Foreword

The Calendaring and Scheduling Consortium (“CalConnect”) is a global non-profit organization with the aim to facilitate interoperability of collaborative technologies and tools through open standards.

CalConnect works closely with international and regional partners, of which the full list is available on our website (https://www.calconnect.org/about/liaisons-and-relationships).

The procedures used to develop this document and those intended for its further maintenance are described in the CalConnect Directives.

In particular the different approval criteria needed for the different types of CalConnect documents should be noted. This document was drafted in accordance with the editorial rules of the CalConnect Directives.

Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. CalConnect shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be provided in the Introduction.

Any trade name used in this document is information given for the convenience of users and does not constitute an endorsement.

This document was prepared by Technical Committee UnitsML.

Introduction

General

UnitsML is a suite of technologies that facilitate digital processing of units of measurement and measurement quantities.

AsciiUnits is a text-based syntax for representing units of measurement in a compact and human-readable format, designed to be fully compatible with and interoperable with the UnitsML XML language.

Purpose

Key features of AsciiUnits include:

Automatic prefix parsing: Prefixes attached to unit codes are automatically recognized and processed.
Automatic dimension calculation: Dimensions are automatically calculated based on the prefixes and units represented.
Mathematical operations: Support for mathematical operations on units and dimensions such as addition, subtraction, division, exponentiation, square root, grouping and parentheses.
Metadata support: Ability to add additional information such as quantity references, custom symbols, and names.
UnitsML UnitsDB integration: The UnitsML UnitsDB database is the basis used by AsciiUnits to define the units and prefixes available for various applications.

Differences with other technologies

UCUM

UCUM is a compact coding system for the representation of units of measurement, primarily designed for electronic data interchange in clinical and healthcare contexts. Originally developed by the Regenstrief Institute and adopted by HL7, UCUM is widely used in electronic health records and clinical information systems.

While AsciiUnits and UCUM (Unified Code for Units of Measure) are both text-based systems for representing units of measurement, they differ in several key aspects.

Table 1
Feature	AsciiUnits	UCUM
Syntax approach	Uses intuitive mathematical operators (`*`, `/`, `^`)	Uses special characters (`.` for multiplication, `/` for division)
Readability	Designed for human readability and ease of typing	Optimized for compactness and machine processing
XML integration	Direct mapping to UnitsML XML	No built-in XML representation
Metadata support	Supports additional metadata (quantity references, custom symbols)	No metadata support
Dimensional analysis	Automatic dimension calculation	No built-in dimensional analysis

EXAMPLE — Example comparison between AsciiUnits and UCUM

Unit	AsciiUnits	UCUM
Meter	`m`	`m`
Square meter	`m^2`	`m2`
Meter per second	`m/s`	`m/s`
Kilogram meter per second squared	`kg*m/s^2`	`kg.m/s2`
Millimole per liter	`mmol/L`	`mmol/l`

QUDT

QUDT (Quantities, Units, Dimensions and Types) is a semantic web framework for representing quantities, units, dimensions, and types in a consistent and interoperable manner based on RDF (Resource Description Framework) and OWL (Web Ontology Language).

QUDT is designed to support the semantic web and linked data applications, enabling interoperability between different systems and applications.

While AsciiUnits and QUDT share some similarities, they differ in their focus and implementation.

QUDT relies on RDF and OWL for its representation and does not provide a text-based syntax for representing units of measurement. AsciiUnits allows for accessible encoding of units in documents and text files, making it possible for authors to represent measurement quantities in a clear and concise manner.

AsciiUnits being a text-based syntax, is allows for conversion to and from UnitsML XML, which is a more structured and machine-readable format. This makes AsciiUnits suitable for applications where human readability and ease of typing are important, while QUDT is more focused on semantic web applications.

Digital SI framework

The Digital SI framework is an initiative to modernize the International System of Units (SI) by providing a digital representation of SI units and their relationships. It aims to enhance the accessibility and usability of SI units in digital applications, including scientific computing, data analysis, and machine learning.

The Digital SI framework provides a comprehensive set of tools and resources for representing SI units, including a digital representation of SI base and derived units, as well as a set of guidelines for using SI units in digital applications.

While AsciiUnits is designed to be a text-based syntax for representing units of measurement, the Digital SI framework focuses on providing a digital representation of SI units and their relationships. AsciiUnits can be used in conjunction with the Digital SI framework to provide a human-readable representation of SI units and their relationships, making it easier for users to work with SI units in digital applications.

Other technologies

AsciiUnits is designed to be compatible with and interoperable with other technologies for representing units of measurement, including UCUM, QUDT, and the Digital SI framework. It can be used in conjunction with these technologies to provide a comprehensive solution for representing units of measurement in scientific and engineering applications.

AsciiUnits syntax specification

1. Scope

AsciiUnits syntax is designed to be simple and intuitive, allowing users to express complex unit expressions using a straightforward notation.

2. Normative references

The following documents are referred to in the text in such a way that some or all of their content constitutes requirements of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.

UnitsML XML schema, https://www.unitsml.org

W3C TR mathml-units, https://www.w3.org/TR/mathml-units/

3. Terms and definitions

For the purposes of this document, the following terms and definitions apply.

3.1. unit

standard quantity used to express a physical quantity

3.2. basic unit

unit not derived from other units

3.3. composite unit

unit derived from two or more basic units conjoined by multiplication or division

3.4. prefix

modifier to a unit with a fixed factor

3.5. quantity

type of measurement that is represented by a unit

EXAMPLE

distance, moment of force, temperature, energy.

3.6. dimension

basic physical quantity that can be expressed in terms of fundamental quantities

EXAMPLE

length, mass, time, plane angle.

3.7. unit name

name of a unit

3.8. unit symbol

abbreviated representation of a unit

3.9. unit reference

reference identifier to a unit in a database

3.10. unit dimensions

fundamental quantities to which the unit applies

4. Principles

AsciiUnits follows the following key principles to ensure clarity and usability in representing units.

Typability: Symbols are meant to be easily typed on standard keyboards, avoiding complex characters or symbols.
Non-ambiguous: Unit representation and composure syntax should lead to exactly one interpretation, ensuring that each expression has an unambiguous meaning.
Extensible: New units and new prefixes can be added to the system without changing the syntax.

5. Representable elements

5.1. General

AsciiUnits allows for the encoding of the following types of elements:

Unit systems
Units
Dimensions
Quantities
Prefixes in conjunction with units

5.2. Unit systems

A unit system is a collection of units of measure that forms a coherent framework for converting between different units of measurement.

UnitsML defines the following unit systems.

Table 3 — UnitsML unit systems
ID	Name	Acceptable to be used in the SI system
`SI_base`	SI	true
`SI_derived_special`	SI-derived special	true
`SI_derived_non-special`	SI-derived non-special	true
`non-SI_acceptable`	non-SI but acceptable to be used together	true
`non-SI_not_acceptable`	non-SI and unacceptable to be used together	false

Each unit in UnitsML belongs to exactly one unit system.

5.3. Units

The AsciiUnits syntax is designed to represent a wide range of units of measurement and their combinations.

It includes:

Basic units: Simple units and prefixed units
Composite units: Units formed by combining basic units through mathematical operations

Basic units available to AsciiUnits are sourced from UnitsDB.

UnitsDB is a comprehensive database of units of measurement and their relationships, which is used to define the units and prefixes available in AsciiUnits for various applications.

NOTE UnitsDB is a part of the UnitsML project and is maintained by CalConnect TC UNITS.

5.4. Prefixes

A prefix is a modifier to a unit that indicates a fixed factor by which the unit is multiplied.

Prefixes can be used in conjunction with units to create prefixed units.

