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{{Math-m}}

{{Math-m}}

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>.

A '''complex number''' is a [[number]] composed of two parts, or components&mdash;a ''[[Real number|real]]'' component and an ''imaginary'' component.  Each of these components is an ordinary (that is, real) number.  The complex numbers form an "extension" of the real numbers:  If the imaginary component of a complex number is zero, that number is essentially identical to the real number that is its real component.

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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.

<!-- This paragraph taken out.  The closure stuff is treated below.  The ordering stuff, while true, doesn't seem relevant to what we are saying.  The matrix material seems unmotivated and unclear.  (In any case, that's not the "necessity of complex numbers" -- the necessity arose from finding roots of polynomials.)  If someone can put the material about equations of motion / eigenvectors / evolution / amplitudes into a focused and clear explanation, please do so, but it probably wouldn't belong on this page.

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]]. One notable consequence, and a very natural way of seeing the necessity of complex numbers is the fact that all matrices of full rank over a vector space over real numbers repesent transformations, which, after a base transformation, are equivalent to a diagonal matrix of the same size with complex entries on the diagonal. Thus, any linear linear equation of motion of arbitrary order and dimension of real numbers can be represented in this way and be decomposed into eigenvectors (or modes). The evolution of the system is fully described by the complex amplitudes.

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]]. One notable consequence, and a very natural way of seeing the necessity of complex numbers is the fact that all matrices of full rank over a vector space over real numbers repesent transformations, which, after a base transformation, are equivalent to a diagonal matrix of the same size with complex entries on the diagonal. Thus, any linear linear equation of motion of arbitrary order and dimension of real numbers can be represented in this way and be decomposed into eigenvectors (or modes). The evolution of the system is fully described by the complex amplitudes.

 

The complex numbers are defined as a 2-dimensional vector space over the [[real number]]s.  That is, a complex number is an ordered pair of numbers: (a, b).  The familiar real numbers constitute the complex numbers with second component zero.  That is, x corresponds to (x, 0).

The complex numbers are defined as a 2-dimensional vector space over the [[real number]]s.  That is, a complex number is an ordered pair of numbers: (a, b).  The familiar real numbers constitute the complex numbers with second component zero.  That is, x corresponds to (x, 0).

==Other functions==

==Other functions==

The higher mathematical functions (often call "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series.  They can be extended to handle complex arguments in the completely natural way, so these functions are defined over the complex plane.  They are in fact "complex [[analytic function]]s".  Just about any normal function one can think of can be extended to the complex numbers, and is complex analytic.  Since the power series coefficients of the common functions are real, they work naturally with conjugates.  For example:

The higher mathematical functions (often called "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series ([[Taylor series]]).  They can be extended to handle complex arguments in the completely natural way, so these functions are defined over the complex plane.  They are in fact "complex [[analytic function]]s".  Just about any normal function one can think of can be extended to the complex numbers, and is complex analytic.  Since the power series coefficients of the common functions are real, they work naturally with conjugates.  For example:

:<math>\sin(\overline{z}) = \overline{\sin(z)}\,</math>

:<math>\sin(\overline{z}) = \overline{\sin(z)}\,</math>

:<math>\log(\overline{z}) = \overline{\log(z)}\,</math>

:<math>\log(\overline{z}) = \overline{\log(z)}\,</math>

==Applications==

==Applications==