The set of complex numbers is denoted
C
{\displaystyle \mathbb {C} }
. A complex number
z
∈
C
{\displaystyle z\in \mathbb {C} }
can be written in Cartesian coordinates as
z
=
a
+
i
b
{\displaystyle z=a+ib}
where
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
.
a
{\displaystyle a}
is called the 'real part' of
z
{\displaystyle z}
and
b
{\displaystyle b}
is called the 'imaginary part' of
z
{\displaystyle z}
. These can also be written in a trigonometric polar form, as
z
=
r
(
cos
(
ϑ
)
+
i
sin
(
ϑ
)
)
{\displaystyle z=r(\cos(\vartheta )+i\sin(\vartheta ))}
where
r
∈
R
{\displaystyle r\in \mathbb {R} }
is the 'magnitude' of
z
{\displaystyle z}
and
ϑ
∈
[
−
π
,
π
)
{\displaystyle \vartheta \in [-\pi ,\pi )}
is called the 'argument' of
z
{\displaystyle z}
. These two forms are related by the equations
r
=
a
2
+
b
2
ϑ
=
arctan
(
b
a
)
.
{\displaystyle {\begin{aligned}r&={\sqrt {a^{2}+b^{2}}}\\\vartheta &=\arctan \left({\frac {b}{a}}\right).\end{aligned}}}
The trigonometric polar form can also be written as
z
=
r
e
i
ϑ
{\displaystyle z=re^{i\vartheta }}
by using Euler's Identity
e
i
ϑ
=
cos
(
ϑ
)
+
i
sin
(
ϑ
)
.
{\displaystyle e^{i\vartheta }=\cos(\vartheta )+i\sin(\vartheta ).}
Coordination
z
=
x
+
y
i
{\displaystyle z=x+yi}
in Cartesian form,
z
=
|
z
|
(
cos
ϑ
+
i
sin
θ
)
{\displaystyle z=|z|(\cos \vartheta +i\sin \theta )}
in trigonometric polar form,
z
=
|
z
|
e
i
ϑ
{\displaystyle z=|z|e^{i\vartheta }\,}
in polar exponential form.
A complex number
z
∗
∈
C
{\displaystyle z^{*}\in \mathbb {C} }
is a complex conjugate of a number
z
∈
C
{\displaystyle z\in \mathbb {C} }
if and only if
z
⋅
z
∗
=
1.
{\displaystyle z\cdot z^{*}=1.}
If a complex number
z
{\displaystyle z}
is written as
z
=
a
+
i
b
{\displaystyle z=a+ib}
, then the conjugate is
z
∗
=
a
−
i
b
.
{\displaystyle z^{*}=a-ib.}
Equivalently in polar form if
z
=
r
e
i
ϑ
{\displaystyle z=re^{i\vartheta }}
then
z
∗
=
r
e
−
i
ϑ
=
r
(
cos
(
ϑ
)
−
i
sin
(
ϑ
)
)
.
{\displaystyle z^{*}=re^{-i\vartheta }=r(\cos(\vartheta )-i\sin(\vartheta )).}
Operation on 2 different complex numbers [ edit | edit source ]
Addition
(
a
+
i
b
)
+
(
c
+
i
d
)
=
(
a
+
c
)
+
i
(
b
+
d
)
{\displaystyle (a+ib)+(c+id)=(a+c)+i(b+d)}
Subtraction
(
a
+
i
b
)
−
(
c
+
i
d
)
=
(
a
−
c
)
+
i
(
b
−
d
)
{\displaystyle (a+ib)-(c+id)=(a-c)+i(b-d)}
Multiplication
(
a
+
i
b
)
⋅
(
c
+
i
d
)
=
(
a
c
−
b
d
)
+
i
(
a
d
+
b
c
)
{\displaystyle (a+ib)\cdot (c+id)=(ac-bd)+i(ad+bc)}
Division
(
a
+
i
b
)
(
c
+
i
d
)
=
(
a
+
i
b
)
(
c
+
i
d
)
(
c
−
i
d
)
(
c
−
i
d
)
=
(
a
c
+
b
d
)
+
i
(
b
c
−
a
d
)
c
2
+
d
2
{\displaystyle {\frac {(a+ib)}{(c+id)}}={\frac {(a+ib)}{(c+id)}}{\frac {(c-id)}{(c-id)}}={\frac {(ac+bd)+i(bc-ad)}{c^{2}+d^{2}}}}
Operation on complex numbers and its conjugate [ edit | edit source ]
Addition
(
a
+
i
b
)
+
(
a
−
i
b
)
=
2
a
{\displaystyle (a+ib)+(a-ib)=2a}
Subtraction
(
a
+
i
b
)
−
(
a
−
i
b
)
=
2
i
b
{\displaystyle (a+ib)-(a-ib)=2ib}
Multiplication
(
a
+
i
b
)
(
a
−
i
b
)
=
a
2
+
i
a
b
−
i
a
b
+
b
2
=
a
2
+
b
2
{\displaystyle (a+ib)(a-ib)=a^{2}+iab-iab+b^{2}=a^{2}+b^{2}}
Division
(
a
+
i
b
)
(
a
−
i
b
)
=
(
a
+
i
b
)
(
a
−
