February 28, 2020

Chapters

## Introduction

**Complex numbers**are multi-part, 2-Dimensional numbers of the form

where** a** is any Real number and is known as** the Real part** of the Complex number,** **,

and** b** is any Real number multiplied by and is known as** the Imaginary part **of the Complex number, .

#### Example

If , then and , which is multiplied by

The set of all** Complex Numbers** is denoted by .

If in the Complex number , then the number reduces to an ordinary Real number

#### Example

A number of the form is known as a pure Imaginary number

#### Example

### Equality of 2 Complex Numbers

## Complex Numbers in the Complex Plane

Complex numbers are multi-part because of the inclusion of both the Real and Imaginary parts in the whole Complex Number.

As for being 2-Dimensional, we equate the ordered pair with a point in the 2-Dimensional Complex or Argand plane

with and

### The Argand Plane

The Complex plane is a plane similar to the -plane, with 2 axes and 4 quadrants. The is treated as an independent dimension and so is the , which has all of its members multiplied by .

In the Complex plane, the is the Real axis and the is the Imaginary axis.

A Real number is represented by a number on the and an Imaginary number is represented by a number on the .

A Complex Number is 2-Dimensional and is represented by a point in the plane.

#### Example

is the Complex number represented by the point in the 1st Quadrant of the Argand plane

#### Example

is the Complex number represented by the point in the 3rd Quadrant of the plane

## Complex Numbers as Vectors

Multidimensionality gives us the ability to graphically represent a Complex number with a vector quantity and an associated directed angle that the vector makes as measured counterclockwise from the positive , with measured in *radians *and

The vector associated with each unique Complex number is a vector emanating from the origin and directed to the point in the plane.

#### Example

is not only the point in the plane, it is also the vector emanating from the origin directed to the point .

We can see geometrically that along this vector, and the associated angle would be or , because a line coincident with the vector cuts the plane into 2 parts at a angle.

## Argument of a Complex Number

The angle is called the ** argument **of the Complex number and is denoted by .

We can find the value of the argument by taking the *inverse tangent* or *arctangent* of the angle , which is the angle that has a *tangent* value of

#### Example

To check that the angle from the previous example is equivalent to , we need to find the angle whose *tangent* is equal to , because and and their ratio is also equal to

### Range of the Argument

All angles will be measured in *radians* and we limit the range for the directed angle to so as to have a unique angle measure for each vector associated with a unique point, because the angle can have an infinite number of equivalent values past .

### Reference Angles

The argument is always a reference angle between and and if the angle is larger than we determine the reference angle by subtracting multiples of from the original angle.

#### Example

If , we must subtract from it to find the argument

#### Example

If , we know the point is located in the 4th Quadrant because and .

We must always be careful to keep track of minus signs and where exactly the point is located, so as not to confuse angles or quadrants.

### Negative Arguments

An angle is an angle measured clockwise from the positive and it has an equivalent but opposite range from .

An angle measure that is negative, , is equivalent to an angle measure of .

## Modulus of a Complex Number

We can resolve any vector into its respective and components and use the *Pythagorean Theorem* to find its length.

The length of a vector is the distance from the origin to the tip and is known as its *magnitude*. The magnitude of the vector associated with each Complex number is known as its ** modulus **or

*and is found by taking the square root of the sum of the squares of its components*

**absolute value**

The modulus is always a non-negative Real number

with and and

#### Example

and

#### Example

and

#### Example

and

### Magnitude of a Complex Number's Opposite

The magnitude of a Complex number and its negative or opposite are equal in value because the square of a number and the square of its negative are always equal