Complex number
A complex number is a number composed of two parts - a real component and and imaginary component, of the form 
, where a and b are real numbers and 
.
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 algebraically closed 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 proofs more rigorous.
In popular culture
In Yvgeny Zamyatin's satirical novel We, the narrator's psychological distress at contemplating the concept of complex numbers becomes a metaphor for the limitations of totalitarian systems of thought.
