Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, distance must meet more rigorous criteria.

In most cases there is symmetry and "distance from A to B" is interchangeable with "distance between B and A".

Geometry

In neutral geometry, the minimum distance between two points is the length of the line segment between them.

In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. The distance between (x₁, y₁) and (x₂, y₂) is given by

d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\,

Similarly, given points (x₁, y₁, z₁) and (x₂, y₂, z₂) in three-space, the distance between them is

d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}.

Which is easily proven by constructing a right triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the plane that contains the 1st triangle) and applying the Pythagorean theorem.

In the study of complicated geometries, we call this (most common) type of distance Euclidean distance, as it is derived from the Pythagorean theorem, which does not hold in Non-Euclidean geometries. This distance formula can also be expanded into the arc-length formula.

Distance in Euclidean space

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 sometimes used instead.

For a point (x₁, x₂, ...,x_n) and a point (y₁, y₂, ...,y_n), the Minkowski distance of order p (p-norm distance) is defined as: ^n \left| x_i - y_i \right| |- | 2-norm distance || = \left( \sum_{i=1}^n \left| x_i - y_i \right|^2 \right)^{1/2} |- | p-norm distance || = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p} |- | infinity norm distance || = \lim_{p \to \infty} \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p} |- | || = \max \left(|x_1 - y_1|, |x_2 - y_2|, \ldots, |x_n - y_n| \right). |}

p need not be an integer, but it cannot be less than 1, because otherwise the triangle inequality does not hold.

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.