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==Mathematical Definition==

==Mathematical Definition==

A real vectorspace ''V'' said to be ''n''-dimensional if ''V'' has a basis of size ''n''. Note that this implies that ''V'' is isomorphic to n-dimensional Euclidean space. Thus, in agreement with the heuristic definition of dimension given above, 'V'' is ''n''-dimensional if every point of ''V'' is uniquely determined by ''n'' real numbers.

A real vectorspace ''V'' said to be ''n''-dimensional if ''V'' has a basis of size ''n''. Note that this implies that ''V'' is isomorphic to n-dimensional Euclidean space. Thus, in agreement with the heuristic definition of dimension given above, ''V'' is ''n''-dimensional if every point of ''V'' is uniquely determined by ''n'' real numbers.

A manifold ''M'' is said to be ''n''-dimensional if the tangent space to ''M'' at every point is an ''n''-dimensional vectorspace. Note that this implies that ''M'' can be covered by local coordinate systems where each coordinate system looks like a region of n-dimensional Euclidean space. In other words, locally, every point on ''M'' can be specified using ''n'' independent parameters.

A manifold ''M'' is said to be ''n''-dimensional if the tangent space to ''M'' at every point is an ''n''-dimensional vectorspace. Note that this implies that ''M'' can be covered by local coordinate systems where each coordinate system looks like a region of n-dimensional Euclidean space. In other words, locally, every point on ''M'' can be specified using ''n'' independent parameters.