UTR #25: Unicode and Mathematics

Unicode Technical Report #25

Unicode Support for Mathematics

Authors	Barbara Beeton (bnb@ams.org), Asmus Freytag (asmus@unicode.org), Murray Sargent III (murrays@microsoft.com)
Date	2003-08-31
This Version	http://www.unicode.org/unicode/reports/tr25/tr25-6.html
Previous Version	http://www.unicode.org/unicode/reports/tr25/tr25-5.html
Latest Version	http://www.unicode.org/unicode/reports/tr25
Revision	6

Summary

Starting with version 3.2, Unicode includes virtually all of the standard characters used in mathematics. This set supports a variety of math applications on computers, including document presentation languages like TeX, math markup languages like MathML and OpenMath, internal representations of mathematics in systems like Mathematica, Maple, and MathCAD, computer programs, and plain text. This technical report describes the Unicode mathematics character groups and gives some of their imputed default math properties.

Status

This document has been reviewed by Unicode members and other interested parties, and has been approved by the Unicode Technical Committee as a Unicode Technical Report. This is a stable document and may be used as reference material or cited as a normative reference by other specifications.

A Unicode Technical Report (UTR) contains informative material. Conformance to the Unicode Standard does not imply conformance to any UTR. Other specifications, however, are free to make normative references to a UTR.

Please submit corrigenda and other comments with the online reporting form [Feedback]. Related information that is useful in understanding this document is found in the References section. For the latest version of the Unicode Standard see [Unicode]. See [Reports] for a list of current Unicode Technical Reports. For more information about versions of the Unicode Standard, see [Versions].

Contents

1. Overview
   
2. Mathematical Character Repertoire
   2.1    Mathematical Alphanumeric Symbols Block
   2.2    Mathematical Alphabets
   2.3    Fonts Used for Mathematical Alphabets
   2.4    Locating Mathematical Characters
   2.5    Duplicated Characters
   2.6    Accented Characters
   2.7    Operators
   2.8    Superscripts and Subscripts
   2.9    Arrows
   2.10   Delimiters
   2.11   Geometrical Shapes
   2.12   Other Symbols
   2.13   Symbol Pieces
   2.14   Invisible Operators
   2.15   Fraction Slash
   2.16   Other Characters
   2.17   Negations
   2.18   Variation Selectors
   2.19   Novel Symbols not yet in Unicode
   
3. Mathematical Character Properties
   3.1    Classification by degree of Mathematical Usage
   • 3.1.1    Strongly Mathematical Characters
   • 3.1.2    Weakly Mathematical Characters
   • 3.1.3    Other
   3.2    Classification by Typographical Behavior
   • 3.2.1    Alphabetic
   • 3.2.2    Operators
   • 3.2.3    Large Operators
   • 3.2.4    Digits
   • 3.2.5    Delimiters
   • 3.2.6    Fences
   • 3.2.7    Combining Marks
4. Implementation Guidelines
   4.1    Use of Normalization with Mathematical Text
   4.2    Input of Mathematical and Other Unicode Characters
   4.3    Use of Math Characters in Computer Programs
   4.4    Recognizing Mathematical Expressions
   4.5    Examples of Mathematical Notation
5. Mathematical Character Classification

    References
    Acknowledgements
    Modifications
  

1 Overview

All of Science and technology uses formulas, equations and mathematical notation as part of the language of the subject. This report presents a discussion of the mathematics character repertoire of the Unicode Standard [Unicode] as used for mathematics, but it is intended that this discussion apply to mathematical notation in general. 

As described in the forthcoming Unicode Character Property Model [PropMod], each Unicode character has associated character properties. This report describes the properties relevant to the mathematics character repertoire, including a number of properties that are not yet part of the Unicode Standard, and details character classifications by usage and by typography. In addition, this report gives some implementation guidelines for input methods and use of Unicode math characters in programming languages.

Some of the text of the character block descriptions in the Unicode Standard was based on early drafts of this report; as a result there is significant overlap, although the focus of the presentation is different. As always, wherever there is a discrepancy, the text of the Standard has precedence.

The notational conventions follow the use in [Unicode]. Due to limitations of the plain HTML format of this report, examples of mathematical formulas are shown in larger size than would be typical for a mathematical paper, and their layout, spacing and vertical alignment are merely approximations of the correct appearance.

2 Mathematical Character Repertoire

The Unicode Standard provides a quite complete set of standard math characters to support  publication of mathematics on and off the web. In addition to the over three hundred math-specific symbols already encoded in Unicode 3.0, Unicode 3.1 introduced almost a thousand new alphanumeric symbols and Unicode 3.2 introduced six hundred new symbols,  for a total of around 2000 mathematical symbols, including the addition of an alphanumeric symbol and one other symbol in Unicode 4.0. This repertoire is the result of input from many sources, notably from the STIX Project (Scientific and Technical Information Exchange) [STIX], a collaborative project of scientific and technical publishers. The STIX collection includes, but is not limited to, symbols gleaned from mathematical publications by experts from the American Mathematical Society (AMS) and symbol sets provided by Elsevier Publishing and by the American Physical Society. The new repertoire enables the display of virtually all standard mathematical symbols. Nevertheless no collection of mathematical symbols can ever be considered complete; mathematicians and other scientists are continually inventing new mathematical symbols and the plan is to add them as they become widely accepted in the scientific communities.

Mathematical Markup Language (MathML™) [MathML], an XML application [XML], is a major beneficiary of the increased repertoire for mathematical symbols. The W3C Math Working Group, which developed MathML, lobbied in favor of the inclusion of the new characters. In addition, the new characters lend themselves to direct plain text encoding of mathematics for various purposes  which can be much more compact than MathML or T_E X, the typesetting language and program designed by Donald Knuth [TeX] (see Section 4, Implementation Guidelines).

2.1 Mathematical Alphanumeric Symbols Block

The Mathematical Alphanumeric Symbols block (U+1D400—U+1D7FF) contains a large collection of letterlike symbols for use in mathematical notation, typically for variables. The characters in this block are intended for use only in mathematical or technical notation; they are not intended for use in non-technical text. When used with markup languages, for example with MathML the characters are expected to be used directly, instead of indirectly via entity references or by composing them from base letters and style markup.

