complex number

mathematics 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 Arganddiagram, 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 Theoremof 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.

(1995-04-10)