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Writing unit symbols and names, and expressing the values of quantities

General principles for the writing of unit symbols and numbers were first given by the 9th CGPM (1948, Resolution 7). These were subsequently elaborated by ISO, IEC, and other international bodies. As a consequence, there now exists a general consensus on how unit symbols and names, including prefix symbols and names, as well as quantity symbols should be written and used, and how the values of quantities should be expressed. Compliance with these rules and style conventions, the most important of which are presented in this chapter, supports the readability of scientific.

and technical papers.

1. Unit symbols

Unit symbols are printed in roman (upright) type regardless of the type used in the surrounding text. They are printed in lower-case letters unless they are derived from a proper name, in which case the first letter is a capital letter.

Example:

m, metre;

s, second;

Pa, pascal;

Ω, ohm.

An exception, adopted by the 16th CGPM (1979, Resolution 6), is that either capital L or lower-case l is allowed for the litre, in order to avoid possible confusion between the numeral 1 (one) and the lower-case letter l (el).

Example:

L or l, litre.

A multiple or sub-multiple prefix, if used, is part of the unit and precedes the unit symbol without a separator. A prefix is never used in isolation, and compound prefixes are never used.

Unit symbols are mathematical entities and not abbreviations. Therefore, they are not followed by a period except at the end of a sentence, and one must neither use the plural nor mix unit symbols and unit names within one expression, since names are not mathematical entities.

Example:

nm, not mμm;

It is 75 cm long, not 75 cm. long;

l = 75 cm, not 75 cms;

coulomb per kilogram, not coulomb per kg.

In forming products and quotients of unit symbols the normal rules of algebraic multiplication or division apply. Multiplication must be indicated by a space or a half-high (centred) dot (·), since otherwise some prefixes could be misinterpreted as a unit symbol. Division is indicated by a horizontal line, by a solidus (oblique stroke, /) or by negative exponents. When several unit symbols are combined, care should be taken to avoid ambiguities, for example by using brackets or negative exponents. A solidus must not be used more than once in a given expression without brackets to remove ambiguities. It is not permissible to use abbreviations for unit symbols or unit names, such as sec (for either s or second), sq. mm (for either mm2 or square millimetre), cc (for either cm3 or cubic centimetre), or mps (for either m/s or metre per second). The use of the correct symbols for SI units, and for units in general, as listed in earlier chapters of this Brochure, is mandatory. In this way ambiguities and misunderstandings in the values of quantities are avoided.

Example:

N m or N · m for a newton metre;

m/s or s m or m s–1, for metre per second;

ms, millisecond;

m s, metre times second;

m kg/(s3 A), or m kg s–3 A–1, but not m kg/s3/A, nor m kg/s3 A

2. Unit names

Unit names are normally printed in roman (upright) type, and they are treated like ordinary nouns. In English, the names of units start with a lower-case letter (even when the symbol for the unit begins with a capital letter), except at the beginning of a sentence or in capitalized material such as a title. In keeping with this rule, the correct spelling of the name of the unit with the symbol 0C is ″degree Celsius″ (the unit degree begins with a lower-case d and the modifier Celsius begins with an upper-case C because it is a proper name).

Example:

unit namesymbol
jouleJ
hertzHz
metrem
seconds
ampereA
wattW

 

Although the values of quantities are normally expressed using symbols for numbers and symbols for units, if for some reason the unit name is more appropriate than the unit symbol, the unit name should be spelled out in full.

Example:

2.6 m/s, or 2.6 metres per second;

When the name of a unit is combined with the name of a multiple or sub-multiple prefix, no space or hyphen is used between the prefix name and the unit name. The combination of prefix name plus unit name is a single word.

Example:

milligram, but not milli-gram;

kilopascal, but not kilo-pascal.

In both English and in French, however, when the name of a derived unit is formed from the names of individual units by multiplication, then either a space or a hyphen is used to separate the names of the individual units.

Example:

pascal second, or pascal-second.

In both English and in French modifiers such as ″squared″ or ″cubed″ are used in the names of units raised to powers, and they are placed after the unit name. However, in the case of area or volume, as an alternative the modifiers ″square″ or ″cubic″ may be used, and these modifiers are placed before the unit name, but this applies only in English.

Example:

metre per second squared;

square centimeter;

cubic millimeter;

ampere per square metre;

kilogram per cubic metre.

3. Rules and style conventions for expressing values of Quantities

3.1. Value and numerical value of a quantity, and the use of quantity calculus

The value of a quantity is expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit. The numerical value of a quantity depends on the choice of unit. Thus the value of a particular quantity is independent of the choice of unit, although the numerical value will be different for different units.