Words Used as Variables. In some specialties, whole words are used as variables, not just single letters. For these cases, style markup is preferred because in ordinary mathematical notation the juxtaposition of variables generally implies multiplication, or some other composition, not word formation as in ordinary text. Markup not only provides the necessary scoping in these cases, it also allows the use of a more extended alphabet.

2.2  Mathematical Alphabets

Basic Set of Alphanumeric Characters. Mathematical notation uses a basic set of mathematical alphanumeric characters which consists of:

• the set of basic Latin digits (0 - 9) (U+0030..U+0039)
• the set of basic upper- and lowercase Latin letters (a - z, A - Z)
• the uppercase Greek letters Α - Ω (U+0391..U+03A9), plus the nabla ∇ (U+2207) and the variant of theta Θ given by U+03F4
• the lowercase Greek letters α - ω (U+03B1..U+03C9), plus the partial differential sign ∂ (U+2202) and the six glyph variants of ε, θ, κ, φ, ρ, and π, given by U+03F5, U+03D1, U+03F0, U+03D5, U+03F1, and U+03D6.

For some characters in the basic set of Greek characters, two variants of the same character are included. This is because they can appear in the same mathematical document with different meanings, even though they would have the same meaning in Greek text.

Mathematical Accents. The diacritics, or accents, in mathematical text usually have special semantic significance different from that of changing the pronunciation of a letter, as is the case for text accents. Because the use of text accents such as the acute accent would interfere with common mathematical diacritics, only unaccented forms of the letters are used for mathematical notation. Examples of common mathematical diacritics that can be confused with text accents are the circumflex, macron, or the single or double dot above, the latter two of which are commonly used in physics to denote derivatives with respect to the time variable. 

Mathematical symbols with diacritics are always represented by combining character sequences, except as required by normalization. See Unicode Standard Annex #15, Unicode Normalization Forms [Normalization] for more information. Note that normalization leaves all characters in the Mathematical Alphanumeric Symbols and Letterlike Symbols blocks unaffected. These blocks contain nearly all alphabetic characters used as math symbols.

Additional Characters. In addition to this basic set, mathematical notation also uses the four Hebrew-derived characters (U+2135..U+2138), for example in ℵ₀ for the first transfinite cardinal. Occasional uses of other alphabetic and numeric characters are known. Examples include U+0428 cyrillic capital letter sha, U+306E hiragana letter no , the ideograph U+4E2D 中 and Eastern Arabic-Indic digits (U+06F0..U+06F9). However, unlike the characters in the mathematical alphabets, these characters are only used in a single, basic form.

Semantic Distinctions. Mathematics has need for a number of Latin and Greek alphabets that on first thought appear to be mere font variations of one another. For example the letter H can appear as plain or upright (H), bold (H), italic (H ), and script H. However in any given document, these characters have distinct, and usually unrelated mathematical semantics. For example, a normal H represents a different variable from a bold H, etc. If these attributes are dropped in plain text, the distinctions are lost and the meaning of the text is altered. Without the distinctions, the well-known Hamiltonian formula:

_,

turns into the integral equation in the variable H:

_.

By encoding a separate set of alphabets, it is possible to preserve such distinctions in plain text.

Mathematical Alphabets. The alphanumeric symbols encountered in mathematics are given in the following table:

Table 2.1 Mathematical Alphabets

* Some of these alphabets have characters in the BMP as noted in the following section.

The plain letters have been unified with the existing characters in the Basic Latin and Greek blocks. There are 24 double-struck, italic, Fraktur and script characters that already exist in the Letterlike Symbols block (U+2100—U+214F). These are explicitly unified with the characters in this block and corresponding holes have been left in the mathematical alphabets.

Compatibility Decompositions. All mathematical alphanumeric symbols have compatibility decompositions to the base Latin and Greek letters—folding away such distinctions, however, is usually not desirable as it loses the semantic distinctions for which these characters were encoded. See Unicode Standard Annex #15, Unicode Normalization Forms [Normalization] for more information.

2.3 Fonts Used for Mathematical Alphabets

Mathematicians place strict requirements on the specific fonts being used to represent mathematical variables. Readers of a mathematical text need to be able to distinguish single letter variables from each other, even when they do not appear in close proximity. They must be able to recognize the letter itself, whether it is part of the text or is a mathematical variable, and lastly which mathematical alphabet it is from.

Fraktur. The black letter style is often referred to as Fraktur or Gothic in various sources. Technically, Fraktur and Gothic typefaces are distinct designs from black letter, but any of several font styles similar in appearance to the forms shown in the charts can be used.

Math italics. Mathematical variables are most commonly set in a form of italics, but not all italic fonts can be used successfully. In common text fonts, the italic letter v and Greek letter nu are not very distinct. A rounded italic letter v is therefore preferred in a mathematical font, as long as it is distinct from the Greek upsilon. There are other characters, which sometimes have similar shapes and require special attention to avoid ambiguity. Examples are shown in the table below.

Theorems are commonly printed in a text italic font. A font intended for mathematical variables should support clear visual distinctions so that variables can be reliably separated from italic text in a theorem. Some languages have common single letter words (English a, Scandinavian i, etc.), which can otherwise be easily confused with common variables.

Hard-to-distinguish Letters. Not all sans-serif fonts allow an easy distinction between lowercase l, and uppercase I and not all monospaced (fixed width) fonts allow a distinction between the letter l and the digit 1. Such fonts are not usable for mathematics. In Fraktur, the letters I and J in particular must be made distinguishable. Overburdened Black Letter forms like I and J are inappropriate. Similarly, the digit zero must be distinct from the uppercase letter O, and the empty set ∅ must be distinct from the letter o with stroke for all mathematical alphanumeric sets. Some characters are so similar that even mathematical fonts do not attempt to provide distinguished glyphs for them. Their use is normally avoided in mathematical notation unless no confusion is possible in a given context, for example uppercase A and uppercase Alpha (A).

Font Support for Combining Diacritics. Mathematical equations require that characters be combined with diacritics (dots, tilde, circumflex, or arrows above are common), as well as followed or preceded by super- or subscripted letters or numbers. This requirement leads to designs for italic styles that are less inclined, and script styles that have smaller overhangs and less slant than equivalent styles commonly used for text such as wedding invitations.