The BIPM SI prefixes are used in AsciiUnits, which include the following:

Table 4 — SI prefixes in base 10
Prefix	Symbol	Factor	Representations
Prefix	Symbol	Factor	ASCII	MathML	HTML	LaTeX	Unicode
quetta	`<mi>Q</mi>`	Q	10^30	`Q`	`Q`	`Q`	`Q`
ronna	`<mi>R</mi>`	R	10^27	`R`	`R`	`R`	`R`
yotta	`<mi>Y</mi>`	Y	10^24	`Y`	`Y`	`Y`	`Y`
zetta	`<mi>Z</mi>`	Z	10^21	`Z`	`Z`	`Z`	`Z`
exa	`<mi>E</mi>`	E	10^18	`E`	`E`	`E`	`E`
peta	`<mi>P</mi>`	P	10^15	`P`	`P`	`P`	`P`
tera	`<mi>T</mi>`	T	10^12	`T`	`T`	`T`	`T`
giga	`<mi>G</mi>`	G	10^9	`G`	`G`	`G`	`G`
mega	`<mi>M</mi>`	M	10^6	`M`	`M`	`M`	`M`
kilo	`<mi>k</mi>`	k	10^3	`k`	`k`	`k`	`k`
hecto	`<mi>h</mi>`	h	10^2	`h`	`h`	`h`	`h`
deka	`<mi>da</mi>`	da	10^1	`da`	`da`	`da`	`da`
deci	`<mi>d</mi>`	d	10^-1	`d`	`d`	`d`	`d`
centi	`<mi>c</mi>`	c	10^-2	`c`	`c`	`c`	`c`
milli	`<mi>m</mi>`	m	10^-3	`m`	`m`	`m`	`m`
micro	`<mi>u</mi>`	u	10^-6	`u`	`µ`	$mu$	`μ`
nano	`<mi>n</mi>`	n	10^-9	`n`	`n`	`n`	`n`
pico	`<mi>p</mi>`	p	10^-12	`p`	`p`	`p`	`p`
femto	`<mi>f</mi>`	f	10^-15	`f`	`f`	`f`	`f`
atto	`<mi>a</mi>`	a	10^-18	`a`	`a`	`a`	`a`
zepto	`<mi>z</mi>`	z	10^-21	`z`	`z`	`z`	`z`
yocto	`<mi>y</mi>`	y	10^-24	`y`	`y`	`y`	`y`
ronto	`<mi>r</mi>`	r	10^-27	`r`	`r`	`r`	`r`
quecto	`<mi>q</mi>`	q	10^-30	`q`	`q`	`q`	`q`

Table 5 — SI prefixes in binary
Prefix	Symbol	Factor	Representations
Prefix	Symbol	Factor	ASCII	MathML	HTML	LaTeX	Unicode
kibi	`<mi>Ki</mi>`	Ki	2^10	`Ki`	`Ki`	`Ki`	`Ki`
mebi	`<mi>Mi</mi>`	Mi	2^20	`Mi`	`Mi`	`Mi`	`Mi`
gibi	`<mi>Gi</mi>`	Gi	2^30	`Gi`	`Gi`	`Gi`	`Gi`
tebi	`<mi>Ti</mi>`	Ti	2^40	`Ti`	`Ti`	`Ti`	`Ti`
pebi	`<mi>Pi</mi>`	Pi	2^50	`Pi`	`Pi`	`Pi`	`Pi`
exbi	`<mi>Ei</mi>`	Ei	2^60	`Ei`	`Ei`	`Ei`	`Ei`
zebi	`<mi>Zi</mi>`	Zi	2^70	`Zi`	`Zi`	`Zi`	`Zi`
yobi	`<mi>Yi</mi>`	Yi	2^80	`Yi`	`Yi`	`Yi`	`Yi`

5.5. Dimensions

A dimension is a physical quantity that can be expressed in terms of fundamental quantities.

Basic dimensions are described with only one quantity. More complex dimensions require multiple quantities to be expressed.

Dimensions are represented in AsciiUnits using the dim_ prefix followed by the dimension name.

The collection of dimensions is defined in the UnitsDB database. Each dimension is represented by a unique identifier in the database, of the pattern NISTd{n}.

5.6. Quantities

A quantity is a type of measurement that is represented by a unit.

Each quantity is associated with one or more quantity names, exactly one dimension, and one or more units of measurement.

The collection of quantities is defined in the UnitsDB database. Each quantity is represented by a unique identifier in the database, of the pattern NISTq{n}.

6. Architecture

6.1. General

AsciiUnits is a text-based syntax for representing units of measurement.

There are two types of representable forms in AsciiUnits:

Units
Values of units

6.2. Statement

An AsciiUnits statement is a sequence of characters that represents a unit or a value of a unit.

EXAMPLE

kg # kilogram
um # micrometer (meter + prefix of micro)
degK # degree kelvin

6.3. Units

A unit statement represented in the AsciiUnits language is composed of up to 3 types of elements in the UnitsML model structure.

Unit, which represents the unit itself
Dimension, which represents the dimension of the unit
Quantity, which represents the quantity being described

6.4. Values of units

A value of a unit is a number that represents a measurement in that unit.

A value consists of a numerical component and a unit. For example, in “2.3 kg,” “2.3” is the value representing the quantity and “kg” is the unit.

EXAMPLE

1.5 m # 1.5 meters
2.3 kg # 2.3 kilograms
3.0 m/s # 3.0 meters per second

AsciiUnits 1.0 does not support encoding of values of units. This may be added in a future version.

7. AsciiUnits prefixes

AsciiUnits statements support the use of prefixes as standalone elements.

Prefixes elements are indicated with a prefix of - after the prefix code.

Syntax:

{prefix-code}-

Figure 1

Where,

{prefix-code}: is the prefix code
-: indicates that there is no unit associated with the prefix

EXAMPLE 1

AsciiUnits statement (for deci):

d-

XML:

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="-1" xml:id="NISTp10_-1">
  <PrefixName xml:lang="en">deci</PrefixName>
  <PrefixSymbol type="ASCII">d</PrefixSymbol>
  <PrefixSymbol type="unicode">d</PrefixSymbol>
  <PrefixSymbol type="LaTeX">d</PrefixSymbol>
  <PrefixSymbol type="HTML">d</PrefixSymbol>
</Prefix>

EXAMPLE 2

AsciiUnits statement (for deca):

da-

XML:

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="1" xml:id="NISTp10_1">
  <PrefixName xml:lang="en">deka</PrefixName>
  <PrefixSymbol type="ASCII">da</PrefixSymbol>
  <PrefixSymbol type="unicode">da</PrefixSymbol>
  <PrefixSymbol type="LaTeX">da</PrefixSymbol>
  <PrefixSymbol type="HTML">da</PrefixSymbol>
</Prefix>

8. AsciiUnits dimensions

8.1. General

AsciiUnits statements support the use of dimensions as standalone elements, that are not associated with any unit.

8.2. Basic dimensions

Dimensions elements are indicated with a prefix of dim_.

Syntax:

dim_{dimension-code}

Figure 2

Where,

{dimension-code}: is the dimension code

EXAMPLE 1

dim_L       # length dimension
dim_Theta   # temperature dimension
dim_T       # time dimension

EXAMPLE 2

AsciiUnits statement (for length):

dim_L       # length dimension

XML:

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

8.3. Dimensions with mathematical operations

Dimension elements can be expressed with mathematic operation adjustments.

The supported operations are:

* for addition
/ for subtraction
^ for power
sqrt() for square root

EXAMPLE 1

dim_L^2         # squared length dimension
dim_Theta^10    # temperature dimension to power 10
sqrt(dim_Theta) # square root of temperature dimension

EXAMPLE 2

AsciiUnits statement (for squared length):

dim_L^2

XML:

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2">
  <Length symbol="L" powerNumerator="2"/>
</Dimension>

EXAMPLE 3

AsciiUnits statement (for square root of temperature):

sqrt(dim_Theta)

XML:

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_Theta0.5">
  <ThermodynamicTemperature symbol="Theta" powerNumerator="0.5"/>
</Dimension>

8.4. Composed dimensions

Composed dimensions are formed by combining more than one basic dimensions through mathematical operations.

EXAMPLE 1

dim_Theta*dim_L^2

EXAMPLE 2

AsciiUnits statement (for temperature times squared length):

dim_Theta*dim_L^2

XML:

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2Theta">
  <Length symbol="L" powerNumerator="2"/>
  <ThermodynamicTemperature symbol="Theta" powerNumerator="1"/>
</Dimension>

9. AsciiUnits units

9.1. General

AsciiUnits supports a wide range of units, including basic units, prefixed units, and composite units.

9.2. Basic units

Basic units are the fundamental units of measurement that are not derived from other units.

These units have the following characteristics:

They are defined as individual entries in the UnitsDB database.
Their RootUnit element contains a single EnumeratedRootUnit element, which represents the unit itself, with the unit attribute set to the unit name, with powerNumerator and powerDenominator set to 1.

Syntax:

{unit-code}

Figure 3

Where,

{unit-code}: is the unit code

EXAMPLE 1 — Examples of basic units

m       # meter
s       # second
Hz      # hertz
K       # kelvin
rad     # radian

The XML representation of an AsciiUnits basic unit is demonstrated below.

Basic units will not have a <Prefix> element since they are not prefixed units.
The <Dimension> element is automatically generated based on the unit code.

EXAMPLE 2 — AsciiUnits representation for m

AsciiUnits statement:

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu1" dimensionURL="#NISTd1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter</UnitName>
  <UnitSymbol type="HTML">m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd1">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

EXAMPLE 3 — AsciiUnits representation for Hz

AsciiUnits statement:

Hz

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu31" dimensionURL="#NISTd24">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">hertz</UnitName>
  <UnitSymbol type="HTML">Hz</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">Hz</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd24">
  <Time symbol="T" powerNumerator="-1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq45" quantityType="base" dimensionURL="#NISTd24">
  <QuantityName xml:lang="en-US">frequency</QuantityName>
</Quantity>

9.3. Prefixed units

A prefixed unit is a unit that has a prefix attached to it, indicating a fixed factor by which the unit is multiplied.

