Editing Dimension (section)

====Complex dimension====
{{main|Complex dimension}}
[[File:Riemann Sphere.gif|right|thumb|The complex plane can be mapped to the surface of a sphere, called the Riemann sphere, with the complex number 0 mapped to one pole, the unit circle mapped to the equator, and a [[point at infinity]] mapped to the other pole.]]
The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the [[real numbers]], it is sometimes useful in the study of [[complex manifold]]s and [[dimension of an algebraic variety|algebraic varieties]] to work over the [[complex numbers]] instead. A complex number (''x'' + ''iy'') has a [[real part]] ''x'' and an [[imaginary part]] ''y'', in which x and y are both real numbers; hence, the complex dimension is half the real dimension.

Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional [[sphere|spherical surface]], when given a complex metric, becomes a [[Riemann sphere]] of one complex dimension.<ref>{{cite book |first1=Shing-Tung |last1=Yau |first2=Steve |last2=Nadis |chapter=4. Too Good to be True |title=The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions |chapter-url=https://books.google.com/books?id=vlA4DgAAQBAJ&pg=PT60 |date=2010 |publisher=Basic Books |isbn=978-0-465-02266-3 |pages=60–}}</ref>