Download 5.4 Complex Numbers

Warm Up
Simplify
1. 3 12
2.
6
2
11
Solve
3. 2 x 2  1  17
4. 7  10 x 2  1
SAT Review question
Dana walks from home to school at a rate of 5 mph. It takes her 2 hours
longer to walk home from school than it did to walk to school. If her total
walking time to and from school was 8 hours, what was Dana’s rate of
speed walking home from school?
a) 3
b) 4
c) 5
d) 8
e) 15
Section 5.4 Complex Numbers
Essential Question
• What is an imaginary number?
Imaginary Unit
• Until now, you have always been told
that you can’t take the square root of
a negative number. If you use
imaginary units, you can!
• The imaginary unit is i.
• i= 1
• It is used to write the square root of
a negative number.
Property of the square root
of negative numbers
• If r is a positive real number, then
r  i r
Examples:
3  i 3
4  i 4 
2i
If i  - 1, then
i i
5
i  1
2
i  i
3
i 1
4
i  1
6
i  i
7
i 1
8
etc.
*For larger exponents,
divide the exponent by
4, then use the
remainder as your
exponent instead.
Example:
i ?
23
23
 5 with a remainder of 3
4
3
So, use i which  -i
i  i
23
Examples
2
1. (i 3 )
 i 2 ( 3)2
 1( 3 * 3 )
 1(3)
 3
2. Solve 3x  10  26
2
3 x  36
2
x  12
2
x   12
x  i 12
x  2i 3
2
Complex Numbers
• A complex number has a real part &
an imaginary part.
• Standard form is:
a  bi
Real part
Example: 5+4i
Imaginary part
Adding and Subtracting complex numbers
1. add or subtract the real parts
2. add or subtract the imaginary parts
Ex: (1  2i)  (3  3i)
 (1  3)  (2i  3i )
 2  5i
Ex: (2  8)  (5  50)
You try!
(13  2i )  (5  6i)
(8  18)  (4  3i 2)
Multiplying
1. Treat the i’s like a variable
2. Change any that are not to the
first power
Ex:
5 * 10
 i 5 * i 10
 i 2 50
 1(5 2)
 5 2
Ex: (2  3i )( 6  2i )
 12  4i  18i  6i 2
 12  22i  6(1)
 12  22i  6
 6  22i
You try!
(6  2i )(2  3i)
8i (9  4i )
Conjugates
• The conjugate of a complex number
has the same real part and the
opposite imaginary part
• Ex. Find the conjugate of 5 + 3i
5 – 3i
Ex. Find the conjugate of 3 – 2i
3 + 2i
Imaginary numbers in the
denominator
• i’s cannot be in the denominator (like
radicals)
• To get rid of the i’s, multiply
numerator and denominator by the
conjugate
• If there is only an imaginary part in
the denominator, multiply by the
same imaginary number
Example
14

2i
14 2i
 *
2i 2i
28i
 2
4i
28i

4
7i
3  11i
Ex :
1  2i
3  11i 1  2i

*
1  2i 1  2i
(3  11i )( 1  2i )

(1  2i )( 1  2i )
 3  6i  11i  22i

1  2i  2i  4i 2
 3  5i  22(1)

1  4(1)
 3  5i  22

1 4
2
 25  5i

5
 25 5i


5
5
 5  i
You try!
5
i
5
1 i
Assignment
Pg. 277: #17-21(odd),
29-33(odd), 37-41(odd),
47-51(odd), 57-61(odd),
65-69(odd), 92
Assessment
• Concept circles