers

*Z _{A} = 3 + j 2 *

*Z _{B} = -3 + j 3 *

*Z _{(A+B)} = (3 + (-3)) + (j 2 + j 3)*

* = j 5*

### Complex numbers on the Polar form

A complex number on the polar form can be expressed as

*Z = r (cosθ + j sinθ) (3)*

*where *

*r = modulus (or magnitude) of Z - and is written as "mod Z" or |Z |*

*θ *= argument (or amplitude) of Z - and is written as "arg Z"

r can be determined using Pythagoras' theorem

*r = (a ^{2} + b^{2})^{1/2} (4)*

*θ* can be determined by trigonometry

* θ = tan ^{-1}(b / a) (5) *

*(3)* can also be expressed as

*Z = r e ^{j θ} (6)*

As we can se from (1), (3) and (6) - a complex number can be written in three different ways.

#### Example - Complex number on the Polar form

The complex number

*Z _{a} = 3 + j 2 *

can be expressed on the polar form by calculating the modulus and the argument.

The "modulus" can be calculated by using eq. *(4)*:

*r = (3 ^{2} + 2^{2})^{1/2} *

* = 3.606*

The "argument" can be calculated by using eq. *(5)*:

*θ = tan ^{-1}(2 / 3)*

* = 33.69 ^{o }*

The complex number on polar form (3):

*Z _{a} = 3.606 (cos(33.69) + j sin(33.69)) *

Or alternatively (6)

*Z _{a} = 3.606 e^{j 33.69} *

### Adding or Subtraction of Complex Numbers

#### Adding Complex Numbers

*Z _{a} = a + j b*

*Z _{b} = c + j d*

*Z _{a} + Z_{b} = *(a + j b)

*+ (c + j d)*

* = (a + c) + j(b + d) (6)*

or alternative

*Z _{a} = r_{a} (cosθ_{a} + j sinθ_{a}) *

*Z _{b} = r_{b} (cosθ_{b} + j sinθ_{b}) *

*Z _{a} + Z_{b} = *

*+**r*_{a}(cosθ_{a}+ j sin*)**θ*_{a}*r*_{b}(cosθ_{b}+ j sin*)**θ*_{b}* = (r_{a} cosθ_{a} + r_{b} cosθ_{b}) + j (r_{a} sinθ_{a} + r_{b} sinθ_{b}) (6b)*

or alternatively

*Z _{a} = r_{a} e^{jθa} *

*Z _{b} = r_{b} e^{jθb} *

*Z _{a} + Z_{b} =*

*+**r*_{a}e^{jθa}*r*_{b}e^{jθb}* = (r_{a} cosθ_{a} + r_{b} cosθ_{b}) + j (r_{a} sinθ_{a} + r_{b} sinθ_{b}) (6c)*

Example - Adding Complex Numbers

*Z _{a} = 3 + j 2*

*Z _{b} = 5 - j 4*

*Z _{a} + Z_{b}* = (3 + j 2)

*+ (5 - j 4)*

* = (3 + 5) + j(2 + (-4))*

* = 8 - j 2*

##### Example - Adding Complex Numbers

*Z _{a} = 3 (cos 35 + j sin 35) *

*Z _{b} = 2 (cos 15 + j sin 15) *

*Z _{a} + Z_{b}* =

*(**3 cos 35+ 2**j (3**cos 15) +**sin 35**+ 2**sin**)**15** = 4.38 - j 2.2 *

#### Subtracting Complex Numbers

*Z _{a} = a + j b*

*Z _{b} = c + j d*

*Z _{a} - Z_{b} = (a + j b) - (c + j d) *

* = (a - c) + j(b - d) (7)*

##### Example - Subtracting Complex Numbers

*Z _{a} = 3 (cos 35 + j sin 35) *

*Z _{b} = 2 (cos 15 + j sin 15) *

*Z _{a} - Z_{b} = 3 (cos 35 + j sin 35) - 2 (cos 15 + j sin 15) *

* = (3 cos 35 - 2 cos 15) + j (3 sin 35 - 2 sin 15)*

* = 0.52 + j 1.2*

### Multiplication of Complex Numbers

Z_{a} = *a + j b*

Z_{b} = *c + j d*

*Z _{a} Z_{b} = (a + j b) (c + j d) *

* = a c + a (j d) + (j b) c + (j b) (j d)*

* = a c + j a d + j b c + j ^{2} b d (8)*

*Since **j ^{2} = -1* -

*(8)*can be transformed to

*Z _{a} Z_{b} =* (a + j b)

*(c + j d)*

* = (a c - b d) + j (a d + b c) (8b)*

Example - Multiplying Complex Numbers

Z_{a} = *3 + j 2*

*Z _{b} = 5 - j 4*

* Z_{a} Z_{b} =* (3 + j 2)

*(5 - j 4)*

* = (3 5 - 2 (-4)) + j(3 (-4) + 2 5)*

* = *23 - j 2

### Complex Conjugate

The complex conjugate of *(a + jb)* is *(a - jb)*.

Multiplying a complex number with its complex conjugate results in a real number like

*Z _{a} = a + jb*

*Z _{a}^{*} = a - jb*

*Z _{a} Z_{a}^{*} = (a + jb)*

*(a - jb)*

* = a ^{2} - j a b + j a b - j^{2} b^{2}*

* = a^{2} - (- b^{2})*

* = a^{2} + b^{2} (9)*

##### Example - Multiplying a Complex Number with its Conjugate

*Z _{a} = 3 + j 2*

*Z _{a}^{*} = 3 - j 2*

*Z _{a} Z_{a}^{*} =* (3 + j 2)

*(3 - j 2)*

* = 3 ^{2} + 2^{2}*

* = 1*3

### Division of Complex Numbers

Division of complex numbers can be done with the help of the denominators conjugate:

*Z _{a} = a + jb*

*Z _{b} = c + j d*

*Z _{a} / Z_{b} = (a + j b) / (c + j d)*

*= ( (a + j b) / (c + j d)) ((c - j d) / (c - j d)) *

*= (a c + j a d + j b c + j ^{2} b d) / (c^{2} + d^{2}) (10)*

Multiplying both the nominator and the denominator with the conjugate of the denominator is called *rationalizing*.