Download Complex Numbers - Souderton Math

Complex Numbers
Objective: To define and use
complex numbers
The imaginary unit i
• The numbers that we use everyday are part of the
real number system. There is another number system
used to define a negative square root. These are
called imaginary numbers, defined as
i  1
i 2  1
The imaginary unit i
• A real number together with an imaginary number is
called a complex number.
• Each complex number can be written in the form
a  bi
• where a is the real part and b is the imaginary part. If
a = 0 (no real part), it is called a pure imaginary
number. If b = 0 (no imaginary part), it is a real
number.
Equality of Complex Numbers
• Two complex numbers written in standard form
a  bi  c  di
• are equal to each other if and only if
a = c and b = d.
• This means that the real parts are the same and the
imaginary parts are the same.
Simplifying Negative Square Roots
• Simplify:
 20
Simplifying Negative Square Roots
• Simplify:
 20
 20   1 4 5  2i 5
You Try
• Simplify:
 48
 72
 40
You Try
• Simplify:
 48
 1 16 3
4i 3
 72
 40
 1 36 2
 1 4 10
6i 2
2i 10
Operations with Complex Numbers
• Add:
(4  7i )  (1  6i )
• Put the real part with the real part, the imaginary
with the imaginary.
(4  1)  (7i  6i )
Operations with Complex Numbers
• Add:
(4  7i )  (1  6i )
• Put the real part with the real part, the imaginary
with the imaginary.
(4  1)  (7i  6i )
5i
You Try
• Add:
(2  11i )  (5  7i )
You Try
• Add:
(2  11i )  (5  7i )
• Put the real part with the real part, the imaginary
with the imaginary.
(2  5)  (11i  7i )
7  4i
Operations with Complex Numbers
• Subtract:
(1  2i )  (4  2i )
• Again, put the real with the real, imaginary with the
imaginary.
Operations with Complex Numbers
• Subtract:
(1  2i )  (4  2i )
(1  4)  (2i  2i )
3
Operations with Complex Numbers
• Multiply:
4( 2  3i )
• We will use the distributive property and treat i like
any other variable.
Operations with Complex Numbers
• Multiply:
4( 2  3i )
• We will use the distributive property and treat i like
any other variable.
4(2  3i )  8  12i
Operations with Complex Numbers
• Multiply:
• We will FOIL.
(2  i )( 4  3i )
Operations with Complex Numbers
• Multiply:
(2  i )( 4  3i )
8  6i  4i  3i
2
Operations with Complex Numbers
• Multiply:
(2  i )( 4  3i )
8  6i  4i  3i
2
8  2i  3(1)
8  2i  3  11  2i
You Try
• Multiply:
(3  2i)(3  6i )
You Try
• Multiply:
(3  2i)(3  6i )
9  18i  6i  12i
2
9  24i  12(1)
9  24i 12  3  24i
You Try
• Multiply:
(3  2i)
2
You Try
• Multiply:
(3  2i)
2
(3  2i )(3  2i )
9  6i  6i  4i 2
9  12i  4(1)  5  12i
Complex Conjugates
• Pairs of complex numbers of the form
a  bi
a  bi
• are called complex conjugates. When we multiply
them, we have an interesting result.
Complex Conjugates
• Pairs of complex numbers of the form
a  bi
a  bi
• are called complex conjugates. When we multiply
them, we have an interesting result.
• When multiplying complex conjugates, the result is a
real number.
Multiplying Complex Conjugats
• Multiply:
(3  2i )(3  2i )
9  6i  6i  4i
2
Multiplying Complex Conjugates
• Multiply:
(3  2i )(3  2i )
9  6i  6i  4i
9  4(1)  13
2
Multiplying Complex Conjugates
• You Try:
• Multiply:
(2  4i )( 2  4i )
Multiplying Complex Conjugates
• You Try:
• Multiply:
(2  4i )( 2  4i )
4  8i  8i  16i
4  16(1)  20
2
Use of Complex Conjugates
• We don’t like the denominator of a fraction to be a
complex number. We need to rationalize the
denominator by using complex conjugates.
Rationalize
• Rationalize the following:
2  3i
4 2i
Rationalize
• Rationalize the following:
2  3i
4 2i
• We multiply the denominator by the complex
conjugate.
2  3i
4 2i

4 2i
4 2i
Rationalize
• Rationalize the following:
2  3i
4 2i
• We multiply the denominator by the complex
conjugate.
2  3i
4 2i

4  2 i 8  4i  12i  6i 2

4 2i
16  4i 2
Rationalize
• Rationalize the following:
2  3i
4 2i
• We multiply the denominator by the complex
conjugate.
2  3i
4 2i

4  2 i 8  4i  12i  6i 2 2  16i 1  8i



2
4 2i
16  4i
20
10
You Try
• Rationalize the following:
1 2i
3 3i
You Try
• Rationalize the following:
1 2i
3 3i
• We multiply the denominator by the complex
conjugate.
1 2 i
3  3i

33i 3  3i  6i  6i 2  3  9i  1  3i



2
33i
9  9i
18
6
Writing Complex numbers in
Standard Form
• We will always write a complex number in standard
form after performing the operation.
Other Operations
• Perform the given operation.
 3  12
 48   27
Other Operations
• Perform the given operation.
 3  12  i 3  i 12  i 2 36  6
 48   27  4i 3  3i 3  i 3
Other Operations
• Perform the given operation.
• You Try:
6 8
 48   12
Other Operations
• Perform the given operation.
• You Try:
 6  8  i 6  i 8  i 2 48  1 16 3  4 3
 48   12  4i 3  2i 3  2i 3
Quadratic Formula
• Solve using the quadratic formula.
3x 2  2 x  5  0
Quadratic Formula
• Solve using the quadratic formula.
3x 2  2 x  5  0
 (2)  (2) 2  4(3)(5) 2   56 2  2i 14 1  i 14



2(3)
6
6
3
Homework
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1-9 odd
17-23 odd
27,29,33,37,39,47,49