Typestyle for Script Characters. In some instances, a deliberate unification with a non-mathematical symbol has been undertaken; for example, U+2133 script capital m is unified with the pre-1949 symbol for the German currency unit Mark. This unification restricts the range of glyphs that can be used for this character in the charts. Therefore the font used for the reference glyphs in the code charts uses a simplified ‘English Script’ style, as recommended by the American Mathematical Society. For consistency, other script characters in the Letterlike Symbols block are now shown in the same typestyle.

The two characters U+2113 script small l, and U+2118 script capital p, are not regular script characters, despite their character names. The latter is the symbol for the Weierstrass elliptic function, a calligraphic letter shape based on the small p, and the former is derived from a special italic letter shape called an 'ell', and is unified with the common non-SI symbol for the liter [SI]. The characters U+1D4C1 mathematical scripts small l and U+1D4A8 mathematical script capital p are the preferred characters for the script style.

Double-struck Characters. The double-struck glyphs shown in earlier editions of the standard attempted to match the design used for all the other Latin characters in the standard, which is based on Times. The current set of fonts for use in the character code charts was prepared after consultation with the American Mathematical Society and leading publishers of mathematics, and shows much simpler forms that are derived from the forms written on a blackboard. However, this font represents just one possible representation of double-struck characters; both serifed and non-serifed forms can be used in mathematical texts, and inline fonts are found in works published by certain publishers. Some fonts differ in which strokes of a glyph to double, for example the left or right leg of the uppercase A. There is no intention to support any of these stylistic preferences via character encoding, therefore only one set of double-struck mathematical alphanumeric symbols have been encoded.

2.3.1 Representative Glyphs for Greek Phi

With Unicode 3.0 and the concurrent second edition of ISO/IEC 10646-1, the representative glyphs for U+03C6 greek letter small phi and U+03D5 greek phi symbol were exchanged. In ordinary Greek text, the character U+03C6 is used exclusively, although this character has considerable glyphic variation, sometimes represented with a glyph more like the representative glyph shown for U+03C6 (the "loopy" form) and less often with a glyph more like the representative glyph shown for U+03D5 (the “straight“ form). See the Greek table in the character code charts [Charts].

For mathematical and technical use, the straight form of the small phi is an important symbol and needs to be consistently distinguishable from the loopy form. The straight form phi glyph is used as the representative glyph for the symbol phi at U+03D5 to satisfy this distinction.

The assignment of representative glyphs had been the reverse in versions of the Unicode Standard prior to Unicode 3.0. This resulted in the problem that the character explicitly identified as the mathematical symbol did not have the straight form of the character that is the preferred glyph for that use. Furthermore, it made it unnecessarily difficult for general purpose fonts supporting ordinary Greek text to also add support for Greek letters used as mathematical symbols. This resulted from the fact that many of those fonts already used the loopy form glyph for U+03C6, as preferred for Greek body text; to support the phi symbol as well, they would have had to disrupt glyph choices already optimized for Greek text.

When mapping symbol sets or SGML entities to the Unicode Standard, it is important to make sure that codes or entities, such as phi1, that require the straight form of the phi symbol be mapped to U+03D5 and not to U+03C6. Mapping to the latter should be reserved for codes or entities that represent the small phi as used in ordinary Greek text.

Fonts used primarily for Greek text may use either glyph form for U+03C6, but fonts that also intend to support technical use of the Greek letters should use the loopy form to ensure appropriate contrast with the straight form used for U+03D5.

2.3.2 Representative Glyphs for 2278 and 2279

In Unicode 3.2 the representative glyphs for 2278 neither less-than nor greater-than and 2279 neither greater-than nor less-than were changed from using a vertical cancellation to using a slanted cancellation. This change was made in order to match their long standing canonical decompositions for these characters, which use 0338 combining long solidus overlay. Irrespective of this change to the representative glyphs, the symmetric forms using the vertical stroke remain acceptable glyph variants. Using 2275 or 2276 followed by 20D2 combining long vertical line overlay will encode these upright variants explicitly.

Except for those fonts created with the intention to add support for both forms (via combination of 2275 or 2276 with 20D2 for the upright forms) there is no need to revise the glyphs for 2278 and 2279: the glyphic range implied by using these character codes encompasses both shapes.

2.4  Locating Mathematical Characters

Mathematical characters can be located by looking in the code charts [Charts] at the blocks listed below or by checking the Unicode MATH property, which is assigned to characters that naturally appear in mathematical contexts (see Section 3 Mathematical Character Properties). Mathematical characters can be found in the following blocks:

Table 2.2 Locations of Mathematical Characters

Block Name	Range	Character Types
Basic Latin	U+0021–U+007E	Variables, operators, digits*
Greek	U+0370–U+03FF	Variables*
General Punctuation	U+2000–U+206F	Spaces, Invisible operators*
Letterlike Symbols	U+2100–U+214F	Variables*
Arrows	U+2190–U+21FF	Arrows, arrow-like operators
Mathematical Operators	U+2200–U+22FF	Operators
Miscellaneous Technical Symbols	U+2300–U+23FF	Braces, operators*
Geometrical Shapes	U+25A0–U+25FF	Symbols
Misc. Mathematical Symbols-A	U+27C0–U+27EF	Symbols and operators
Supplemental Arrows-A	U+27F0–U+27FF	Arrows, arrow-like operators
Supplemental Arrows-B	U+2900–U+297F	Arrows, arrow-like operators
Misc. Mathematical Symbols-B	U+2980–U+29FF	Braces, symbols
Suppl. Mathematical Operators	U+2A00–U+2AFF	Operators
Misc. Symbols and Arrows	U+2B00-U+2BFF	Arrows, operators or symbols
Mathematical Alphanumeric Symbols	U+1D400–U+1D7FF	Variables and digits
Other blocks	…	Characters for occasional use

*This block contains non-mathematical characters as well.

2.5  Duplicated Characters

Some Greek letters are re-encoded as technical symbols. These include U+00B5 µ micro sign, U+2126 Ω ohm sign, and several characters among the APL functional symbols in the Miscellaneous Technical block. U+03A9 Ω greek letter capital omega is the canonical equivalent of U+2126 and its use is preferred. Micro sign is included in several parts of ISO/IEC 8859, and therefore supported in many legacy environments where U+03BC greek letter small mu is not available. Because of that, implementations need to be able to recognize it, even though mu is the preferred character in a Unicode context.