Example:

The same value of a speed v = dx/dt of a particle might be given by either of the expressions v = 25 m/s = 90 km/h, where 25 is the numerical value of the speed in the unit metres per second, and 90 is the numerical value of the speed in the unit kilometres per hour.

Symbols for quantities are generally single letters set in an italic font, although they may be qualified by further information in subscripts or superscripts or in brackets.

Thus C is the recommended symbol for heat capacity, Cm for molar heat capacity, Cm, p for molar heat capacity at constant pressure, and Cm,V for molar heat capacity at constant volume.

Recommended names and symbols for quantities are listed in many standard references, such as the ISO Standard 31 Quantities and Units, the IUPAP SUNAMCO Red Book Symbols, Units and Nomenclature in Physics, and the IUPAC Green Book Quantities, Units and Symbols in Physical Chemistry. However, symbols for quantities are recommendations (in contrast to symbols for units, for which the use of the correct form is mandatory). In particular circumstances authors may wish to use a symbol of their own choice for a quantity, for example in order to avoid a conflict arising from the use of the same symbol for two different quantities. In such cases, the meaning of the symbol must be clearly stated. However, neither the name of a quantity, nor the symbol used to denote it, should imply any particular choice of unit.

Symbols for units are treated as mathematical entities. In expressing the value of a quantity as the product of a numerical value and a unit, both the numerical value and the unit may be treated by the ordinary rules of algebra. This procedure is described as the use of quantity calculus, or the algebra of quantities. For example, the equation T = 293 K may equally be written T/K = 293. It is often convenient to write the quotient of a quantity and a unit in this way for the heading of a column in a table, so that the entries in the table are all simply numbers. For example, a table of vapour pressure against temperature, and the natural logarithm of vapour pressure against reciprocal temperature, may be formatted as shown below.

T/K103 K/Tp/MPaln(p/MPa)
216.554.61790.5180−0.6578
273.153.66103.4853 1.2486
304.193.28747.38151.9990

 

The axes of a graph may also be labelled in this way, so that the tick marks are labelled only with numbers, as in the graph below.

Algebraically equivalent forms may be used in place of 103 K/T, such as kK/T, or 103 (T/K) −1.

3.2. Quantity symbols and unit symbols

Just as the quantity symbol should not imply any particular choice of unit, the unit symbol should not be used to provide specific information about the quantity, and should never be the sole source of information on the quantity. Units are never qualified by further information about the nature of the quantity; any extra information on the nature of the quantity should be attached to the quantity symbol and not to the unit symbol.

Example:

The maximum electric potential difference is Umax = 1000 V but not U = 1000 Vmax.

The mass fraction of copper in the sample of silicon is w(Cu) = 1.3 × 10−6 but not 1.3 × 10−6 w/w.

3.3. Formatting the value of a quantity

The numerical value always precedes the unit, and a space is always used to separate the unit from the number. Thus the value of the quantity is the product of the number and the unit, the space being regarded as a multiplication sign (just as a space between units implies multiplication). The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle, °, ′, and ″, respectively, for which no space is left between the numerical value and the unit symbol.

Example:

m = 12.3 g where m is used as a symbol for the quantity mass, but φ = 30° 22′ 8″, where φ is used as a symbol for the quantity plane angle.

This rule means that the symbol °C for the degree Celsius is preceded by a space when one expresses values of Celsius temperature t.

Example:

t = 30.2 °C, but not t = 30.2°C, nor t = 30.2° C

Even when the value of a quantity is used as an adjective, a space is left between the numerical value and the unit symbol. Only when the name of the unit is spelled out would the ordinary rules of grammar apply, so that in English a hyphen would be used to separate the number from the unit.

Example:

a 10 kΩ resistor;

a 35-millimetre film.

In any one expression, only one unit is used. An exception to this rule is in expressing the values of time and of plane angles using non-SI units. However, for plane angles it is generally preferable to divide the degree decimally. Thus one would write 22.20° rather than 22° 12′ , except in fields such as navigation, cartography, astronomy, and in the measurement of very small angles.

Example:

l = 10.234 m, but not l = 10 m 23.4 cm

3.4. Formatting numbers, and the decimal marker

The symbol used to separate the integral part of a number from its decimal part is called the decimal marker. Following the 22nd CGPM (2003, Resolution 10), the decimal marker ″shall be either the point on the line or the comma on the line.″ The decimal marker chosen should be that which is customary in the context concerned.

If the number is between +1 and −1, then the decimal marker is always preceded by a zero.

Example:

−0.234, but not −.234

Following the 9th CGPM (1948, Resolution 7) and the 22nd CGPM (2003, Resolution 10), for numbers with many digits the digits may be divided into groups of three by a thin space, in order to facilitate reading. Neither dots nor commas are inserted in the spaces between groups of three. However, when there are only four digits before or after the decimal marker, it is customary not to use a space to isolate a single digit. The practice of grouping digits in this way is a matter of choice; it is not always followed in certain specialized applications such as engineering drawings, financial statements, and scripts to be read by a computer.

