sparknotes
Complex Numbers
Terms and Formulae
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.
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
.
Formulae
| 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
.
|
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