Complex Number

algebra complex-numbers

Definition

Complex Number

The set of complex numbers is defined as:

where is called the imaginary unit.

The structure forms a field.

Motivation

Algebraically, we defined the set of complex numbers to resolve the fact that certain polynomial equations have no solutions in the set of real numbers .

In , the equation has no solution because the squares of any real number is non-negative. To address this, we introduce an imaginary unit , defined by the property:

A complex number is then defined as an expression of the form:

where . The value is the real part , and is the imaginary part .

Gaussian Plane

The Gaussian plane represents a complex number as a point or vector in the plane.

The horizontal axis is the real part and the vertical axis is the imaginary part.

Forms

The reason there are three different representation (Cartesian, Polar, Matrix) is that each highlights a different mathematical personality of complex numbers: arithmetic, geometry, and linear transformations.

Cartesian (Component) Form

The Cartesian (component) form is:

It treats complex numbers as a 2D vector space over .

Polar Form

Based on the Euler’s formula , complex numbers can be viewed as points in a plane defined by a distance from the origin () and an angle ().

Every non-zero admits a polar representation:

where is called the phase factor of . This identifies multiplication in with a product of a scaling by and a rotation by in the complex plane.

Matrix Form

There’s a structural isomorphism between and a specific subset of real matrices. Thus, we can represent a complex number as:

We map to the identity matrix and to a rotation matrix :

Using that mapping, we can represent with and :

More generally, and can be represented by any pair of matrices and satisfying

Arithmetic

Conjugation

For , the complex conjugate is

It is the unique non-trivial automorphism of over .

Moreover,

Modulus

The modulus is defined by

It satisfies

Addition

For and ,

Addition is performed component-wise.

Additive Inverse: For ,

This satisfies

Multiplication

For and ,

This follows from distributivity and the relation .

Multiplicative Inverse: For ,

This satisfies