Download The product of a complex number and its conjugate

Subtraction:
Notes
 4  4 1  4.  1  2i
(a+bi)-(c+di )=(a-c)+(b - d)i
Multiplication: ( a + b i )( c + d i ) = ( ac - bd ) + ( ad + bc ) i
 9  9 1  9.  1  3i
Argand Diagram
The properties of the complex conjugate
y (imaginary axis)
(a,b)
b
a + bi
O
A = a + bi
a
Let z = a + b i , the complex conjugate, or briefly conjugate of
z is defined by
z= a–bi.
x (real axis)
corresponds to (a,b)
Equality of Complex Numbers
Given that
x and y .
( 5x – y ) + ( x - y ) i = 9 + i . Find the value of
The product of a complex number and its conjugate
( a + b i )( a – b i ) = a2 + b2
(A real number)
For any complex numbers z1 , z2 , z3 , we have the following
algebraic properties of the conjugate operation:
(i) z1  z 2  z1  z 2
(ii) z1  z 2  z1  z 2
(iii) z1 z 2  z1  z 2
z
(iv)  1
 z2

z
  1 , provided z2 0
z2

Example
Properties of Complex Numbers
Addition:
( a + b i )+ ( c + d i ) = ( a + c ) + ( b + d ) i
Write
4  3i
in the standard form a + bi.
2  5i
y
De Moivre’s Theorem
r
y
If z = x + i y = r ei  , and n is a natural number, then

x
x
zn = (x + i y)n = rn en i
In term of trigonometric form
Trigonometric Form
z = x
+ i y
= r cos  + i
= r ( cos  + i sin  )
Exponential Form
z =
x
zn = [r( cos  + i sin )]n = rn [cos n + i sin n].
r sin 
Theorem 2
Given any nonzero complex number z = rei , the equation
wn  z has precisely n solutions given by
+ i y
= r ( cos  + i sin  )
= r e i
    2k  
wk  n r ei  n   , k = 0, 1, 2,…., n-1


Theorem 1
or
i 1
i 2
If z1  r1e and z2  r2e , the products and the quotients of
complex numbers in exponential form are
1. z1 z2  r1ei1 r2ei 2  r1r2ei 1  2 
z1 r1ei1 r1 i 1  2 

 e
z2 r2ei 2 r2
    360o k 
   360o k 
  i sin 
 , k = 0,
wk  n r cos
n
n



 
1, 2,…., n-1
or
    2k 
   2k 
wk  n r cos
  i sin 
 , k = 0, 1,
 n 
  n 
2,…., n-1
Forms of Complex Numbers
1. Express the following complex numbers in the exponential and
trigonometric
form.
a. –1 – 4i
b.  3  i
c. 3  4i
d. 2 + 3i
2.
Write in the form of a + bi
a. e
i
b. 3 e

i
2
Multiplication and division in exponential form
1. If
z1  4e
5
i
18
and
a. z1.z2
2.
z2  e
17
i
9
, find
b.
z1
z2
Use the above identities to find z1.z2 and
z1
if
z2
a. z1 = 2 (cos 90o + i sin 90o) and z2 = 4 (cos 30o + i sin 30o)
Powers of Complex Numbers
1. Use De Moivre’s Theorem to find (1 + 2i)8. Leave your answer in the
exponential form.
2. Use De Moivre’s Theorem to find ( 3 - 3i)5. Write your answer in
the
i.
trigonometric form
ii.
standard form
3. By using De Moivre’s Theorem, write each expression in the standard
form.
a. [ 3 (cos 20o + i sin20o) ]3
b.
[
2 (cos
5
5 4
+ i sin
)]
8
8
c.
(-
3 - i)6
Roots of Complex Numbers
1. Find the fourth roots of 3  7i Leave your answers in the exponential
form.
2. Find all the complex cube roots of 27i. Leave your answers in the
trigonometric form.
3. Find all the five distinct roots of - 4 + 3i. Leave your answers in the
standard form.
4. Find all the sixth distinct roots of -16. Leave your answers in the
standard form.
5. Evaluate all the values of
form.
4
 5  5i . Leave your results in the standard