For numbers in a table, the format used should not vary within one column.

Example:

43 279.168 29, but not 43,279.168,29

either 3279.1683 or 3 279.168 3

Expressing the measurement uncertainty in the value of a quantity The uncertainty that is associated with the estimated value of a quantity should be evaluated and expressed in accordance with the Guide to the Expression of Uncertainty in Measurement [ISO, 1995]. The standard uncertainty (i.e. estimated standard deviation, coverage factor k = 1) associated with a quantity x is denoted by u(×). A convenient way to represent the uncertainty is given in the following example:
mn = 1.674 927 28 (29) × 10–27 kg.
where mn is the symbol for the quantity (in this case the mass of a neutron), and the number in parenthesis is the numerical value of the combined standard uncertainty of the estimated value of mn referred to the last two digits of the quoted value; in this case u(mn) = 0.000 000 29 × 10−27 kg. If any coverage factor, k, different from one, is used, this factor must be stated.

3.6. Multiplying or dividing quantity symbols, the values of quantities, or numbers

When multiplying or dividing quantity symbols any of the following methods may be used: ab, a b, a · b, a × b, a/b,, a b−1.

When multiplying the value of quantities either a multiplication sign, ×, or brackets should be used, not a half-high (centred) dot. When multiplying numbers only the multiplication sign, ×, should be used.

When dividing the values of quantities using a solidus, brackets are used to remove ambiguities.

Example:

F = ma for force equals mass times acceleration (53 m/s) × 10.2 s or (53 m/s)(10.2 s)

25 × 60.5 but not 25 · 60.5;

(20 m)/(5 s) = 4 m/s;

(a/b)/c, not a/b/c.

3.7. Stating values of dimensionless quantities, or quantities of dimension one

The coherent SI unit for dimensionless quantities, also termed quantities of dimension one, is the number one, symbol 1. Values of such quantities are expressed simply as numbers. The unit symbol 1 or unit name ″one″ are not explicitly shown, nor are special symbols or names given to the unit one, apart from a few exceptions as follows. For the quantity plane angle, the unit one is given the special name radian, symbol rad, and for the quantity solid angle, the unit one is given the special name steradian, symbol sr. For the logarithmic ratio quantities, the special names neper, symbol Np, bel, symbol B, and decibel, symbol dB, are use.

Example:

n = 1.51, but not n = 1.51 × 1

where n is the quantity symbol for refractive index.

Because SI prefix symbols can neither be attached to the symbol 1 nor to the name ″one″, powers of 10 are used to express the values of particularly large or small dimensionless quantities.

In mathematical expressions, the internationally recognized symbol % (percent) may be used with the SI to represent the number 0.01. Thus, it can be used to express the values of dimensionless quantities. When it is used, a space separates the number and the symbol %. In expressing the values of dimensionless quantities in this way, the symbol % should be used rather than the name ″percent″.

In written text, however, the symbol % generally takes the meaning of ″parts per hundred″.

Phrases such as ″percentage by mass″, ″percentage by volume″, or ″percentage by amount of substance″ should not be used; the extra information on the quantity should instead be conveyed in the name and symbol for the quantity.

Example:

xB = 0.0025 = 0.25 %;

where xB is the quantity symbol for amount fraction (mole fraction) of entity B;

The mirror reflects 95 % of the incident photons;

φ = 3.6 %, but not φ = 3.6 % (V/V);

where φ denotes volume fraction.

In expressing the values of dimensionless fractions (e.g. mass fraction, volume fraction, relative uncertainties), the use of a ratio of two units of the same kind is sometimes useful.

Example:

xB = 2.5 ×10−3= 2.5 mmol/mol ur(U) = 0.3 μV/V;

where ur(U) is the relative uncertainty of the measured voltage U.

The term ″ppm″, meaning 10−6 relative value, or 1 in 106, or parts per million, is also used. This is analogous to the meaning of percent as parts per hundred. The terms ″parts per billion″, and ″parts per trillion″, and their respective abbreviations ″ppb″, and ″ppt″, are also used, but their meanings are language dependent. For this reason the terms ppb and ppt are best avoided. (In English-speaking countries, a billion is now generally taken to be 109 and a trillion to be 1012; however, a billion may still sometimes be interpreted as 1012 and a trillion as 1018. The abbreviation ppt is also sometimes read as parts per thousand, adding further confusion.)

When any of the terms %, ppm, etc., are used it is important to state the dimensionless quantity whose value