ISO 31-0

ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the SI. It is intended for use in all fields of science and technology and is augmented by more specialized conventions defined in other parts of the ISO 31 standard. ISO 31-0 was withdrawn on 17 November 2009. It is superseded by ISO 80000-1. Other parts of ISO 31 have also been withdrawn and replaced by parts of ISO 80000.


ISO 31 covers only physical quantities used for the quantitative description of physical phenomena. It does not cover conventional scales (e.g., Beaufort scale, Richter scale, colour intensity scales), results of conventional tests, currencies, or information content. The presentation here is only a brief summary of some of the detailed guidelines and examples given in the standard.

Quantities and units

Physical quantities can be grouped into mutually comparable categories. For example, length, width, diameter and wavelength are all in the same category, that is they are all quantities of the same kind. One particular example of such a quantity can be chosen as a reference quantity, called the unit, and then all other quantities in the same category can be expressed in terms of this unit, multiplied by a number called the numerical value. For example, if we write

the wavelength is λ = 6.982 × 10−7 m

then "λ" is the symbol for the physical quantity (wavelength), "m" is the symbol for the unit (metre), and "6.982 × 10−7" is the numerical value of the wavelength in metres.

More generally, we can write

A = {A} ⋅ [A]

where A is the symbol for the quantity, {A} symbolizes the numerical value of A, and [A] represents the corresponding unit in which A is expressed here. Both the numerical value and the unit symbol are factors, and their product is the quantity. A quantity itself has no inherent particular numerical value or unit; as with any product, there are many different combinations of numerical value and unit that lead to the same quantity (e.g., A = 300 ⋅ m = 0.3 ⋅ km = ...). This ambiguity makes the {A} and [A] notations useless, unless they are used together.

The value of a quantity is independent of the unit chosen to represent it. It must be distinguished from the numerical value of the quantity that occurs when the quantity is expressed in a particular unit. The above curly-bracket notation could be extended with a unit-symbol index to clarify this dependency, as in {λ}m = 6.982 × 10−7 or equivalently {λ}nm = 698.2. In practice, where it is necessary to refer to the numerical value of a quantity expressed in a particular unit, it is notationally more convenient to simply divide the quantity through that unit, as in

λ/m = 6.982 × 10−7

or equivalently

λ/nm = 698.2.

This is a particularly useful and widely used notation for labelling the axes of graphs or for the headings of table columns, where repeating the unit after each numerical value can be typographically inconvenient.

Typographic conventions

Symbols for quantities

  • Quantities are generally represented by a symbol formed from single letters of the Latin or Greek alphabet.
  • Symbols for quantities are set in italic type, independent of the type used in the rest of the text.
  • If in a text different quantities use the same letter symbol, they can be distinguished via subscripts.
  • A subscript is only set in italic type if it consists of a symbol for a quantity or a variable. Other subscripts are set in upright (roman) type. For example, write Vn for a "nominal volume" (where "n" is just an abbreviation for the word "nominal"), but write Vn if n is a running index number.

Names and symbols for units

  • If an internationally standardized symbol exists for a unit, then only that symbol should be used. See the SI articles for the list of standard symbols defined by the International System of Units. Note that the distinction between uppercase and lowercase letters is significant for SI unit symbols. For example, "k" is the prefix kilo and "K" stands for the unit kelvin. The symbols of all SI units named after a person or a place start with an uppercase letter, as do the symbols of all prefixes from mega on upwards. All other symbols are lowercase; the only exception is litre, where both l and L are allowed. However, it is stated that the CIPM will examine whether one of the two may be suppressed.
  • Symbols for units should be printed in an upright (roman) typeface.


See Sect. 3.3 of the Standard text.

  • Numbers should be printed in upright (roman) type.
  • ISO 31-0 (after Amendment 2) specifies that "the decimal sign is either the comma on the line or the point on the line". This follows resolution 10[1] of the 22nd CGPM, 2003.[2]
For example, one divided by two (one half) may be written as 0.5 or 0,5.
  • Numbers consisting of long sequences of digits can be made more readable by separating them into groups, preferably groups of three, separated by a small space. For this reason, ISO 31-0 specifies that such groups of digits should never be separated by a comma or point, as these are reserved for use as the decimal sign.
For example, one million (1000000) may be written as 1 000 000.
  • For numbers whose magnitude is less than 1, the decimal sign should be preceded by a zero.
  • The multiplication sign is either a cross or a half-height dot, though the latter should not be used when the dot is the decimal separator.


