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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>The Unified Code for Units of Measure</title>
<link rel="stylesheet" type="text/css" href="v3m00.css">
<style type="text/css"></style>
<link rel="stylesheet" type="text/css" href="xml-verbatim.css">
<script type="text/javascript" src="xml-verbatim.js"></script>
</head>
<body>
<h1>The Unified Code for Units of Measure</h1>
<table border="0">
<tr>
<td>Gunther Schadow<br>Pragmatic Data LLC
</td>
</tr>
<tr>
<td>Clement J. McDonald<br>National Library of Medicine, Lister Hill
</td>
</tr>
</table>
<p>Version: 2.2</p>
<p>Revision: N/A</p>
<p>Date: 2024-06-17</p>
<p class="copyright">
<tr>
<td valign="top">Copyright © 1998-2024, Regenstrief Institute, Inc. and the UCUM Organization. All rights
reserved.</td>
</tr>
</p>
<hr title="Separator for header">
<h2><a name="contents">Table of contents</a></h2>1 <a href="#section-introduction">Introduction</a><br>2 <a
href="#section-Grammar-of-Units-and-Unit-Terms">Grammar of Units and Unit Terms</a><br> 2.1
<a href="#section-Character-Set-and-Lexical-Rules">Character Set and Lexical Rules</a><br> 2.2
<a href="#section-Syntax-Rules">Syntax Rules</a><br> 2.3 <a
href="#section-The-Predicate-%E2%80%9CMetric%E2%80%9D">The Predicate “Metric”</a><br> 2.4
<a href="#section-Style">Style</a><br>3 <a href="#section-Semantics">Semantics</a><br> 3.1 <a
href="#section-Special-Units-on-non-ratio-Scales">Special Units on non-ratio
Scales</a><br> 3.2 <a href="#section-Arbitrary-Units">Arbitrary Units</a><br>4 <a
href="#section-Tables-of-Terminal-Symbols">Tables of Terminal Symbols</a><br> 4.1 <a
href="#section-Prefixes">Prefixes</a><br> 4.2 <a href="#section-Base-Units">Base
Units</a><br> 4.3 <a href="#section-Derived-Unit-Atoms">Derived Unit
Atoms</a><br> 4.4 <a href="#section-Customary-Unit-Atoms">Customary Unit
Atoms</a><br> 4.5 <a href="#section-Other-Legacy-Units">Other Legacy
Units</a><br> 4.6 <a
href="#section-Prefixes-and-Units-Used-in-Information-Technology">Prefixes and Units Used in Information
Technology</a><br>
<h3>Appendices</h3>A <a href="#section-Examples-for-some-Non-Units.">Examples for some Non-Units.</a><br>B <a
href="#section-Summary-of-Conflicts">Summary of Conflicts</a><br>C <a href="#section-Alphabetic-Index">Alphabetic
Index</a><br> C.1 <a href="#section-Alphabetic-Index-By-Name">Alphabetic Index By
Name</a><br> C.2 <a href="#section-Alphabetic-Index-By-Symbol">Alphabetic Index By
Symbol</a><br> C.3 <a href="#section-Alphabetic-Index-By-Kind-Of-Quantity">Alphabetic Index
By Kind Of Quantity</a><br>D <a href="#datyp2apdxatblxmp">Example Unit Terms</a><br>E <a href="#license">Copyright
and License Terms of Use</a><br>
<hr>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h2><a name="section-introduction"></a>1
</h2>
</td>
<td valign="top">
<h2>Introduction</h2>
</td>
</tr>
</table>
<p><em>The Unified Code for Units of Measure</em> is a code system intended to include <em>all</em> units of
measures being contemporarily used in international science,
engineering, and business. The purpose is to facilitate unambiguous
electronic communication of quantities together with their units. The
focus is on electronic communication, as opposed to communication
between humans. A typical application of <em>The Unified Code for Units of Measure</em> are electronic data
interchange (EDI) protocols, but there is nothing that prevents it
from being used in other types of machine communication.
</p>
<p><em>The Unified Code for Units of Measure</em> is inspired by and heavily based on ISO 2955-1983, ANSI
X3.50-1986, and HL7's extensions called “ISO+”. The
respective ISO and ANSI standards are both entitled
“Representation of [...] units in systems with limited
character sets” where ISO 2955 refers to SI and other units
provided by ISO 1000-1981, while ANSI X3.50 extends ISO 2955 to
include U.S. customary units. Because these standards carry the
restriction of “limited character sets” in their names
they seem to be of less value today, when graphical user interfaces and
laser printers are in wide-spread use. For this reason, the European
standard ENV 12435 in its clause 7.3 declares ISO 2955 obsolete.
</p>
<p>
ENV 12435 is dedicated exclusively to the communication of
measurements between humans in display and print, and does not provide
codes that can be used in communication between systems. It does not
even provide a specification that would allow communication of units
from one system to the screen or printer of another system. The issue
about displaying units in the common style defined by the 9th
<em>Conférence Générale des Poids et Mesures</em>
(CGPM) in 1947 is not just the character set. Although the
<em>Unicode</em> standard and its predecessor ISO/IEC 10646 is
the richest character set ever, it is still not enough to specify the
presentation of units, because there are important typographical
details such as superscripts, subscripts, roman and
italics.<a name="fn-src1"></a><a href="#fn1"><sup style="font-size: smaller">1</sup></a>
</p>
<p>
The real value of the restriction on the character set and
typographical details, however, is not to cope with legacy systems and
less powerful technology, but to facilitate unambiguous communication
and interpretation of the meaning of units from one computer system to
another. In this respect, ISO 2955 and ANSI X3.50 are not
obsolete because there is no other standard that would fill in for
inter-systems communication of units. However, ISO 2599 and ANSI
X3.50 currently have severe defects:
</p>
<ol>
<li>
ISO 2955 and ANSI X3.50 contain numerous name conflicts,
both direct conflicts (e.g., “<code>a</code>” being used
for both “year” and “are”) and conflicts
that are generated through combination of unit symbols with prefixes
(e.g., “<code>cd</code>” means candela and centi-day and
“<code>PEV</code>” means peta-volt and pico-electronvolt.)
