A number of the form x+iy where i is the square root of -1, and x and y are real numbers, known as the "real" and "imaginary" part. Complex numbers can be plotted as points on a two-dimensional plane, known as an argand diagram, where x and y are the cartesian coordinates.

An alternative, polar notation, expresses a complex number as (r e^it) where e is the base of natural logarithms, and r and t are real numbers, known as the magnitude and phase. The two forms are related:

        r e^it = r cos(t) + i r sin(t)
               = x + i y

where

        x = r cos(t)
        y = r sin(t)

All solutions of any polynomial equation can be expressed as complex numbers. This is the so-called fundamental theorem of algebra, first proved by Cauchy.

Complex numbers are useful in many fields of physics, such as electromagnetism because they are a useful way of representing a magnitude and phase as a single quantity.