Prefixes are defined in the UnitsDB database and are represented in the XML representation of the unit.

NOTE AsciiUnits only support prefixes defined by the BIPM in the SI Brochure, as described in Clause 5.4.

Prefixes are automatically parsed in AsciiUnits.

Syntax:

{prefix-code}{unit-code}

Figure 4

Where,

{prefix-code}: is the prefix code
{unit-code}: is the unit code

kg      # kilogram
mm      # millimeter
mW      # milliwatt

Figure 5

The XML representation of an AsciiUnits prefixed unit is demonstrated below.

Prefixed units will have a <Prefix> element since they are prefixed.
The <Dimension> element is automatically generated based on the unit and the prefix.

EXAMPLE — AsciiUnits representation for kg

AsciiUnits statement:

kg

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu2" dimensionURL="#NISTd2">
  <UnitSystem name="SI" type="SI_base" xml:lang="en-US"/>
  <UnitName xml:lang="en">kilogram</UnitName>
  <UnitSymbol type="HTML">kg</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">kg</mi>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="gram" prefix="k"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd2">
  <Mass symbol="M" powerNumerator="1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq2" quantityType="base" dimensionURL="#NISTd2">
  <QuantityName xml:lang="en-US">mass</QuantityName>
</Quantity>

9.4. Composite units

9.4.1. General

Composite units are units that are derived from two or more basic units conjoined by a mathematical operation.

Mathematical operations are represented in AsciiUnits using the following symbols:

* for multiplication
/ for division
^ for exponentiation
sqrt() for square root
() for grouping

The XML representation of an AsciiUnits composite unit:

Prefixed units inside the composite will have a <Prefix> element.
The <Dimension> element is automatically generated based on the final mathematical evaluation of the involved units and prefixes.

9.4.2. Multiplication

Units are multiplied using *

EXAMPLE 1

kg*m    # kilogram times meter

EXAMPLE 2

AsciiUnits statement (for ampere times coulomb):

A*C

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_A.C" dimensionURL="#D_M3I4">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">A*C</UnitName>
  <UnitSymbol type="HTML">A&#x22c5;C</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">A</mi>
      <mo>&#x22c5;</mo>
      <mi mathvariant="normal">C</mi>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="ampere"/>
    <EnumeratedRootUnit unit="coulomb" powerNumerator="1"/>
  </RootUnits>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_M3I4">
  <Mass symbol="M" powerNumerator="1"/>
  <ElectricCurrent symbol="I" powerNumerator="4"/>
</Dimension>

9.4.3. Division

Units are divided using /

EXAMPLE

AsciiUnits statement (for GHz per volt):

GHz//V

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_GHz.V-1" dimensionURL="#D_L-2M-1T2I">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">GHz*V^-1</UnitName>
  <UnitSymbol type="HTML">GHz&#x22c5;V<sup>&#x2212;1</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">GHz</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">V</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="hertz" prefix="G"/>
    <EnumeratedRootUnit unit="volt" powerNumerator="-1"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="9" xml:id="NISTp10_9">
  <PrefixName xml:lang="en">giga</PrefixName>
  <PrefixSymbol type="ASCII">G</PrefixSymbol>
  <PrefixSymbol type="unicode">G</PrefixSymbol>
  <PrefixSymbol type="LaTeX">G</PrefixSymbol>
  <PrefixSymbol type="HTML">G</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L-2M-1T2I">
  <Length symbol="L" powerNumerator="-2"/>
  <Mass symbol="M" powerNumerator="-1"/>
  <Time symbol="T" powerNumerator="2"/>
  <ElectricCurrent symbol="I" powerNumerator="1"/>
</Dimension>

9.4.4. Grouping

Parentheses define order of operations

K/(kg*m)    # kelvin divided by (kilogram times meter)
J/(kg*K)    # joule per (kilogram kelvin)

Figure 6

EXAMPLE

AsciiUnits statement (for joule per kilogram kelvin):

unitsml(m^(-2))

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_m-2" dimensionURL="#D_L-2">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">m^-2</UnitName>
  <UnitSymbol type="HTML">m<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">m</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L-2">
  <Length symbol="L" powerNumerator="-2"/>
</Dimension>

9.4.5. Exponents

Syntax:

{unit-code}^{exponent}

Figure 7

Where,

{unit-code}: is the unit code
{exponent}: is the exponent

EXAMPLE 1

m^2     # square meter

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_m2" dimensionURL="#D_L2">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">m^2</UnitName>
  <UnitSymbol type="HTML">m<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">m</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2">
  <Length symbol="L" powerNumerator="2"/>
</Dimension>

Negative exponents are allowed.

EXAMPLE 2

kg^-2     # per kilogram
kg^(-2)   # per kilogram with group

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_kg-2" dimensionURL="#D_M-2">
  <UnitSystem name="SI" type="SI_base" xml:lang="en-US"/>
  <UnitName xml:lang="en">kg^-2</UnitName>
  <UnitSymbol type="HTML">kg<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">kg</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="gram" prefix="k" powerNumerator="-2"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_M-2">
  <Mass symbol="M" powerNumerator="-2"/>
</Dimension>

9.4.6. Square root

Using sqrt() function.

EXAMPLE 1 — Statement with square root on a basic unit

AsciiUnits statement (for square root of hertz):

sqrt(Hz)

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_Hz0.5" dimensionURL="#D_T-0.5">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">Hz^0.5</UnitName>
  <UnitSymbol type="HTML">Hz<sup>0.5</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">Hz</mi>
        </mrow>
        <mrow>
          <mn>0.5</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_T-0.5">
  <Time symbol="T" powerNumerator="-0.5"/>
</Dimension>

EXAMPLE 2 — Statement with square root on a prefixed unit

AsciiUnits statement (for square root of millimeter):

sqrt(mm)

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_mm0.5" dimensionURL="#D_L0.5">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">mm^0.5</UnitName>
  <UnitSymbol type="HTML">mm<sup>0.5</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">mm</mi>
        </mrow>
        <mrow>
          <mn>0.5</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="meter" prefix="m" powerNumerator="0.5"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="-3" xml:id="NISTp10_-3">
  <PrefixName xml:lang="en">milli</PrefixName>
  <PrefixSymbol type="ASCII">m</PrefixSymbol>
  <PrefixSymbol type="unicode">m</PrefixSymbol>
  <PrefixSymbol type="LaTeX">m</PrefixSymbol>
  <PrefixSymbol type="HTML">m</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L0.5">
  <Length symbol="L" powerNumerator="0.5"/>
</Dimension>

9.4.7. Realized units

Realized composite units are formed by combining more than one basic unit through the above mentioned operations.

EXAMPLE

AsciiUnits statement (for joule per kilogram kelvin):

J/kg*K

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu13.u27p10_3e-1/1.u5e-1/1" dimensionURL="#D_L2M0T-2Theta-1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">joule per kilogram kelvin</UnitName>
  <UnitSymbol type="HTML">J&#x22c5;kg<sup>&#x2212;1</sup>&#x22c5;K<sup>&#x2212;1</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">J</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">kg</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">K</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="joule"/>
    <EnumeratedRootUnit unit="gram" prefix="k" powerNumerator="-1"/>
    <EnumeratedRootUnit unit="kelvin" powerNumerator="-1"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd40">
  <Length symbol="L" powerNumerator="2"/>
  <Time symbol="T" powerNumerator="-2"/>
  <ThermodynamicTemperature symbol="Theta" powerNumerator="-1"/>
</Dimension>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2M0T-2Theta-1">
  <Length symbol="L" powerNumerator="2"/>
  <Mass symbol="M" powerNumerator="0"/>
  <Time symbol="T" powerNumerator="-2"/>
  <ThermodynamicTemperature symbol="Theta" powerNumerator="-1"/>
</Dimension>

9.5. Symbol ambiguity resolution

Multiple units can be represented by the same symbol. This is often differentiated according to historical or contextual usage, and in some cases they cannot be disambiguated by the context of the unit.

EXAMPLE 1

m can represent meter or mole.
B can represent both bel (as in decibel) and byte.

These units that are associated with multiple symbols are considered to have:

a primary unit symbol
one or more alternative unit symbols.

To ensure unambiguous identification of units, AsciiUnits uses unique identifiers for these ambiguous symbols accross units.

When using a potentially ambiguous symbol, AsciiUnits will attempt to resolve the ambiguity based on context where possible.

