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See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic
In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of scalars to each point in an n- dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. More generally, co-ordinates may sometimes be taken from rings or other ring-like algebraic structures.
Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. By convention the origin of the coordinate system in Cartesian coordinates is the point (0,0,...,0), which may be assigned to any given point of Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers. Other coordinate systems do not, however, have a clear notion of origin. For example polar coordinates (r,θ) assign the point (x,y) = (0,0) the value r = 0 but θ any angle.
An example of a coordinate system is to describe a point P in the Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers Rn by an n-tupleSee also tuple (music) as in duple and triple. In mathematics, a tuple is a finite sequence of objects (an ordered list of a limited number of objects). An infinite sequence is a family. Tuples are used by mathematicians to describe mathematical objects t
of real numbers
These numbers r1,...,rn are called the coordinates of the point P.
If a subset S of a Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers is mapped continuouslyIn mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output, the function is said to be onto another topological spaceTopological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies, this defines coordinates in the image of S. That can be called a parametrization of the image, since it assigns numbers to points. That correspondence is unique only if the mapping is bijective.
The system of assigning longitude and latitude to geographical locations is a coordinate system. In this case the parametrization fails to be unique at the north and south poles.