Terms
Argument

The angle created by the positive real axis and the segment connecting the
origin to the plot of a complex number in the complex plane.
Complex Conjugate

The complex conjugate of a given complex number
a + bı
is
a  bı
.
Complex Number

The set of all numbers of the form
a + bı
, where
a
and
b
are real
numbers. The real numbers are all complex numbers.
Complex Plane

A plane with two perpendicular axes, the real axis and the imaginary
axis, on which a complex number
a + bı
is plotted at the
coordinate
(a, b)
. It is customary for the real axis to coincide with the
x
axis of the rectangular coordinate
system, and for the
imaginary axis to coincide with the
y
axis of the rectangular coordinate
system.
Imaginary Axis

The axis in the complex plane that customarily coincides with the
y
axis
of the rectangular coordinate
system, and on
which the imaginary part
bı
of the complex number
a + bı
is plotted.
Imaginary Number

A number that can be expressed in the form
bı
, where
b
is a real
number.
Imaginary Part

The
bı
term in every complex number
a + bı
.
Imaginary Unit

The imaginary unit is
ı
.
ı =
.
Modulus

for a complex number
a + bı
. In the complex
plane, it is the distance between the plot of a complex number and the
origin.
Polar Form of a Complex Number

The polar form of a complex number
z = a + bı
is this:
z = r(cos(θ) + ısin(θ))
, where
r =  z
and
θ
is the
argument of
z
.
Real Axis

The axis in the complex plane that typically coincides with the
x
axis of
the rectangular coordinate
system, and on
which the real part
a
of a complex numbers
a + bı
is plotted.
Real Part

In a complex number
a + bı
,
a
.
Standard Form

For a complex number,
a + bı
.
De Moivre's Theorem

Let
z = r(cos(θ) + ısin(θ).Thenz
^{n} = [r(cos(θ) + ısin(θ)]^{n} = r
^{n}(cos(nθ) + ısin(nθ)
, where
n
is
any positive integer.

Roots of a Complex Number

A complex number
z = r(cos(θ) + ısin(θ)
has exactly
n
n
th roots given by the equation
[cos() + ısin()]
, where
n
is a positive integer, and
k = 0, 1, 2,..., n  2, n  1
.
