Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas 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"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between strings of text ) or a degree of separation (as exemplified by distance between people in a social network ). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space .
31-475: Length is a measure of distance . In the International System of Quantities , length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre . Length is commonly understood to mean the most extended dimension of
62-519: A graph , the distance between two vertices is measured by the length of the shortest edge path between them. For example, if the graph represents a social network , then the idea of six degrees of separation can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the Erdős number and the Bacon number —the number of collaborative relationships away
93-506: A ruler , or indirectly with a radar (for long distances) or interferometry (for very short distances). The cosmic distance ladder is a set of ways of measuring extremely long distances. The straight-line distance between two points on the surface of the Earth is not very useful for most purposes, since we cannot tunnel straight through the Earth's mantle . Instead, one typically measures
124-450: A solid rectangular box (such as a plank of wood ) is often described as length × height × depth. The perimeter of a polygon is the sum of the lengths of its sides . The circumference of a circular disk is the length of the boundary (a circle ) of that disk. In other geometries, length may be measured along possibly curved paths, called geodesics . The Riemannian geometry used in general relativity
155-416: A statistical manifold . The most elementary is the squared Euclidean distance , which is minimized by the least squares method; this is the most basic Bregman divergence . The most important in information theory is the relative entropy ( Kullback–Leibler divergence ), which allows one to analogously study maximum likelihood estimation geometrically; this is an example of both an f -divergence and
186-459: A weighted graph , it may instead be the sum of the weights of the edges that it uses. Length is used to define the shortest path , girth (shortest cycle length), and longest path between two vertices in a graph. In measure theory, length is most often generalized to general sets of R n {\displaystyle \mathbb {R} ^{n}} via the Lebesgue measure . In
217-547: A Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are flat manifolds in the corresponding geometry, allowing an analog of the Pythagorean theorem (which holds for squared Euclidean distance) to be used for linear inverse problems in inference by optimization theory . Other important statistical distances include the Mahalanobis distance and
248-471: A ball thrown straight up, or the Earth when it completes one orbit . This is formalized mathematically as the arc length of the curve. The distance travelled may also be signed : a "forward" distance is positive and a "backward" distance is negative. Circular distance is the distance traveled by a point on the circumference of a wheel , which can be useful to consider when designing vehicles or mechanical gears (see also odometry ). The circumference of
279-422: A degree of difference or separation between similar objects. This page gives a few examples. In statistics and information geometry , statistical distances measure the degree of difference between two probability distributions . There are many kinds of statistical distances, typically formalized as divergences ; these allow a set of probability distributions to be understood as a geometrical object called
310-426: A fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include height , which is vertical length or vertical extent, width, breadth, and depth. Height is used when there is a base from which vertical measurements can be taken. Width and breadth usually refer to a shorter dimension than length . Depth
341-469: A person is from prolific mathematician Paul Erdős and actor Kevin Bacon , respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations. In psychology , human geography , and the social sciences , distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience. For example, psychological distance
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#1732780154418372-517: A reference frame that is moving relative to the first frame. This means the length of an object varies depending on the speed of the observer. In Euclidean geometry, length is measured along straight lines unless otherwise specified and refers to segments on them. Pythagoras's theorem relating the length of the sides of a right triangle is one of many applications in Euclidean geometry. Length may also be measured along other types of curves and
403-446: Is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality". In sociology , social distance describes the separation between individuals or social groups in society along dimensions such as social class , race / ethnicity , gender or sexuality . Most of the notions of distance between two points or objects described above are examples of
434-412: Is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are". The distance travelled by an object is the length of a specific path travelled between two points, such as the distance walked while navigating a maze . This can even be a closed distance along a closed curve which starts and ends at the same point, such as
465-416: Is an example of such a geometry. In spherical geometry , length is measured along the great circles on the sphere and the distance between two points on the sphere is the shorter of the two lengths on the great circle, which is determined by the plane through the two points and the center of the sphere. In an unweighted graph , the length of a cycle , path , or walk is the number of edges it uses. In
496-667: Is computed using the Pythagorean theorem . The distance between points ( x 1 , y 1 ) and ( x 2 , y 2 ) in the plane is given by: d = ( Δ x ) 2 + ( Δ y ) 2 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . {\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.} Similarly, given points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) in three-dimensional space,
