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Complex Numbers
Complex Numbers
• Complex number is a number in the form z = a+bi, where a and b are real
numbers and i is imaginary. Here a is the real part and b is the imaginary
part of z.
• Where
r  i r
If r is a positive real number, then
• Examples:
 4  i 4  2i
3  i 3
i 2  1
i 3  i 2  i  i
i 4  i 2  i 2  1   1  1
i5  i 4  i  1 i  i
...
Geometry
Plot the points 3 + 4i and –2 – 2i in the complex plane.
Imaginary axis
4
(3, 4)
or
3 + 4i
2
Real
axis
–2
(– 2, – 2)
or
– 2 – 2i
2
–2
Operations on Complex Numbers
(2  3i )( 6  2i )  12  4i  18i  6i 2
2. Solve 3x  10  26
2
3 x 2  36
x  12
2
x   12
2
3i
5  2i
3i


5  2i 5  2i 5  2i
15i  6i 2

25  4i 2
6  15i
6 15



i
29
29 29
x  i 12
x  2i 3
 i (3  i )  3i  i 2
 3i  (1)
 1 3i
Which one is true?
16  49  16  49  4  7  28
or
 12  22i  6(1)
 12  22i  6
 6  22i
Absolute Value
The absolute value of the complex number z = a + bi is
the distance between the origin (0, 0) and the point (a, b).
| a  bi |  a  b
2
2
Example:
Plot z = 3 + 6i and find its absolute value.
Imaginary axis
z  3 6
2
8
z = 3 + 6i
 9  36
6
4
–4
2
3 5 units
–2
4
2
Real
axis
 45
3 5
Trigonometric Representation of Complex Number
To write a complex number z = x + yi in trigonometric form, let  be the
angle from the positive real axis (measured counter clockwise) to the line
segment connecting the origin to the point (x, y).
Imaginary axis
x = r cos 
y = r sin 
z = (x, y)
y
r

x
Let z1  r1 cos1  i sin 1 
Real axis
r=
tan = 𝑦𝑥
𝑥2 + 𝑦2
z = x + yi = r (cos  + i sin  )
The number r is the modulus of z, and  is the
argument of z.
Example: z  12 cos 3  i sin 3

modulus
and
z2  r2 cos2  i sin 2 
z1 z2   r1  cos 1  i sin 1    r2  cos  2  i sin  2  
 r1r2 cos1  2   i sin 1  2 

argument
If z2  0, then
z1 r1
 cos1  2   i sin1  2 
z2 r2
Example
Write the complex number z = –7 + 4i in trigonometric form.
2
2
z  r  7  4i  (7)  4  65
Imaginary
axis
tan = 𝑦𝑥
𝐼𝑚𝑝𝑙𝑖𝑒𝑠
 
𝑦
 = arctan (𝑥 )
tan 1  4  29.74
7
z = –7 + 4i
  180  29.74  150.26
z  65
150.2
6° Real
axis
z  r  cos θ  i sin θ 
 65(cos150.26  i sin150.26)
Example:
Write the complex number 3.75 cos 3  i sin 3
4
4
in standard form a + bi.

cos 3   2
4
2
sin 3  2
4
2


2
2
z  3.75  

i
2 
 2
 15 2  15 2 i Standard form
8
8

De Moivre’s Theorem
Expanding a power of a complex
number in rectangular form is tedious.
 a  bi 
 a  bi  a  bi  a  bi  ...  a  bi 
n
The best way to expand one
of these is using Pascal’s
triangle and binomial
expansion.
It’s much nicer in trig form. We
just raise the r to the power and
multiply theta by the exponent.
z  r  cos   i sin  
z n  r n  cos n  i sin n 
Example
z  5  cos 20  i sin 20 
z 3  53  cos 3  20  i sin 3  20 
z 3  125  cos 60  i sin 60 
Nth Root of a Complex Number
z  r  cos   i sin  
n
  360k
  360k 

z  n r  cos
 i sin
 or
n
n


n
  2 k
  2 k 

r  cos
 i sin
 ; k =0,1,…,n-1
n
n


Find the 4th root of z  81cos80  i sin80
𝑟 = 81 𝑠𝑜,
4
𝑟 = 3 and θ = 80 n=4
Put K=0,1,2,3 into the above equation we get 4 roots as follows:
z1  3  cos 20  i sin 20 
z2  3  cos  20  90   i sin  20  90    3  cos110  i sin110 
z3  3  cos 110  90   i sin 110  90    3  cos 200  i sin 200 
z4  3  cos  200  90   i sin  200  90    3  cos 290  i sin 290 
Assignment: Find the 6th root of unity. That is, solve z 6 = 1
Hint: 1=1+i.0=1(cos0+isin0)