, Calculus
Lesson
63
Complex
Numbers
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Back to Dr. Nandor's Calculus Page
Recall that ,
, 
This lends itself to a method of representing 
and
in terms of 
. We need to alternate
between a pair of positive
values and a pair of negative values. This suggests that we use
complex numbers! In fact,
If we break this down into two parts, we see that all of the imaginary parts
kept together are exactly 
and
all of the real parts are exactly 
.
So, we have that 
.
The other relationships that are
important (and that we can easily derive) are 
,
, and 
. We can take
derivatives of 
and 
to show that they make
sense. We can also
take derivatives of 
to
show that it also makes sense. You may see a
short-hand notation to represent the equality: 
.
So, if we are looking at some complex number, 
, we can see
that there are two ways to graph it since 
.
Some
items of note: 
Note one of the most wonderful equations in mathematics: 
Multiplying complex numbers also becomes very easy now:
instead of 
, we now
transform each number into radial coordinates:
so that the product is 
dividing, we would have 
Now, this seems like it might be more work, but given the ease of calculation,
most number are put in the radial form anyway, so that is how they
will be given!
Others: Powers:
Roots: 
BUT,
don't forget that there are other solutions!
Example: 
Also do 
, 
, 
, 
, 
,
, which is real!!!!