complex number
A number of the form x:iy where i is the square
root of -1, and x and y are {real number}s, 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.
(1995-04-10)