  • Unit symbols follow the numerical value in the expression of a quantity.
  • Numerical value and unit symbol are separated by a space. This rule also applies to the symbol "°C" for degrees Celsius, as in "25 °C". It also applies to the percent sign ("%"). The only exceptions are the symbols for the units of plane angle: degree, minute of arc, and second of arc - which follow the numerical value without a space in between (for example "30°").
  • Where quantities are added or subtracted, parenthesis can be used to distribute a unit symbol over several numerical values, as in
T = 25 °C − 3 °C = (25 − 3) °C
P = 100 kW ± 5 kW = (100 ± 5) kW
(but not: 100 ± 5 kW)
d = 12 × (1 ± 10−4) m
  • Products can be written as ab, a b, ab, or a×b. The sign for multiplying numbers is a cross (×) or a half-height dot (⋅). The cross should be used adjacent to numbers if a dot on the line is used as the decimal separator, to avoid confusion between a decimal dot and a multiplication dot.
  • Division can be written as , a/b, or by writing the product of a and b−1, for example ab−1. Numerator or denominator can themselves be products or quotients, but in this case, a solidus (/) should not be followed by a multiplication sign or division sign on the same line, unless parentheses are used to avoid ambiguity.

Mathematical signs and symbols

A comprehensive list of internationally standardized mathematical symbols and notations can be found in ISO 31-11.

See also


  1. ^ "Resolution 10", 22nd General Conference on Weights and Measures, BIPM.
  2. ^ Baum, Michael (22 November 2006). " Brief reference to the history | Decimals Score a Point on International Standards ". NIST. Archived from the original (html) on 27 November 2006. Retrieved 17 November 2018. Until recently, the rule at the International Organization for Standardization (ISO—the world's largest developer of standards) and the International Electrotechnical Commission (IEC—the leading global electrical and electronic standards organization) was that all numbers with a decimal part must be written in formal documents with a comma decimal separator, the prevailing fashion in Europe. The constant pi, for example, starts 3,141 592 653.


Decimal separator

A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form.

Different countries officially designate different symbols for the decimal separator. The choice of symbol for the decimal separator also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article.

Any such symbol can be called a decimal mark, decimal marker or decimal sign. But symbol-specific names are also used; decimal point and decimal comma refer to an (either baseline or middle) dot and comma respectively, when it is used as a decimal separator; these are the usual terms used in English, with the aforementioned generic terms reserved for abstract usage.In many contexts, when a number is spoken, the function of the separator is assumed by the spoken name of the symbol: comma or point in most cases. In some specialized contexts, the word decimal is instead used for this purpose (such as in ICAO-regulated air traffic control communications).

In mathematics the decimal separator is a type of radix point, a term that also applies to number systems with bases other than ten.

Dimension (metadata)

In metadata, dimension is a set of equivalent units of measure, where equivalence between two units of measure is determined by the existence of a quantity preserving one-to-one correspondence between values measured in one unit of measure and values measured in the other unit of measure, independent of context, and where characterizing operations are the same.

The equivalence defined here forms an equivalence relation on the set of all units of measure. Each equivalence class corresponds to a dimensionality. The units of measure "temperature in degrees Fahrenheit" and "temperature in degrees Celsius" have the same dimensionality, because given a value measured in degrees Fahrenheit there is a value measured in degrees Celsius with the same quantity, and vice versa. Quantity preserving one-to-one correspondences are the well-known equations Cº = (5/9)*(Fº − 32) and Fº = (9/5)*(Cº) + 32.

Units of measure are not limited to physical categories. Examples of physical categories are: linear measure, area, volume, mass, velocity, time duration.Examples of non-physical categories are: currency, quality indicator, colour intensity.

Quantities may be grouped together into categories of quantities which are mutually comparable. Lengths, diameters, distances, heights, wavelengths and so on would constitute such a category. Mutually comparable quantities have the same dimensionality. ISO 31-0 calls these quantities of the same kind.

Guobiao standards

GB standards are the Chinese national standards issued by the Standardization Administration of China (SAC), the Chinese National Committee of the ISO and IEC. GB stands for Guobiao (simplified Chinese: 国标; traditional Chinese: 國標; pinyin: Guóbiāo), Chinese for national standard.

Mandatory standards are prefixed "GB". Recommended standards are prefixed "GB/T" (T from Chinese language 推荐; tuījiàn; 'recommended'). A standard number follows "GB" or "GB/T".

GB standards are the basis for the product testing which products must undergo during the China Compulsory Certificate (CCC) certification. If there is no corresponding GB Standard, CCC is not required.

ISO/IEC 80000

ISO 80000 or IEC 80000 is an international standard promulgated jointly by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC).

The standard introduces the International System of Quantities (ISQ). It is a style guide for the use of physical quantities and units of measurement, formulas involving them, and their corresponding units, in scientific and educational documents for worldwide use. In most countries, the notations used in mathematics and science textbooks at schools and universities follow closely the guidelines in this standard.The ISO/IEC 80000 family of standards was completed with the publication of Part 1 in November 2009.

ISO 31

ISO 31 (Quantities and units, International Organization for Standardization, 1992) is a deprecated international standard for the use of physical quantities and units of measurement, and formulas involving them, in scientific and educational documents. It is superseded by ISO/IEC 80000.

