Complex Numbers

z=a+bi

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

3+0i=3

A number of the form

is known as a pure Imaginary number

Example

4i

sqrt{-25}=sqrt{25i^{2}}=5i

Equality of 2 Complex Numbers

and

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

0leqthetaleq2pi

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

theta=tan^{-1}(frac{y}{x})=arctan(frac{y}{x})

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

|z|=sqrt {x^{2}+y^{2}}

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