Reciprocal proportionality

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 ² ${\displaystyle A}$

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: ${\displaystyle x}$ ${\displaystyle y}$
${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$

${\displaystyle y={\frac {A}{x}}={\frac {x_{0}\cdot y_{0}}{x}}}$.

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. ${\displaystyle y=A/x}$

Further reciprocal relationships may be mentioned:

• The average speed is inversely proportional to the duration of the trip.
• According to Ohm's law , the electrical current strength is inversely proportional to the resistance .

Reciprocal representation

Upper scale linearly divided into lower scale reciprocally divided into divided ${\displaystyle 1/x}$
${\displaystyle x}$

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

${\displaystyle k=k_{0}\cdot \mathrm {e} ^{-{\frac {E_{\mathrm {A} }}{R\cdot T}}}}$

With

${\displaystyle k}$	Reaction rate constant
${\displaystyle \mathrm {e} }$	Euler's number
${\displaystyle E_{\mathrm {A} }}$	Activation energy
${\displaystyle R}$	universal gas constant
${\displaystyle T}$	absolute temperature

The equation can be rewritten as

${\displaystyle \quad \ln \left({\frac {k}{k_{0}}}\right)=-{\frac {E_{\mathrm {A} }}{R}}\cdot {\frac {1}{T}}}$.

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 . ${\displaystyle \ln(k/k_{0})}$ ${\displaystyle 1/T}$ ${\displaystyle (-E_{\mathrm {A} }/R)}$

Notation

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

${\displaystyle a\sim {\frac {1}{b}}}$   or   ${\displaystyle \displaystyle a\propto {\frac {1}{b}}}$

Web links

Wikibooks: ${\displaystyle {\begin{smallmatrix}{\mathbf {MATHE} \mu \alpha T\mathbb {R} ix}\end{smallmatrix}}}$  - Mathematics for school

Individual evidence

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