# Complex Numbers

## Complex numbers are used in alternating current theory and in 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) *

#### 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:

*Z = 3.606 (cos(33.69) + j sin(33.69)) *

### Adding and Subtraction of Complex Numbers

#### Adding Complex Numbers

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

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

Example - Adding Complex Numbers

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

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

* = 8 - j 2*

#### Subtracting Complex Numbers

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

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

### Multiplication of Complex Numbers

*(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

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

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

Example - Multiplying Complex Numbers

*(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

*(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

*(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:

*(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*.

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