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Section 2.4
Complex
Numbers
Complex Numbers
Solve x2 + 1 = 0
x2  1  0
x 2  1
x 2   1
But what is
1 ?
Remember in Algebra II, this was defined as the imaginary unit
or i.
i  1
Therefore, in our problem the final solution is
x  i
Complex Numbers
With the addition of the imaginary unit, a new set of numbers
was formed – the complex numbers.
Definition of a Complex Number
If a and b are real numbers, the number a + bi is a complex
number, and it is said to be written in standard form. If
b = 0, the number a + bi = a is a real number. If b  0, the
number a + bi is called an imaginary number. A number
of the form bi, where b  0, is called a pure imaginary
number.
Complex
Numbers
Real
Numbers
Imaginary
Numbers
Complex Numbers
Equality of Complex Numbers
Two complex numbers a + bi and c + di, written in standard form,
are equal to each other
a + bi = c + di
if and only if a = c and b = d.
Example: Solve for x and y.
x – 33) + (44 – y)i
(x
y = 9 – 6i
6
a
=
x = 12
b
a
b
=
y = 10
Complex Numbers
Operations with Complex Numbers
Addition and Subtraction of Complex Numbers
If a + bi and c + di are two complex numbers written in standard form,
Their sum and differences are defined as follows.
Sum: (a + bi) + (c + di) = (a + c) + (b + d)i
Difference: (a + bi) – (c + di) = (a – c) + (b – d)i
Examples: (4 + 2i) + (7 – 6i) = (4 + 7) + (2 – 6)i
= 11 – 4i
(7 – 2i) – (3 – 5i) = (7 – 3) + (– 2 + 5)i
= 4 + 3i
Additive Inverse:
Additive Identity
(a + bi) + [ -(a + bi)] = 0
(a + bi) + 0 = a + bi
Complex Numbers
Powers of i
2
i2 = 1 = -1
i3 = i 2  i = (-1)i = – i
i4 = i 2  i 2 = – 1  – 1 = 1
i5 = i 4  i = 1(i) = i
i6 = i 4  i 2 = 1(– 1) = – 1
i7 = i 4  i 3 = 1(– i ) = – i
4
4
i8 = i  i = 1(1) = 1


Every power of i can be
simplified into one of
four choices:
i
–1
–i
1
What is i98?
Think of i98 as (i4)24  i2
i98 = (i4)24  i2
= (1)24  – i
i98 = – i
Complex Numbers
Powers of i (cont.)
Is there a “quicker” way to simplify a power of i? YES!
Since i4 = 1, divide the power of i by 4 and find the remainder.
It is the remainder that gives one the simpler power of i to simplify.
Example: Simplify i115
Step 1:
28
4 115
112
3
REMAINDER
Step 2: Rewrite i115 as
its equivalence
using the
remainder as
the new power
of i
i115 = i3
Step 3: Simplify the
equivalent
power of i
i3 = – i
Therefore, i115 = – i
Complex Numbers
Multiplication of complex numbers
When multiplying follow the same procedures as in algebra,
Except simplify all negatives under the square root symbol first
And simplify all powers of i.
Example:
Solution:

4

16

 4  16 
i 4 i 16 
 2i  4i 
8i 2
8  1
8
Example:
Solution:
3i  6  4i 
3i  6  4i 
18i  12i 2
18i  12  1
18i  12
12  18i
Complex Numbers
Multiplication of complex numbers (cont.)
Example:  3  2i 1  5i 
Example:  2  4i  2  4i  Example:  3  5i 
Solution:  3  2i 1  5i 
Solution:  2  4i  2  4i  Solution:  3  5i 
3  15i  2i  10i
3  13i  10  1
2
4  8i  8i  16i 2
4  16  1
3  13i  10
4  16
13  13i
20
2
2
 3  5i  3  5i 
9  15i  15i  25i 2
9  30i  25  1
9  30i  25
16  30i
Complex Numbers
Rationalizing the Denominator
To rationalize a denominator, remember you are multiplying by a
form of 1. That form of one consists of the denominator’s conjugate.
Remember a conjugate is identical to its partner, but holds the
opposite operation.
For example, the conjugate of 2 + 3i is 2 – 3i.
Complex Numbers
2  3i
Example: Express the following in a + bi form.
1  5i
2  3i 1  5i
1  5i 1  5i
2  13i  15
1  25
2  10i  3i  15i 2
1  5i  5i  25i 2
13  13i
26
2  13i  15  1
1  i
2
1  25  1
1 1
  i
2 2
Complex Numbers
Try these
1.
 a  1   b  3 i  5  8i
a = 6, b = 5 5.
8  3i 1  4i 
20 + 29i
Solve for a and b.
2.
 1 
 
3.
 4  6i   3  2 9
8  8  50

4. 4i  5  2i 
8 – 20i


7  3i 2
7
6.
 6  7i  6  7i 
7.
2  i
8. 4  5i
4  5i
2
85
3 + 4i
9 20
 i
41 41
Complex Numbers
Graphing a complex number
When graphing a complex number in the complex plane, a corresponds
to the x-axis and bi corresponds to the y-axis. The x-axis is then
referred to the Real axis and the y-axis is referred to as the Imaginary
Imaginary
axis.
axis
– 3 + 4i
Graph –3 + 4i
a
+
bi
4i
–3
Real
axis
Complex Numbers
What you should know
1. How to simplify a power of i
2. How to add and subtract complex numbers.
3. How to multiply complex numbers
4. How to use complex conjugates to write the quotient of two
complex numbers in standard form
5. How to graph a complex number