## Complex Roots

As a simple example, let , , and , *i.e.*,

*complex numbers*, that is, numbers of the form

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 *always* factor
polynomials completely over the field of complex numbers while we cannot do
this over the reals (as we saw in the example
).

**Next Section:**

Fundamental Theorem of Algebra

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The Quadratic Formula