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 + jb                         (1)

where

Z = complex number

a = real part

jb = 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 + j2                              (2a)

Z_B = -3 + j3                             (2b)

Z_C = -2 - j2                             (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 + j2    

Z_B = -3 + j3  

Z_(A+B) = (3 + (-3)) + (j2 + j3)

         = j5

Complex numbers on the Polar form

A complex number on the polar form can be expressed as

Z = r (cos(θ) + jsin(θ))                   (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^jθ                    (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 + j2    

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 + jb

Z_b = c + jd

Z_a + Z_b = (a + jb) + (c + jd)   

   = (a + c) + j(b + d)                            (6)

or alternative

Z_a = r_a (cos(θ_a) + jsin(θ_a))

Z_b = r_b (cos(θ_b) + jsin(θ_b))

Z_a + Z_b = r_a (cos(θ_a) + jsin(θ_a) + r_b (cos(θ_b) + jsin(θ_b))

   = (r_a cos(θ_a) + r_b cos(θ_b)) + j(r_a sin(θ_a) +  r_b sin(θ_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 cos(θ_a) + r_b cos(θ_b)) + j(r_a sin(θ_a) +  r_b sin(θ_b))                            (6c)

Example - Adding Complex Numbers

Z_a = 3 + j2

Z_b = 5 - j4

Z_a + Z_b = (3 + j2) + (5 - j4)

  = (3 + 5) + j(2 + (-4))

  = 8 - j2

Example - Adding Complex Numbers

Z_a = 3 (cos(35) + jsin(35))

Z_b = 2 (cos(15) + jsin(15))

Z_a + Z_b = (3 cos(35)+ 2 cos(15)) + j(3 sin(35) +  2 sin(15))

  = 4.38 - j2.2 

Subtracting Complex Numbers

Z_a = a + jb

Z_b = c + jd

Z_a - Z_b = (a + jb) - (c + jd)

    = (a - c) + j(b - d)                           (7)

Example - Subtracting Complex Numbers

Z_a = 3 (cos(35) + jsin(35))

Z_b = 2 (cos(15) + jsin(15))

Z_a - Z_b = 3 (cos(35) + jsin(35)) - 2 (cos(15) + jsin(15))

     = (3 cos(35) - 2 cos(15)) + j(3 sin(35) - 2 sin(15))

     =  0.52 + j1.2

Multiplication of Complex Numbers

Z_a = a + jb

Z_b = c + jd

Z_a Z_b = (a + jb) (c + jd)

   = a c + a (jd) + (jb) c + (jb) (jd)

   = 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 + jb) (c + jd)

    = (a c - b d) + j(a d + b c)                          (8b)

Example - Multiplying Complex Numbers

Z_a = 3 + j2

Z_b = 5 - j4

Z_a Z_b = (3 + j2) (5 - j4)

  = (3 5 - 2 (-4)) + j(3 (-4) + 2 5)

  = 23 - j2

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) - j²b²

   = a² - (- b²)

   = a² + b²                                 (9)

Example - Multiplying a Complex Number with its Conjugate

Z_a = 3 + j2

Z_a^* = 3 - j2

Z_a Z_a^* = (3 + j2) (3 - j2)

  = 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 + jd

Z_a / Z_b = (a + jb) / (c + jd)

= ((a + jb) / (c + jd)) ((c - jd) / (c - jd))

= (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.