Complex Numbers
Complex numbers are used in alternating current theory and mechanical vector analysis.
There are two main forms of complex numbers
 Cartesian
 Polar
Complex numbers on the Cartesian form
A complex number consists of a real part and an imaginary part and can be expressed on the Cartesian form as
Z = a + j b (1)
where
Z = complex number
a = real part
j b = imaginary part (it is common to use i instead of j)
A complex number can be represented in a Cartesian axis diagram with an real and an imaginary axis  also called the Argand diagram:
Example  Complex numbers on the Cartesian form
The complex numbers
Z_{A} = 3 + j 2 (2a)
Z_{B} = 3 + j 3 (2b)
Z_{C} = 2  j 2 (2c)
can be represented in the Argand diagram:
Addition and Subtraction of Complex numbers
Complex numbers are added/subtracted by adding/subtracting the separately the real parts and the imaginary parts of the number.
Example  Adding two Complex numbers
Z_{A} = 3 + j 2
Z_{B} = 3 + j 3
Z_{(A+B)} = (3 + (3)) + (j 2 + j 3)
= j 5
Complex numbers on the Polar form
A complex number on the polar form can be expressed as
Z = r (cosθ + j sinθ) (3)
where
r = modulus (or magnitude) of Z  and is written as "mod Z" or Z
θ = argument (or amplitude) of Z  and is written as "arg Z"
r can be determined using Pythagoras' theorem
r = (a^{2} + b^{2})^{1/2} (4)
θ can be determined by trigonometry
θ = tan^{1}(b / a) (5)
(3) can also be expressed as
Z = r e^{j θ} (6)
As we can se from (1), (3) and (6)  a complex number can be written in three different ways.
Example  Complex number on the Polar form
The complex number
Z_{a} = 3 + j 2
can be expressed on the polar form by calculating the modulus and the argument.
The "modulus" can be calculated by using eq. (4):
r = (3^{2} + 2^{2})^{1/2}
= 3.606
The "argument" can be calculated by using eq. (5):
θ = tan^{1}(2 / 3)
= 33.69^{o }
The complex number on polar form (3):
Z_{a} = 3.606 (cos(33.69) + j sin(33.69))
Or alternatively (6)
Z_{a} = 3.606 e^{j 33.69}
Adding or Subtraction of Complex Numbers
Adding Complex Numbers
Z_{a} = a + j b
Z_{b} = c + j d
Z_{a} + Z_{b} = (a + j b) + (c + j d)
= (a + c) + j(b + d) (6)
or alternative
Z_{a} = r_{a} (cosθ_{a} + j sinθ_{a})
Z_{b} = r_{b} (cosθ_{b} + j sinθ_{b})
Z_{a} + Z_{b} = r_{a} (cosθ_{a} + j sinθ_{a}) + r_{b} (cosθ_{b} + j sinθ_{b})
= (r_{a} cosθ_{a} + r_{b} cosθ_{b}) + j (r_{a} sinθ_{a} + r_{b} sinθ_{b}) (6b)
or alternatively
Z_{a} = r_{a} e^{jθa}
Z_{b} = r_{b} e^{jθb}
Z_{a} + Z_{b} = r_{a} e^{jθa} + r_{b} e^{jθb}
= (r_{a} cosθ_{a} + r_{b} cosθ_{b}) + j (r_{a} sinθ_{a} + r_{b} sinθ_{b}) (6c)
Example  Adding Complex Numbers
Z_{a} = 3 + j 2
Z_{b} = 5  j 4
Z_{a} + Z_{b} = (3 + j 2) + (5  j 4)
= (3 + 5) + j(2 + (4))
= 8  j 2
Example  Adding Complex Numbers
Z_{a} = 3 (cos 35 + j sin 35)
Z_{b} = 2 (cos 15 + j sin 15)
Z_{a} + Z_{b} = (3 cos 35+ 2 cos 15) + j (3 sin 35 + 2 sin 15)
= 4.38  j 2.2
Subtracting Complex Numbers
Z_{a} = a + j b
Z_{b} = c + j d
Z_{a}  Z_{b} = (a + j b)  (c + j d)
= (a  c) + j(b  d) (7)
Example  Subtracting Complex Numbers
Z_{a} = 3 (cos 35 + j sin 35)
Z_{b} = 2 (cos 15 + j sin 15)
Z_{a}  Z_{b} = 3 (cos 35 + j sin 35)  2 (cos 15 + j sin 15)
= (3 cos 35  2 cos 15) + j (3 sin 35  2 sin 15)
= 0.52 + j 1.2
Multiplication of Complex Numbers
Z_{a} = a + j b
Z_{b} = c + j d
Z_{a} Z_{b} = (a + j b) (c + j d)
= a c + a (j d) + (j b) c + (j b) (j d)
= a c + j a d + j b c + j^{2} b d (8)
Since j^{2} = 1  (8) can be transformed to
Z_{a} Z_{b} = (a + j b) (c + j d)
= (a c  b d) + j (a d + b c) (8b)
Example  Multiplying Complex Numbers
Z_{a} = 3 + j 2
Z_{b} = 5  j 4
Z_{a} Z_{b} = (3 + j 2) (5  j 4)
= (3 5  2 (4)) + j(3 (4) + 2 5)
= 23  j 2
Complex Conjugate
The complex conjugate of (a + jb) is (a  jb).
Multiplying a complex number with its complex conjugate results in a real number like
Z_{a} = a + jb
Z_{a}^{*} = a  jb
Z_{a} Z_{a}^{*} = (a + jb) (a  jb)
= a^{2}  j a b + j a b  j^{2} b^{2}
= a^{2}  ( b^{2})
= a^{2} + b^{2} (9)
Example  Multiplying a Complex Number with its Conjugate
Z_{a} = 3 + j 2
Z_{a}^{*} = 3  j 2
Z_{a} Z_{a}^{*} = (3 + j 2) (3  j 2)
= 3^{2} + 2^{2}
= 13
Division of Complex Numbers
Division of complex numbers can be done with the help of the denominators conjugate:
Z_{a} = a + jb
Z_{b} = c + j d
Z_{a} / Z_{b} = (a + j b) / (c + j d)
= ((a + j b) / (c + j d)) ((c  j d) / (c  j d))
= (a c + j a d + j b c + j^{2} b d) / (c^{2} + d^{2}) (10)
Multiplying both the nominator and the denominator with the conjugate of the denominator is called rationalizing.
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