i
b
)
(
a
+
i
b
)
(
a
+
i
b
)
=
(
a
+
i
b
)
2
a
2
+
b
2
{\displaystyle {\frac {(a+ib)}{(a-ib)}}={\frac {(a+ib)}{(a-ib)}}{\frac {(a+ib)}{(a+ib)}}={\frac {(a+ib)^{2}}{a^{2}+b^{2}}}}
Operation on complex number and its conjugate
z
⋅
z
∗
=
|
z
|
∠
e
i
ϑ
⋅
|
z
|
∠
e
−
i
ϑ
=
|
z
|
2
e
i
(
ϑ
−
ϑ
)
=
|
z
|
2
e
0
=
|
z
|
2
{\displaystyle z\cdot z^{*}=|z|\angle e^{i\vartheta }\cdot |z|\angle e^{-i\vartheta }=|z|^{2}e^{i(\vartheta -\vartheta )}=|z|^{2}e^{0}=|z|^{2}}
z
z
∗
=
|
z
|
e
i
ϑ
|
z
|
e
−
i
ϑ
=
e
2
i
ϑ
{\displaystyle {\frac {z}{z^{*}}}={\frac {|z|e^{i\vartheta }}{|z|e^{-i\vartheta }}}=e^{2i\vartheta }}
Operation on 2 different complex numbers
z
1
⋅
z
2
=
|
z
1
|
e
i
ϑ
1
⋅
|
z
1
|
e
i
ϑ
2
=
|
z
1
z
2
|
e
i
(
ϑ
1
+
ϑ
2
)
{\displaystyle z_{1}\cdot z_{2}=|z_{1}|e^{i\vartheta _{1}}\cdot |z_{1}|e^{i\vartheta _{2}}=|z_{1}z_{2}|e^{i(\vartheta _{1}+\vartheta _{2})}}
z
1
z
2
=
|
z
1
|
e
i
ϑ
1
|
z
2
|
e
i
ϑ
2
=
|
z
1
z
2
|
e
i
(
ϑ
1
−
ϑ
2
)
{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {|z_{1}|e^{i\vartheta _{1}}}{|z_{2}|e^{i\vartheta _{2}}}}=\left|{\frac {z_{1}}{z_{2}}}\right|e^{i(\vartheta _{1}-\vartheta _{2})}}
A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous
e
i
θ
=
cos
θ
+
i
sin
θ
{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \,}
of which there is the famous case (for θ = π):
e
i
π
=
−
1
{\displaystyle e^{i\pi }=-1\,}
More generally,
x
+
y
i
=
r
(
cos
θ
+
i
sin
θ
)
=
r
e
i
θ
{\displaystyle x+yi=r(\cos \theta +i\sin \theta )=re^{i\theta }\,}
(
cos
(
x
)
+
i
sin
(
x
)
)
n
=
cos
(
n
x
)
+
i
sin
(
n
x
)
{\displaystyle (\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)}
for any real
x
{\displaystyle x}
and integer
n
{\displaystyle n}
. This result is known as .
The higher mathematical functions (often called "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series (Taylor series). They can be extended to handle complex arguments in a completely natural way, so these functions are defined over the complex plane. They are in fact "complex analytic functions". Many standard functions can be extended to the complex numbers, and may well be analytic (the most notable exception is the logarithm). Since the power series coefficients of the common functions are real, they work naturally with conjugates. For example:
sin
(
z
¯
)
=
sin
(
z
)
¯
{\displaystyle \sin({\overline {z}})={\overline {\sin(z)}}\,}
log
(
z
¯
)
=
log
(
z
)
¯
{\displaystyle \log({\overline {z}})={\overline {\log(z)}}\,}
Complex number
z
=
x
+
y
i
{\displaystyle z=x+yi}
. In Rectangular plane
z
=
z
∠
θ
{\displaystyle z=z\angle \theta }
. In Polar plane
z
=
z
(
cos
θ
+
i
sin
θ
)
{\displaystyle z=z(\cos \theta +i\sin \theta )}
. In trigonometry
z
=
z
e
j
θ
{\displaystyle z=ze^{j\theta }\,}
. In Complex plane
Complex conjugate number
z
∗
=
x
−
y
i
{\displaystyle z^{*}=x-yi}
. In Rectangular plane
z
∗
=
z
∠
−
θ
{\displaystyle z^{*}=z\angle -\theta }
. In Polar plane
z
∗
=
z
∗
(
cos
θ
−
i
sin
θ
)
{\displaystyle z^{*}=z^{*}(\cos \theta -i\sin \theta )}
. In trigonometry angle
z
∗
=
z
∗
e
j
θ
{\displaystyle z^{*}=z^{*}e^{j\theta }\,}
. In Complex plane
"Complex Numbers" .