</li><br><br>
<li>
Neither ISO 2955 nor ANSI X3.50 cover all units that are
currently used in practice. There are many more units in use than what
is allowed by the <em>Système International
d'Unités</em> (SI) and accompanying standards. For example,
the older CGM-units dyne and erg are still used in the science of
physiology. Although ANSI X3.50 extends ISO 2955 with some
U.S. customary units, it is still not complete in this respect. For
example it does not define the degree Fahrenheit.
</li><br><br>
<li>
ANSI X3.50 is semantically ambiguous with respect to customary
units, even if we do not consider the history and international
aspects of customary units. Three systems of mass units are used in
the U.S., avoirdupois used generally, apothecaries' used by
pharmacists, and troy used in trade with Gold and other precious
metals. ANSI X3.50 has no way to select any one of those
specifically, which is bad in medicine, where both apothecaries' and
avoirdupois weights are being used frequently.
</li><br><br>
</ol>
<p>
ISO 2955 and all standards that do only look for the resolutions
and recommendations of the CGPM and the <em>Comité
International des Poids et Mesures</em> (CIPM) as published by the
<em>Bureau International des Poids et Mesures</em> (BIPM) and various
ISO standards (ISO 1000 and ISO 31) fail to recognize that
the needs in practice are often different from the ideal propositions
of the CGPM. Although not allowed by the CGPM and related ISO
standards, many other units are used in international sciences,
healthcare, engineering, and business, both meaningfully and some
units of questionable meaning. A coding system that is to be useful in
practice must cover the requirements and habits of the
practice—even some of the bad habits.
</p>
<p>
None of the current standards attempt to specify a semantics of units
that can be deployed in information systems with moderate
requirements. Metrological standards such as those published by the
BIPM are dedicated to maximal scientific correctness of reproducible
definitions of units. These definitions make sense only to human
specialists and can hardly be deployed to their full extent by any
information system that is not dedicated to metrology. On the other
hand, ISO 2955 and ANSI X3.50 provide no semantics at all for the
codes they define.
</p>
<p><em>The Unified Code for Units of Measure</em> provides a single coding system for units that is complete,
free of all ambiguities, and that assigns to each defined unit a
concise semantics. In communication it is not only important that all
communicating parties have the same repertoire of symbols, but also that
all attach the same meaning to the symbols they exchange. The common
meaning must be computationally verifiable. <em>The Unified Code for Units of Measure</em> assumes a
semantics for units based on dimensional analysis.<a name="fn-src2"></a><a href="#fn2"><sup
style="font-size: smaller">2</sup></a></p>
<p>
In short, each unit is defined relative to a system of base units by a
numeric factor and a vector of exponents by which the base units
contribute to the unit to be defined. Although we can reflect all the
meaning of units covered by dimensional analysis with this vector
notation, the following tables do not show these vectors. One reason
is that the vectors depend on the base system chosen and even on the
ordering of the base units. The other reason is that these vectors are
hard to understand to human readers while they can be easily derived
computationally. Therefore we define new unit symbols using algebraic
terms of other units. Those algebraic terms are also valid codes of
<em>The Unified Code for Units of Measure</em>.
</p>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h2><a name="section-Grammar-of-Units-and-Unit-Terms"></a>2
</h2>
</td>
<td valign="top">
<h2>Grammar of Units and Unit Terms</h2>
</td>
</tr>
</table>
<p name="preliminaries"><strong><a name="para-1"></a>§1 preliminaries</strong>
<strong><sup> ■1</sup></strong><em>The Unified Code for Units of Measure</em> consists of a basic set
of terminal symbols for units, called
<em>atomic unit symbols</em> or <em>unit atoms</em>, and multiplier
prefixes. It also consists of an expression syntax by which these
symbols can be combined to yield valid units.
<strong><sup> ■2</sup></strong>
The tables of terminal symbols are fixed as of every revision of
<em>The Unified Code for Units of Measure</em>, additions, deletions or changes are <em>not</em> allowed.
<strong><sup> ■3</sup></strong>
All expression that can be derived from these terminal symbols and the
expression syntax are valid codes. Any expression of <em>The Unified Code for Units of Measure</em> has a
precisely defined semantics.
</p>
<p class="comment">
The expression syntax of <em>The Unified Code for Units of Measure</em> generates an infinite number of codes
with the consequence that it is impossible to compile a table of all
valid units.
</p>
<p class="comment">
That the tables of terminal symbols may not be extended does not mean
that missing symbols will never be available in <em>The Unified Code for Units of Measure</em>. Suggestions
for additions of new symbols are welcome and revisions of
<em>The Unified Code for Units of Measure</em> will be released as soon as a change request has been approved.