527-491: Is referred to as arclength . In a triangle , the length of an altitude , a line segment drawn from a vertex perpendicular to the side not passing through the vertex (referred to as a base of the triangle), is called the height of the triangle. The area of a rectangle is defined to be length × width of the rectangle. If a long thin rectangle is stood up on its short side then its area could also be described as its height × width. The volume of
558-512: Is the shortest possible path. This is the usual meaning of distance in classical physics , including Newtonian mechanics . Straight-line distance is formalized mathematically as the Euclidean distance in two- and three-dimensional space . In Euclidean geometry , the distance between two points A and B is often denoted | A B | {\displaystyle |AB|} . In coordinate geometry , Euclidean distance
589-432: Is used for the measure of a third dimension . Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed). Measurement has been important ever since humans settled from nomadic lifestyles and started using building materials, occupying land and trading with neighbours. As trade between different places increased,
620-510: The energy distance . In computer science , an edit distance or string metric between two strings measures how different they are. For example, the words "dog" and "dot", which differ by just one letter, are closer than "dog" and "cat", which have no letters in common. This idea is used in spell checkers and in coding theory , and is mathematically formalized in a number of different ways, including Levenshtein distance , Hamming distance , Lee distance , and Jaro–Winkler distance . In
651-493: The inch (in), the foot (ft), the yard (yd), and the mile (mi). A unit of length used in navigation is the nautical mile (nmi). 1.609344 km = 1 miles Units used to denote distances in the vastness of space, as in astronomy , are much longer than those typically used on Earth (metre or kilometre) and include the astronomical unit (au), the light-year , and the parsec (pc). Units used to denote sub-atomic distances, as in nuclear physics , are much smaller than
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#1732780154418682-629: The theory of relativity , because of phenomena such as length contraction and the relativity of simultaneity , distances between objects depend on a choice of inertial frame of reference . On galactic and larger scales, the measurement of distance is also affected by the expansion of the universe . In practice, a number of distance measures are used in cosmology to quantify such distances. Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics: Many abstract notions of distance used in mathematics, science and engineering represent
713-754: The distance between them is: d = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 . {\displaystyle 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}}}.} This idea generalizes to higher-dimensional Euclidean spaces . There are many ways of measuring straight-line distances. For example, it can be done directly using
744-441: The length of some common object. In the International System of Units (SI), the base unit of length is the metre (symbol, m), now defined in terms of the speed of light (about 300 million metres per second ). The millimetre (mm), centimetre (cm) and the kilometre (km), derived from the metre, are also commonly used units. In U.S. customary units , English or imperial system of units , commonly used units of length are
775-401: The mathematical idea of a metric . A metric or distance function is a function d which takes pairs of points or objects to real numbers and satisfies the following rules: As an exception, many of the divergences used in statistics are not metrics. There are multiple ways of measuring the physical distance between objects that consist of more than one point : The word distance
806-426: The millimetre. Examples include the fermi (fm). Distance In the social sciences , distance can refer to a qualitative measurement of separation, such as social distance or psychological distance . The distance between physical locations can be defined in different ways in different contexts. The distance between two points in physical space is the length of a straight line between them, which
837-469: The need for standard units of length increased. And later, as society has become more technologically oriented, much higher accuracy of measurement is required in an increasingly diverse set of fields, from micro-electronics to interplanetary ranging. Under Einstein 's special relativity , length can no longer be thought of as being constant in all reference frames . Thus a ruler that is one metre long in one frame of reference will not be one metre long in
868-464: The one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely, the length of an open interval is first defined as so that the Lebesgue outer measure μ ∗ ( E ) {\displaystyle \mu ^{*}(E)} of a general set E {\displaystyle E} may then be defined as In
899-400: The physical sciences and engineering, when one speaks of units of length , the word length is synonymous with distance . There are several units that are used to measure length. Historically, units of length may have been derived from the lengths of human body parts, the distance travelled in a number of paces, the distance between landmarks or places on the Earth, or arbitrarily on
930-415: The shortest path along the surface of the Earth , as the crow flies . This is approximated mathematically by the great-circle distance on a sphere. More generally, the shortest path between two points along a curved surface is known as a geodesic . The arc length of geodesics gives a way of measuring distance from the perspective of an ant or other flightless creature living on that surface. In
961-417: The wheel is 2π × radius ; if the radius is 1, each revolution of the wheel causes a vehicle to travel 2π radians. The displacement in classical physics measures the change in position of an object during an interval of time. While distance is a scalar quantity, or a magnitude , displacement is a vector quantity with both magnitude and direction . In general, the vector measuring