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A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and and [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.

A '''complex number''' is a [[number]] composed of two parts - a [[Real number|real]] component and an [[Imaginary number|imaginary]] component, of the form <math>a + bi</math>, where ''a'' and ''b'' are real numbers and <math>i^2 = -1</math>.

Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate'').

Whereas the real numbers can be represented as all the possible points on an infinitely extended [[number line]], to represent all the complex numbers requires the use of a two dimensional coordinate system, usually with the real components on the horizontal axis (the ''abscissa'') and the imaginary components on the vertical axis (the ''ordinate'').

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|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 elemantary 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.

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.

===In popular culture===

===In popular culture===