</p>
<p name="full and limited conformance"><strong><a name="para-2"></a>§2 full and limited conformance</strong>
<strong><sup> ■1</sup></strong>
The semantics of <em>The Unified Code for Units of Measure</em> implies equivalence classes such that
different expressions may have the same meaning.
<strong><sup> ■2</sup></strong>
Programs that declare <em>full conformance</em> with <em>The Unified Code for Units of Measure</em> must
compare unit expressions by their semantics, i.e. they must detect
equivalence for different expressions with the same meaning.
<strong><sup> ■3</sup></strong>
Programs with <em>limited conformance</em> may compare unit expressions
literally and thus may not detect equivalence of unit expressions.
</p>
<p class="comment">
The option for “limited conformance” allows <em>The Unified Code for Units of Measure</em> to be adopted
even by less powerful systems that can not or do not want to deal with
the full semantics of units. Those systems typically have a table of
fixed unit expression literals that may be related to other literals
with fixed conversion factors. Although these systems will have
difficulties to receive unit expressions from various sources, they
will at least send out valid expressions of <em>The Unified Code for Units of Measure</em>, which is an
important step towards a commonly used coding scheme for units.
</p>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h3><a name="section-Character-Set-and-Lexical-Rules"></a>2.1
</h3>
</td>
<td valign="top">
<h3>Character Set and Lexical Rules</h3>
</td>
</tr>
</table>
<p name="character set"><strong><a name="para-3"></a>§3 character set</strong>
<strong><sup> ■1</sup></strong> All expressions of <em>The Unified Code for Units of Measure</em> shall
be built from characters of
the 7-bit US-ASCII character set exclusively.
<strong><sup> ■2</sup></strong> Terminal unit symbols can consist of all ASCII characters in
the range of 33–126 (0x21–0x7E) excluding
double quotes (‘<code>"</code>’),
parentheses (‘<code>(</code>’ and ‘<code>)</code>’),
plus sign (‘<code>+</code>’'),
minus sign (‘<code>-</code>’'),
period (‘<code>.</code>’'),
solidus (‘<code>/</code>’'),
equal sign (‘<code>=</code>’'),
square brackets (‘<code>[</code>’
and ‘<code>]</code>’), and
curly braces (‘<code>{</code>’ and ‘<code>}</code>’),
which have special meaning.
<strong><sup> ■3</sup></strong> A terminal unit symbol can not consist of only digits
(‘<code>0</code>’–‘<code>9</code>’) because
those digit strings are interpreted as positive integer
numbers. However, a symbol “<code>10*</code>” is allowed
because it ends with a non-digit allowed to be part of a symbol.
<strong><sup> ■4</sup></strong> For every terminal symbol there is a case insensitive variant
defined, to be used when there is a risk of upper and lower case to be
confused. Although upper and lower case can be mixed in case
insensitive symbols there is no meaning to the case. Case insensitive
symbols are incompatible to the case sensitive symbols.
</p>
<p class="comment">
The 7-bit US-ASCII character code is the greatest common denominator
that can be expected to be available in any communication environment.
Only very few units normally require symbols from the Greek alphabet
and thus the cost of requiring Unicode does not outweigh the benefit.
As explained above, the real issue about writing unit terms naturally
is not the character set but the ability to write subscripts and
superscripts and distinguish roman letters from italics.
</p>
<p class="comment">
Some computer systems or programming languages still have the
requirement of case insensitivity and some humans who are not familiar
with SI units tend to confuse upper and lower case or can not
interpret the difference in upper and lower case correctly. For this
reason the case insensitive symbols are defined. Although <em>The Unified Code for Units of Measure</em>
does not encourage use of case insensitive symbols where not
absolutely necessary, in some circumstances the case insensitive
representation may be the greatest common denominator. Thus some
systems that can handle case sensitivity may end up using case
insensitive symbols in order to communicate with less powerful
systems.
</p>
<p class="comment">
ISO 2955 and ANSI X3.50 call case sensitive symbols “mixed
case” and case insensitive symbols “single case” and
list two columns for “single case” symbols, one for upper
case and one for lower case. In <em>The Unified Code for Units of Measure</em> all units can be written in
mixed upper and lower case, but in the case insensitive variant the
mixing of case does not matter.
</p>
<p class="comment">
White space is not recognized in a a unit term and should generally
not occur. UCUM implementations may flag whitespace as an error
rather than ignore it. Whitespace is not used as a separator of
otherwise ambiguous parts of a unit term.
</p>
<p name="prefixes"><strong><a name="para-4"></a>§4 prefixes</strong>
<strong><sup> ■1</sup></strong>Metric units (cf. <a class="termref" href="#para-11">§11</a>) may be
combinations of a unit symbol with a prefix symbol.
<strong><sup> ■2</sup></strong>The unit symbol to be combined with the prefix must not itself
contain a prefix. Such a prefix-less unit symbol is called <em>unit
atom</em>.
<strong><sup> ■3</sup></strong>Prefix and atom are connected immediately without any
delimiter. Separation of an optional prefix from the atom occurs on
the lexical level by finding a matching combination of an optional
prefix and a unit atom.
