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# metric

A metric on a set of objects defines a measure of distance on the set, that is, for each pair of objects it defines the distance between those objects. Formally a metric is a function $$d$$ from pairs of elements of the set to the non-negative real numbers satisfying,

1. $$d(a,b) = 0$$ if and only if $$a=b$$,
2. $$d(a,b)=d(b,a)$$, and
3. $$d(a,b)+d(b,c)\geq d(a,c)$$

for all $$a$$, $$b$$, and $$c$$ in the set. The third condition is known as the triangle inequality, and captures the idea that you can't draw a triangle with three lengths if one is longer than the other two combined.

The distance formula in the Cartesian plane is an example of a metric, but there are many other possible metrics on the Cartesian plane, and in more abstract settings the concept of a metric is central, for instance in vector spaces and Hilbert spaces.

Citation Info

• [MLA] “metric.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 28 Mar 2013. Web. 28 Mar 2013. <http://platonicrealms.com/>
• [APA] metric (28 Mar 2013). Retrieved 28 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/metric/