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A metric (from the Greek word for ‘measure’), in mathematics is a generalized idea of the distance between two points. The archetypal metric is the distance function in n dimensional Cartesian co-ordinates, which in one dimension is just the absolute value (value made positive by negating if necessary) of the difference between the co-ordinates of the two points; in two dimensions, the sum of the squares of the differences between the x co-ordinates and the y co-ordinates is taken and then square rooted (which gives the answer you would expect, because of Pythagoras\' theorem).
In general, a metric is a function which takes a pair of points and maps them to a positive real number; the function maps the pair to zero only when they are both the same point, and satisfies the ‘triangle inequality’: d(a,b) + d(b,c) > = d(a,c) for any three points a,b,c.
A set with a metric is called a metric space, and such objects continue to have many of the properties of real numbers. This is particularly true of properties involving distance and limits. For this reason, metric spaces are important in the study of continuity and other similar properties. The ideas are further generalized in topology. SMcL |
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