Latin letters duplicated include U+212A K kelvin sign and U+212B Å angstrom sign. As in the case of  the ohm sign, the corresponding regular Latin letters, are the canonical equivalents and therefore their use is preferred.

The left and right angle brackets at U+2328 and U+2329 have long been canonically equivalent with the CJK punctuation characters at U+3008 and U+3009, which implies that the use of the latter code points is preferred and that the characters are ‘wide’ characters. See Unicode Standard Annex #11, East Asian Width [EAW]. Unicode 3.2 added two new mathematical angle bracket characters (U+27E8 and U+27E9) that are unequivocally intended for mathematical use. 

2.6  Accented Characters

Mathematical characters are often enhanced via use of combining marks in the ranges U+0300..U+036F and the combining marks for symbols in the range U+20D0..U+20FF. These characters follow the base characters as in non-mathematical Unicode text. This section discusses these characters and preferred ways of representing accented characters in mathematical expressions. If a span of characters is enhanced by a combining mark, e.g., a tilde over AB, typically some kind of higher-level markup is needed as is done in [MathML]. Unicode does include some combining marks that are designed to be used for pairs of characters, e.g., U+0360..U+0362. However, their use for mathematical text is not encouraged.

For some mathematical characters, such as many negated relations, there are multiple ways of expressing the character: as precomposed or as a sequence of base character and combining mark (see also Section 2.17 Negations). Having only a single way to represent any given character, would simplify recognizing the character in searches and other manipulations. Selecting a unique representation among multiple equivalent representations is called normalization. Unicode Standard Annex #15 Unicode Normalization Forms [Normalization] discusses the subject in detail; however, due to requirements of non-mathematical software, not all the normalization forms presented there are ideal from the perspective of mathematics.

Ideally, one always uses the shortest form of a math operator symbol wherever possible. So U+2260 should be used for the not equal sign instead of the combining sequence U+003D U+0338. If a negated operator is needed that does not have a precomposed form, the characters U+0338 combining long solidus overlay or U+20D2 CombininG Vertical LONG OVERLAY can be used to indicate negation. This approach concurs with Normalization Form C (NFC), which is also the preferred normalization form for use on the web. 

On the other hand, for accented alphabetic characters used as variables, ideally only decomposed sequences are used. This is because mathematics uses a multitude of combining marks that greatly exceeds the predefined composed characters in Unicode. Accordingly, it is better to have the math display facility handle all of these cases uniformly to give a consistent look between characters that happen to have a fully composed Unicode character and those that do not. The combining character sequences also typically have semantics as a group, so it is handy to be able to manipulate and search for them individually without having to have special tables to decompose characters for this purpose. Since there are no precomposed math alphanumeric symbols, this approach concurs with Normalization Form C, except for the upright alphabetic characters (ASCII letters). 

To facilitate interchange on the web, accented characters should conform to NFC when interchanged. However, to achieve consistent results, a mathematical display system should transiently decompose any precomposed upright letters when used in mathematical expressions and use a single algorithm to place embellishments.

Normalization Form D (NFD) uses the opposite approach from NFC. It works naturally for mathematical use of alphabetic characters, but would not use the shortest encoding of math operator symbols, making it less attractive. The other two normalization forms NFKC and NFKD remove the distinction between math alphanumeric alphabets, mapping all of them to plain ASCII or Greek characters. As a result they would destroy the semantics of many mathematical expressions, preventing their use with mathematical texts.

2.7  Operators

The Mathematical Operators (U+2200—U+22FF) and Supplementary Mathematical Operators (U+2A00—U+2AFF) blocks contain many mathematical operators, relations, geometric symbols and other symbols with special usages confined largely to mathematical contexts. In addition to the characters in these blocks, mathematical operators are also found in the Basic Latin (ASCII) and Latin-1 Supplement Blocks. A few of the symbols from the Miscellaneous Technical block and characters from General Punctuation are also used in mathematical notation. The allocation of any operator to a particular block is rarely significant.

Semantics. Mathematical operators often have more than one meaning in different subdisciplines or different contexts. For example, the "+" symbol normally denotes addition in a mathematical context, but might refer to concatenation in a computer science context dealing with strings, or incrementation, or have any number of other functions in given contexts. Therefore The Unicode Standard only encodes a single character for a single symbolic form. There are numerous other instances in which several semantic values can be attributed to the same Unicode value. For example, U+2218 ring operator may be the equivalent of white small circle or composite function or apl jot. The Unicode Standard does not attempt to distinguish all possible semantic values that may be applied to mathematical operators or relational symbols. It is up to the application or user to distinguish such meanings according to the appropriate context. Where information is available about the usage (or usages) of particular symbols, it has been indicated in the character annotations in Chapter 16, Code Charts in The Unicode Standard, Version 4.0 [Unicode] and in the online code charts [Charts].

Similar glyphs. The Standard does include many characters that appear to be quite similar to one another, but that may well convey different meaning in a given context. On the other hand, mathematical operators, and especially relation symbols, may appear in various standards, handbooks, and fonts with a large number of purely graphical variants. Where variants were recognizable as such from the sources, they were not encoded separately.

For relation symbols, the choice of a vertical or forward-slanting stroke typically seems to be an aesthetic one, but both slants might appear in a given context. However, a back-slanted stroke almost always has a distinct meaning compared to the forward-slanted stroke. See Section 2.18 Variation Selector for more information on some particular variants.

Unifications. Mathematical operators such as implies and if and only if  have been unified with the corresponding arrows (U+21D2 ⇒ rightwards double arrow and U+2194 ↔ left right arrow, respectively) in the Arrows block.

The operator U+2208 element of is occasionally rendered with a taller shape than shown in the code charts. Mathematical handbooks and standards consulted treat these characters as variants of the same glyph. U+220A small element of is a distinctively small version of the element of that originates in mathematical pi fonts.

The operators U+226B much greater-than and U+226A much less-than are some­times rendered in a nested shape, but the Unicode Standard provides a single encoding for each operator.

A large class of unifications applies to variants of relation symbols involving equality, simi­larity, and/or negation. Variants involving one- or two-barred equal signs, one- or two-tilde similarity signs, and vertical or slanted negation slashes and negation slashes of different lengths are not separately encoded. Thus, for example, U+2288 neither a subset of nor equal to, is the archetype for at least six different glyph variants noted in various collections.

