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Complex Numbers
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.
Definition of pure
imaginary numbers:
i  1
2
i  1
i is not a variable
it is a symbol for a specific
number
Simplify each expression.
81 1  9i
1. 81 
2. 121x  121x 1 x
2
 11x i x
5
4
3. 200x  100 1 2x
 10i 2x
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
Definition of Complex
Numbers
Any number in form
a+bi, where a and b are
real numbers and i is
imaginary unit.
Definition of Equal
Complex Numbers
Two complex numbers are
equal if their real parts are
equal and their imaginary
parts are equal.
If a + bi = c + di,
then a = c and b = d
When adding or subtracting
complex numbers, combine like
terms.
Ex: 8  3i  2  5i 
8  2  3i  5i
10  2i
Solving quadratic functions
with complex numbers.
x  2  12
2
-2
-2
x   10
2
subtract 2
Take the square root
of both sides
x   1  10  i 10
simplify
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 L2
3018i  10i  6i
30 28i  6
24 28i
CONJUGATES
Each imaginary unit has a conjugate.
Two imaginary units are conjugates
if and only if their products are a
real number.
3i
2  4i
 2i
 3i
2  4i
2i
(3i )( 3i )  9i  9(1)  9
2
(2  4i )( 2  4i )  4  8i  8i  16i  4  16  20
2
(2i )( 2i )  4i  (4)( 1)  4
2
Dividing complex numbers.
To divide complex numbers
you must multiply the
numerator and denominator by
the conjugate of the
denominator.
4i
2i
What’s the conjugate of the
denominator?
4i 2  i
8i  4i
8i  4



2
2  i 2  i 4  2i  2i  i
5
2
Your turn. Write in standard
form by dividing.
 2  3i
6i
Graphing complex numbers
y-axis is the
imaginary axis.
x-axis is the
real numbers
Identify the numbers plotted.