A comprehensive table of ambiguous unit symbols and their disambiguated identifiers is provided in Appendix C.

In UnitsDB, there is a notion of a “prefixed unit”, which is a unit that has a prefix attached to it (marked by prefixed: true in the UnitsDB entry).

In case of ambiguity, the interpretation with no prefix is prioritised over the interpretation as a unit.

Exceptionally, kg is decomposed into “kilo-gram” (a unit) rather than treated as a basic unit, for consistency with other prefixes of grams.

EXAMPLE 2

ct is interpreted as hundredweight, rather than centi-ton.

Units with multiple symbols have those symbols encoded separately in the UnitsDB Units’ database.

NOTE While different symbols can be used to reference the same Unit in UnitsML, the default unit symbol will still apply during presentation of the unit unless explicitly specified using Clause 10.

10. Unit metadata

10.1. General

Metadata to a unit’s usage can be supplied after a comma after an AsciiUnits statement.

Syntax:

{unit-code}, {metadata}

Figure 8

Where,

{unit-code}: is the unit code
{metadata}: is the metadata

EXAMPLE

m, quantity: NISTq103               # meter with quantity reference
cal_th/cm^2, name: langley          # calorie thermal per square centimeter with name
cm*s^-2, symbol: cm cdot s^-2       # custom symbol representation
cm*s^-2, multiplier: xx             # custom multiplier

10.2. Quantity

10.2.1. General

A unit of measurement is used to describe a quantity, and one unit can be used to measure different quantities.

EXAMPLE

The unit “meter” (m) can be used to measure length, wavelength, radius, height, displacement.

For interoperability between systems and value conversions, it is important to specify the quantity being measured. This ensures that the unit is used correctly and consistently in scientific and engineering contexts.

10.2.2. Default quantity

An AsciiUnit unit is associated with a default quantity.

If a single quantity is associated with the unit in UnitsDB, that quantity is added automatically.
Otherwise, no quantity is added unless explicitly specified as in Clause 10.2.3.

EXAMPLE 1

The unit m (meter) is associated with the quantities length, diameter, length of path, and so on in the UnitsDB database, with the length quantity being listed first. Therefore, the default quantity for m is “length”.

The XML representation of an AsciiUnits unit always includes the default quantity association in the <Quantity> element unless a different quantity is specified.

EXAMPLE 2

AsciiUnits statement:

m # meter

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu1" dimensionURL="#NISTd1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter</UnitName>
  <UnitSymbol type="HTML">m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd1">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq1" quantityType="base" dimensionURL="#NISTd1">
  <QuantityName xml:lang="en-US">length</QuantityName>
</Quantity>

10.2.3. Specific quantity

AsciiUnits allows specifying the quantity a unit measures using the quantity metadata parameter, which provides the UnitsDB identifier for the quantity being measured.

Syntax:

{unit-code}, quantity: {quantity-code}

Figure 9

Where,

{unit-code}: is the unit code
{quantity-code}: is the reference to a quantity identifier or its shortname in the UnitsDB database

EXAMPLE 1

The AsciiUnits statement of s, quantity: NISTq109 indicates that the second is used to measure period duration.

EXAMPLE 2 — Examples of units with quantity references

kg, quantity: NISTq2                # kilogram with mass quantity reference
kg, quantity: mass                  # kilogram with mass quantity reference
s, quantity: NISTq3                 # second with time quantity reference
s, quantity: duration               # second with time quantity reference
m, quantity: NISTq103               # meter with length quantity reference
m, quantity: position_vector        # meter with length quantity reference
l, quantity: NISTq105               # liter with volume quantity reference
l, quantity: volume                 # liter with volume quantity reference

The XML representation of an AsciiUnits unit with quantity reference includes the specific <Quantity> element that provides information about the quantity being measured.

EXAMPLE 3 — AsciiUnits representation for kilogram with mass quantity reference

AsciiUnits statement:

kg [NISTq2]

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu1" dimensionURL="#NISTd1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter</UnitName>
  <UnitSymbol type="HTML">m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd1">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq2" quantityType="base" dimensionURL="#NISTd1">
  <QuantityName xml:lang="en-US">mass</QuantityName>
</Quantity>

EXAMPLE 4 — AsciiUnits representation for meter with position vector quantity reference

AsciiUnits statement:

m [NISTq103]

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu1" dimensionURL="#NISTd1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter</UnitName>
  <UnitSymbol type="HTML">m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd1">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq103" quantityType="base" dimensionURL="#NISTd1">
  <QuantityName xml:lang="en-US">position vector</QuantityName>
</Quantity>

10.2.4. Inferred quantity

In composite units, when a quantity is not explicitly specified, AsciiUnits will attempt to infer the appropriate quantity based on the unit’s dimension. This inference is based on the standard quantities defined in the UnitsDB database.

EXAMPLE 1

A unit with dimension L (length) will be detected to be associated with the quantity “length”.

A unit with dimension L*T^-1 will be detected to be associated with the quantity “velocity”.

EXAMPLE 2 — Inferred quantity based on UnitsDB lookup

AsciiUnits statement:

m/s

Given that the unit m/s is a composite unit, AsciiUnits can detect the dimension L*T^-1 (NISTd11), which allows an inference to the quantity “velocity” (NISTq12).

XML (with inferred quantity):

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTu1.u3e-1/1" dimensionURL="#NISTd11">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter per second</UnitName>
  <UnitSymbol type="HTML">m&#x22c5;s<sup>&#x2212;1</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">s</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="meter"/>
    <EnumeratedRootUnit unit="second" powerNumerator="-1"/>
  </RootUnits>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd11">
  <Length symbol="L" powerNumerator="1"/>
  <Time symbol="T" powerNumerator="-1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq12" quantityType="derived" dimensionURL="#NISTd11">
  <QuantityName xml:lang="en-US">velocity</QuantityName>
  <QuantityName xml:lang="en-US">speed</QuantityName>
</Quantity>

Composite units that lead to a detected of dimension associated with multiple quantities will always infer the first quantity in the UnitsDB database.

NOTE Composite units that lead to a dimension that is not associated with any quantity in the UnitsDB database will not be inferred.

EXAMPLE 3 — Inferred quantity using composite units with detected dimension and quantity from UnitsDB

AsciiUnits statement:

cd*sr*m^(-2)

Given that the unit cd*sr*m^(-2) is a composite unit, AsciiUnits can detect the dimension L^-2*J (NISTd27), which allows an inference to the quantity “luminous flux” (NISTq47) or “luminance” (NISTq56), and the first detected quantity is always inferred since the exact quantity type is not specified.

XML (with inferred quantity):

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_cd.sr.m-2" dimensionURL="#NISTd27">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">cd*sr*m^-2</UnitName>
  <UnitSymbol type="HTML">cd&#x22c5;sr&#x22c5;m<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">cd</mi>
      <mo>&#x22c5;</mo>
      <mi mathvariant="normal">sr</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">m</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="candela"/>
    <EnumeratedRootUnit unit="steradian"/>
    <EnumeratedRootUnit unit="meter" powerNumerator="-2"/>
  </RootUnits>
</Unit>

<!--
  The dimension NISTd27 is detected given the length and luminous intensity
  power numerators of L = -2 and J = 1.
-->
<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd27">
  <Length symbol="L" powerNumerator="-2"/>
  <LuminousIntensity symbol="J" powerNumerator="1"/>
</Dimension>

<!--
  The dimension NISTd27 is used in two UnitsDB quantities: NISTq47 (luminous
  flux) and NISTq56 (luminance), detection always infers the first.
-->
<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq47" quantityType="derived" dimensionURL="#NISTd27">
  <QuantityName xml:lang="en-US">luminous flux</QuantityName>
</Quantity>

10.3. Symbol representation

10.3.1. General

The symbol representation of a unit can be customized using the symbol metadata parameter. This allows for the specification of a custom symbol representation for the unit.

10.3.2. Default symbol

The default symbol representation is the first symbol entry as associated to the unit in the UnitsDB database.

EXAMPLE

The unit s (NISTu3, second) is associated with the symbols s, "_s, and " in the context of time measurement.

Therefore, the default symbol for s is “s”.

10.3.3. Custom symbol

The custom symbol representation is specified using the symbol metadata parameter.

This allows for the specification of a custom symbol representation for the unit in AsciiUnits format.

A custom symbol can be assigned to an AsciiUnits unit whether it is a unit of measure or a composite unit.

Syntax:

{unit-code}, symbol: {custom_symbol}

Figure 10

Where,

{unit-code}: is the unit code
{custom_symbol}: is the custom symbol representation in AsciiUnits format

NOTE While the unit-code gives the canonical representation of the unit, custom_symbol is what will be rendered.

