Proportionality (mathematics)

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, also called the coefficient of proportionality.

For other uses, see Proportionality.

This definition is commonly extended to related varying quantities, which are often called variables. This meaning of variable is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons.

Two functions $f(x)$ and $g(x)$ are proportional if their ratio ${\textstyle {\frac {f(x)}{g(x)}}}$ 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., $a / b = x / y = \dots = k$ (for details see Ratio). Proportionality is closely related to linearity.

If an object travels at a constant , then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.

speed

The of a circle is directly proportional to its diameter, with the constant of proportionality equal to $π$ .

circumference

On a 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.

map

The , 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.

force

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, , is the classical mass of the object.

Newton's second law

U+221D ∝ PROPORTIONAL TO (&prop;, &Proportional;, &propto;, &varpropto;, &vprop;)

U+007E ~ TILDE

U+2237 ∷ PROPORTION

U+223C ∼ TILDE OPERATOR (&sim;, &thicksim;, &thksim;, &Tilde;)

U+223A ∺ GEOMETRIC PROPORTION (&mDDot;)

The Unicode characters for proportionality are the following:

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

Ya. B. Zeldovich, : Higher math for beginners, p. 34–35.

I. M. Yaglom

Brian Burrell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, 9780877796213, p. 85–101.

ISBN

Lanius, Cynthia S.; Williams Susan E.: . Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.

PROPORTIONALITY: A Unifying Theme for the Middle Grades

Seeley, Cathy; Schielack Jane F.: . Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.

A Look at the Development of Ratios, Rates, and Proportionality

Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : . Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.