Proportionality (mathematics)

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.

If one variable is always the product of the other and a constant, the two are said to be directly proportional. x and y are directly proportional if the ratio ⁠y/x⁠ is constant.
If the product of the two variables is always a constant, the two are said to be inversely proportional. x and y are inversely proportional if the product xy is 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 y ∝ x.

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.^{[dubious – discuss]} 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

y=kx.

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

The relation is often denoted, using the ∝ or ~ symbol, as

y\propto x

and the constant ratio

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

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

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 drawn to scale, the distance between any two points on the map is directly proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
The force acting on a certain object due to gravity is directly proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.

Properties

Since

y=kx

is equivalent to

x=\left({\frac {1}{k}}\right)y,

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

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

y=\left({\frac {k}{x}}\right)

or equivalently $xy=k.$ 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.

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. pp. 34-35
Brian Burell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, pp. 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), pp. 392-96 (JSTOR)
Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, pp. 140-42 (JSTOR)
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, pp. 187-211 (JSTOR)

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

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

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