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| In [[mathematics]], ''n''-dimensional '''complex space''' is a multi-dimensional generalisation of the [[complex number]]s, which have both [[real number|real]] and [[imaginary number|imaginary]] parts or dimensions. The ''n''-dimensional complex space can be seen as ''n'' [[cartesian product]]s of the complex numbers with itself:
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| :<math> \C^n = \underbrace{\C \times \C \times \cdots \times \C}_{n-\text{times}}</math> | |
| The ''n''-dimensional complex space consists of ordered ''n''-tuples of complex numbers, called [[coordinate]]s:
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| :<math> \C^n = \{ (z_1,\ldots,z_n) : z_i \in \C \ \text{for all} \ 1 \le i \le n\}</math>
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| The real and imaginary parts of a complex number may be treated as separate dimensions. With this interpretation, the space <math>\C^n</math> of ''n'' complex numbers can be seen as having <math>2 \times n</math> dimensions represented by <math>2 \times n</math>-tuples of real numbers. The two different interpretations can cause confusion about the dimension of a complex space.
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| The study of complex spaces, or [[complex manifold]]s, is called [[complex geometry]].
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| ==One dimension==
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| The [[complex line]] <math>\C^1</math> has one real and one imaginary dimension. It is analogous in some ways to two-dimensional real space, and may be represented as an [[Argand diagram]] in the real plane.
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| In [[projective geometry]], the [[complex projective line]] includes a point at infinity in the Argand diagram and is an example of a [[Riemann sphere]].
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| ==Two dimensions==
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| The term "complex plane" can be confusing. It is sometimes used to denote <math>\C^2</math>, and sometimes to denote the <math>\C^1</math> space represented in the [[Argand diagram]] (with the Riemann sphere referred to as the "extended complex plane"). In the present context of <math>\C^n</math>, it is understood to denote <math>\C^2</math>.
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| An intuitive understanding of the [[complex projective plane]] is given by Edwards (2003), which he attributes to [[Von Staudt]].
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| ==References== | |
| *Djoric, M. & Okumura, M.; ''CR Submanifolds of Complex Projective Space'', Springer 2010
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| *Edwards, L.; ''Projective geometry'' (2nd Ed), Floris, 2003.
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| *Lindenbaum, S.D.; ''Mathematical methods in physics'', World Scientific, 1996
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| ==See also==
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| *[[Calabi-Yau space]]
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| *[[Complex polytope]]
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| *[[Riemann surface]]
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| *[[Several complex variables]]
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| [[Category:Complex manifolds]]
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| {{Geometry-stub}}
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