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 and its reciprocal is known as constant of normalization. Two sequences are inversely proportional if corresponding elements have a constant product.
Two functions and are proportional if their ratio 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., .
Proportionality is closely related to linearity.
Direct proportionality
Given an independent variable and a dependent variable, is to if there is a positive constant such that:The relation is often denoted using the symbols or, with exception of Japanese texts, where is reserved for intervals:
For the proportionality constant can be expressed as the ratio:
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 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
Hence the constant is the product of and.
The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the and values of each point on the curve equals the constant of proportionality. Since neither nor can equal zero, the graph never crosses either axis.
Direct and inverse proportion contrast as follows: in direct proportion the variables increase or decrease together. With inverse proportion, an increase in one variable is associated with a decrease in the other. For instance, in travel, a constant speed dictates a direct proportion between distance and time travelled; in contrast, for a given distance, the time of travel is inversely proportional to speed:.