Numerical equation

The numerical equation is an equation between observable quantities, e.g. B. physical quantities or financial engineering sizes, in which only the numerical values of the respective sizes occur, but not their units . A numerical equation must be identified as such and the units of measurement applicable in each case must be specified (separately).

The use of numerical value equations is systematically unsatisfactory, which is why these have been considered obsolete since the 1930s, should no longer be used and must also be identified as numerical value equations according to DIN 1313 and ISO 80000-1 . The units to be used must therefore be specified for all sizes. In addition to the numerical value equations, there are the size equations , which are generally to be preferred, as they express the relationship between physical quantities and apply regardless of the choice of certain units.

In the technical and craft sector, numerical value equations are widespread for purely practical reasons, whereby a uniform choice of units of measurement is assumed. Error-prone conversions of the units customary in technical practice to physically correct base units are then eliminated.

term

The terms "numerical value equation", "size equation" and "tailored size equation" go back to Julius Wallot and are dealt with in the DIN 1313 standard "sizes" (first edition 1931: "Notation of physical equations").

Inadequacies

Numerical value equations alone are not meaningful, since the additional information is always required as to the units or unit sizes in which the numerical values ​​to be used must be available. Different numerical value equations of a size equation can exist, which differ from one another through numerical factors. If this information about the units used is missing, a numerical equation is useless and may lead to completely wrong results.

By using size equations, on the other hand, it is ensured that the quantities contained are SI units or coherent SI units, or at least within the formula itself it is clearly described which quantities are to be used.

By consistently dispensing with numerical value equations, misunderstandings and misuse are avoided.

Examples

Ohm's law

For example, the following equation from Ohm's law is written in numerical notation:

${\displaystyle \{U\}=10^{-3}\{R\}\cdot \{I\}\qquad {\text{mit}}\qquad \{U\}{\text{ in Kilovolt, }}\{R\}{\text{ in Ohm, }}\{I\}{\text{ in Ampere}}}$

If misunderstandings are excluded, simply G can be written for the numerical values ​​{ G } of a quantity G for reasons of clarity . This results in the following alternative numerical notation for Ohm's law:

${\displaystyle U=10^{-3}R\cdot I\qquad {\text{mit}}\qquad U{\text{ in Kilovolt, }}R{\text{ in Ohm, }}I{\text{ in Ampere}}}$

There used to be another spelling in which the units were put in square brackets.

${\displaystyle U\lbrack \mathrm {kV} \rbrack =10^{-3}\cdot R\lbrack \Omega \rbrack \cdot I\lbrack \mathrm {A} \rbrack }$.

Today, this notation often leads to misunderstandings, because today the square brackets around the symbol are written with the meaning "unit of ...". According to DIN 1313, "square brackets [...] must not be placed around unit symbols". In the outdated form of the numerical value equation, however, the unit is bracketed.

Mechanical performance formula

In technical mechanics are the most common sizes

• Power in kilowatts; ${\displaystyle P}$ ${\displaystyle [P]=\mathrm {kW} }$
• Speed in revolutions per minute; ${\displaystyle n}$ ${\displaystyle [n]=\mathrm {min} ^{-1}}$
• Torque in newton meters; ${\displaystyle M}$ ${\displaystyle [M]=\mathrm {Nm} }$

about the equation

${\displaystyle M={\frac {P}{2\pi \ n}}}$

connected. If two values ​​are known, the third value can be sufficiently precise with the numerical value equation

${\displaystyle \{M\}={\frac {9550\cdot \{P\}}{\{n\}}}\qquad {\text{mit}}\qquad \{M\}{\text{ in Nm, }}\{P\}{\text{ in kW, }}\{n\}\mathrm {\ in\ min^{-1}} }$

be calculated. The new international standardization summarizes the notation

${\displaystyle \{M\}_{\mathrm {Nm} }={\frac {9550\cdot \{P\}_{\mathrm {kW} }}{\{n\}_{\mathrm {min^{-1}} }}}}$

is there

${\displaystyle \{M\}_{\mathrm {Nm} }={\frac {M}{\mathrm {Nm} }}}$ etc.

• For the performance there is a changeover ${\displaystyle P}$

${\displaystyle \{P\}={\frac {\{M\}\cdot \{n\}}{9550}}\qquad {\text{mit}}\qquad \{M\}{\text{ in Nm, }}\{P\}{\text{ in kW, }}\{n\}\mathrm {\ in\ min^{-1}} }$

• The speed is determined from ${\displaystyle n}$

${\displaystyle \{n\}={\frac {9550\cdot \{P\}}{\{M\}}}\qquad {\text{mit}}\qquad \{M\}{\text{ in Nm, }}\{P\}{\text{ in kW, }}\{n\}\mathrm {\ in\ min^{-1}} }$

Derivation

The rounded numerical value 9550 is just an easy-to-remember constant for the calculation. There is no need to convert the units commonly used in technical practice to SI units .

In addition to the factor, the constant includes the conversion of minutes into seconds, kilowatts into watts and the identity . This results in: ${\displaystyle {\tfrac {1}{2\pi }}}$ ${\displaystyle \mathrm {1\;Ws=1\;Nm} }$

${\displaystyle {\frac {1}{2\pi }}={\frac {60\,\mathrm {\frac {s}{min}} \cdot 1000\,\mathrm {\frac {W}{kW}} }{2\,\cdot \pi }}={\frac {30\,000}{\pi }}\ \mathrm {min} ^{-1}\,\mathrm {\frac {Nm}{kW}} }$

The agreed units are then left out in numerical value equations.

The deviation of the approximate value from the exact value is only 0.007%. ${\displaystyle 9550}$ ${\displaystyle {\tfrac {30\,000}{\pi }}}$

Tailored size equation

The size equation of the numerical value equation mentioned above looks like this:

${\displaystyle U=R\cdot I}$

If you only want to build this equation with numerical values, the tailored size equation can be used, in which the physical quantities are divided by their units:

${\displaystyle {\frac {U}{\left[U\right]}}={\frac {R}{\left[R\right]}}\cdot {\frac {I}{\left[I\right]}}}$

This follows from the notation of a physical quantity :

${\displaystyle U=\left\{U\right\}\cdot [U]}$

(i.e. the voltage U is the numerical value of U times the unit of U )

Please note: the formula symbol is in square brackets, not the unit itself, since the formula symbol in square brackets represents the unit itself, ie [U] = V!

The above equation resolved according to the numerical value gives:

${\displaystyle \left\{U\right\}={\frac {U}{\left[U\right]}}}$

The numerical equation in today's form follows directly from this:

${\displaystyle \left\{U\right\}=\left\{R\right\}\cdot \left\{I\right\}}$

so an equation that only consists of numbers!

However, this equation only leads to a correct result if, as already indicated above, the associated units were used for the calculation.

The above equation therefore looks as a tailored size equation as follows:

${\displaystyle {\frac {U}{\mathrm {kV} }}=10^{-3}\cdot {\frac {R}{\Omega }}\cdot {\frac {I}{\mathrm {A} }}}$.

A quotient of the size and the desired unit is written in each case . This representation can therefore be read as users of numerical equations are used to. However, it is valid and understandable according to the usual conventions today.

Individual evidence

1. ^ Klaus H. Blankenburg ( ITG Technical Committee 9.1): The correct handling of quantities, units and equations. December 1999.
2. ↑ DIN 1313 December 1998 - sizes. P. 13.
3. ↑ DIN 1313 December 1998 - sizes. P. 5.
4. ↑ German version as DIN EN ISO 80000-1: 2013 Sizes and units - Part 1: General. Cape. 6.3.