In two instances, essentially stylistic variants are separately encoded: U+2265 greater-than or equal to is distinguished from U+2267 greater-than over equal to; the same distinction applies to U+2264 less-than or equal to and U+2266 less-than over equal to. This exception to the general rule regarding variation results from requirements for character mapping to some Asian standards that distinguish the two forms.

Several mathematical operators derived from Greek characters have been given separate encodings since they are used differently than the corresponding letters. These operators may occasionally occur in context with Greek-letter variables. They include U+2206 increment, U+220F n-ary product, and U+2211 n-ary summation. The latter two are large operators that take limits. Some typographical aspects of operators are discussed in Section 3.2 Classification by Typographical Behavior. For example, the n-ary operators are distinguished from letter variables by their larger size and the fact that they take limit expressions.

The unary and binary minus sign is preferably represented by U+2212 minus sign rather than by the ASCII-derived U+002D hyphen-minus, both because the former is unambiguous and because it is rendered with a more desirable length. (For a complete list of dashes in the Unicode Standard, see Table 6-2 in [TUS]).

Miscellaneous Symbols.  The symbol U+2205 empty set is distinct from the letters U+00D8 Ø and U+00F8 ø, even though historically derived from the letter forms. A widespread alternate symbol for the empty set is a slashed digt zero. This can be encoded as U+0030 digit zero followed by U+0338 combining long solidus overlay.

U+22EE.. U+22F1 are a set of ellipses used in matrix notation.

2.8  Superscripts and Subscripts

The Unicode block U+2070.. U+209F plus U+00B2, U+00B3, and U+00B9 contain sequences of superscript and subscript digits and punctuation that can be useful in mathematics. If they are used, it is recommended that they be displayed with the same font size as other subscripts and superscripts at the corresponding nested script level. For example, a² and a<super>2</super> should be displayed the same. However, these subscript/superscript characters are not used in MathML or T_EX and their use with XML documents is discouraged, see Unicode Technical Report #20, Unicode in XML and other Markup Languages [UXML].

2.9  Arrows

Arrows are used for a variety of purposes in mathematics and elsewhere, such as to imply directional relation, to show logical derivation or implication, and to represent the cursor control keys. Accordingly Unicode includes a fairly extensive set of arrows (U+2190..U+21FF and U+2900..U+297F), many of which appear in mathematics. It does not attempt to encode every possible stylistic variant of arrows separately, especially where their use is mainly decorative. For most arrow variants, the Unicode Standard provides encodings in the two horizontal directions, often in the four cardinal directions. For the single and double arrows, the Unicode Standard provides encodings in eight directions.

Unifications. Arrows expressing mathematical relations have been encoded in the arrows block as well as in Supplemental Arrows-A and Supplemental Arrows-B. An example is U+21D2 rightwards double arrow, which may be used to denote implies. Where available, such usage information is indicated in the annotations to individual characters in the Unicode Standard 4.0 [U4.0], Chapter 16, Code Charts, and in the online code charts [Charts].

Long Arrows. The long arrows encoded in the range U+27F5..U+27FF map to standard SGML entity sets supported by MathML. Long arrows represent distinct semantics from their short counterparts, rather than mere stylistic glyph differences. For example, the shorter forms of arrows are often used in connection with limits, whereas the longer ones are associated with mappings. The use of the long arrows is so common that they were assigned entity names in the ISOAMSA entity set, one of the suite of mathematical symbol entity sets covered by the Unicode Standard.

2.10 Delimiters

The mathematical white square brackets, angle brackets, and double angle brackets encoded at U+27E6..U+27EB are intended for ordinary mathematical use of these particular bracket types. They are unambiguously narrow, for use in mathematical and scientific notation, and should be distinguished from the corresponding wide forms of white square brackets, angle brackets, and double angle brackets used in CJK typography. (See the CJK Symbols and Punctuation block.)

Note especially that the angle brackets formerly aliased as "bra" and "ket", U+2329 left-pointing ANGLE BRACKET and U+232A right-pointing angle bracket, are now deprecated for use with mathematics because of their canonical equivalence to CJK angle brackets, which is likely to result in unintended spacing problems when used in mathematical formulae.

However, the set of lenticular and tortoise-shell brackets in the CJK Punctuation block have not been duplicated and are intended for both CJK and mathematical use. For these characters, implementations for mathematical use need to use fonts without white space padding, i.e. containing the 'narrow' glyphs.

2.11  Geometrical Shapes

The basic geometric shapes (circle, square, triangle, diamond, and lozenge) are used for a variety of purposes in mathematical texts. Because their shapes are distinct and they are easily available in multiple sizes from a variety of widely available fonts, they are also often used in an ad-hoc manner.

Ideal sizes. Mathematical usage requires at least four distinct sizes of simple shapes, and sometimes more. The size gradation must allow each size to be recognized, even when it occurs in isolation. In other words shapes of the same size should ideally have roughly the same visual "impact" as opposed to same nominal height or width. For mathematical usage simple shapes ideally share a common center. The following diagram shows which size relationship across shapes of the same nominal size is considered ideal.

Please note that neither the current set of glyphs in the standard nor the glyphs from many commonly available non-mathematical fonts achieves this ideal.

Actual sizes. The sizes of existing characters and their names are not always consistent. For mathematical usage, therefore, the MEDIUM SMALL SQUARE should be used together with the MEDIUM size of the other basic shapes, and correspondingly for the other sizes. (The basic shapes from the Zapf Dingbats font match the unmarked size for triangle, diamond and circle and the MEDIUM size for the square.) To achieve the correct size relation, mathematical fonts may need to deviate in minor amounts from the sizes shown in the character charts.  The table below summarizes the available sizes for a given symbol. 

Most simple geometrical shapes exist in both black and outline (white) form in all sizes. For circles and lozenges separate images and code points are provided in the table; for many sizes, both white and black forms exist, but are not encoded under matching names or close together. Squares and triangles exist in both black and white forms at all sizes, except large. Triangles have size as well as direction. The Unicode Standard encodes the four principal directions. Note that there is no tiny white circle, nor are there tiny sizes in the other shapes. At that small a size, it is difficult to distinguish the symbol shapes making a shape distinction less useful for notational purposes.

