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 (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant). Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality.
This definition is commonly extended to related varying quantities, which are often called variables. This meaning of variable is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons.
f(x)
g(x)
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., (for details see Ratio).Proportionality is closely related to linearity.
See also: Equals sign. Given an independent variable x and a dependent variable y, y is directly proportional to x[1] if there is a positive constant k 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\proptox
y\simx
For
x\ne0
k=
| 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
\{(a,b)\inA x B:a=kb\}.
A direct proportionality can also be viewed as a linear equation in two variables with a y-intercept of and a slope of k > 0, which corresponds to linear growth.
Two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion)[2] if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant.[3] 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 | |
| x |
xy=k
The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the x and y values of each point on the curve equals the constant of proportionality (k). Since neither x nor y can equal zero (because k is non-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 constant), the time of travel is inversely proportional to speed: s × t = d.
See main article: Hyperbolic coordinates. The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.
The Unicode characters for proportionality are the following: