## Complex Roots

As a simple example, let , , and ,

*i.e.*,

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

^{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