Sizes of derived shapes. Circled and squared operators and similar derived shapes are more constrained in their usage than "plain" geometric shapes. They tend to occur in two generic sizes based on function: a smaller size for binary operators and large size for n-ary operators.

Positioning. For a mathematical font, the centerline should go through the middle of a parenthesis, which should go from bottom of descender to top of ascender. This is the same level as the minus or the middle of the plus and equal signs. For correct positioning, the glyph will descend below the baseline for the larger sizes of the basic shapes as in the following schematic diagram:

The standard triangles used for mathematics are also center aligned. This is different from the positioning for the representative glyphs shown in the charts, which are often based on existing non-mathematical fonts. Mathematical fonts may need to deviate in positioning of these triangles.

2.12  Other Symbols

Other symbols of use in mathematics are contained in the Miscellaneous Technical block (U+2300—U+23FF), the Geometric Shapes block (U+25A0—U+25FF), the Miscellaneous Symbols block (U+2600—U+267F), and the General Punctuation block (U+2000—U+206F).

Generally any easily recognized and distinct symbol is fair game for mathematicians faced with the need of creating notations for new fields of mathematics. For example, the card suits, U+2665 ♥ black heart suit, U+2600 ♠ black spade suit, etc. can be found as operators and as subscripts.

2.13  Symbol Pieces

The characters from the Miscellaneous Technical block in the range U+239B—U+23B3, plus U+23B7, comprise a set of bracket and other symbol fragments for use in mathematical typesetting. These pieces originated in older font standards, but have been used in past mathematical processing as characters in their own right to make up extra-tall glyphs for enclosing multi-line mathematical formulae. Mathematical fences are ordinarily sized to the content that they enclose. However, in creating a large fence, the glyph is not scaled proportionally; in particular the displayed stem weights must remain compatible with the accompanying smaller characters. Thus, simple scaling of font outlines cannot be used to create tall brackets. Instead, a common technique is to build up the symbol from pieces. In particular, the characters U+239B LEFT PARENTHESIS UPPER HOOK through U+23B3 SUMMATION BOTTOM represent a set of glyph pieces for building up large versions of the fences (, ), [, ], {, and }, and of the large operators ∑ and ∫. These brace and operator pieces are compatibility characters. They should not be used in stored mathematical text, but are often used in the data stream created by display and print drivers.

The following table shows which pieces are intended to be used together to create specific symbols.

Table 2.4 Use of Symbol Pieces

For example, an instance of U+239B can be positioned relative to instances of U+239C and U+239D to form an extra-tall (three or more line) left-parenthesis. The center sections encoded here are meant to be used only with the top and bottom pieces encoded adjacent to them, since the segments are usually graphically constructed within the fonts so that they match perfectly when positioned at the same x coordinates.

2.14  Invisible Operators

In mathematics some operators or punctuation are often implied, but not displayed. U+2063 invisible separator or invisible comma is intended for use in index expressions and other mathematical notation where two adjacent variables form a list and are not implicitly multiplied. In mathematical notation, commas are not always explicitly present, but need to be indicated for symbolic calculation software to help it disambiguate a sequence from a multiplication. For example, the double _ij subscript in the variable a_ij means a_{i, j} — that is, the i and j are separate indices and not a single variable with the name ij or even the product of i and j. Accordingly to represent the implied list separation in the subscript _ij one can insert a non-displaying invisible separator between the i and the j. In addition, use of the invisible comma would hint to a math layout program to set a small space between the variables.

Similarly an expression like mc² implies that the mass m multiplies the square of the speed c. To unambiguously represent the implied multiplication in mc², one inserts a non-displaying U+2062 invisible times between the m and the c. A related case is the use of U+2061 function application for an implied function dependence as in f(x + y). To indicate that this is the function f of the quantity x + y and not the expression fx + fy, one can insert the non-displaying function application symbol between the f and the left parenthesis.

Another example is the expression f ^ij(cos(ab)), which means the same as f ^i,j(cos(a×b)), where × represents multiplication, not the cross product. Note that the spacing between characters may also depend on whether the adjacent variables are part of a list or are to be concatenated, that is, multiplied.

2.15  Fraction Slash

U+2044 fraction slash is used to build up simple fractions in running text. It applies to the immediately adjacent sequences of decimal digits, that is characters with the general category=Nd. In general mathematical use a more general method for layout of fractions is needed, however parsers of mathematical texts should be prepared to handle U+2044 when it is being received from other sources.

2.16  Other Characters

These include all remaining Unicode characters. They may appear in mathematical expressions, typically in spelled-out names for variables in fractions or simple formulae, but they most commonly appear in ordinary text. An English example is the equation

distance = rate × time,

which uses ordinary ASCII letters to aid in recognizing sequences of letters as words instead of products of individual symbols. Such usage corresponds to identifiers, discussed elsewhere.

2.17  Negations

Many negated forms, particularly of relations, can be encoded by using the base symbol, together with a combining overlay. Occasionally, both a vertical and a slanted negation are used; which one is often a matter of style. Sometimes the negation is only indicated for part of a symbol. In these cases, the negated relations are encoded directly, and variants can be accessed via the variation selector method described in the next section.

The following table lists those of the currently encoded negated mathematical relations for which a variant can be realized via composition, by using U+20D2 combining long vertical line overlay together with a base character. In the table, the part of the description in small caps is the character name of the corresponding standard character; the part in lower case indicates the variation in appearance.

Table 2.5 Negated relations using vertical line overlay

Std Symbol	Alternate Symbol	Description of alternate symbol
	2209		2208,20D2	not an element of with vertical stroke
	220C		220B,20D2	does not contain as member with vertical stroke
	2241		223C,20D2	not tilde with vertical stroke
	2244		2243,20D2	not asymptotically equal to with vertical stroke
	2247		2245,20D2	neither approximately nor actually equal to with vertical stroke
	2249		2248,20D2	not almost equal to with vertical stroke
	2260		003D,20D2	not equal to with vertical stroke
	2262		2261,20D2	not identical to with vertical stroke
	226D		224D,20D2	not equivalent to with vertical stroke
	226E		003C,20D2	not less-than with vertical stroke
	226F		003E,20D2	not greater-than with vertical stroke
	2270		2264,20D2	neither less-than nor equal to with vertical stroke
	2271		2265,20D2	neither greater-than nor equal to with vertical stroke
	2278		2278, 20D2	neither less-than nor greater-than with vertical stroke (*)
	2279		2279, 20D2	neither greater-than nor less-than with vertical stroke (*)
	2280		227A,20D2	does not precede with vertical stroke
	2281		227B,20D2	does not succeed with vertical stroke
	2284		2282,20D2	not a subset of with vertical stroke
	2285		2283,20D2	not a superset of with vertical stroke
	2288		2286,20D2	neither a subset of nor equal to with vertical stroke
	2289		2287,20D2	neither a superset of nor equal to with vertical stroke
	22E0		227C,20D2	does not precede or equal with vertical stroke
	22E1		227D,20D2	does not succeed or equal with vertical stroke

* The representative glyphs shown in the code charts [Charts] have been revised in Unicode 4.0 to show the slanted forms - this matches their existing decomposition using U+0338 combining long solidus overlay (see Section 2.32, Representative Glyphs for 2278 and 2279 for more information).

