A complex number z can be expressed as:

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

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


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

Important Properties

Arithmetic Relations:

Complex Conjugates: Complex Numbers Related Calculator

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