Given an
independent variable and a dependent variable , is '''''' to if there is a positive constant 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 \quad\text{or}\quad y \sim x. For 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 , the proportionality
relation with proportionality constant between two sets and is the
equivalence relation defined by \{(a, b) \in A \times B : a = k b\}. A direct proportionality can also be viewed as a
linear equation in two variables with a
-intercept of and a
slope of , 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|. • 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 ==