<strong><sup> ■4</sup></strong> The prefix is the longest leading substring that matches a
valid prefix where the remainder is a valid metric unit atom. If no
such prefix can be matched, the unit atom is without prefix and may be
both metric or non-metric.<span class="small">[1–3: ISO 1000, 3; ISO 2955-1983, 3.7;
ANSI X3.50-1986, 3.7 (Rule No. 6).]</span></p>
<p name="square brackets"><strong><a name="para-5"></a>§5 square brackets</strong>
<strong><sup> ■1</sup></strong> Square brackets (‘<code>[</code>’ and
‘<code>]</code>’) may be part of a
unit atom at any place but only as matched pairs. Square brackets are
lexical elements and not separate syntactical tokens. <strong><sup> ■2</sup></strong> Within a matching
pair of square brackets the full range of
characters 33–126 can be used.<a name="fn-src3"></a><a href="#fn3"><sup
style="font-size: smaller">3</sup></a> <strong><sup> ■3</sup></strong> Square brackets do
<em>not</em> determine the boundary between
prefix and unit atom, but they never span the boundary of unit atoms.
<strong><sup> ■4</sup></strong>
Square brackets must not be nested.
</p>
<p class="comment">
For example
%
“<code>[abc+ef]</code>”,
“<code>ab[c+ef]</code>”,
“<code>[abc+]ef</code>”, and
“<code>ab[c+ef]</code>”
%
could all be valid symbols if defined in the tables.
In “<code>ab[c+ef]</code>” either
“<code>a</code>” or “<code>ab</code>”
could be defined as a prefix, but not “<code>ab[c</code>”.
</p>
<p class="comment">
Square brackets take on one task of round parentheses in HL7's
“ISO+” code, where one use of parentheses is to augment
unit symbols with suffixes, as in “<code>mm(Hg)</code>”.
Another use is to enclose one full unit symbol into parentheses, as
“<code>(ka_u)</code>” (for the King-Armstrong unit of
catalytic amount of phosphatase). Apparently, in a unit symbol such
enclosed one is supposed not to expect a prefix. Thus, even if
“<code>a_u</code>” would have been defined,
“<code>(ka_u)</code>” should not be matched against
kilo-<code>a_u</code>.
</p>
<p class="comment">
Parentheses, however, were also used for the nesting of terms since
HL7 version 2.3. At this point it became ambiguous whether parentheses
are part of the unit symbol or whether they are syntactic tokens. For
instance, “<code>(ka_u)</code>” could mean a nested
“<code>ka_u</code>” (where “<code>k</code>”
could possibly be a prefix), but also the proper symbol
“<code>(ka_u)</code>” that happens to have parentheses as
part of the symbol. <em>The Unified Code for Units of Measure</em> uses parentheses for the usual meaning of
term nesting and uses square brackets where HL7's “ISO+”
assumes parentheses to be part of the unit symbol.
</p>
<p name="curly braces"><strong><a name="para-6"></a>§6 curly braces</strong>
<strong><sup> ■1</sup></strong> The full range of characters 33–126 can be used within a
pair of curly braces (‘<code>{</code>’ and
‘<code>}</code>’). The material enclosed in curly braces is
called <em>annotation</em>.
<strong><sup> ■2</sup></strong>
Annotations do not contribute to the semantics of the unit but are
meaningless by definition. Therefore, any fully conformant parser must
discard all annotations. Parsers of limited conformance <em>should</em>
not value annotations in comparison of units.
<strong><sup> ■3</sup></strong>
Annotations do, however, signify the end of a unit symbol.
<strong><sup> ■4</sup></strong> An annotation without a leading symbol implies the default
unit 1 (the unity).
<strong><sup> ■5</sup></strong>
Curly braces must not be nested.
</p>
<p class="comment">
Curly braces are here because people want annotations and deeply
believe that they need annotations. Especially in chemistry and
biomedical sciences, there are traditional habits to write annotations
at units or instead of units, such as “%vol.”,
“RBC”, “CFU”, “kg(wet tis.)”, or
“mL(total)”. These habits are hard to overcome. Any
attempt of a coding scheme to restrict this perceived expressiveness
will ultimately result in the coding scheme not being adopted, or just
“half-way” adopted (which is as bad as not adopted).
</p>
<p class="comment">
Two alternative responses to this reality exist: either give in to the
bad habits and blow up of the code with dimension- and meaningless
unit atoms, or canalize this habit so that it does no harm. <em>The Unified Code for Units of Measure</em>
canalizes this habit using curly braces. Nevertheless we do continuing
efforts to upgrade doubtful units to genuine units of <em>The Unified Code for Units of Measure</em> by
defining and linking them to the other units as good as
possible. Thus, “<code>g%</code>” is a valid metric unit
atom (so that “<code>mg%</code>” is a valid unit too.)
A <em>drops</em>, although quite imprecise, is a valid unit of volume
“<code>[drp]</code>”. Even HPF and LPF (the so called
“high-” and “low power field” in the
microscope) have been defined so that at least they relate to each
other.
</p>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h3><a name="section-Syntax-Rules"></a>2.2
</h3>
</td>
<td valign="top">
<h3>Syntax Rules</h3>
</td>
</tr>
</table>
<p name="algebraic unit terms"><strong><a name="para-7"></a>§7 algebraic unit terms</strong>
<strong><sup> ■1</sup></strong> All units can be combined in an algebraic term using the
operators for multiplication (period ‘<code>.</code>‘) and
division (solidus ‘<code>/</code>’). <strong><sup> ■2</sup></strong> The multiplication operator is
mandatory it must not be
omitted or replaced by a space. The multiplication operator is a
strict binary operator that may occur only <em>between two</em> unit
terms. <strong><sup> ■3</sup></strong> The division operator can be used as a binary and unary
operator, i.e. a leading solidus will invert the unit that directly
follows it. <strong><sup> ■4</sup></strong> Terms are evaluated from left to right with the period and
the
solidus having the same operator precedence. Multiple division
operators are allowed within one term. <span class="small">[ISO 1000, 4.5.2; ISO 2955-1983, 3.3f;
ANSI X3.50-1986, 3.3f
(Rule No. 2f).]</span></p>
<p class="comment">
The use of the period instead of the asterisk
(‘<code>*</code>’) as a multiplication operator continues a
tradition codified in ISO 1000 and maintained in ISO 2955. Because
floating point numbers may not occur in unit terms the period is not
ambiguous. A period in a unit term has no other meaning than to be the
multiplication operator.
