Complex Numbers
Complex numbers arise naturally from an attempt to solve the equation
It's easy enough to write such an equation down, but how would you
solve it? The answer is
We let the symbol 
represent 
, so that 
. Then
any number of the form
where 
and 
are real, is called a complex number.
Let's take some other complex number to be 
, where 
and 
are real. Then the two complex numbers are equal,
which is to say
if and only if
We refer to as the real part of the complex number 
and
as the complex part. Sometimes these are written as Re()
and Im(), respectively.
We may restate the equivalence condition as 
if and only if
the real part of is equal to the real part of
and the imaginary part of is equal to the imaginary part of .
Complex numbers are multiplied like any other binomial expression:
where we have used .
The complex conjugate of the complex
number is denoted 
and is given by
One reason for defining this is that a number times its own complex
conjugate is real,
Note that the complex conjugate of the complex conjugate is the
original complex number and
Note that the complex conjugate of a product is the product of complex
conjugates:
We also call this the modulus squared so
that the modulus is
It is often useful to look at a graph for a complex number. The graph
consists of an x-axis for the real part, and a y-axis for the
complex part. This is shown in Fig.~?. In this
figure, it is easily seen that we can think of as a
two-dimensional vector and that the magnitude (length) of the vector
is the modulus of the complex number, 
.
Another useful way to represent this is with polar coordinates. We
can do this by writing
It turns out that
so we could also write
It is often the case that people will write this as
where 
as is usual for polar coordinates. Then,
everything is just like polar coordinates, with the exception of the
inclusion of the factor . (See Fig.~?.)
