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
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:
- 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
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
With
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 .
Notation
For "a is inversely proportional to b" one writes briefly with one of the two proportionality symbols:
- or
Web links
Individual evidence
- ↑ So named in the Bronstein
- ↑ The large table work interactive ISBN 978-3-464-57143-9