</p>
<p class="comment">
Since Resolution 7 of the 9th CGPM in 1948 the myth of ambiguity being
introduced by more than one solidus lives on and is quoted in all
standards concerning the writing of SI units. However, when the strict
left to right rule is followed there is no ambiguity, neither with one
solidus nor with more than one solidus. However, in human practice we
find the tendency to assign a lower precedence to the solidus which
misleads people to write <em>a</em>/<em>b</em>·<em>c</em> when they
really mean <em>a</em>/(<em>b</em>·<em>c</em>). When this is
rewritten as <em>a</em>/<em>b</em>/<em>c</em> there is actually less
ambiguity that in <em>a</em>/<em>b</em>·<em>c</em>. So the real
source of ambiguity is when a multiplication operator follows a
solidus, not when there is more than one solidus in a term. Hence, we
remove the restriction for only one solidus and introduce parentheses
which may be used to remove any perceived ambiguity.
</p>
<p name="integer numbers"><strong><a name="para-8"></a>§8 integer numbers</strong>
<strong><sup> ■1</sup></strong> A positive integer number may appear in place of a simple unit
symbol. <strong><sup> ■2</sup></strong> Only a pure string of decimal digits
(‘<code>0</code>’–‘<code>9</code>’)
is interpreted as a number. If after one or more digits there is any
non-digit character found that is valid for unit atoms, all the
characters (including the digits) will be interpreted as a simple unit
symbol.
</p>
<p class="comment">
For example, the string “<code>123</code>” is a positive
integer number while “<code>12a</code>” is a symbol.
</p>
<p class="comment">
Note that the period is only used as a multiplication operator, thus
“<code>2.5</code>” means 2 × 5 and is not equal to 5/2.
</p>
<p name="exponents"><strong><a name="para-9"></a>§9 exponents</strong>
<strong><sup> ■1</sup></strong> Simple units may be raised to a power. The exponent is an
integer number and is written immediately behind the unit
term. Negative exponents must be preceded by a minus sign
(‘<code>-</code>’ positive exponents may be preceded by an
optional plus sign (‘<code>+</code>’). <strong><sup> ■2</sup></strong> If the simple unit raised to a
power is a combination of a
prefix and a unit atom, both are raised to the power, e.g. “1
<code>cm3</code>” equals “10<sup>-6</sup><code>m3</code>” not “10<sup>-2</sup><code>m3</code>”.
<span class="small">[ISO 2955-1983, 3.5f; ANSI X3.50-1986, 3.5f (Rule
No. 4f).]</span>
</p>
<p class="comment">
ISO 2955 and ANSI X3.50 actually do not allow a plus sign leading a
positive exponent. However, if there can be any perceived ambiguities,
an explicit leading plus sign may be of help sometimes. <em>The
Unified Code for Units of Measures</em> therefore allows such plus signs
at exponents. The plus sign on positive exponents can be used to
delimit exponents from integer numbers used as simple units. Thus,
<code>2+10</code> means 2<sup>10</sup> = 1024.
</p>
<p name="nested terms"><strong><a name="para-10"></a>§10 nested terms</strong>
<strong><sup> ■1</sup></strong> Unit terms with operators may be enclosed in parentheses
(‘<code>(</code>’ and ‘<code>)</code>’) and used
in place of simple units. Normal left-to-right evaluation can be
overridden with parentheses. <strong><sup> ■2</sup></strong> Parenthesized terms are <em>not</em>
considered unit atoms
and hence must not be preceded by a prefix.
</p>
<p class="comment">
Up until revision 1.9 there was a third clause
“Since a unit term in parenthesis can be used in place of
a simple unit, an exponent may follow on a closing parenthesis which
raises the whole term within the parentheses to the power.”
However this feature was inconsistent with any BNF or other syntax
description ever provided, was never used and seems to have no
relevant use case. For this reason this clause has been stricken.
This is a <em>tentative</em> change. Users who have used this
feature in the past, should please comment on this deprecation.
If we receive indication that this feature was used by anyone, we
would undo the deprecation. If no comments are received, the
deprecation continues to take effect.
</p>
<table id="" width="90%">
<caption xmlns:u="http://aurora.regenstrief.org/UCUM">Exhibit 1:
The complete syntax in the Backus-Naur Form.
