Complex Numbers

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 = (a² + b²)^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 = (3² + 2²)^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 + j² b d                           (8)

Since j² = -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)

   = a² - j a b + j a b - j² b²

   = a² - (- b²)

   = a² + b²                                 (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)

  = 3² + 2²

  = 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 + j² b d) / (c² + d²)                                  (10)

Multiplying both the nominator and the denominator with the conjugate of the denominator is called rationalizing.