Note that, the use of a base character together with the slanted U+0338 combining long solidus overlay would be equivalent to the use of the precomposed negation (see also the discussion in Section 2.6 Accented Characters). For those symbols for which only a partial vertical stroke is used, use of U+20D2 would not give the intended result; instead U+FE00 variation selector-1, is used as described in Section 2.18 Variation Selector. 

The following table lists some of the negated forms of mathematical relations that can only be encoded by using either U+0338 combining long solidus overlay or U+20D2 combining long vertical line overlay. Depending on the overlay used, the negation has a diagonal or vertical stroke. The part of the description that is in small caps reflects the Unicode character name of the non-negated symbol. Since these are not glyph variants of existing characters, the word "negated" is used instead of "NOT" as in the list above, to indicate that the negation is expressed by the combining character sequence, and not inherent in the character.

Table 2.6 Using vertical line or solidus overlay

Glyph / Sequence	Glyph / Sequence	Description
	220A,0338		220A,20D2	negated small element of
	220D,0338		220D,20D2	negated small contains as member
	2242,0338		2242,20D2	negated minus tilde
	2263,0338		2263,20D2	negated strictly equivalent to
	2266,0338		2266,20D2	negated less-than over equal to
	2267,0338		2267,20D2	negated greater-than over equal to
	22F7,0338		22F7,20D2	negated element of with overbar
	22FE,0338		22FE,20D2	negated small contains with overbar
	2A6C,0338		2A6C,20D2	negated similar minus similar
	2A70,0338		2A70,20D2	negated approximately equal or equal to
	2A7D,0338		2A7D,20D2	negated less-than or slanted equal to
	2A7E,0338		2A7E,20D2	negated greater-than or slanted equal to
	2A95,0338		2A95,20D2	negated slanted equal to or less-than
	2A96,0338		2A96,20D2	negated slanted equal to or greater-than
	2A99,0338		2A99,20D2	negated double-line equal to or less-than
	2A9A,0338		2A9A,20D2	negated double-line equal to or greater-than
	2AC5,0338		2AC5,20D2	negated subset of above equals sign
	2AC6,0338		2AC6,20D2	negated superset of above equals sign

Note that in some cases, as seen in the two preceding tables, simply using the generic glyph for the vertical overlay will not give the correct appearance. 2286 or 2A99 are examples of characters that may require a taller stroke. Similarly, the generic position of the solidus overlay as shown for 2AC6 above is not ideal.

2.18  Variation Selector

The variation selector VS1 is used to represent well-defined variants of particular math symbols. The variations include: different slope of the cancellation element in some negated symbols, changed orientation of an equating or tilde operator element, and some well-defined different shapes. These mathematical variants are all produced with the addition of U+FE00 variation selector 1 (VS1) to mathematical operator base characters. To select one of the predefined variations, follow the base character with the variation selector. 

The following table lists only those of the currently defined combinations that are of interest in the context of mathematics. In the table, the part of the description in small caps is the character name of the corresponding standard character; the part in lower case indicates the variation in appearance. The full set of valid, recognized combinations is listed, complete with glyphs, in the table of standardized variants [StdVar] in the Unicode Character Database. All combinations not listed there are unspecified and are reserved for future standardization; no conformant process may interpret them as standardized variants. For more information, see Section 15.9, Variation Selectors, in Unicode 4.0 [U4.0].

Table 2.7 Variants of Mathematical Symbols using VS1

Sequence	Description
2229 + VS1	INTERSECTION with serifs
222A + VS1	UNION with serifs
2268 + VS1	LESS-THAN BUT NOT EQUAL TO - with vertical stroke
2269 + VS1	GREATER-THAN BUT NOT EQUAL TO - with vertical stroke
2272 + VS1	LESS-THAN OR EQUIVALENT TO - following the slant of the lower leg
2273 + VS1	GREATER-THAN OR EQUIVALENT TO - following the slant of the lower leg
228A + VS1	SUBSET OF WITH NOT EQUAL TO - variant with stroke through bottom members
228B + VS1	SUPERSET OF WITH NOT EQUAL TO - variant with stroke through bottom members
2293 + VS1	SQUARE CAP with serifs
2294 + VS1	SQUARE CUP with serifs
2295 + VS1	CIRCLED PLUS with white rim
2297 + VS1	CIRCLED TIMES with white rim
229C + VS1	CIRCLED EQUALS - equal sign inside and touching the circle
22DA + VS1	LESS-THAN slanted EQUAL TO OR GREATER-THAN
22DB + VS1	GREATER-THAN slanted EQUAL TO OR LESS-THAN
2A3C + VS1	INTERIOR PRODUCT - tall variant with narrow foot
2A3D + VS1	RIGHTHAND INTERIOR PRODUCT - tall variant with narrow foot
2A9D + VS1	SIMILAR OR LESS-THAN - following the slant of the upper leg - or less-than
2A9E + VS1	SIMILAR OR GREATER-THAN - following the slant of the upper leg - or greater-than
2AAC + VS1	SMALLER THAN OR slanted EQUAL
2AAD + VS1	LARGER THAN OR slanted EQUAL
2ACB + VS1	SUBSET OF ABOVE NOT EQUAL TO - variant with stroke through bottom members
2ACC + VS1	SUPERSET OF ABOVE NOT EQUAL TO - variant with stroke through bottom members

Using a variation selector allows users and font designers to make a distinction between alternate glyph shapes both of which are ordinarily acceptable glyphs for generic, non-distinguishing usage of the standalone character code. This situation is somewhat analogous to the variants of Greek letterforms, which are not distinguished when used in text, but must be distinguished when used as symbols. See Section 2.31, Representative Glyphs for Greek phi. However, unlike the Greek symbols, which are encoded using distinct character codes, the Unicode Standard considers the distinctions expressed via the variation selector as somewhat optional. Processes or fonts that cannot support it should yield acceptable results by ignoring the variation selector.