</caption>
<tr>
<td class="exhibit">
<table xmlns:u="http://aurora.regenstrief.org/UCUM" class="plain" border="0">
<tr valign="top">
<td><<em>sign</em>>
</td>
<td>::=</td>
<td>“<code>+</code>” | “<code>-</code>”
</td>
</tr>
<tr valign="top">
<td><<em>digit</em>>
</td>
<td>::=</td>
<td>“<code>0</code>” | “<code>1</code>” | “<code>2</code>” |
“<code>3</code>” | “<code>4</code>” | “<code>5</code>” |
“<code>6</code>” | “<code>7</code>” | “<code>8</code>” |
“<code>9</code>”
</td>
</tr>
<tr valign="top">
<td><<em>digits</em>>
</td>
<td>::=</td>
<td><<em>digit</em>><<em>digits</em>>
| <<em>digit</em>>
</td>
</tr>
<tr valign="top">
<td><<em>factor</em>>
</td>
<td>::=</td>
<td><<em>digits</em>>
</td>
</tr>
<tr valign="top">
<td><<em>exponent</em>>
</td>
<td>::=</td>
<td><<em>sign</em>><<em>digits</em>>
| <<em>digits</em>>
</td>
</tr>
<tr valign="top">
<td><<em>simple-unit</em>>
</td>
<td>::=</td>
<td><<em>ATOM-SYMBOL</em>><br>
| <<em>PREFIX-SYMBOL</em>><<em>ATOM-SYMBOL[metric]</em>>
</td>
</tr>
<tr valign="top">
<td><<em>annotatable</em>>
</td>
<td>::=</td>
<td><<em>simple-unit</em>><<em>exponent</em>><br>
| <<em>simple-unit</em>>
</td>
</tr>
<tr valign="top">
<td><<em>component</em>>
</td>
<td>::=</td>
<td><<em>annotatable</em>><<em>annotation</em>><br>
| <<em>annotatable</em>><br>
| <<em>annotation</em>><br>
| <<em>factor</em>><br>
| “<code>(</code>”<<em>term</em>>“<code>)</code>”
</td>
</tr>
<tr valign="top">
<td><<em>term</em>>
</td>
<td>::=</td>
<td><<em>term</em>>“<code>.</code>”<<em>component</em>><br>
| <<em>term</em>>“<code>/</code>”<<em>component</em>><br>
| <<em>component</em>>
</td>
</tr>
<tr valign="top">
<td><<em>main-term</em>>
</td>
<td>::=</td>
<td>“<code>/</code>”<<em>term</em>><br>
| <<em>term</em>>
</td>
</tr>
<tr valign="top">
<td><<em>annotation</em>>
</td>
<td>::=</td>
<td>“<code>{</code>”<<em>ANNOTATION-STRING</em>>“<code>}</code>”
</td>
</tr>
</table>
</td>
</tr>
</table>
<div class="descriptive"><img
src="https://raw.githubusercontent.com/ucum-org/ucum/main/assets/images/ucum-state-automaton.gif"
alt="Pushdown-state automaton describing the syntax." class="graphic"></div>
<div class="descriptive">
<p>
Figure
1: Pushdown-state automaton describing the syntax.
</p>
</div>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h3><a name="section-The-Predicate-%E2%80%9CMetric%E2%80%9D"></a>2.3
</h3>
</td>
<td valign="top">
<h3>The Predicate “Metric”</h3>
</td>
</tr>
</table>
<p id="para-metric" name="metric and non-metric unit atoms"><strong><a name="para-11"></a>§11 metric and non-metric
unit atoms</strong>
<strong><sup> ■1</sup></strong> Only metric unit atoms may be combined with a prefix.
<strong><sup> ■2</sup></strong> To be metric or not to be metric is a predicate assigned to
each unit atom where that unit atom is defined.
<strong><sup> ■3</sup></strong> All base units are metric. No non-metric unit can be part of
the basis.
<strong><sup> ■4</sup></strong> A unit must be a quantity on a ratio scale in order to be
metric.
</p>
<p class="comment">
The metric predicate accounts for the fact that there are units that
are prefixed and others that are not. This helps to disambiguate the
parsing of simple units into prefix and atom.
</p>
<p class="comment">
To determine whether a given unit atom is metric or not is not
trivial. It is a cultural phenomenon, subject to change, just like
language, the meaning of words and how words can be used. At one time
we can clearly tell right or wrong usage of words, but these
decisions may need to be revised with the passage of time.
</p>
<p class="comment">
Generally, metric units are those defined “in the spirit”
of the metric system, that emerged in France of the 18th century and
was rapidly adopted by scientists. Metric units are usually based on
reproducible natural phenomena and are usually not part of a system of
comparable units with different magintudes, especially not if the
ratios of these units are not powers of 10. Instead, metric units use
multiplier prefixes that magnify or diminish the value of the unit
by powers of ten.
</p>
<p class="comment">
Conversely, customary units are in the spirit of the middle age as
most of them can be traced back into a time around the 10th century,
some are even older from the Roman and Babylonian empires. Most
customary units are based on the average size of human anatomical or
botanic structures (e.g., foot, ell, fathom, grain, rod) and come in
series of comparable units with ratios 1/2, 1/4, 1/12, 1/16, and
others. Thus all customary units are non-metric
</p>
<p class="comment">
Not all units from ISO 1000 are metric as degree, minute and second of
plane angle are non-metric as well as minute, hour, day, month, and
year. The second is a metric unit because it is a part of the SI
basis, although it used to be part of a series of customary units
(originating in the Babylonian era).
