In mathematics, two varying quantities are said to be in a relation of proportionality, if they are multiplicatively connected to a constant; that is, when either their 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 is equal to a constant, then the variable in the numerator of the ratio is the 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 expressing the equality of these ratios is called a proportion, e.g., .
Direct proportionality
Given two variables x and y, y is directly proportional to x if there is a non-zero constant k such that
See also: Equals sign
The relation is often denoted using the symbols "∝" or "~": For the proportionality constant can be expressed as the ratio 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. This corresponds to linear growth.
Examples
If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
On a map of a sufficiently small geographical area, drawn to scale distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by that points; the constant of proportionality is the scale of the map.
The concept of inverse proportionality can be contrasted with direct proportionality. Consider two variables said to be "inversely proportional" to each other. 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 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 if each of the variables is directly proportional to the multiplicative inverse of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that or equivalently, 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. Since neitherx nor y can equal 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 ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.
