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Proportionality (mathematics)

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In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.

Definition Edit

More formally, the variable y is said to be proportional (or sometimes directly proportional) to the variable x, if there exists a constant non-zero number k such that

y = k \times x.

The relation is often denoted

y \propto x,

and the constant ratio

 k =  y/x\,

is called the proportionality constant or constant of proportionality of the proportionality relation.

Examples Edit

  • On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations the points represent, with the constant of proportionality being the scale of the map.

Properties Edit


y = k \times x

is equivalent to

x = (1/k) \times y,



Inverse proportionality Edit

As noted in the definition above two proportional variables are sometime said to be directly proportional. This is done so as to contrast proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely) if one of the variables is directly proportional with the multiplicative inverse of the other, or equivalently if their product is a constant. It follows, that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

y = {k \over x}.




Exponential and logarithmic proportionality Edit

A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exists a non-zero constant k such that

y = k a^x.\,

Likewise, a variable y is logarithmically proportional to a variable x, if y is directly proportional to the logarithm of x, that is if there exists a non-zero constant k such that

y = k \log_a (x).\,

Experimental determination Edit

To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.

See alsoEdit

de:Proportionalität es:Proporcionalidad fr:Proportionnalité io:Proporcionohe:יחס ישר nl:Recht evenredigno:proporsjonalitetvi:Tỉ lệ nghịch zh:正比


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