*
Section 1-5: Complex Numbers *
*Recall from Algebra
II
that square roots of negative numbers are not real
numbers. The uses of these types of
numbers has become quite common. Therefore, we have the following
basic definition of an imaginary unit!*
*The powers of i
form
a cycle. They repeat the same pattern as above. Thus, i*^{5}
= i, i^{6} = -1
etc.
To make it easy to find higher powers of i, simply divide the power by
4 and the remainder tells you which of the above to use.
Example:
i^{43}
Divide 43
by 4 and get a remainder of 3 thus it is equivalent to i^{3}
= -i!!

Study the sample problems
below
for the basic arithmetic with imaginary numbers!
The above examples
illustrate
that to work with imaginary numbers, separate the imaginary part from
the
real part and simplify the real part.

__Complex numbers__
*Any number in the
form
a + bi
is an imaginary number. The a
is the real
part
and b is the
imaginary
part.*
*Adding complex
numbers
is easy. simply combine like terms!!*
*(5 + 3i) + (4 + 6i)
=
9 + 9i*
*(3 + 2i) - (4 - 3i)
=
-1 + 5i (remember to switch signs when subtracting)*

*Multiplying is also
fairly
easy. Since complex numbers are binomials, to multiply use the
foil
method.*
*(2 + 3i)(1 + 2i) =
2
+ 4i + 3i + 6i*^{2}
=
(2 - 6) + (4i + 3i) = -4 + 7i
*remember i*^{2}
= -1

*Recall that a + bi
and
a - bi are complex conjugates.
Their sum and product will always be a real number!! You can use
this property to rationalize the division of two complex numbers by
multiplying
top and bottom by the complex conjugate of the denominator!! Look
at the example below:*

That's our simple review
of
the complex numbers. This topic should be easily remembered from
last year!
On to quadratics!