Complex number
A complex number is a number composed of two parts - a real component and an 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). This representation is known as the Argand diagram.
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.
Many functions used in real analysis can be extended in to complex numbers using Taylor series. This is the subject of complex analysis.
One should noted that complex numbers do not really exist in nature, for example everything that one can measure in Physics for example weight, energy, pressure and so on are all real numbers. Rather, they are imaginary objects that are used in formal computations. Still, the result of any computation that pertains to the real world, clearly will be a natural number.
Polar notation
The complex number can also be written in the form 
, where
-
is the square of the number's magnitude -
is the phase
If a line is drawn on the complex plane (also known as an 'Argand diagram' or the 'Argand plane') from the origin to a given complex number, the length of that line will be 
and the angle it makes to the real (horizontal) axis will be 
. This leads to a straight-forward geometric interpretation for multiplication by a complex number: multiplying a complex number by 
is equivalent to an anticlockwise rotation through an angle in the complex plane.
Complex Numbers as Matrices
The field F on complex numbers is isomorphic to the field F' of 2x2 matrices of the form
- [a -b]
- [b a],
with mapping as a function f to the above matrix.
We can see that F and F' are isomorphic because:
The function f is clearly 1-to-1 and onto,
,
and 
.
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.
