Proportionality (mathematics)

Variable y is directly proportional to the variable x.

In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called the coefficient of proportionality or proportionality constant.

To express the statement "y is directly proportional to x" mathematically, we write an equation y = cx, where c is the proportionality constant. Symbolically, this is written as yx.

To express the statement "y is inversely proportional to x" mathematically, we write an equation y = c/x. We can equivalently write "y is directly proportional to 1/x".

An equality of two ratios is called a proportion. An example is a/c = b/d where no term is zero.

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 ∝ or ~ symbol, as

and the constant ratio

is called the proportionality constant, constant of variation or constant of proportionality.

Examples

Properties

Since

is equivalent to

it follows that if y is directly proportional to x, with (nonzero) proportionality constant k, then x is also directly proportional to y with proportionality constant 1/k.

If y is directly proportional to x, then the graph of y as a function of x is a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.

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.

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 is the product of x and y.

As an example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.

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

References

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