Proportionality (mathematics)

File:Proportional variables.svg

The variable Template:Mvar is directly proportional to the variable Template:Mvar with proportionality constant ~0.6.

File:Inverse proportionality function plot.gif

The variable Template:Mvar is inversely proportional to the variable Template:Mvar with proportionality constant 1.

In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant). Two sequences are inversely proportional if corresponding elements have a constant product.

Two functions $f(x)$ and $g(x)$ are proportional if their ratio $\frac{f(x)}{g(x)}$ is a constant function.

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., Template:Math (for details see Ratio). Proportionality is closely related to linearity.

Direct proportionality

Given an independent variable Template:Mvar and a dependent variable Template:Mvar, Template:Mvar is Template:Dfn to Template:Mvar if there is a positive constant Template:Mvar such that: $$y=kx.$$

The relation is often denoted using the symbols ∝ (not to be confused with the Greek letter alpha) or ~, with exception of Japanese texts, where ~ is reserved for intervals: $$y\propto x\text{or}y\sim x.$$

For Template:Math the proportionality constant can be expressed as the ratio: $$k=\frac{y}{x}.$$

It is also called the constant of variation or constant of proportionality. Given such a constant Template:Mvar, the proportionality relation ∝ with proportionality constant Template:Mvar between two sets Template:Mvar and Template:Mvar is the equivalence relation defined by $$\{(a,b)\in A\times B:a=kb\}.$$

A direct proportionality can also be viewed as a linear equation in two variables with a [[y-intercept|Template:Mvar-intercept]] of Template:Math and a slope of Template:Math, which 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.
• The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to [[pi|Template:Pi]].
• 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 those points; the constant of proportionality is the scale of the map.
• The 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

File:Inverse proportionality function plot.gif

Inverse proportionality with product Template:Math

Two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse 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. It follows that the variable Template:Mvar is inversely proportional to the variable Template:Mvar if there exists a non-zero constant Template:Mvar such that $$y=\frac{k}{x}⟺xy=k.$$ Hence the constant Template:Mvar is the product of Template:Mvar and Template:Mvar.

The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the Template:Mvar and Template:Mvar values of each point on the curve equals the constant of proportionality Template:Mvar. Since neither Template:Mvar nor Template:Mvar can equal zero (because Template:Mvar is non-zero), the graph never crosses either axis.

Direct and inverse proportion contrast as follows: in direct proportion the variables increase or decrease together. With inverse proportion, an increase in one variable is associated with a decrease in the other. For instance, in travel, a constant speed dictates a direct proportion between distance and time travelled; in contrast, for a given distance (the constant), the time of travel is inversely proportional to speed: Template:Math.

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.

Computer encoding

The Unicode characters for proportionality are the following:

• Template:Unichar
• Template:Unichar
• Template:Unichar
• Template:Unichar
• Template:Unichar

See also

• Linear map
• Correlation
• Eudoxus of Cnidus
• Golden ratio
• Inverse-square law
• Proportional font
• Ratio
• Rule of three (mathematics)
• Sample size
• Similarity
• Trairāśika
• Basic proportionality theorem
• Linear growth
• Hyperbolic growth

References

• Burrell, Brian: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, Template:Isbn, 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.
• Zeldovich, Ya. B., Yaglom, I. M.: Higher math for beginners, p. 34–35.