Distance

Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other.

Overview

In most cases, "distance from A to B" is interchangeable with "distance from B to A". In mathematics, a distance function or metric is a function (a rule that relates an input to an output) that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.

In Physics

In physics or everyday usage, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over"). The distance between two objects is the length of the shortest path connecting them. This is the most common usage, and the one used in everyday life.

In Mathematics

In mathematics, the generalization of the concept of distance is the idea of metric. This is a function defined on a set of points and taking values in the real numbers. The function satisfies the conditions that, for any three points A, B, and C,

• The distance from A to B is zero if and only if A and B are the same point.
• The distance from A to B is the same as the distance from B to A.
• The distance from A to B is less than or equal to the distance from A to C plus the distance from C to B (triangle inequality).

See Also

• Length
• Metric (mathematics)
• Metric space
• Metrizable space
• Topology

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