Proportionality (mathematics)

Proportionality exists between two variable quantities, if they are always in the same proportion to each other.

Basics

Proportional sizes are relatively equal, that is, in quantities proportional to the doubling ( tripling, halving, ...) is of a size always associated with a doubling ( tripling, halving, ...) of different size, or in general terms: one size is shown ( the ratio of the two quantities, called proportionality or proportionality ) from the other by multiplication with a factor always the same.

Examples:

  • The circumference is proportional to the diameter; the proportionality factor is the county number = 3.14159 ...
  • In a sale where the value added is proportional to the net price; the proportionality factor is the value-added tax rate, for example 0.19 ( 19 percent).
  • The mass of a liquid is proportional to its volume (see detailed example below).

Proportionality is a special case of linearity, more precisely, the affinity (see linear function ). Linear in this sense is any correlation between two variables, whose representation in the xy - coordinates is a straight line; Proportionality means that this straight line through the origin ( coordinate origin ) goes.

In contrast to the proportional size is in Antiproportionalität ( reciprocal inverse, reverse or indirect proportionality ) is proportional to the reciprocal of the other size; Thus, instead of the relationship in this case the product of two quantities is constant.

Mathematical definition

Geometric definition

Euclid Elements Book V, Definitions 3-6.

Definition 5 is:

" It is said that sizes are in the same ratio, the first to the second as the third to the fourth, when, in pairs taken at any multiplication equal multiples of the first and third the same multiple of the second and fourth against accordingly, either at the same time greater than or at the same time equal to or are also smaller. "

Definition 6:

" And this ratio -off sizes are standing hot in proportion. "

Arithmetic definition

A proportional function is a homogeneous linear mapping between arguments and their function values ​​:

Proportionality exists if and only if the ratio of all pairs of values ​​are equal, that is, the value of m is constant. The factor m is then the proportionality factor.

The feature is characterized in that its graph is a straight line passing through the origin. The proportionality factor m determines the slope of the line.

The table shows the mass of different volumes of oil:

The three pairs of values ​​in the image ( right) marked as dots. Calculates the quotient y / x, mass / volume, so you always get the same value, 0.8 t/m3, the density of the oil. General indicates the ratio y / x is the slope m of the straight line and is also the proportionality of the assignment. The reverse ratio is a proportionality constant, in this case, the specific volume. In the example, one volume / mass as much volume gets = 1.25 m3 / t, so a ton of oil occupies.

The calculus for the calculation of proportional functions is called the rule of three (formerly also: rule of three ).

Spelling

For "a is proportional to b" is used according to DIN 1302, the tilde character "~":

Also standardized the notation:

The symbol α is derived from the medieval » æ " for Latin aequalis, the predecessor of the equal sign.

Related terms

It is spoken by Überproportionalität between two variables when one is always more changes than the other. Accordingly, one speaks of Unterproportionalität at a systematically weaker change of different size. "Stronger " and " weaker " here mean when it refers to the formulation of the linear equation with an exponent a, that is valid for normal proportionality in Überproportionalität and Unterproportionalität.

  • Proportionality
  • Sequences and series
Volume (thermodynamics)
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