Proportionality (mathematics)
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality 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. Two functions formula_1 and formula_2 are  proportional if their ratio formula_3 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., (for details see Ratio). Proportionality is closely related to "linearity". Direct proportionality. Given two variables "x" and "y", "y" is directly proportional to "x" if there is a non-zero constant "k" such that The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter alpha) or "~": For formula_7 the proportionality constant can be expressed as the ratio It is  also called the constant of variation or constant of proportionality. A direct proportionality can also be viewed as a linear equation in two variables with a "y"-intercept of and a slope of "k". This corresponds to linear growth. Inverse proportionality. The concept of "inverse proportionality" can be contrasted with "direct proportionality". Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality "k") is always the same. As an example , the time taken for a journey is inversely proportional to the speed of travel. Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion) if each of the variables is directly proportional to the multiplicative inverse (reciprocal) 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 or equivalently, formula_10 Hence the constant ""k"" is the product of "x" and "y". 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. 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.