Proportionality (mathematics)

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 ( $y / x$ ) of two variables ( $x$ and $y$ ) is equal to a constant $(k = y / x)$ , then the variable in the numerator of the ratio ( $y$ ) is the product of the other variable and the constant $(y = k \cdot x)$ . In this case $y$ is said to be directly proportional to $x$ with proportionality constant $k$ . Equivalently one may write $x = 1 / k \cdot y$ ; that is, $x$ is directly proportional to $y$ with proportionality constant $1 / k (= x / y)$ . If the term proportional is connected to two variables without further qualification, generally direct proportionality can be assumed.
If the product of two variables $(x \cdot y)$ is equal to a constant $(k = x \cdot y)$ , then the two are said to be inversely proportional to each other with the proportionality constant $k$ . Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant $k$ $(x = k \cdot 1 / y$ and $y = k \cdot 1 / x)$ .

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.

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

y=kx.

Unicode characters
U+221D ∝ PROPORTIONAL TO (HTML `∝` · `&prop;, &Proportional;, &propto;, &varpropto;, &vprop;`) U+007E ~ TILDE (HTML `~`) U+223C ∼ TILDE OPERATOR (HTML `∼` · `&sim;, &thicksim;, &thksim;, &Tilde;`) U+223A ∺ GEOMETRIC PROPORTION (HTML `∺` · `&mDDot;`) See also: Equals sign

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

y\propto x,

or

y\sim x.

For $x\neq 0$ 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.

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

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 $π$ .
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

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

y={\frac {k}{x}},

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 ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.

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Notes

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

References

Ya. B. Zeldovich, I. M. Yaglom: Higher math for beginners, p. 34–35.
Brian Burell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, 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.

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[1] Weisstein, Eric W. "Directly Proportional". MathWorld – A Wolfram Web Resource.

[2] Weisstein, Eric W. "Inversely Proportional". MathWorld – A Wolfram Web Resource.