Download 5.6 Complex Numbers

5.6 – Complex Numbers
What is a Complex Number???
• A complex number
is made up of two
parts – a real
number and an
imaginary number.
• Imaginary numbers
are defined to be the
square root of -1
a
Real Part
+
bi
Imaginary
Part
COMPLEX NUMBERS
• Main Rules
1  i
2
1 i
Where i is imaginary
• Examples
1)  4  i 4  2i
2)  24  i 24  i 4  6  2i 6
The Complex Number Plane
2i
Because a complex number is
made up of a real and an imaginary
value, the complex number plane is
different than an xy coordinate
plane.
i
-2
-1
-i
1
2
Say we want to know
where 2 – 2i would be
-2i
We would go left or right for the
real part and up or down for the
imaginary part.
Finding Absolute Value
• The Absolute Value of
a complex number is
the distance away
from the origin on the
complex number
plane.
Ex: |3 - 4i|
3  (4)
2
9  16
You can find the absolute value by
using the Pythagorean Theorem. In
general,
|a + bi|=
5
a 2  b2
2
Additive Inverses of Complex
Numbers
• Remember that to get the additive inverse
of something, you simply multiply
everything by a negative
Ex: The additive
Inverse of -5 is 5
Therefore, what is the
additive inverse of 5 – 2i?
-5 + 2i
Complex Number Operations
• Combining like terms (adding or subtracting)
(5 + 7i) + (-2 + 6i)
(Hint: treat the imaginary i like
a variable)
3 + 13i
• Multiplying Complex Numbers
(12i)(7i)
84 i2 =
84 (-1) =
-84
You can even FOIL Complex
Numbers!
• (6 – 5i)(4 – 3i) =
24 – 20i -18i + 15i2
24 – 38i + 15(-1)
24 – 15 – 38i
9 – 38i
Now, try a couple on your own:
A)
(2 + 3i)(-3 + 5i)
-21 + i
B)
(4 – 9i)(4 + 3i)
43 – 24i
SOLVING EQUATIONS WITH
COMPLEX SOLUTIONS
4( x  1)  101  20
2
( x  1)  i 9 2
4( x  1)  81
2
( x  1)  814
( x  1)  814
2
( x  1)  i
81
x  1 i 92
1 9i 4
4
( x  1)  i 9 2
1 9i 4