Proportionality (mathematics)

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.

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).

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: 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: For

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.

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. 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.