A complex number can be expressed as:

where is the imaginary unit. Both and are real numbers. They are called the real and imaginary parts of , respectively.

The complex number can also be expressed in exponential and polar forms:

where:

In the above equations, is called the modulus of and is the argument or phase angle of . It can easily be shown that:

Arithmetic Relations:

Complex Conjugates:

The complex conjugates will not affect common operators:

De Moivre's Theorem:

Roots of Complex Numbers:

where is an integer.

Complex Numbers and Trigonometric Functions:

Trigonometric functions can be expressed in terms of complex numbers in exponential forms, e.g., . (See further detail.)