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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 
5
121x 1 x
2
 11x i x
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
Cycl e of "i"
i 1
1
i i
0
i  1
3
i  i
2
i 1
4
i  i
6
i  1
7
i  i
5
Simplify.
i
12
To figure out where we
are in the cycle divide the
exponent by 4 and look at
the remainder.
12 4 = 3 w ith remainder 0
So i
12
i 1
0
Simplify.
i
17 Divide the exponent by 4
and look at the remainder.
17 4 = 4 w ith remainder 1
So i
17
i i
1
Simplify.
i
26
Divide the exponent by 4
and look at the remainder.
26 4 = 6 with remainder 2
So i
26
 i  1
2
Simplify.
i
11
Divide the exponent by 4
and look at the remainder.
11 4 = 2 w ith remainder 3
So i  i  i
11
3
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
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
The Habitat for humanity project utilizes
volunteers to help build house for low – income
families who might not be able to afford the
purchase of a home. At a recent site, Habitat
workers built a small storage shed attached to
the house. The electrical blueprint for the shed
called for two AC circuits connected in series
with a total voltage of 220 volts. One of the
circuits must have an impedance of 7-10j ohms,
and the other needs to have an impedance of
9+5j ohms. According to the building codes,
the impedance cannot exceed 20-5j ohms. Will
the circuits, as designed, meet the code?