Editing Proportionality (mathematics) (section)

== Direct proportionality ==
{{See also|Equals sign}}
Given an [[Variable (mathematics)#Dependent and independent variables|independent variable]] ''x'' and a dependent variable ''y'', ''y'' is '''directly proportional''' to ''x''<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/DirectlyProportional.html "Directly Proportional"]. ''MathWorld'' – A Wolfram Web Resource.</ref> if there is a positive constant ''k'' such that:

: <math>y = kx</math>

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:
: <math>y \propto x</math> (or <math>y \sim  x</math>)

For <math>x \ne 0</math> the '''proportionality constant''' can be expressed as the ratio:

: <math> k = \frac{y}{x}</math>

It is also called the '''constant of variation''' or '''constant of proportionality'''. 
Given such a constant ''k'', the proportionality [[Binary relation|relation]] ∝ with proportionality constant ''k'' between two sets ''A'' and ''B'' is the [[equivalence relation]] defined by <math>\{(a, b) \in A \times B : a = k b\}.</math>

A direct proportionality can also be viewed as a [[linear equation]] in two variables with a [[y-intercept|''y''-intercept]] of {{math|0}} and a [[slope]] of ''k'' > 0, 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|{{pi}}]].
* On a [[map]] of a sufficiently small geographical area, drawn to [[scale (map)|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 (physics)|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.