EXAMPLE 1

unitsml(cal_th/cm^2, name: langley, symbol: La) assigns the custom symbol La to the unit cal_th/cm^2 (calorie thermal per square centimeter).

unitsml(mm*s^-2, symbol: mm cdot s^-2) assigns the custom symbol mm cdot s^{-2` to the unit `mm*s}-2 (millimeter per second squared).

EXAMPLE 2

AsciiUnits statement (for candela steradian per square meter), with a custom symbol representation:

cd*sr*m^(-2),symbol:cd cdot sr cdot m^(-2)

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_cd.sr.m-2" dimensionURL="#D_L-2J">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">cd*sr*m^-2</UnitName>
  <UnitSymbol type="HTML">cd&#x22c5;sr&#x22c5;m<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">cd</mi>
      <mo>&#x22c5;</mo>
      <mi mathvariant="normal">sr</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">m</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="candela"/>
    <EnumeratedRootUnit unit="steradian"/>
    <EnumeratedRootUnit unit="meter" powerNumerator="-2"/>
  </RootUnits>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd27">
  <Length symbol="L" powerNumerator="-2"/>
  <LuminousIntensity symbol="J" powerNumerator="1"/>
</Dimension>

10.4. Name

10.4.1. General

The name of a unit can be customized using the name metadata parameter. This allows for the specification of a custom name for the unit.

10.4.2. Default name

The default name is the first name entry as associated to the unit in the UnitsDB database.

EXAMPLE

The unit metric_ton (NISTu88, metric ton) is associated with the unit names “metric ton” and “ton”. The default name for metric_ton is the first entry, “metric ton”.

10.4.3. Custom name

The custom name representation is specified using the name metadata parameter, allowing for the specification of a custom name representation for the unit.

This is especially relevant if the unit is a composite unit, as the custom name representation can be used to provide a more meaningful description of the unit, and when its name is not already available from UnitsDB.

Syntax:

{unit-code}, name: {custom_name}

Figure 11

Where,

{unit-code}: is the unit code
{custom_name}: is the custom name representation in AsciiUnits format
name: NAME: is an optional name for the unit, if one is not already available from UnitsDB.

EXAMPLE 1

unitsml(cal_th/cm^2, name: langley) assigns the custom name “langley” to the unit cal_th/cm^2 (calorie thermal per square centimeter).

EXAMPLE 2

The AsciiUnits statement (for the “metric ton” unit with a custom name “ton”), with a custom name representation:

metric_ton, name: ton

XML:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTu88" dimensionURL="#NISTd2">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">ton</UnitName>
  <UnitSymbol type="HTML">t</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant='normal'>t</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd2">
  <Mass symbol="M" powerNumerator="1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq2" quantityType="base" dimensionURL="#NISTd2">
  <QuantityName xml:lang="en-US">mass</QuantityName>
</Quantity>

10.5. Multiplier

10.5.1. General

A multiplier is the conjoining symbol that is used to combine multiple units in a composite unit.

It is typically represented as a multiplication sign (×) or a dot (·). The choice of symbol can depend on the context in which the unit is used and the conventions of the field of study.

10.5.2. Default multiplier

The default AsciiUnits multiplier adopts the specified default from MathML Units, the middle dot (&#xB7) symbol.

This default multiplier symbol is rendered in various formats as:

ASCII: a single empty space
Unicode: the invisible times symbol (⁢)
MathML: middle dot (&#xB7)
LaTeX: a middle dot (\cdot)
HTML: a non-breaking space

10.5.3. Custom multiplier

The multiplier of a unit can be specified using the multiplier metadata parameter.

Syntax:

{unit-code}, multiplier: {custom_multiplier}

Figure 12

Where,

{unit-code}: is the unit code
{custom_multiplier}: is the custom multiplier representation in AsciiUnits format. It accepts the following values:
:default: is the default multiplier.
:space: a spacing invisible times.
:nospace: a non-spacing invisible times
custom_multiplier: is an arbitrary UTF-8 string that can be supplied instead; it will be encoded as XML entities.

The :space value is rendered as:

ASCII: a single empty space
Unicode: the invisible times symbol (⁢)
MathML: spacing invisible times (<mo rspace='thickmathspace'>⁢</mo>)
LaTeX: a non-breaking space (~)
HTML:   (non-breaking space)

The :nospace value is rendered as:

ASCII: no space
Unicode: the invisible times symbol (⁢)
MathML: non-spacing invisible times (<mo>⁢</mo>)
LaTeX: no space
HTML: no space

If the value is not one of the above, it will be treated as a custom multiplier, where an arbitrary UTF-8 string can be supplied. It will be rendered as:

ASCII: the UTF-8 string
Unicode: the UTF-8 string
MathML: the UTF-8 string with XML entities
LaTeX: the UTF-8 string
HTML: the UTF-8 string

EXAMPLE

cm*s^-2, multiplier: xx             # custom multiplier

The xx character is the AsciiMath explicit multiplication sign (×). This statement will render the unit as cm xx s^-2 in AsciiUnits format..

11. Using AsciiUnits in AsciiMath

11.1. General

AsciiUnits can be embedded within AsciiMath using the unitsml() expression. This provides a way to incorporate UnitsML markup within mathematical content.

The Plurimath library converts AsciiMath containing AsciiUnits into MathML that incorporates UnitsML expressions complying with https://www.w3.org/TR/mathml-units/, with unique identifiers for each distinct unit, prefix, and dimension.

Units expressions are identified in MathML as <mtext>unitsml(...)</mtext>, which in turn can be identified in AsciiMath as "unitsml(…)".

11.2. Notation

The unitsml(...) expression consists of a unit string following the AsciiUnits syntax.

Syntax:

unitsml(unit-string{, unit-metadata})

Figure 13

Where,

unit-string: is the unit string, which can be a single unit or a combination of units, prefixes, and dimensions.
unit-metadata: is an optional comma-separated list of unit metadata parameters, described in Clause 10.

EXAMPLE

The AsciiMath with AsciiUnits expression for “millimeters per second squared” is represented as:

"unitsml(mm*s^-2)"

The "…" is an AsciiMath expression for string, which ensures backwards compatibility with AsciiMath parsers that are not UnitsML aware.

The unitsml() expression can take additional parameters as defined in Clause 10.

11.3. Representation in UnitsML in MathML

AsciiUnits in AsciiMath can be converted into a combination of UnitsML in MathML.

MathML unit expressions are expressed as <mi> (in accordance with W3C TR mathml-units) that cross-references to UnitsML definitions embedded in the XML that contains MathML.

The following MathML Units convention are adopted:

inserting a spacing invisible times operator (<mo rspace='thickmathspace'>⁢</mo>) between any numbers (<mn>) and unit expressions in MathML;
representing units in MathML as non-italic variables (<mi mathvariant='normal'>).

No whitespace is inserted between a number and a unit expression, when that unit expression wholly consists of punctuation.

EXAMPLE 1

1 m renders as 1 m.

1 deg_C renders as 1 °C.

However,

9 arc_degree 7 arc_minute 22 arc_second renders as 9° 7′ 22″.

EXAMPLE 2

The AsciiUnits in AsciiMath expression:

9 "unitsml(C^3*A)"

is converted into:

<math xmlns='http://www.w3.org/1998/Math/MathML'>
  <mrow>
    <mn>9</mn>
    <mo rspace='thickmathspace'>&#x2062;</mo>
    <mrow xref='U_C3.A'>
      <msup>
        <mrow>
          <mi mathvariant='normal'>C</mi>
        </mrow>
        <mrow>
          <mn>3</mn>
        </mrow>
      </msup>
      <mo>&#xB7;</mo>
      <mi mathvariant='normal'>A</mi>
    </mrow>

    <Unit xmlns='http://unitsml.nist.gov/2005' xml:id='U_C3.A' dimensionURL='#D_T3I4'>
      <UnitSystem name='SI' type='SI_derived' xml:lang='en-US'/>
      <UnitName xml:lang='en'>C^3*A</UnitName>
      <UnitSymbol type='HTML'>C<sup>3</sup> &#xB7; A</UnitSymbol>
      <UnitSymbol type='MathML'>
        <math xmlns='http://www.w3.org/1998/Math/MathML'>
          <mrow>
            <msup>
              <mrow>
                <mi mathvariant='normal'>C</mi>
              </mrow>
              <mrow>
                <mn>3</mn>
              </mrow>
            </msup>
            <mo>&#xB7;</mo>
            <mi mathvariant='normal'>A</mi>
          </mrow>
        </math>
      </UnitSymbol>
      <RootUnits>
        <EnumeratedRootUnit unit='coulomb' powerNumerator='3'/>
        <EnumeratedRootUnit unit='ampere'/>
      </RootUnits>
    </Unit>
    <Dimension xmlns='http://unitsml.nist.gov/2005' xml:id='D_T3I4'>
      <Time symbol='T' powerNumerator='3'/>
      <ElectricCurrent symbol='I' powerNumerator='4'/>
    </Dimension>

  </mrow>
</math>

11.4. Representing alternative unit symbols

As described in Clause 9.5, a unit can be associated with alternative unit symbols, and can be referenced by any of those symbols.

