Reciprocal proportionality

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Reciprocal proportionality , indirect proportionality, inverse proportionality or anti- proportionality exists between two quantities if one is proportional to the reciprocal of the other, or equivalently, the product of the quantities is constant. One variable is then a reciprocally proportional (also anti- proportional ) function of the other variable. The doubling (tripling, halving, ...) of one is associated with halving (thirding, doubling, ...) the other. The function graph is a hyperbola that asymptotically approaches the coordinate axes .

Reciprocal relationships

Function graph of a reciprocal proportional relationship: height and width of rectangles with area = 4 cm

The constant product of two quantities and is known from a value pair ( , ). Then one size can be specified as a function of the other:


Example: Given is a rectangle, 8 cm wide and 0.5 cm high. We are looking for a rectangle with the same area and a width of 5 cm.
The constant product is 8 cm x 0.5 cm = 4 cm 2 .
The required height is 4 cm 2 / (5 cm) = 0.8 cm.

The diagram opposite shows the two pairs of values ​​as marked points. At the hyperbola you can read off other rectangles of equal area, e.g. B. 1 cm wide, 4 cm high.

Further reciprocal relationships may be mentioned:

Reciprocal representation

Upper scale linearly divided into lower scale reciprocally divided into divided

The representation of reciprocal relationships in a Cartesian coordinate system often uses axis labeling , in which the numerical value of a quantity to be represented is not plotted in a linear division , but the reciprocal of its numerical value. Such a representation is particularly helpful when there is a proportionality between the dependent and the reciprocal of the independent variable . This creates a straight curve in a line diagram .

Processes of chemical kinetics of the first order , whose rate constant depends on the temperature according to the Arrhenius equation, should serve as an example


Reaction rate constant
Euler's number
  Activation energy
universal gas constant
absolute temperature

The equation can be rewritten as


If a process actually proceeds in accordance with the Arrhenius equation as a first order reaction, it can be seen that, in a representation in the above is coated with linear partitions, a straight line is formed, see Arrhenius plot . The activation energy for this straight line results from its rise .


For "a is inversely proportional to b" one writes briefly with one of the two proportionality symbols:


Web links

Individual evidence

  1. So named in the Bronstein
  2. The large table work interactive ISBN 978-3-464-57143-9