[<< wikibooks] Arithmetic/Types of Numbers/Complex Number
== Complex Number ==
Complex Number is a number that can be expressed mathematically as a sum of a Real Number and an Imaginary Number

Z
=
A
+
j
B

{\displaystyle Z=A+jB}

Z
=

|

Z

|

∠
θ

{\displaystyle Z=|Z|\angle \theta }

|

Z

|

=

A

2

+

B

2

{\displaystyle |Z|={\sqrt {A^{2}+B^{2}}}}

θ
=
T
a

n

−

1

B
A

{\displaystyle \theta =Tan^{-}1{\frac {B}{A}}}

== Complex Conjugate Number ==

Z
=
A
−
j
B

{\displaystyle Z=A-jB}

Z
=

|

Z

|

∠
−
θ

{\displaystyle Z=|Z|\angle -\theta }

|

Z

|

=

A

2

+

B

2

{\displaystyle |Z|={\sqrt {A^{2}+B^{2}}}}

θ
=
−
T
a

n

−

1

B
A

{\displaystyle \theta =-Tan^{-}1{\frac {B}{A}}}

== Rules ==
If there are two Complex Numbers

Z

1

=
A
+
j
B

{\displaystyle Z_{1}=A+jB}

Z

2

=
C
+
j
D

{\displaystyle Z_{2}=C+jD}
(A + jB) + (C + jD) = (A + C) + j (B + D)
(A + jB) - (C + jD) = (A - C) + j (B - D)
(A + jB) x (C + jD) = (AC + BD) + j (AD + BC)

(
A
+
j
B
)

(
C
+
j
D
)

{\displaystyle {\frac {(A+jB)}{(C+jD)}}}
=

(
A
+
j
B
)
(
C
−
j
D
)

(
C
+
j
D
)
(
C
−
j
D
)

{\displaystyle {\frac {(A+jB)(C-jD)}{(C+jD)(C-jD)}}}
=

(
A
C
+
B
D
)
+
j
(
B
C
−
A
D
)

C

2

+

D

2

Z

1

×

Z

2

=

|

Z

1

|

|

Z

2

|

∠
(

θ

1

+

θ

2

)

{\displaystyle Z_{1}\times Z_{2}=|Z_{1}||Z_{2}|\angle (\theta _{1}+\theta _{2})}

Z

1

/

Z

2

=

|

Z

1

|

|

Z

2

|

∠
(

θ

1

−

θ

2

)

{\displaystyle Z_{1}/Z_{2}={\frac {|Z_{1}|}{|Z_{2}|}}\angle (\theta _{1}-\theta _{2})}