Complex Number [ edit ] Notation
Z
=
A
+
i
B
=
|
Z
|
∠
θ
=
B
2
+
A
2
∠
tan
−
1
B
A
=
|
Z
|
e
i
θ
{\displaystyle Z=A+iB=|Z|\angle \theta ={\sqrt {B^{2}+A^{2}}}\angle \tan ^{-1}{\frac {B}{A}}=|Z|e^{i\theta }\,}
Coordination
z
=
x
+
y
i
{\displaystyle z=x+yi}
z
=
z
∠
θ
{\displaystyle z=z\angle \theta }
z
=
z
(
cos
θ
+
i
sin
θ
)
{\displaystyle z=z(\cos \theta +i\sin \theta )}
z
=
z
e
j
θ
{\displaystyle z=ze^{j\theta }\,}
Complex conjugate Number [ edit ]
Z
∗
=
A
−
i
B
=
|
Z
|
∠
−
θ
=
B
2
+
A
2
∠
−
tan
−
1
B
A
=
|
Z
|
e
−
i
θ
{\displaystyle Z^{*}=A-iB=|Z|\angle -\theta ={\sqrt {B^{2}+A^{2}}}\angle -\tan ^{-1}{\frac {B}{A}}=|Z|e^{-i\theta }\,}
z
∗
=
x
−
y
i
{\displaystyle z^{*}=x-yi}
z
∗
=
z
∠
−
θ
{\displaystyle z^{*}=z\angle -\theta }
z
∗
=
z
∗
(
cos
θ
−
i
sin
θ
)
{\displaystyle z^{*}=z^{*}(\cos \theta -i\sin \theta )}
z
∗
=
z
∗
e
j
θ
{\displaystyle z^{*}=z^{*}e^{j\theta }\,}
Mathematical Operations [ edit ] Operation on 2 different complex numbers [ edit ]
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)}
Multilication
(
A
+
i
B
)
(
C
+
i
D
)
=
(
A
C
+
B
D
)
+
i
(
A
D
+
B
C
)
{\displaystyle (A+iB)(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
(
A
D
+
B
C
)
(
C
+
i
D
)
2
{\displaystyle {\frac {(A+iB)}{(C+iD)}}={\frac {(A+iB)}{(C+iD)}}{\frac {(C+iD)}{(C+iD)}}={\frac {(AC+BD)+i(AD+BC)}{(C+iD)^{2}}}}
Operation on complex numbers and its conjugate [ edit ]
Addition
(
A
+
i
B
)
+
(
A
−
i
B
)
=
2
A
{\displaystyle (A+iB)+(A-iB)=2A}
Subtraction
(
A
+
i
B
)
−
(
A
−
i
B
)
=
i
2
B
{\displaystyle (A+iB)-(A-iB)=i2B}
Multiplication
(
A
+
i
B
)
(
A
−
i
B
)
=
A
2
−
B
2
{\displaystyle (A+iB)(A-iB)=A^{2}-B^{2}}
Division
(
A
+
i
B
)
(
A
−
i
B
)
=
(
A
+
i
B
)
(
A
−
i
B
)
(
A
−
i
B
)
(
A
−
i
B
)
=
A
2
−
B
2
(
A
−
i
B
)
2
{\displaystyle {\frac {(A+iB)}{(A-iB)}}={\frac {(A+iB)}{(A-iB)}}{\frac {(A-iB)}{(A-iB)}}={\frac {A^{2}-B^{2}}{(A-iB)^{2}}}}
In Polar form [ edit ] Operation on complex number and its conjugate
Z
×
Z
∗
=
|
Z
|
∠
θ
×
|
Z
|
∠
−
θ
=
|
Z
|
2
∠
(
θ
−
θ
)
=
|
Z
|
2
{\displaystyle Z\times Z^{*}=|Z|\angle \theta \times |Z|\angle -\theta =|Z|^{2}\angle (\theta -\theta )=|Z|^{2}}
Z
Z
∗
=
|
Z
|
∠
θ
|
Z
|
∠
−
θ
=
1
∠
2
θ
{\displaystyle {\frac {Z}{Z^{*}}}={\frac {|Z|\angle \theta }{|Z|\angle -\theta }}=1\angle 2\theta }
Operation on 2 different complex numbers
Z
1
×
Z
2
=
|
Z
1
|
∠
θ
1
×
|
Z
1
|
∠
θ
2
=
|
Z
1
|
×
|
Z
2
|
∠
(
θ
1
+
θ
2
)
{\displaystyle Z_{1}\times Z_{2}=|Z_{1}|\angle \theta _{1}\times |Z_{1}|\angle \theta _{2}=|Z_{1}|\times |Z_{2}|\angle (\theta _{1}+\theta _{2})}
Z
1
Z
2
=
|
Z
1
|
∠
θ
1
|
Z
2
|
∠
θ
2
=
Z
1
Z
2
∠
θ
1
−
θ
2
{\displaystyle {\frac {Z_{1}}{Z_{2}}}={\frac {|Z_{1}|\angle \theta _{1}}{|Z_{2}|\angle \theta _{2}}}={\frac {Z_{1}}{Z_{2}}}\angle \theta _{1}-\theta _{2}}
Complex power [ edit ] A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous
Euler formula [ edit ]
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 }\,}
de Moivre's formula [ edit ]
(
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}
n
{\displaystyle n}
Transcendental functions [ edit ] 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 the completely natural way, so these functions are defined over the complex plane. They are in fact "complex analytic functions". Just about any normal function one can think of can be extended to the complex numbers, and is complex analytic. 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)}}\,}
Summary [ edit ] Complex number
z
=
x
+
y
i
{\displaystyle z=x+yi}
z
=
z
∠
θ
{\displaystyle z=z\angle \theta }
z
=
z
(
cos
θ
+
i
sin
θ
)
{\displaystyle z=z(\cos \theta +i\sin \theta )}
z
=
z
e
j
θ
{\displaystyle z=ze^{j\theta }\,}
Complex conjugate number
z
∗
=
x
−
y
i
{\displaystyle z^{*}=x-yi}
z
∗
=
z
∠
−
θ
{\displaystyle z^{*}=z\angle -\theta }
z
∗
=
z
∗
(
cos
θ
−
i
sin
θ
)
{\displaystyle z^{*}=z^{*}(\cos \theta -i\sin \theta )}
z
∗
=
z
∗
e
j
θ
{\displaystyle z^{*}=z^{*}e^{j\theta }\,}
References [ edit ]
See Also [ edit ] "Complex Numbers" .
