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{{Math-m}}

{{Math-m}}

A '''complex number''' is a [[number]] composed of two parts, or components—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.

The '''complex numbers''' are a set of numbers which have important applications in the analysis of periodic, oscillatory, or wavelike phenomena.  They are the 5^th item in this hierarchy of types of [[number]]s:

 

*The "[[natural number]]s", 1, 2, 3, ...  (There is controversy about whether zero should be included.  It doesn't matter.)

*The "[[real number]]s", including irrational numbers

 

A complex number is composed of two parts, or components—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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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.

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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 second component is called the ''imaginary part''.  Its unit basis vector is called <math>i</math>.  The first component is called the ''real part''.  Its unit basis vector is just 1.  Thus, the complex number <math>(a, b)\,</math> can also be written <math>a + bi\,</math>.  Numbers with real part of zero are sometimes called "pure imaginary", with the term "complex" reserved for numbers with both components nonzero.

The second component is called the ''imaginary part''.  Its unit basis vector is called <math>i</math>.  The first component is called the ''real part''.  Its unit basis vector is just 1.  Thus, the complex number <math>(a, b)\,</math> can also be written <math>a + bi\,</math>.

While the "invisible" nature of the imaginary component may be disconcerting at first (and the word "imaginary" may be an unfortunate term for it), the complex numbers are just as genuine as the Dedekind cuts and Cauchy sequences that are used in the definition of the "real" numbers.

The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.

The complex numbers form a [[Field_(mathematics)|field]], with the mathematical operations defined as shown below.

If <math>z\,</math> is a root of a real polynomial, then, since zero is its own conjugate, <math>\overline{z}\,</math> is also a root.  This is often expressed as "Non-real roots of real polynomials come in conjugate pairs."  We saw that above for the cube roots of 1&mdash;two of the roots are complex and are conjugates of each other.  The third root is its own conjugate.

If <math>z\,</math> is a root of a real polynomial, then, since zero is its own conjugate, <math>\overline{z}\,</math> is also a root.  This is often expressed as "Non-real roots of real polynomials come in conjugate pairs."  We saw that above for the cube roots of 1&mdash;two of the roots are complex and are conjugates of each other.  The third root is its own conjugate.

==Other functions==

==Transcendental functions==

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:

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>

::algebra (including finding roots of polynomials)

::algebra (including finding roots of polynomials)

::linear algebra&mdash;vector spaces, inner products, Hermitian and unitary operators, Hilbert spaces, etc.

::linear algebra&mdash;vector spaces, inner products, Hermitian and unitary operators, Hilbert spaces, etc.

::number theory&mdash;This seems improbable, but it is true.  The Riemann zeta function, and the Riemann hypothesis, are very important in number theory.  For a long time, the only proofs of the prime number theorem used complex numbers.  The Wiles/Taylor proof of [[Fermat's Last Theorem]] uses modular forms, which use complex numbers.

::number theory&mdash;This seems improbable, but it is true.  The Riemann zeta function, and the Riemann hypothesis, are very important in number theory.  For a long time, the only proofs of the prime number theorem used complex numbers.  The Wiles/Taylor proof of [[Fermat's Last Theorem]] uses modular forms, which use complex numbers.

*Applied mathematics

*Applied mathematics