When using AsciiUnits in AsciiMath, the unit symbol referenced in the unitsml(...) expression supports the use of alternative unit symbols.

It is crucial to note that if an alternative unit symbol used in the unitsml(...) expression, the unit symbol in the MathML representation will still be the canonical unit symbol unless explicitly specified otherwise using the syntax at Clause 10.3.

EXAMPLE

An expression like 1 unitsml(mL) will be recognised as referring to the “microlitre” unit.

While the expression will be given under its canonical rendering of ml in UnitsML markup, the MathML rendering referencing that UnitsML expression will keep the notation mL.

Appendix A
(normative)
EBNF grammar

A.1. General

This section provides a formal grammar for AsciiUnits syntax using Extended Backus-Naur Form (EBNF).

The grammar is derived from the Ruby implementation of AsciiUnits and the test suite at unitsml-ruby/spec/unitsml/conv/xml_spec.rb.

A.2. EBNF notation

The EBNF notation used in this document follows these conventions:

::= defines a production rule
| represents alternatives
[ ... ] represents optional elements
{ ... } represents elements that can be repeated zero or more times
( ... ) groups elements
" ... " and ' ... ' represent literal strings
/* ... */ represents comments

A.3. Grammar definition

A.3.1. Top-level structure

/* Top-level AsciiUnits expression */
AsciiUnits ::= UnitExpression [ "," Metadata ]

/* Unit expression can be a basic unit, prefixed unit, composite unit, dimension expression, or prefix expression */
UnitExpression ::= BasicUnit | PrefixedUnit | CompositeUnit | DimensionExpression | PrefixExpression

Figure A.1

A.3.2. Basic units

/* Basic unit is a unit code from the UnitsDB database */
BasicUnit ::= UnitCode

/* Unit code is a sequence of letters and digits */
UnitCode ::= Letter { Letter | Digit }

/* Letter is any uppercase or lowercase letter */
Letter ::= "A" | "B" | "C" | ... | "Z" | "a" | "b" | "c" | ... | "z"

/* Digit is any decimal digit */
Digit ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

Figure A.2

A.3.3. Prefixed units

/* Prefixed unit is a prefix code followed by a unit code */
PrefixedUnit ::= PrefixCode UnitCode

/* Prefix code is a sequence of letters */
PrefixCode ::= Letter { Letter }

/* Prefix expression is a prefix code followed by a hyphen */
PrefixExpression ::= PrefixCode "-"

Figure A.3

A.3.4. Composite units

/* Composite unit is a combination of units with mathematical operations */
CompositeUnit ::= MultiplicationExpression | DivisionExpression | ExponentExpression | SqrtExpression | GroupedExpression

/* Multiplication expression combines units with the * operator */
MultiplicationExpression ::= UnitTerm "*" UnitExpression

/* Division expression combines units with the / or // operator */
DivisionExpression ::= UnitTerm "/" UnitExpression | UnitTerm "//" UnitExpression

/* Exponentiation expression raises a unit to a power */
ExponentExpression ::= UnitTerm "^" Power

/* Square root expression takes the square root of a unit */
SqrtExpression ::= "sqrt(" UnitExpression ")"

/* Grouped expression is a unit expression in parentheses */
GroupedExpression ::= "(" UnitExpression ")"

/* Unit term is a basic unit, prefixed unit, or grouped expression */
UnitTerm ::= BasicUnit | PrefixedUnit | GroupedExpression | SqrtExpression

/* Power can be a positive or negative number, or a grouped expression */
Power ::= Integer | "-" Integer | "(" Integer ")" | "(" "-" Integer ")"

/* Integer is a sequence of digits */
Integer ::= Digit { Digit }

Figure A.4

A.3.5. Dimension expressions

/* Dimension expression represents a physical dimension */
DimensionExpression ::= DimensionTerm | DimensionMultiplication | DimensionExponentiation

/* Dimension term is a basic dimension */
DimensionTerm ::= "dim_" DimensionSymbol

/* Dimension symbol is a dimension identifier */
DimensionSymbol ::= "L" | "M" | "T" | "I" | "Theta" | "N" | "J"

/* Dimension multiplication combines dimensions with the * operator */
DimensionMultiplication ::= DimensionTerm "*" DimensionExpression

/* Dimension exponentiation raises a dimension to a power */
DimensionExponentiation ::= DimensionTerm "^" Integer

Figure A.5

A.3.6. Metadata

/* Metadata provides additional information about the unit */
Metadata ::= MetadataItem { "," MetadataItem }

/* Metadata item is a key-value pair */
MetadataItem ::= MetadataKey ":" MetadataValue

/* Metadata key is a sequence of letters */
MetadataKey ::= Letter { Letter | "_" }

/* Metadata value is a sequence of characters */
MetadataValue ::= Character { Character }

/* Character is any printable character */
Character ::= Letter | Digit | Symbol

/* Symbol is any printable symbol */
Symbol ::= "+" | "-" | "*" | "/" | "^" | "(" | ")" | "." | "," | "_" | " " | ...

Figure A.6

A.4. Examples with grammar breakdown

A.4.1. Basic unit

Figure A.7

Grammar breakdown: * AsciiUnits → UnitExpression → BasicUnit → UnitCode → "m"

A.4.2. Prefixed unit

kg

Figure A.8

Grammar breakdown: * AsciiUnits → UnitExpression → PrefixedUnit → PrefixCode UnitCode → "k" "g"

A.4.3. Composite unit with multiplication

kg*m

Figure A.9

Grammar breakdown: * AsciiUnits → UnitExpression → CompositeUnit → MultiplicationExpression → UnitTerm "*" UnitExpression → PrefixedUnit "*" BasicUnit → "kg" "*" "m"

A.4.4. Composite unit with division

m/s

Figure A.10

Grammar breakdown: * AsciiUnits → UnitExpression → CompositeUnit → DivisionExpression → UnitTerm "/" UnitExpression → BasicUnit "/" BasicUnit → "m" "/" "s"

A.4.5. Composite unit with exponentiation

m^2

Figure A.11

Grammar breakdown: * AsciiUnits → UnitExpression → CompositeUnit → ExponentExpression → UnitTerm "^" Power → BasicUnit "^" Integer → "m" "^" "2"

A.4.6. Composite unit with square root

sqrt(Hz)

Figure A.12

Grammar breakdown: * AsciiUnits → UnitExpression → CompositeUnit → SqrtExpression → "sqrt(" UnitExpression ")" → "sqrt(" BasicUnit ")" → "sqrt(" "Hz" ")"

A.4.7. Composite unit with grouping

m^(-2)

Figure A.13

Grammar breakdown: * AsciiUnits → UnitExpression → CompositeUnit → ExponentExpression → UnitTerm "^" Power → BasicUnit "^" "(" "-" Integer ")" → "m" "^" "(" "-" "2" ")"

A.4.8. Unit with metadata

cd*sr*m^(-2),symbol:cd cdot sr cdot m^(-2)

Figure A.14

Grammar breakdown: * AsciiUnits → UnitExpression "," Metadata → CompositeUnit "," MetadataItem → MultiplicationExpression "," MetadataKey ":" MetadataValue → UnitTerm "*" MultiplicationExpression "," "symbol" ":" "cd cdot sr cdot m^(-2)"

A.5. Railroad diagrams

Railroad diagrams provide a visual representation of the grammar rules. Below are railroad diagrams for the main components of the AsciiUnits grammar:

@startuml
skinparam monochrome true
skinparam shadowing false
skinparam defaultFontName "Arial"
skinparam defaultFontSize 14

start
:UnitExpression;
if (Metadata?) then (yes)
  :,;
  :Metadata;
endif
stop
@enduml

Figure A.15 — AsciiUnits Expression

@startuml
skinparam monochrome true
skinparam shadowing false
skinparam defaultFontName "Arial"
skinparam defaultFontSize 14

start
split
  :BasicUnit;
split again
  :PrefixedUnit;
split again
  :CompositeUnit;
split again
  :DimensionExpression;
split again
  :PrefixExpression;
end split
stop
@enduml

Figure A.16 — Unit Expression

@startuml
skinparam monochrome true
skinparam shadowing false
skinparam defaultFontName "Arial"
skinparam defaultFontSize 14

start
split
  :MultiplicationExpression;
split again
  :DivisionExpression;
split again
  :ExponentExpression;
split again
  :SqrtExpression;
split again
  :GroupedExpression;
end split
stop
@enduml

Figure A.17 — Composite Unit

A.6. Grammar validation

The EBNF grammar defined in this section has been validated against the test suite at unitsml-ruby/spec/unitsml/conv/xml_spec.rb. All test cases in the suite conform to the grammar rules.

