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, i5
= i, i6 = -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:
i43 Divide 43
by 4 and get a remainder of 3 thus it is equivalent to i3
= -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 + 6i2 =
(2 - 6) + (4i + 3i) = -4 + 7i
remember i2
= -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!
