Proportionality (mathematics)

Two functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are proportional if their ratio $\frac{f(x)}{g(x)}$ 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 = ⋯ = k (for details see Ratio). Proportionality is closely related to linearity.

Direct proportionality

Given an independent variable x and a dependent variable y, y is directly proportional to x if there is a positive constant k such that: $${\displaystyle y=kx.}$$

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: $${\displaystyle y\propto x\text{or}y\sim x.}$$

For x ≠ 0 the proportionality constant can be expressed as the ratio: $${\displaystyle k=\frac{y}{x}.}$$

It is also called the constant of variation or constant of proportionality. Given such a constant k, the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by $${\displaystyle \{(a,b)\in A\times B:a=kb\}.}$$

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 > 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 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.