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{{Math-m}}

{{Math-m}}

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 '''complex numbers''' are a set of numbers which have important applications in the analysis of periodic, oscillatory, or wavelike phenomena.  Mathematicians denote the set of complex numbers with an ornate capital letter: <math>\mathbb{C}</math>.  They are the 5th 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 "[[natural number]]s", 1, 2, 3, ...  (There is controversy about whether zero should be included.  It doesn't matter.)

*The "[[rational number]]s", or [[fraction]]s, like 355/113

*The "[[rational number]]s", or [[fraction]]s, like 355/113

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

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

*The "complex numbers, which give solutions to polynomial equations

*The "complex numbers", which give solutions to polynomial equations

A complex number is 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.

A complex number is composed of two parts, or components&mdash;a ''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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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.

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.

==Arithmetic operations==

==Arithmetic operations==

==Applications==

==Applications==

It may seem improbable that an artificial construction like this, which does not arise in ordinary measurements, has extensive applications.  But complex numbers in fact are extremely important in many areas of pure and applied mathematics and physics.  Basically, any description of oscillatory phenomena can benefit from a formulation in terms of complex numbers.

Complex numbers are extremely important in many areas of pure and applied mathematics and physics.  Basically, any description of oscillatory phenomena can benefit from a formulation in terms of complex numbers.

The applications include these:

The applications include these:

*Theoretical mathematics:

*Theoretical mathematics:

::algebra (including finding roots of polynomials)

::'''algebra'''&mdash; (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.

::eigenvalue/eigenvector problems<!-- Yes, listing it twice-->

::'''eigenvalue/eigenvector problems'''<!-- Yes, listing it twice-->

::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.

::'''trigonometry'''&mdash;finding arcsine and arccosine functions of numbers with [[absolute values]] more than 1 or arcsecant and arccosecant functions with absolute values between 0 and 1.

*Applied mathematics

*Applied mathematics

::eigenvalue/eigenvector problems

::eigenvalue/eigenvector problems

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Complex analysis]]

[[Category:Complex Analysis]]