ISO 80000-1

ISO 80000-1:2009 is a standard describing scientific and mathematical quantities and their units. The standard, whose full name is Quantities and units Part 1: General was developed by the International Organization for Standardization (ISO), superseding ISO 31-0. It provides general information concerning quantities and units and their symbols, especially the International System of Quantities and the International System of Units, and defines these quantities and units. It is a part of a group of standards called ISO/IEC 80000.

ISO 8601

ISO 8601 Data elements and interchange formats – Information interchange – Representation of dates and times is an international standard covering the exchange of date- and time-related data. It was issued by the International Organization for Standardization (ISO) and was first published in 1988. The purpose of this standard is to provide an unambiguous and well-defined method of representing dates and times, so as to avoid misinterpretation of numeric representations of dates and times, particularly when data are transferred between countries with different conventions for writing numeric dates and times.

In general, ISO 8601 applies to representations and formats of dates in the Gregorian (and potentially proleptic Gregorian) calendar, of times based on the 24-hour timekeeping system (with optional UTC offset), of time intervals, and combinations thereof. The standard does not assign any specific meaning to elements of the date/time to be represented; the meaning will depend on the context of its use. In addition, dates and times to be represented cannot include words with no specified numerical meaning in the standard (e.g., names of years in the Chinese calendar) or that do not use characters (e.g., images, sounds).In representations for interchange, dates and times are arranged so the largest temporal term (the year) is placed to the left and each successively smaller term is placed to the right of the previous term. Representations must be written in a combination of Arabic numerals and certain characters (such as "-", ":", "T", "W", and "Z") that are given specific meanings within the standard; the implication is that some commonplace ways of writing parts of dates, such as "January" or "Thursday", are not allowed in interchange representations.

Italic type

In typography, italic type is a cursive font based on a stylized form of calligraphic handwriting. Owing to the influence from calligraphy, italics normally slant slightly to the right. Italics are a way to emphasise key points in a printed text, to identify many types of creative works, or, when quoting a speaker, a way to show which words they stressed. One manual of English usage described italics as "the print equivalent of underlining".The name comes from the fact that calligraphy-inspired typefaces were first designed in Italy, to replace documents traditionally written in a handwriting style called chancery hand. Aldus Manutius and Ludovico Arrighi (both between the 15th and 16th centuries) were the main type designers involved in this process at the time. Different glyph shapes from Roman type are usually used – another influence from calligraphy – and upper-case letters may have swashes, flourishes inspired by ornate calligraphy. An alternative is oblique type, in which the type is slanted but the letterforms do not change shape: this less elaborate approach is used by many sans-serif typefaces.

Kilowatt hour

The kilowatt hour (symbol kWh, kW⋅h or kW h) is a unit of energy equal to 3.6 megajoules. If energy is transmitted or used at a constant rate (power) over a period of time, the total energy in kilowatt hours is equal to the power in kilowatts multiplied by the time in hours. The kilowatt hour is commonly used as a billing unit for energy delivered to consumers by electric utilities.

Parts-per notation

In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement. Commonly used are ppm (parts-per-million, 10−6), ppb (parts-per-billion, 10−9), ppt (parts-per-trillion, 10−12) and ppq (parts-per-quadrillion, 10−15). This notation is not part of the SI system and its meaning is ambiguous.

Percent sign

The percent (per cent) sign (%) is the symbol used to indicate a percentage, a number or ratio as a fraction of 100. Related signs include the permille (per thousand) sign ‰ and the permyriad (per ten thousand) sign ‱ (also known as a basis point), which indicate that a number is divided by one thousand or ten thousand respectively. Higher proportions use parts-per notation.


In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number (pure number).

Scientific notation

Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.

In scientific notation all numbers are written in the form

m × 10n(m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number. The integer n is called the

order of magnitude and the real number m is called the significand or mantissa. However, the term "mantissa" may cause confusion because it is the name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m (as in ordinary decimal notation). In normalized notation, the exponent is chosen so that the absolute value of the coefficient is at least one but less than ten.

Decimal floating point is a computer arithmetic system closely related to scientific notation.

Typographical conventions in mathematical formulae

Typographical conventions in mathematical formulae provide uniformity across mathematical texts and help the readers of those texts to grasp new concepts quickly.

Mathematical notation includes letters from various alphabets, as well as special mathematical symbols. Letters in various fonts often have specific, fixed meanings in particular areas of mathematics. A mathematical article or a theorem typically starts from the definitions of the introduced symbols, such as: "Let G = (V, E) be a graph with the vertex set V and edge set E...". Theoretically it is admissible to write "Let X = (a, q) be a graph with the vertex set a and edge set q..."; however, this would decrease readability, since the reader has to consciously memorize these unusual notations in a limited context.

Usage of subscripts and superscripts is also an important convention. In the early days of computers with limited graphical capabilities for text, subscripts and superscripts were represented with the help of additional notation. In particular, n2 could be written as n^2 or n**2 (the latter borrowed from FORTRAN) and n2 could be written as n_2.

ISO standards by standard number

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