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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, two varying quantities are said to be in a relation of proportionality, multiplicatively connected to a constant; that is, when either their
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant. *If the ''ratio'' () of two variables ( and ) is equal to a constant , then the variable in the numerator of the ratio () can be product of the other variable and the constant . In this case is said to be ''directly proportional'' to with proportionality constant . Equivalently one may write ; that is, is directly proportional to with proportionality constant . If the term ''proportional'' is connected to two variables without further qualification, generally direct proportionality can be assumed. *If the ''product'' of two variables is equal to a constant , then the two are said to be ''inversely proportional'' to each other with the proportionality constant . Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant and . If several pairs of variables share the same direct proportionality constant, the
equation In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are connected by the equals sign "=". The word ''equation'' and its cognates in other languages may h ...
expressing the equality of these ratios is called a proportion, e.g., (for details see
Ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

Ratio
).


Direct proportionality

Given two variables ''x'' and ''y'', ''y'' is directly proportional to ''x'' if there is a non-zero constant ''k'' such that : y = kx. The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter
alpha
alpha
) or "~": : y \propto x, or y \sim x. For x \ne 0 the proportionality constant can be expressed as the ratio : k = \frac. It is also called the constant of variation or constant of proportionality. A direct proportionality can also be viewed as a linear equation in two variables with a ''y''-intercept of and a slope of ''k''. This corresponds to
linear growth Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...
.


Examples

* If an object travels at a constant
speed In everyday use and in kinematics, the speed (commonly referred to as v) of an object is the magnitude (mathematics), magnitude of the change of its Position (vector), position; it is thus a Scalar (physics), scalar quantity. The average speed ...
, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality. * The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to pi, . * On a map of a sufficiently small geographical area, drawn to scale (map), scale distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map. * The force (physics), force, acting on a small object with small mass by a nearby large extended mass due to gravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as gravitational acceleration. * The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, Newton's second law, is the classical mass of the object.


Inverse proportionality

The concept of ''inverse proportionality'' can be contrasted with ''direct proportionality''. Consider two variables said to be "inversely proportional" to each other. Ceteris paribus, If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality ''k'') is always the same. As an example, the time taken for a journey is inversely proportional to the speed of travel. Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion, in reciprocal proportion) if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their Product (mathematics), product is a constant.Weisstein, Eric W
"Inversely Proportional"
''MathWorld'' – A Wolfram Web Resource.
It follows that the variable ''y'' is inversely proportional to the variable ''x'' if there exists a non-zero constant ''k'' such that : y = \frac, or equivalently, xy = k. Hence the constant "''k''" is the product of ''x'' and ''y''. The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the ''x'' and ''y'' values of each point on the curve equals the constant of proportionality (''k''). Since neither ''x'' nor ''y'' can equal zero (because ''k'' is non-zero), the graph never crosses either axis.


Hyperbolic coordinates

The concepts of ''direct'' and ''inverse'' proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular line (mathematics)#Ray, ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.


See also

* Linear map * Correlation * Eudoxus of Cnidus * Golden ratio * Inverse-square law * Proportional font *
Ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

Ratio
* Rule of three (mathematics) * Sample size * Similarity (geometry), Similarity * Basic proportionality theorem * Mathematical Operators (Unicode block)#Block, ∷ the ''a'' is to ''b'' as ''c'' is to ''d'' symbol (U+2237 ''PROPORTION'')


Growth

* Linear growth * Hyperbolic growth


Notes


References

* Ya. B. Zeldovich, Isaak Yaglom, I. M. Yaglom: ''Higher math for beginners''
p. 34–35
* Brian Burrell: ''Merriam-Webster's Guide to Everyday Math: A Home and Business Reference''. Merriam-Webster, 1998,
p. 85–101
* Lanius, Cynthia S.; Williams Susan E.
''PROPORTIONALITY: A Unifying Theme for the Middle Grades''
Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396. * Seeley, Cathy; Schielack Jane F.
''A Look at the Development of Ratios, Rates, and Proportionality''
Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142. * Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven
''Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions''
Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211. {{DEFAULTSORT:Proportionality (Mathematics) Mathematical terminology Ratios