Proportionality (mathematics)

Ratio Product (mathematics) Multiplicative inverse
The variable y is directly proportional to the variable x with proportionality constant ~0.6.
The variable y is inversely proportional to the variable x with proportionality constant 1.

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 several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., a/b = x/y = ... = k (for details see Ratio).

Direct proportionality

Given two variables x and y, y is directly proportional to x[1] if there is a non-zero constant k such that

Unicode characters
  • U+221D PROPORTIONAL TO (HTML ∝ · ∝, ∝, ∝, ∝, ∝)
  • U+007E ~ TILDE (HTML ~)
  • U+223C TILDE OPERATOR (HTML ∼ · ∼, ∼, ∼, ∼)
  • U+223A GEOMETRIC PROPORTION (HTML ∺ · ∺)

See also: Equals sign

The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter alpha) or "~":

 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 0 and a slope of k. This corresponds to linear growth.

Examples

Inverse proportionality

Inverse proportionality with a function of y = 1/x

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 (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 is a constant.[2] 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 (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 ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.

See also

Growth

Notes

  1. ^ Weisstein, Eric W. "Directly Proportional". MathWorld – A Wolfram Web Resource.
  2. ^ Weisstein, Eric W. "Inversely Proportional". MathWorld – A Wolfram Web Resource.