Luttrell 2012
126
Name: ______________________
Date: _____
8m –Complex Numbers
Imaginary Numbers are comple
x
numbers that are square roots of negative numbers. The
y
are
called imaginar
y
because at the time of their discover
y
, no one could imagine such numbers
e
x
isting. But the
y
do!
Y
ou’ll encounter them in engineering.
The unit for an imaginar
y
number is
i
which is equal to
−
1
. Any imaginar
y
number can be
written as a multiple of
i
. Take
−
4
as an e
x
ample. It can be written as a product,
−
1 4
which can be written as 2
i
. Of course the e
x
ample could be irreducible like
−
=
13
13
i
.
Imaginar
y
numbers can be plotted on the “imaginar
y
” plane. It looks ver
y
similar to the real
number line! Now if
y
ou intersect both the real and imaginar
y
lines at zero.
Y
ou should have the
comple
x
plane. The comple
x
plane contains all numbers. It’s onl
y
a matter of finding them on
the plane. A comple
x
number has the form
a + bi
, where
a
is the real number and
bi
is the
imaginar
y
number.
Y
ou plot the number as if
y
ou would in the cartesian coordinate s
y
stem.
Y
ou
go left/right from the origin as man
y
units as
y
our real number (
a
) and up/down the number of
b
units.
The graphs of 2 + i, 3 - 2i, and 3i are plotted
to the right. (in their approximate spot)
●3i
i
● 2 + i
5
●3-2i
Y
ou can combine comple
x
numbers like an
y
real number, e
x
cept
y
ou can onl
y
add reals together
and imaginaries together. (Like Terms:)
Simplif
y
the following. Graph the results.
1. (-3 - 2
i
) + (-3 + 2
i
)
2. 3 - 2
i -
(3 - 2
i
)
3. 5 -
i+
3
- 3i
4. -3-2
i
-(10-12
i
)
Solve the following and simplify in terms of its complex solution:
5. 4x
2
+8x +9 = 0
6. x
2
+x + 1 = 0
