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Interdivisional Committee on Terminology, Nomenclature and Symbols

The following summaries are intended to help in the preparation of manuscript:

> Quantity calculus
> Percents and per mils

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Instructions for Authors

Quantity calculus
T. Cvitas, February 2002

Each symbol of a physical quantity (single letter italic) in an equation stands for the value of the quantity which is

(quantity) = (numerical value) (unit)

In this way the equations hold for any units as we believe the laws of nature should. Units are a matter of human choice and no law in nature should depend on it.


force = mass acceleration

or with symbols

F = m a

irrespective of what units we choose.

Equations should therefore be written in a form not implying certain units.

In applications with many repetitive calculations it is often convenient to write equations with numerical values in certain units. Then, however, different symbols should be used.

Equation (2) can for a certain purpose be written in the form



where {F}N = F/N is the numerical value of the force in newtons, etc. (This notation is recommended by ISO 31-0: 1992.) Eq. (3) can be derived from (2) by division of both sides by N = kg m s-2.

If we measure the mass in pounds and acceleration in inches per second squared and we are still interested in the force in newtons, we can divide equation (2) by (lb in s-2) = 0.545 kg 0.0254 m s-2 = 0.0115 N obtaining

or in a more convenient form


This is also the way in which we would write computer programs. However, this does not mean that we are allowed to write

F = 86.7 m a

which obviously only holds if the symbols denote numerical values of quantities in a special choice of units. Even worse, if mass happens to be 1 lb always in our experiments, this does not allow us to write

F = 86.7 a

See also: Quantities, Units and Symbols in Physical Chemistry, 2nd edition (The Green Book), Mills, I.; Cvitas, T.; Homann, K.; Kallay, N. and Kuchitsu, K. Blackwell Science, 1993 [ISBN 0-63203-5838]

Percents and per mils
T. Cvitas, February 2002

Although these are not units in the same sense as the units of dimensioned quantities, they can be treated as such. The symbols % and are printed with a space between the numerical value and the unit:

25 %, 3.2

The sign % represents a unit symbol equal to the submultiple 0.01 of the coherent SI unit 1. The notation including a space between the numerical value and the unit is recommended by ISO 31-0: 1992.

Great care should be taken when using these symbols in compound derived units. While some expressions, such as % s-1 appear unambiguous, more complex ones involving powers ()2 or (%)-1 should be avoided. When listing percents in a table a column heading such as 100 w for mass fractions in percents is preferred to w / %.

Unit symbols should never be modified, as for instance: w/w %, vol. %, ppmv, atom %, mol. %, ... . All % are equal in the same sense as all metres are equal irrespective of whether we express heights, depths, lengths, widths, diameters, ... .

We have to name the quantity we measure:

mass fraction w = 96 %
volume fraction j = 21 %
amount fraction x = 2.4 %

These are not concentrations in different types of percents, but different types of fractions expressed by the same unit: % = 0.01 = 1/100 = 10-2.

Abbreviations such as ppb, ppt, ... are geography dependent (a billion is 1012 in Europe and 109 in America) and should be avoided or at least their meaning should be defined.

See also: T. Cvitas, Quantities describing compositions of mixtures, Metrologia 33 (1996) 35-39. > access to Metrologia

Questions or comments regarding this document, please contact Prof. T. Cvitas

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