= Complex Numbers =

'''Complex numbers''' have a real and imaginary part.

<<TableOfContents>>

----



== Description ==

A complex number has a '''real part''' and an '''imaginary part'''. It can be expressed as ''a + bi'' where ''a'' and ''b'' are from the set of real numbers.

The defining characteristic of ''i'' is that ''i^2^ = -1''.



=== Coordinates ===

Complex numbers can be expressed as being in a 2-dimensional plane: given ''z = x + yi'', ''z = (x,y)''.

It follows that they can also be expressed in [[Calculus/CoordinateSystem|polar coordinates]]. In this case, they are expressed either as ''z = r(cosθ + i sinθ)'' or ''z = r e^θi^''. This reveals a relation between complex numbers and rotation in a coordinate system.

To convert the above ''z'' into polar coordinates:
 * ''r = |z| = √(x^2^ + y^2^)''
 * ''θ = tan^-1^(y/x)''

To convert back:
 * ''x = r cosθ''
 * ''y = r sinθ''



=== Vectors ===

Complex numbers are sometimes expressed as a 2-dimensional vector with a real first member and an imaginary second member. In this case they are expressed as ''z = [a b]'' where ''z'' is in ''R^2^'' space.

----



== Complex Conjugates ==

For a complex number as ''z = a + bi'', there is a '''complex conjugate''' notated and evaluated as ''z̅ = a - bi''.



=== Properties ===

The double conjugate of a complex number is the original complex number: ''a̿ = a''.

Conjugation is distributive.
 * If ''c = ab'', then ''c̅ = a̅b̅''.
 * If ''c = a + b'', then ''c̅ = a̅ + b̅''.
 * If ''c = a - b'', then ''c̅ = a̅ - b̅''.
 * If ''d = exp(a)'', then ''d̅ = exp(a̅)''
 * If ''d = ln(a)'', then ''d̅ = ln(a̅)''



----
CategoryRicottone