The grammar provides a formal specification of the AsciiUnits syntax, which can be used to:

Implement parsers and generators for AsciiUnits
Validate AsciiUnits expressions
Document the AsciiUnits syntax in a precise and unambiguous way
Ensure interoperability between different implementations of AsciiUnits

Appendix B
(informative)
Fixed symbols

B.1. General

Some symbols used in AsciiUnits are fundamental symbols that while exist in UnitsDB as individual entries, they are considered constant entries because of the fundamental nature of their values.

B.2. Dimension symbols

Dimensions are represented by the following symbols.

Table B.1
Dimension	UnitsDB symbol	UnitsDB ID
Length	`dim_L`	`NISTd1`
Mass	`dim_M`	`NISTd2`
Time	`dim_T`	`NISTd3`
Electric current	`dim_I`	`NISTd4`
Thermodynamic temperature	`dim_Theta`	`NISTd5`
Amount of substance	`dim_N`	`NISTd6`
Luminous intensity	`dim_J`	`NISTd7`
Plane angle (dimensionless)	`dim_phi`	`NISTd9`

EXAMPLE

unitsml(dim_I) for the dimension of electric current, 𝖨.

B.3. Non-SI units who values in SI units that are experimentally determined

This table maps from NIST SP 811, Table 7.

Table B.2 — Non-SI Units accepted for use with the SI by the CIPM and NIST SP 811 Table 7
Name	UnitsDB symbol	UnitsDB name	UnitsDB ID
electronvolt	`eV`	electronvolt	`NISTu240`
astronomical unit	`ua`	astronomical unit	`NISTu183`
unified atomic mass unit	`u`	unified atomic mass unit	`NISTu241`
dalton	`Da`	dalton	`NISTu289`

B.4. Natural units and atomic units

Natural units and atomic units are represented by the following symbols.

This table maps from NIST SP 811, Table 8.

Table B.3
Quantity kind	UnitsDB symbol	Description	UnitsDB name	UnitsDB ID
speed	`c`	Speed of light in vacuum (n.u.)	natural unit of velocity	`NISTu242`
action	`h-bar`	Planck constant divided by 2π (n.u.)	natural unit of action	`NISTu243`
mass	`m_e`	electron rest mass (n.u. and a.u.)	natural unit of action	`NISTu244`
electric charge	`e`	elementary charge (a.u.)	atomic unit of charge	`NISTu246`
length	`a_0`	Bohr radius (a.u.)	atomic unit of length	`NISTu249`
energy	`e_h`	Hartree energy (a.u.)	atomic unit of energy	`NISTu250`
time	`h/E_h`	ratio of action to energy (a.u.)	natural unit of time	`NISTu248`

Appendix C
(informative)
Ambiguous unit symbols

The symbols used for units can be highly ambiguous. In order to guarantee accurate parsing, the symbols used to data enter units are unambiguous in UnitsDB’s Units database.

EXAMPLE

B is ambiguous between bel (as in decibel) and byte.

These units are kept unambiguous by using bel_B and byte_B to refer to them, although they will still both be rendered as B.

The following table lists ambiguous symbols and how they are disambiguated in the symbol IDs.

Table C.1 — Ambiguous symbols supported in UnitsDB
Symbol	Unit + ID	Unit + ID	Unit + ID	Unit + ID	Unit + ID
′	minute (minute of arc): `'`	foot: `'_ft`	minute: `'_min`	minute of arc: `prime`	foot: `prime_ft`
″	second (second of arc): `"`	second: `"_s`	inch: `"_in`	second of arc: `dprime`	second: `dprime_s`
″Hg	conventional inch of mercury: `"Hg`	conventional inch of mercury: `dprime_Hg`	inch of mercury (32 degF): `"Hg_32degF`	inch of mercury (60 degF): `"Hg_60degF`	inch of mercury (32 degF): `dprime_Hg_32degF`
hp	horsepower: `hp`	horsepower (UK): `hp_UK`	horsepower, water: `hp_water`	horsepower, metric: `hp_metric`	horsepower, boiler: `hp_boiler`
Btu	British thermal unit_IT: `Btu`	British thermal unit (mean): `Btu_mean`	British thermal unit (39 degF): `Btu_39degF`	British thermal unit (59 degF): `Btu_59degF`	British thermal unit (60 degF): `Btu_60degF`
a	are: `a`	year (365 days): `a_year`	year, tropical: `a_tropical_year`	year, sidereal: `a_sidereal_year`
d	day: `d`	darcy: `darcy`	day, sidereal: `d_sidereal`
inHg	conventional inch of mercury: `inHg`	inch of mercury (32 degF): `inHg_32degF`	inch of mercury (60 degF): `inHg_60degF`
inH₂O	conventional inch of water: `inH_2O`	inch of water (39.2 degF): `inH_2O_39degF`	inch of water (60 degF): `inH_2O_60degF`
min	minute: `min`	minim: `minim`	minute, sidereal: `min_sidereal`
pc	parsec: `pc`	pica (printer’s): `pica_printer`	pica (computer): `pica_computer`
t	metric ton: `t`	long ton: `ton_long`	short ton: `ton_short`
B	bel: `bel_B`	byte: `byte_B`
cmHg	conventional centimeter of mercury: `cmHg`	centimeter of mercury (0 degC): `cmHg_0degC`
cmH₂O	conventional centimeter of water: `cmH_2O`	centimeter of water (4 degC): `cmH_2O_4degC`
cup	cup (US): `cup`	cup (FDA): `cup_label`
D	debye: `D`	darcy: `Darcy`
ft	foot: `ft`	foot (based on US survey foot): `ft_US_survey`
ftH₂O	conventional foot of water: `ftH_2O`	foot of water (39.2 degF): `ftH_2O_39degF`
gi	gill (US): `gi`	gill [Canadian and UK (Imperial)]: `gi_imperial`
h	hour: `h`	hour, sidereal: `h_sidereal`
′Hg	conventional foot of mercury: `'Hg`	conventional foot of mercury: `prime_Hg`
ℏ	natural unit of action: `h-bar`	atomic unit of action: `h-bar_atomic`
m_e	natural unit of mass: `m_e`	atomic unit of mass: `m_e_atomic`
in	inch: `in`	inch (based on US survey foot): `in_US_survey`
K	kelvin: `K`	kayser: `kayser`
L	liter: `L`	lambert: `Lambert`

Appendix D
(normative)
Test suite

D.1. General

This test suite is used for:

Validating implementations of AsciiUnits parsers and generators
Ensuring compatibility between different implementations
Providing examples for users of AsciiUnits
Serving as a reference for the expected behavior of AsciiUnits

Implementers of AsciiUnits parsers and generators should ensure that their implementations can correctly process all the examples in this test suite.

This comprehensive test suite for AsciiUnits is based on the Ruby implementation’s test suite at unitsml-ruby/spec/unitsml/conv/xml_spec.rb.

The test suite includes a variety of AsciiUnits statements and their corresponding XML representations, serving as a reference for implementers and users of AsciiUnits.

D.2. Basic units

D.2.1. Meter (m)

AsciiUnits statement:

Figure D.1

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu1" dimensionURL="#NISTd1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter</UnitName>
  <UnitSymbol type="HTML">m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd1">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

Figure D.2

D.2.2. Hertz (hz)

AsciiUnits statement:

Hz

Figure D.3

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu31" dimensionURL="#NISTd24">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">hertz</UnitName>
  <UnitSymbol type="HTML">Hz</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">Hz</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd24">
  <Time symbol="T" powerNumerator="-1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq45" quantityType="base" dimensionURL="#NISTd24">
  <QuantityName xml:lang="en-US">frequency</QuantityName>
</Quantity>

Figure D.4

D.2.3. Radian (rad)

AsciiUnits statement:

rad

Figure D.5

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu9" dimensionURL="#D_L0">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">radian</UnitName>
  <UnitSymbol type="HTML">rad</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">rad</mi>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd9">
  <PlaneAngle symbol="phi" powerNumerator="1"/>
</Dimension>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L0">
  <Length symbol="L" powerNumerator="0"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq9" dimensionURL="#NISTd9" quantityType="base">
  <QuantityName xml:lang="en-US">plane angle</QuantityName>
  <QuantityName xml:lang="en-US">angle</QuantityName>
</Quantity>

Figure D.6

D.3. Prefixed units

D.3.1. Kilogram (kg)

