# Proportionality

There is proportionality between two variable quantities if they are always in the same relationship to one another.

## Basics

Proportional sizes are proportional, that is, with proportional sizes, doubling (tripling, halving, ...) of one size is always associated with doubling (tripling, halving, ...) of the other size, or in general: one size goes out of the others by multiplying by a factor that is always the same. The ratio of the two quantities is called the proportionality factor or proportionality constant .

Examples:

• The circumference is proportional to the diameter of the circle; the proportionality factor is the circle number = 3.14159 ...${\ displaystyle \ pi}$
• On a purchase, VAT is proportional to the net price; the proportionality factor is the VAT rate, for example 0.19 (= 19%).
• The mass of a liquid is proportional to its volume (all other things being equal) (see detailed example below).

Proportionality is a special case of linearity , more precisely: affinity (see linear function ). For a real linear function, every relationship between two quantities is linear , the representation of which in a Cartesian coordinate system is a straight line. Proportionality here means that this straight line goes through the zero point (coordinate origin ) ( straight line through the origin ); the proportionality factor determines its slope.

Occasionally one speaks of direct proportionality as opposed to indirect, inverse, inverse or reciprocal proportionality , in which one quantity is proportional to the reciprocal of the other; instead of the ratio, the product of the two quantities is constant. The graph is a hyperbola and does not go through the zero point.

The calculus of the rule of three assumes a proportional function.

## Mathematical definition

### Historical definition

Euclid , Elements Book V, Definitions 3-6.

Definition 5 is:

“It is said that sizes are in the same relationship, the first to the second as the third to the fourth, if, with any multiplication, the equal multiples of the first and third compared to the equal multiples of the second and fourth, taken in pairs, either at the same time greater or at the same time equal or are smaller at the same time. "

Definition 6:

"And the sizes with this ratio should be called in proportion."

### Current definition

A proportional function is a homogeneous linear assignment between arguments and their function values : ${\ displaystyle x}$${\ displaystyle y}$

${\ displaystyle y = m \ cdot x}$

with a constant proportionality factor . The factor does not make sense here. ${\ displaystyle m}$${\ displaystyle m = 0}$

Since it is in proportionality equivalent whether the size of the size by multiplying apparent with an ever same factor, or vice versa from further applies ${\ displaystyle y}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle y}$

${\ displaystyle x = {\ frac {1} {m}} \ cdot y}$ ;

the factor is not allowed. ${\ displaystyle m = 0}$

Two variables for which the ratio of related values and is constant are called proportional to one another ${\ displaystyle x_ {i}}$${\ displaystyle y_ {i}}$

${\ displaystyle {\ frac {y_ {i}} {x_ {i}}} = m}$ .

Accordingly, proportionality is present precisely when this ratio is constant; if it is real, it can be positive or negative . ${\ displaystyle m}$

## example

### density

Function graph for a proportional relationship

The table indicates the mass of different volumes of oil:

Volume in m 3${\ displaystyle x}$ Mass in t ${\ displaystyle y}$
1 0.8
3 2.4
7th 5.6

The three value pairs are marked as points in the picture (right). If you calculate the quotient , mass / volume, you always get the same value 0.8 t / m 3 . In general, the quotient indicates the gradient of the straight line and is also the proportionality factor of the assignment, here with the meaning of the density of the oil. The reverse quotient is also a constant of proportionality, in this case with the meaning of the specific volume . In the example you get ${\ displaystyle y / x}$${\ displaystyle y / x}$${\ displaystyle x / y}$

Volume / mass = 1.25 m 3 / t

### Air pressure change

The air pressure depends on the height above sea level. In low earth layers, the change in pressure is proportional to the change in height with ${\ displaystyle \ Delta p}$${\ displaystyle \ Delta h}$

${\ displaystyle {\ frac {\ Delta p} {\ Delta h}} = c}$

and with the constant of proportionality for these changes , see barometric height formula . ${\ displaystyle c = -0 {,} 12 \, \ mathrm {\ tfrac {hPa} {m}}}$

The minus sign means: When climbing a staircase (positive ) the pressure decreases (negative ). ${\ displaystyle \ Delta h}$${\ displaystyle \ Delta p}$

## Notation

For "a proportional to b" use the tilde symbol "~":

${\ displaystyle a \ sim b}$

The spelling is also standardized:

${\ displaystyle a \ propto b}$

The sign is derived from the medieval »æ« for Latin aequalis , the predecessor of the equal sign . ${\ displaystyle \ propto}$

character HTML TeX Unicode ASCII
~ ~ or &#126; \sim U + 007E 126
&sim; or &#8764; U + 223C -
&prop; or &#8733; \propto U + 221D -

## Related terms

There is talk of disproportionality between two sizes when one changes more and more than the other. Correspondingly, one speaks of underproportionality in the case of a systematically weaker change in the other variable. "Stronger" and "weaker" mean here, if you refer to the formulation with the equation with an exponent , that applies to normal proportionality , overproportionality and underproportionality . ${\ displaystyle y = mx ^ {a}}$${\ displaystyle a}$${\ displaystyle a = 1}$${\ displaystyle a> 1}$${\ displaystyle a <1}$