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The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations.  They are the algebraic closure of the [[real numbers]].

The complex numbers form an [[algebraic closure|algebraically closed]] [[field (mathematics)|field]] but do not permit a non-trivial ordering that is preserved under operations.  They are the algebraic closure of the [[real numbers]].

Computations with complex numbers can easily lead to contradictions, like equating positive and negative one. This problem is usually "solved" by declaring that certain elementary arithmetic rules (in particular of exponentiation) do not apply to complex numbers. It is however unproven that this actually solves all contradictions, which is why most mathematicians consider [[elementary proof]]s more rigorous.

Many functions used in real analysis can be extended in to complex numbers using [[Taylor series]].

===In popular culture===

===In popular culture===