AsciiUnits statement:

kg

Figure D.7

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu2" dimensionURL="#NISTd2">
  <UnitSystem name="SI" type="SI_base" xml:lang="en-US"/>
  <UnitName xml:lang="en">kilogram</UnitName>
  <UnitSymbol type="HTML">kg</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">kg</mi>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="gram" prefix="k"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd2">
  <Mass symbol="M" powerNumerator="1"/>
</Dimension>

<Quantity xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTq2" quantityType="base" dimensionURL="#NISTd2">
  <QuantityName xml:lang="en-US">mass</QuantityName>
</Quantity>

Figure D.8

D.3.2. Micrometer (um)

AsciiUnits statement:

um

Figure D.9

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_um" dimensionURL="#NISTd1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">um</UnitName>
  <UnitSymbol type="HTML">&micro;m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant='normal'>µm</mi>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="meter" prefix="u"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="-6" xml:id="NISTp10_-6">
  <PrefixName xml:lang="en">micro</PrefixName>
  <PrefixSymbol type="ASCII">u</PrefixSymbol>
  <PrefixSymbol type="unicode">μ</PrefixSymbol>
  <PrefixSymbol type="LaTeX">$mu$</PrefixSymbol>
  <PrefixSymbol type="HTML">&micro;</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd1">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

Figure D.10

D.3.3. Millibar (mbar)

AsciiUnits statement:

mbar

Figure D.11

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu362">
  <UnitSystem name="not_SI" type="not_SI" xml:lang="en-US"/>
  <UnitName xml:lang="en">millibar</UnitName>
  <UnitSymbol type="HTML">mbar</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">mbar</mi>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="bar" prefix="m"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="-3" xml:id="NISTp10_-3">
  <PrefixName xml:lang="en">milli</PrefixName>
  <PrefixSymbol type="ASCII">m</PrefixSymbol>
  <PrefixSymbol type="unicode">m</PrefixSymbol>
  <PrefixSymbol type="LaTeX">m</PrefixSymbol>
  <PrefixSymbol type="HTML">m</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd14">
  <Length symbol="L" powerNumerator="-1"/>
  <Mass symbol="M" powerNumerator="1"/>
  <Time symbol="T" powerNumerator="-2"/>
</Dimension>

Figure D.12

D.4. Composite units

D.4.1. Multiplication (kg*m)

AsciiUnits statement:

kg*m

Figure D.13

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_kg.m" dimensionURL="#D_LM">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">kg*m</UnitName>
  <UnitSymbol type="HTML">kg&#x22c5;m</UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">kg</mi>
      <mo>&#x22c5;</mo>
      <mi mathvariant="normal">m</mi>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="gram" prefix="k"/>
    <EnumeratedRootUnit unit="meter"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_LM">
  <Length symbol="L" powerNumerator="1"/>
  <Mass symbol="M" powerNumerator="1"/>
</Dimension>

Figure D.14

D.4.2. Division (m/s)

AsciiUnits statement:

m/s

Figure D.15

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_m.s-1" dimensionURL="#NISTd11">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">meter per second</UnitName>
  <UnitSymbol type="HTML">m&#x22c5;s<sup>&#x2212;1</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">m</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">s</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="meter"/>
    <EnumeratedRootUnit unit="second" powerNumerator="-1"/>
  </RootUnits>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="NISTd11">
  <Length symbol="L" powerNumerator="1"/>
  <Time symbol="T" powerNumerator="-1"/>
</Dimension>

Figure D.16

D.4.3. Exponentiation (m^2)

AsciiUnits statement:

m^2

Figure D.17

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_m2" dimensionURL="#D_L2">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">m^2</UnitName>
  <UnitSymbol type="HTML">m<sup>2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">m</mi>
        </mrow>
        <mrow>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2">
  <Length symbol="L" powerNumerator="2"/>
</Dimension>

Figure D.18

D.4.4. Negative exponentiation (kg^-2)

AsciiUnits statement:

kg^-2

Figure D.19

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_kg-2" dimensionURL="#D_M-2">
  <UnitSystem name="SI" type="SI_base" xml:lang="en-US"/>
  <UnitName xml:lang="en">kg^-2</UnitName>
  <UnitSymbol type="HTML">kg<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">kg</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="gram" prefix="k" powerNumerator="-2"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_M-2">
  <Mass symbol="M" powerNumerator="-2"/>
</Dimension>

Figure D.20

D.4.5. Square root (sqrt(hz))

AsciiUnits statement:

sqrt(Hz)

Figure D.21

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_Hz0.5" dimensionURL="#D_T-0.5">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">Hz^0.5</UnitName>
  <UnitSymbol type="HTML">Hz<sup>0.5</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <msup>
        <mrow>
          <mi mathvariant="normal">Hz</mi>
        </mrow>
        <mrow>
          <mn>0.5</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_T-0.5">
  <Time symbol="T" powerNumerator="-0.5"/>
</Dimension>

Figure D.22

D.4.6. Complex composite (j/kg*k)

AsciiUnits statement:

J/kg*K

Figure D.23

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_NISTu13.u27p10_3e-1/1.u5e-1/1" dimensionURL="#D_L2M0T-2Theta-1">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">joule per kilogram kelvin</UnitName>
  <UnitSymbol type="HTML">J&#x22c5;kg<sup>&#x2212;1</sup>&#x22c5;K<sup>&#x2212;1</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">J</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">kg</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">K</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>1</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="joule"/>
    <EnumeratedRootUnit unit="gram" prefix="k" powerNumerator="-1"/>
    <EnumeratedRootUnit unit="kelvin" powerNumerator="-1"/>
  </RootUnits>
</Unit>

<Prefix xmlns="https://schema.unitsml.org/unitsml/1.0" prefixBase="10" prefixPower="3" xml:id="NISTp10_3">
  <PrefixName xml:lang="en">kilo</PrefixName>
  <PrefixSymbol type="ASCII">k</PrefixSymbol>
  <PrefixSymbol type="unicode">k</PrefixSymbol>
  <PrefixSymbol type="LaTeX">k</PrefixSymbol>
  <PrefixSymbol type="HTML">k</PrefixSymbol>
</Prefix>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2M0T-2Theta-1">
  <Length symbol="L" powerNumerator="2"/>
  <Mass symbol="M" powerNumerator="0"/>
  <Time symbol="T" powerNumerator="-2"/>
  <ThermodynamicTemperature symbol="Theta" powerNumerator="-1"/>
</Dimension>

Figure D.24

D.5. Dimension expressions

D.5.1. Basic dimension (dim_l)

AsciiUnits statement:

dim_L

Figure D.25

XML representation:

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L">
  <Length symbol="L" powerNumerator="1"/>
</Dimension>

Figure D.26

D.5.2. Dimension multiplication (dim_theta*dim_l^2)

AsciiUnits statement:

dim_Theta*dim_L^2

Figure D.27

XML representation:

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L2Theta">
  <Length symbol="L" powerNumerator="2"/>
  <ThermodynamicTemperature symbol="Theta" powerNumerator="1"/>
</Dimension>

Figure D.28

D.6. Units with metadata

D.6.1. Unit with symbol metadata

AsciiUnits statement:

cd*sr*m^(-2),symbol:cd cdot sr cdot m^(-2)

Figure D.29

XML representation:

<Unit xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="U_cd.sr.m-2" dimensionURL="#D_L-2J">
  <UnitSystem name="SI" type="SI_derived" xml:lang="en-US"/>
  <UnitName xml:lang="en">cd*sr*m^-2</UnitName>
  <UnitSymbol type="HTML">cd&#x22c5;sr&#x22c5;m<sup>&#x2212;2</sup></UnitSymbol>
  <UnitSymbol type="MathMl">
    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
      <mi mathvariant="normal">cd</mi>
      <mo>&#x22c5;</mo>
      <mi mathvariant="normal">sr</mi>
      <mo>&#x22c5;</mo>
      <msup>
        <mrow>
          <mi mathvariant="normal">m</mi>
        </mrow>
        <mrow>
          <mo>&#x2212;</mo>
          <mn>2</mn>
        </mrow>
      </msup>
    </math>
  </UnitSymbol>
  <RootUnits>
    <EnumeratedRootUnit unit="candela"/>
    <EnumeratedRootUnit unit="steradian"/>
    <EnumeratedRootUnit unit="meter" powerNumerator="-2"/>
  </RootUnits>
</Unit>

<Dimension xmlns="https://schema.unitsml.org/unitsml/1.0" xml:id="D_L-2J">
  <Length symbol="L" powerNumerator="-2"/>
  <LuminousIntensity symbol="J" powerNumerator="1"/>
</Dimension>

Figure D.30

Bibliography

[1] unitsml/unitsml-ruby, — the Ruby implementation of AsciiUnits from which the grammar was extracted, and the reference parser validated against it

[2] unitsml/asciiunits, — the PARG grammar artifact and conformance corpus repository