</p>
<p class="comment">
Furthermore, for a unit to be metric it must be a quantity on a ratio
scale where multiplication and division with scalars are defined. The
<em>Comité Consultatif d'Unités</em> (CCU) decided
in February 1995 that SI prefixes may be used with the degree
Celsius. This statement has not been made explicitly before. This is
an unfortunate decision because difference-scale units like the degree
Celsius have no multiplication operation, so that the prefix value
could be multiplied with the unit. Instead the prefix at non-ratio
units scales the measurement value. One dekameter is 10 times of a
meter, but there is no meaning to 10 times of 1 °C in the
same way as 30 °C are not 3 times as much as
10 °C. See <a class="termref" href="#para-21">§§21ff</a> on how <em>The Unified Code for Units of
Measure</em> finds a
way to accommodate this different use of prefixes at units such as the
degree Celsius, bel or neper.
</p>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h3><a name="section-Style"></a>2.4
</h3>
</td>
<td valign="top">
<h3>Style</h3>
</td>
</tr>
</table>
<p class="comment">
Except for the rule on curly braces (<a class="termref" href="#para-12">§12</a>), the
rules on style govern the creation of the tables of unit atoms not
their individual use. Users of <em>The Unified Code for Units of Measure</em> need not care about style rules
(<a class="termref" href="#para-13">§§13–15</a>) because users
just use the symbols defined in the tables. Hence, style rules do not
affect conformance to <em>The Unified Code for Units of Measure</em>. New submissions of unit atoms, however,
must conform to the style rules.
</p>
<p id="para-curly" name="curly braces"><strong><a name="para-12"></a>§12 curly braces</strong>
<strong><sup> ■1</sup></strong>
Curly braces may be used to enclose annotations that are often written
in place of units or behind units but that do not have a proper
meaning of a unit and do not change the meaning of a unit.
<strong><sup> ■2</sup></strong>
Annotations have no semantic value.
</p>
<p class="comment">
For example one can write “<code>%{vol}</code>”,
“<code>kg{total}</code>”, or “<code>{RBC}</code>”
(for “red blood cells”) as pseudo-units. However, these
annotations do not have any effect on the semantics, which is why
these example expressions are equivalent to
“<code>%</code>”, “<code>kg</code>”, and
“<code>1</code>” respectively.
</p>
<p id="para-underscore" name="underscore"><strong><a name="para-13"></a>§13 underscore</strong>
<strong><sup> ■1</sup></strong> When in print a unit would have a subscript, an underscore
(‘<code>_</code>’) is used to separate the subscript from
the stem of the unit symbol. <strong><sup> ■2</sup></strong>
The subscript is part of the unit atom.
<strong><sup> ■3</sup></strong> subscripts are used to disambiguate the two units with the
same name but different meanings.
</p>
<p class="comment">
For example when distinguishing the International Table calorie from
the thermochemical calorie, we would use 1 cal<sub>IT</sub> or
1 cal<sub>th</sub> in print. <em>The Unified Code for Units of Measure</em> defines the symbols
“<code>cal_IT</code>” and
“<code>cal_th</code>” with the underscore signifying that
“IT” and “th” are subscripts. Other examples
are the distinctions between the Julian and Gregorian calendar year
from the tropical year or the British imperial gallon from the U.S.
gallon (see <a class="termref" href="#para-31">§31</a> and <a class="termref" href="#para-37">§§37ff</a>).
</p>
<p name="square brackets"><strong><a name="para-14"></a>§14 square brackets</strong>
<strong><sup> ■1</sup></strong> Square brackets enclose suffixes of unit symbols that change
the meaning of a unit stem.
<strong><sup> ■2</sup></strong> All customary units shall be enclosed completely by square
brackets.
<strong><sup> ■3</sup></strong>
Other unit atoms shall be enclosed in square brackets if they are very
rare, if they will conflict with other units, or if they are normally
not used as a unit symbol but do have a proper meaning as a unit in
<em>The Unified Code for Units of Measure</em>.
<strong><sup> ■4</sup></strong>
Square brackets are part of the unit atom.
</p>
<p class="comment">
For example 1 m H<sub>2</sub>O is written as
“<code>m[H2O]</code>” in <em>The Unified Code for Units of Measure</em> because the suffix
H<sub>2</sub>O changes the meaning of the unit atom for meter (length)
to a unit of pressure.
</p>
<p class="comment">
Customary units are defined in <em>The Unified Code for Units of Measure</em> in order to accommodate
practical needs. However metric units are still preferred and the
customary symbols should not interfere with metric symbols in any
way. Thus, customary units are “stigmatized” by enclosing
them into square brackets.
</p>
<p class="comment">
If unit symbols for the purpose of display and print are derived from
<em>The Unified Code for Units of Measure</em> units, the square brackets can be removed. However, display
units are out of scope of <em>The Unified Code for Units of Measure</em>.
</p>
<p id="para-apostrophe" name="apostrophe"><strong><a name="para-15"></a>§15 apostrophe</strong>
<strong><sup> ■1</sup></strong>
The apostrophe (‘<code>'</code>’) is used to separate words
or abbreviated words in a multi-word unit symbol.
<strong><sup> ■2</sup></strong>
Since units are mathematically defined symbols and not abbreviations
of words, multi-word unit symbols should be defined only to reflect
existing habits, not in order to create new ones.
<strong><sup> ■3</sup></strong>
Multi-word units should always be enclosed in square brackets.
</p>
<p class="comment">
For example, such legacy units called “Bodansky unit” or
“Todd unit” have the unit symbols
“<code>[bdsk'U]</code>”, and
“<code>[todd'U]</code>” respectively.
