Download 4.5 Complex Numbers

Objectives:
1.
Write complex numbers in standard form.
2.
Perform arithmetic operations on complex numbers.
3.
Find the conjugate of a complex number.
4.
Simplify square roots of negative numbers.
5.
Find all solutions of polynomial equations.
Imaginary & Complex Numbers
 The imaginary unit is defined as i  1
 Imaginary numbers can be written in the form bi
where b is a real number.
 A complex number is a sum of a real and imaginary
number written in the form a + bi.
 Any real number can be written as a complex number:
Example:
2 = 2 + 0i ,
−3 = −3 + 0i
Example #1
Equating Two Complex Numbers
 Find x and y.
4x  5i  8  6yi
 4x  8
To solve make two equations
equating the real parts and
imaginary parts separately.
x  2
5i  6 yi
5i
5
y

 6i
6
Example #2
Adding, Subtracting, & Multiplying Complex Numbers
 Perform the indicated operation and write the result in
the form a + bi.
A.
(2  3i)  (1  i)
Combine like terms.
B.
 2  1   3i  i 
 3  2i
(2  3i)  (4  2i)  2  3i  4  2i
Distribute the (-)
and then combine
like terms.
 2  4  3i  2i 
 2  5i
Example #2
Adding, Subtracting, & Multiplying Complex Numbers
 Perform the indicated operation and write the result in
the form a + bi.
2



50
i

2
i


1



C. 10i 
5

i

 5 
  50i  2 1


 2  50i
Distribute & Simplify.
Remember:
(3  i)(2  5i)  6  15i  2i  5i 2
Use FOIL, substitute and
 6  13i  5 1
combine like terms.
 11  13i
D.
i  1
i 2  1
Example #3
Products & Powers of Complex Numbers
 Perform the indicated operation and write the result in
the form a + bi.
A.
(4  5i)(4  5i)
 16  20i  20i  25i 2
 16  25 1
 41
B.
Since these groups are the
same but with opposite signs
they are conjugates of each
other. The middle terms
always cancel with conjugates.
(4  i)  4  i 4  i   16  4i  4i  i 2
 16  8i   1  15  8i
2
Powers of i
This pattern of {i, −1, −i, 1} will continue for even higher patterns.
A shortcut to evaluating higher powers requires you to memorize this pattern,
but it is not necessary to evaluate them.
Example #4
Powers of i
Method 1:
 Find the following:
A.
i
73
 i i
2. Rewrite the even exponent as a
power of i2 (divide it by 2).
72

 i
2 36
1. If the exponent is odd, first “break
off” an i from the original term.
i
  1  i
 1 i  i
36
3. Replace the i2 with −1.
4. Evaluate the power on −1. Even
exponents make it positive and odd
exponents keep it negative.
5. Multiply what is left back together.
Example #4
Powers of i
Method 2:
 Find the following:
A.
i
73
18
4 73
4
33
 32
1
1. Use long division and divide the
exponent by 4. Always use 4 which
is the four possible values of the
pattern {i, −1, −i, 1}.
2. The remainder represents the term
number in the sequence.
For this problem the remainder is 1 which means the
answer is the first term in the sequence {i, −1, −i, 1}
which is i.
Example #4
Powers of i
 Find the following:
B.
i
64
This time it isn’t necessary to
“break off” any i because the
exponent is already even.
 
 i
2 32
  1
1
32
16
4 64
4
24
 24
0
If the remainder is 0 this
indicates that the value is
the 4th term in the sequence
{i, −1, −i, 1} since you can’t
have a remainder of 4 when
dividing by 4.
Therefore, the answer is 1.
Example #4
Powers of i
 Find the following:
C. 51
i
This time it is necessary to
“break off” an i because the
exponent is odd.
 i 50 ·i

 i
2 25
·i
  1 ·i
 1·i  i
25
12
4 51
4
11
8
3
If the remainder is 3 this
indicates that the value is
the 3rd term in the sequence
{i, −1, −i, 1}.
Therefore, the answer is −i.
Example #4
Powers of i
 Find the following:
D. i 30

 i
2 15
15


 1
 1
7
4 30
 28
2
If the remainder is 2 this
indicates that the value is
the 2nd term in the sequence
{i, −1, −i, 1}.
Therefore, the answer is −1.
Example #5
Quotients of Two Complex Numbers
 Express each quotient in standard form.
A.
2  5i  2  5i  1  2i 


1  2i
1  2i  1  2i 

 2  4i  5i  10i
1  4i 2
 2  i  10 1

1  4 1
 12  i

5
12 1
  i
5 5
2
Multiply the top and
bottom by the conjugate
of the denominator, FOIL,
and simplify.
Write your final answer as
a complex number of the
form a + bi.
Example #5
Quotients of Two Complex Numbers
 Express each quotient in standard form.
B.
4i
3  3i


4  i    3  3i 
 3  3i   3  3i 
 12  12i  3i  3i 2
9  9i 2
 12  15i  3 1

9  9 1
 9  15i

18
1 5
  i
2 6
Example #6
Square Roots of Negative Numbers
 Write each of the following as a complex number.
A.
5   1  5
 1  5
i 5
B.
2
11
5
2  i 11

5
2
11
 
i
5
5
After removing the i, make
sure to place it out front as
this can be confusing:
5i  5i
This time with the i off to the
side there is no confusion.
Example #6
Square Roots of Negative Numbers
 Write each of the following as a complex number.
C.



1







16 
5






36 


 1  i 16 5  i 36
 1  4i 5  6i 
 5  6i  20i  24i 2
 5  14i  24 1
 29  14i

Be sure to remove the i from each
radical first!
Example #7
Complex Solutions to a Quadratic Equation
 Find all solutions to the following:
3x  5x  13  0
2
5   5  4313
x
23
2
5   131

6
5
131
 
i
6
6
Sum & Difference of Cubes


3
3
2
2




a  b  
a  ba
  ab  b 







3
3
2
2




a  b  
a  ba
  ab  b 





Example #8
Zeros of Unity
 Find all solutions to the following:
A.
x  64
3
x 3  64  0
x 4 0
3
3
x 2  4 x  16  0
x
4
42  4116 
21
 4   48

2
2
( x  4)( x  4 x  16)  0
 4  4i 3
x40

2
x4
 2  2i 3
Example #8
Zeros of Unity
 Find all solutions to the following:
B.
x  625  0
4
x
2


 25 x  25  0
2
x  25  0
2
x  5x  5x 2  25  0 x 2  25
x  5,  5
x    25
x  5i