The distance between two points is the length of a straight line between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious.
Distance as mentioned above is sometimes not symmetric, hence not a metric (see below): this applies to distance by car in the case of one-way streets, and also in the case the distance is expressed in terms of the time to cover it (a road may be more crowded in one direction than in the other, for a ship upstream and downstream makes a difference).
We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B.
In the Euclidean space Rn, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are often used instead. For a point (x1, x2, ... ,xn) and a point (y1, y2, ... ,yn), the distances are defined as:
|1-norm distance||=||∑ |xi-yi||
|2-norm distance||=||(∑ (xi-yi)2)1/2|
|p-norm distance||=||(∑ |xi-yi|p)1/p|
|infinity norm distance||=||limit of the p norm distance as p goes to infinity|
The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean Theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance.
The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets).
If you measure the strength of each of the n links in a chain (where larger numbers mean weaker links), then because a chain is only as strong as its weakest link, the strength of the chain will be the infinity-norm distance from the list of measurements to the origin.
The p norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse.
Closeness or proximity (keeping a small distance) and touching (zero distance) are forms of physical intimacy. What distance is appropriate for a particular social situation depends on culture, in Western culture it tends to be larger. It is also a matter of personal preference. People may feel uncomfortable if the distance is too large (cold) or too small (intrusive).
Similar observations apply to figurative senses of distance, such as emotional distance.
The term proxemics was introduced by researcher E.T. Hall in 1963 when he investigated people's use of personal space. He used four categories for informal space: the intimate distance for embracing or whispering (6-18 inches), the personal distance for conversations among good friends (1.5-4 feet), social distance for conversations among acquaintances (4-12 feet), and public distance used for public speaking (12 feet or more).