proportionality (mathematics)

This article is about proportionality, the mathematical relation. For other uses of the term 'proportionality', see proportionality (disambiguation).

In mathematics, two related quantities x and y are called proportional (or directly proportional) if there exists a functional relationship with a constant, non-zero number k such that

y = k \times x

In this case, k is called the proportionality constant or constant of proportionality of the relation. If y and x are proportional, we often write

y \sim x

y \propto x

For example, if you travel at a constant speed, then the distance you cover and the time you spend are proportional, the proportionality constant being the speed. Similarly, the amount of force acting on a certain object from the gravity of the Earth at sea level is proportional to the object's mass. To test whether x and y are proportional, one performs several measurements and plots the resulting points in a Cartesian coordinate system. If the points lie on (or close to) a straight line passing through the origin (0,0), then the two variables are proportional, with the proportionality constant given by the line's slope. The two quantities x and y are inversely proportional if there exists a non-zero constant k such that

y = {k \over x}

For instance, the number of people you hire to shovel sand is (approximately) inversely proportional to the time needed to get the job done. The functional relationship does not need to be linear, it can also be logarithmic, exponential or otherwise.

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Proportionality (Mathematics)

See also