As shown in Fig.2.1, this is a parabola centered at (where
) and reaching upward to positive infinity, never going below .
It has no real zeros. On the other hand, the quadratic formula says that the
``roots'' are given formally by
. The
square root of any negative number can be expressed as
, so the only new algebraic object is .
Let's give it a name:

Then, formally, the roots of are , and we can formally
express the polynomial in terms of its roots as

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 and are real numbers, and
.

It can be checked that all algebraic operations for real
numbers^{2.2} 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 alwaysfactor
polynomials completely over the field of complex numbers while we cannot do
this over the reals (as we saw in the example
).