</p>
<table border="0" cellpadding="0" cellspacing="0" height="0" width="100%" class="header">
<tr>
<td width="68" nowrap="" valign="top" align="left">
<h2><a name="section-Semantics"></a>3
</h2>
</td>
<td valign="top">
<h2>Semantics</h2>
</td>
</tr>
</table>
<p name="preliminaries"><strong><a name="para-16"></a>§16 preliminaries</strong>
<strong><sup> ■1</sup></strong> The semantics of <em>The Unified Code for Units of Measure</em> is
defined by the algebraic
operations of multiplication, division and exponentiation between
units, by the equivalence relations of equality and commensurability
of units, and by the multiplication of a unit with a scalar.
<strong><sup> ■2</sup></strong> Every expression in <em>The Unified Code for Units of Measure</em> is
mapped to one and only one
semantic element. But every semantic element may have more than one
valid representant in <em>The Unified Code for Units of Measure</em>.
<strong><sup> ■3</sup></strong>
The set of expressions in <em>The Unified Code for Units of Measure</em> is infinite.
</p>
<p name="equality and commensurability"><strong><a name="para-17"></a>§17 equality and commensurability</strong>
<strong><sup> ■1</sup></strong> The set of expressions in <em>The Unified Code for Units of
Measure</em> has two binary, symmetric,
reflexive, and transitive relations (equivalence relations)
“equals” = and “is commensurable with”
~. All expressions that are equal are also commensurable but not
all commensurable expressions are equal.
</p>
<p name="algebra of units"><strong><a name="para-18"></a>§18 algebra of units</strong>
<strong><sup> ■1</sup></strong> The equivalence classes generated by the equality relation =
are called <em>units</em>.
<strong><sup> ■2</sup></strong> The set of units <em>U</em> has a binary multiplication operator
· that is associative and commutative and has the neutral
element <strong>1</strong> (so called <em>the unity</em>). For each unit
<strong>u</strong> ∈ <em>U</em> there is an inverse unit
<strong>u</strong><sup>-1</sup> such that <strong>u</strong> ·
<strong>u</strong><sup>-1</sup> = <strong>1</strong>. Thus, (<em>U</em>, ·) is
an Abelian group.
<strong><sup> ■3</sup></strong> The division operation <strong>u</strong> / <strong>v</strong> is
defined as
<strong>u</strong> · <strong>v</strong><sup>-1</sup>. <strong><sup> ■4</sup></strong> The
exponentiation operation with integer exponents <em>n</em>
is defined as <strong>u</strong><sup><em>n</em></sup> =
<strong>Π</strong><sub>1</sub><sup>n</sup><strong>u</strong>.
<strong><sup> ■5</sup></strong>
The product <strong>u</strong>' = <em>r</em><strong>u</strong> of a real number scalar
with the unit <strong>u</strong> is also a
unit, where <strong>u</strong>' ~ <strong>u</strong>.
</p>
<p id="para-dimension" name="dimension and magnitude"><strong><a name="para-19"></a>§19 dimension and
magnitude</strong>
<strong><sup> ■1</sup></strong> The equivalence classes generated by the commensurability
relation ~ are called <em>dimensions</em>. The set <em>D</em>
of dimensions is infinite in principle, but only a finite subset of
dimensions are used in practice. Thus, implementations of <em>The Unified Code for Units of Measure</em> need
not be able to represent the infinite set of dimensions.
<strong><sup> ■2</sup></strong>
Two commensurable units that are not equal differ only by their
magnitude.
<strong><sup> ■3</sup></strong> The quotient <strong>u</strong> / <strong>v</strong> of any two
commensurable
units <strong>u</strong> ~ <strong>v</strong> is of the same dimension as the
unity (<strong>u</strong> / <strong>v</strong> ~ <strong>1</strong>). This quotient is
also equal to the unity multiplied with a scalar <em>r</em>: <strong>u</strong> / <strong>v</strong> =
<em>r</em><strong>1</strong>,
where <em>r</em> is called the <em>relative magnitude</em> of
<strong>u</strong> regarding <strong>v</strong>.
</p>
<p name="base units"><strong><a name="para-20"></a>§20 base units</strong>
<strong><sup> ■1</sup></strong>
Any system of units is constructed from a finite set <strong>B</strong> of
mutually independent base units <strong>B</strong> = {
<strong>b</strong><sub>1</sub>, <strong>b</strong><sub>2</sub>, ...,
<strong>b</strong><sub><em>n</em></sub> }, on which any other unit <strong>u</strong>
∈ <em>U</em> is defined as <strong>u</strong> =
<em>r</em><sub>1</sub><strong>b</strong><sub>1</sub><sup><em>u</em><sub>1</sub></sup> ·
<em>r</em><sub>2</sub><strong>b</strong><sub>2</sub><sup><em>u</em><sub>2</sub></sup> ·
... ·
<em>r</em><sub><em>n</em></sub><strong>b</strong><sub><em>n</em></sub><sup><em>u</em><sub><em>n</em></sub></sup>,
where <em>r</em> = <em>r</em><sub>1</sub> · <em>r</em><sub>2</sub>
·· · <em>r</em><sub><em>n</em></sub> is called the
<em>magnitude</em> of the unit <strong>u</strong> regarding <strong>B</strong>.
<strong><sup> ■2</sup></strong>
With respect to a basis <strong>B</strong> every unit can thus be
represented as a pair (<em>r</em>, <em>û</em>) of magnitude <em>r</em> and dimension <em>û</em> =
(<em>u</em><sub>1</sub>, <em>u</em><sub>2</sub>, ...,
<em>u</em><sub><em>n</em></sub>).
<strong><sup> ■3</sup></strong>