Proportionality (mathematics)

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.

Two functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are proportional if their ratio ${\textstyle {\frac {f(x)}{g(x)}}}$ 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., ⁠a/b⁠ = ⁠x/y⁠ = ⋯ = k (for details see Ratio). Proportionality is closely related to linearity.

Given an independent variable x and a dependent variable y, y is directly proportional to x if there is a positive constant k such that: $${\displaystyle y=kx.}$$

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: $${\displaystyle y\propto x\quad {\text{or}}\quad y\sim x.}$$

For x ≠ 0 the proportionality constant can be expressed as the ratio: $${\displaystyle k={\frac {y}{x}}.}$$

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 $${\displaystyle \{(a,b)\in A\times B:a=kb\}.}$$

A direct proportionality can also be viewed as a linear equation in two variables with a y-intercept of 0 and a slope of k > 0, which corresponds to linear growth.