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...the ground field has been extended from the real numbers to the field of [[complex number]]s and later to arbitrary [[field]]s. In the second place notions of abstra

...like [[integer]]s, [[rational number|rational]], [[real number|real]] or [[complex number|complex]] numbers. Still it is important to understand that calculations wi

...ath> and <math>G</math> are established, the function can be iterated even complex number of times. ...omorphic function can be iterated any number <math>n</math> of times, even complex number of times.

...d, but applying them requires sophisticated knowledge of arithmetic with [[complex number]]s. There are no analogous formulas for solving polynomial equations of or

Some simple examples of sequences of the natural, [[real numbers|real]], or [[complex number]]s include (respectively)

**The [[rational number|rational]], [[real number|real]] and [[complex number|complex]] numbers each form commutative rings.

In the case of a [[complex number|complex]]-valued function ''ƒ''(&xi;), ''Fourier's theorem'' states that a

However, over the [[complex number]]s we can write

Galois expressed his theory in terms of polynomials and [[complex number|complex numbers]], today Galois theory is usually formulated using general

...ressed in radicals using only real numbers. But finding the cube root of a complex number can be accomplished in polar form, so changing the form of the complex numb

In order to prove the last relation we write a complex number in polar form

...value of <math>f(x)</math> within the [[real number]]s &mdash; but using [[complex number]]s a value can be found, as will be discussed further below. To find the logarithm of a complex number <math>a + bi</math>, it is convenient to express the number in polar coordi

...tended in several directions. Perhaps the most natural extension are the [[complex number]]s, which contain solutions to all [[polynomial]] equations, and, unlike th

for any [[complex number]] <math>\scriptstyle z</math>. Equivalently, it may be defined as the [[inv

...upon frequency, as normally expressed using an impedance represented by a complex number.

...1843 theory of [[quaternions]]. The quaternions are a generalization of [[complex number]]s. They can be added and multiplied and thus form a structure that is now

...math>\mathbb{R}^n</math></font> for any [[natural number]] ''n''; or the [[complex number|complex plane]] or powers of it, <font style = "vertical-align: 13%"><math

...lisations of the rational numbers. Briefly, an ''algebraic number'' is any complex number that is a solution to some polynomial equation <math>\scriptstyle f(x)=0</m

...th>, where <math>\scriptstyle c^*\,</math> is the complex conjugate of the complex number ''c''.

...theory must be recalled. Matrices over the [[field (algebra)|field]] of [[complex number]]s are considered.

...mathbb{Q}</math> (also [[Real numbers|real]] <math>\mathbb{R}</math> and [[Complex number|complex]] <math>\mathbb{C}</math> numbers)

...encountered in applications or linear algebra courses usually use real or complex number scalars, the most general type of vector space is a module over a [[divisio

*<math>\mathbb{C}</math>, the set of [[complex number|complex numbers]]

Newton's method also works in the [[complex number]]s; for example, it can be used to find complex roots of polynomials. Its b

...alled a ''[[signed measure]]'', while such a function with values in the [[complex number]]s is called a ''[[complex measure]]''. Measures that take values in [[Bana

An object parametrized by ''n'' [[complex number]]s may be treated as a point of a complex ''n''-dimensional space. However, ...ce (the latter ''underlies'' the former), since each real number is also a complex number. Linear operations, given in a linear space by definition, lead to such not

...that these form only a real Lie algebra, as multiplication with a general complex number does not preserve skew-hermiticity.

..., "positive" will be understood to mean either a positive real number or a complex number with positive real part.</ref> ...t comes arbitrarily close as <math>z\to-\infty</math>. There is in fact no complex number ''z'' for which <math>\Gamma(z) = 0</math>, and hence the ''reciprocal gamm

...> a_n \cos nx + b_n \sin nx </math>; the latter being the real part of the complex number <math> c_n e^{inx} </math> for <math> c_n = a_n - i b_n </math>, the series

Chose a [[complex number]] <math>c</math> and define function

...]s, allowing their extension to positive and negative values and even to [[complex number]]s. All of these approaches will be presented below. ...n from the series definitions that the sine and cosine functions are the [[complex number|imaginary]] and real parts, respectively, of the [[exponential function|com

...he [[integer]]s, the [[rational number]]s, the [[real number]]s, and the [[complex number]]s under addition, as well as the non-zero rationals, reals, and complex nu

*''z'': generic symbol for a complex number

...alyse them. Many of these properties are also meaningful with respect to [[complex number|complex]] functions and functions of several variables.

...ch are [[irrational number]]s), and negative numbers. He did not accept [[complex number]]s.

...upon frequency, as normally expressed using an impedance represented by a complex number.

...alytic on ''C'' and within the region bounded by the contour ''C'' and the complex number ''a'' is somewhere in this region. The contour integral is taken counter-cl

Using the rules for [[complex number]]s, the magnitude of this gain expression is plotted on the Bode gain plot: where vertical bars denote the magnitude of a complex number (for example, | a + j b | = [ a² + b²]^1/2

...alytic on ''C'' and within the region bounded by the contour ''C'' and the complex number ''a'' is somewhere in this region. The contour integral is taken counter-cl

...zendium.org/wiki?title=Complex_number/Draft&diff=100636845&oldid=100613762 Complex number]

...bold and write <math>\Delta=\left(22\sqrt{-1}\right)^2</math>. '' (from [[Complex number]], try to resize). The point is that, generally, a punctuation mark at the

...are also [[real number]]s. However, the definition above can be used for [[complex number]]s too.

...used to represent continuous quantities. Real numbers are generalised to [[complex number]]s. These are the first steps of a hierarchy of number systems that include ...number]]s|| [[Integer]]s || [[Rational number]]s || [[Real number]]s || [[Complex number]]s

...quantum mechanics, the state of a system at a given time is described by [[complex number|complex]] [[wave functions]] (sometimes referred to as orbitals in the case ...ace" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are point

...y unit]] √−1 ''might'' be an internal question framed in the language of [[complex number]]s about the correct usage of √−1, or it ''might'' be a question about

: It has been pointed out that markup has its advantages, see [[Talk:Complex number#Comments]] (and also [http://forum.citizendium.org/index.php/topic,860.msg6

where ''z'' is chosen to be some complex number which is not an eigenvalue. Then, computing the resolvent amounts to solvin

...have clearly rendered their opinion. This was exactly what took place in [[Complex number]].[[User:Nancy Sculerati|Nancy Sculerati]] 14:29, 16 May 2007 (CDT)

...number]], as directed, the "articles to approve" special page lists [[Talk:Complex number]] as the one needing approval. Code glitch? or did I do it wrong? - [[User: ...have clearly rendered their opinion. This was exactly what took place in [[Complex number]].[[User:Nancy Sculerati|Nancy Sculerati]] 14:29, 16 May 2007 (CDT)