Download 4.8 Complex Numbers

Entry task- Solve two different
ways
4.8
Complex Numbers
Target: I can identify and
perform operations with
complex numbers
-In the set of real numbers, negative numbers do
not have square roots.
-Imaginary numbers were invented so that negative
numbers would have square roots and certain
equations would have solutions.
-These numbers were devised using an imaginary
unit named i.
i  1
Imaginary numbers:
i  1
2
i  1
i is not a variable
it is a symbol for a specific
number
With your a/b partner determine the
values for the cycle of i
1
1
1
i
i
i
-1
-1
-1
-i
-i
1
Definition of Imaginary
Numbers
Any number in form
a+bi, where a and b are
real numbers and i is
imaginary unit.
Definition of Pure
imaginary numbers:
Any positive real number b,
2
2
b  b  1  bi
where i is the imaginary unit
and bi is called the pure
imaginary number.
Simplify the expression.
1. 81 
81 1  9i
Simplify each expression.
4. 8i  3i  24i  24 1
2
2
Remember i  1
 24
5. 5  20 i 5  i 20
Remember that
1  i
 i  100 110 10
2
2
Remember i  1
When adding or subtracting
complex numbers, combine like
terms.
Ex: 8  3i  2  5i 
8  2  3i  5i
10  2i
Simplify.
8 7i 12 11i
8 12 7i  11i
4 18i
Simplify.
9 6i 12 2i 
9 12 6i  2i 
3 8i
Multiplying
complex numbers.
To multiply complex
numbers, you use the
same procedure as
multiplying polynomials.
Simplify.
8 5i2 3i
F
O
I
L
16 24i 10i 15i
16 14i 15
31 14i
2
Simplify.
6 2i 5 3i 
F
O
I
L
3018i  10i  6i
30 28i  6
24 28i
2
-Express these numbers in
terms of i.
1.) 5  1*5  1 5  i 5
2.)  7   1* 7   1 7  i 7
3.)
99  1*99  1 99
 i 9 *11
 3i 11
Conjugates
In order to simplify a fractional
complex number, use a conjugate.
What is a conjugate?
a b  c d and a b  c d
are said to be conjugates of
each other.
Ex: 3 2i  5 and 3 2i  5
Lets do an example:
8i
Ex:
1  3i
8i 1  3i

1  3i 1  3i
Rationalize using
the conjugate
Next
8i  24i
8i  24

19
10
2
4i  12
5
Reduce the fraction
Lets do another example
4i
Ex:
2i
4  i i 4i  i
 
2
2i
2i i
2
Next
4i  1
4i  i

2
2
2i
2
Try these problems.
3
1.
2  5i
3-i
2.
2-i
1.
2  5i
9
7i
2.
5
MULTIPLYING COMPLEX NUMBERS
Multiply
1. 4( 2  3i )
2. (i )( 3i )
3. (2  i )(4  3i )
4. (3  2i )(3  2i )
5. (3  2i )
2
ANSWERS
1. 4(2  3i)  8  12i
2. (i )(3i)  3i  3(1)  3
2
3. (2  i)(4  3i)  8  6i  4i  3i
8  2i  3i
11  2i
2
(-1)
2
9  4i
4. (3  2i)(3  2i) 
2
(-1)
 13
5. (3  2i) (3  2i)(3  2i)
2
= 9  6i  6i  4i 2
 9  12i  4(1)
 5 12i
Use the quadratic formula to solve the following:
2. 3x  2 x  5  0
2
a=3, b= -2, c=5
x  2  (2)  4(3)(5)
2
4
2(3)
x  2  4  60
6
 2  2i 14
6
2   56
x
6
 1  i 14
3
14
Let’s Review
 You need to be able to:
– 1) Recognize what i, i2, i3 ect. is equal to (slide 5)
– 2) Simplify Complex numbers
– 3) Combine like terms (add or subtract)
– 4) Multiply (FOIL) complex numbers
– 5) Divide (multiply by complex conjugates)
Assignment Pg.253 -254
Homework – #9-43 odds, skip
13,15,17
Challenge - 70