As a simple example, let
,
, and
, i.e.,
As shown in Fig.2.1, this is a parabola centered at
Then, formally, the roots of
We can think of these as ``imaginary roots'' in the sense that square roots of negative numbers don't really exist, or we can extend the concept of ``roots'' to allow for complex numbers, that is, numbers of the form
where
It can be checked that all algebraic operations for real
numbers2.2 apply equally well to complex numbers. Both real numbers
and complex numbers are examples of a
mathematical field.2.3 Fields are
closed with respect to multiplication and addition, and all the rules
of algebra we use in manipulating polynomials with real coefficients (and
roots) carry over unchanged to polynomials with complex coefficients and
roots. In fact, the rules of algebra become simpler for complex numbers
because, as discussed in the next section, we can always factor
polynomials completely over the field of complex numbers while we cannot do
this over the reals (as we saw in the example
).