It is important to further note that the variation selector only selects a different appearance of an already encoded character. It is not intended as a general code extension mechanism. At this time the variations encoded with the variation selector are thought to be primarily glyphic variations. Should their usage or interpretation change—over time, or because of better evidence about how these shapes are actually used in mathematical notation—it is likely that another character would be coded so that the distinction in meaning can be kept directly in the character code.

2.19  Novel Symbols not yet in Unicode

Mathematicians are by their nature inventive people and will continue to invent new symbols to express their concepts. Until these symbols are used by a considerable number of people, they should not be standardized. Nevertheless, one needs a way to handle these novel symbols even before they are standardized.

The Private Use Areas (U+E000—U+F8FF, U+F0000—U+FFFFD, and U+100000—U+10FFFD) can be used for such nonstandard symbols. However, that can be a tricky business, since the Private Use Area (PUA) is used for many purposes. Hence when using the PUA, it is a good idea to have higher-level backup to define what kind of characters are involved. If they are used as math symbols, it would be good to assign them a math attribute that is maintained in a rich-text layer parallel to the plain text.

Markup languages also may have other ways of using arbitrary glyphs as 'pseudo-characters'; for instance, MathML [MathML] has an mglyph element.

3  Mathematical Character Properties

Unicode assigns a number of mathematical character properties to aid in the default interpretation and rendering of these characters. Such properties include the classification of characters into operator, digit, delimiter, and variable. These properties may be overridden, or explicitly specified in some environments, such as MathML [MathML], which uses specific tags to indicate how Unicode characters are used, such as <mo> for operator, <mn> for one or more digits comprising a number, and <mi> for identifier. T_E X, [TeX] is a higher-level composition system that uses implicit character semantics. In the following, these properties are described in greater detail.

In particular, many Unicode characters nearly always appear in mathematical expressions and are given the generic mathematics property [Math]. Among others, they include the math operators in the ranges U+2200..U+22FF and U+29B0..U+2AFF, the math combining marks U+20D0..U+20FF, the math alphanumeric characters (some of the Letterlike Symbols and the mathematics alphanumerics range U+1D400..U+1D7FF). Other characters may occur in mathematical usage depending on context. The math useful in heuristics that seek to identify mathematical expressions in plain text.

For more information about character properties, see the draft Unicode Character Property Model [PropMod].

3.1  Classification by degree of Mathematical Usage

Each character in the Unicode Standard is given a General Category. This is one of a set of values that represent a primary feature or function of a character. Characters that are primarily used as mathematical symbols and operators are given the General Category (gc) value of symbol_math (Sm). However, many characters commonly or exclusively used in mathematics, are classified first and foremost by their function as delimiting punctuation, rather than as math symbols. This affects in particular many of the math delimiters. The math property, which is designed to be applied to all characters used primarily or exclusively with mathematical notation is therefore a superset of  the characters with gc = Sm. The difference between the sets of characters that have the math property and those for which gc = Sm, is given by the set of characters that have the other_math property. 

3.1.1  Strongly Mathematical Characters

Strong mathematical characters are all characters that are used primarily or exclusively in mathematical notation. This includes all characters with the math property in Unicode. 

Despite their classification as strongly mathematical characters, many characters can also occur in non-mathematical texts, and the concept of mathematical use is deliberately kept broad. However, the delimiters in the ASCII range, such as parentheses, and brackets are so common in non-mathematical use, that they are considered weakly mathematical characters. For details on the assignment of the math property see the Unicode Character Database [UCD].

Note: The math property in Unicode 4.0 and earlier did include these ASCII characters, and did not include many characters more specifically used for mathematics. The math property was revised in Unicode 4.0.1 [U4.0.1] to match the definition of strong mathematical character presented here.

3.1.2  Weakly Mathematical Characters

These characters commonly appear in mathematical expressions, but they also appear naturally in ordinary text. They include the ASCII letters, punctuation, as well as the arrows and many of the geometric and technical shapes. The ASCII hyphen minus (U+002D) is a weakly mathematical character that may be used for the subtraction operator, but U+2212 is preferred for this purpose and looks better. Geometric shapes are frequently used as mathematical operators, but have other uses as well.

Weakly mathematical characters include the characters listed below. However this list should not be considered as comprehensive. On the other hand, it includes characters that would not be considered proper for mathematical notation by at least some authorities.

Code	Description
0021	Exclamation mark (factorial)
0028..0029	ASCII Parentheses
002A	ASTERISK
002C	SOLIDUS
002D	HYPHEN-MINUS
002E	FULL STOP
0030..0039	Digits
0041..005A	Letters
006E	CIRCUMFLEX ACCENT
005B,005D	Square brackets
005C	Backslash
007B,007D	Curly brackets
007E	TILDE
0061..007A	..
3010..3011	CJK brackets unified with Math use
3014..3019	..
All other arrows in the Arrows block, except 21EA..21F3 which are excepted because they are specifically keyboard symbols as well as 2B00..2B0D.
The following characters are compatibility variants of characters that are weakly mathematical characters. Since the list of characters that have the math property in Unicode includes compatibility variants, the characters below should also be considered weakly math characters:
FE35..FE38	Vertical parens and brackets
FE47..FE48	..
FE59..FE5C	CJK small form of parens and brackets
FF0D	FULLWIDTH HYPHEN-MINUS
FF0F	FULLWIDTH SOLIDUS
FF08..FF09	FULLWIDTH  Parentheses
FF4E	FULLWIDTH CIRCUMFLEX ACCENT
FF3B,FF3D	FULLWIDTH Square brackets
FF3C	FULLWIDTH Backslash
FF5B,FF5D	FULLWIDTH Curly brackets
FF5C	FULLWIDTH Vertical Bar
FF5E	FULLWIDTH TILDE
FFE9..FFEC	Halfwidth arrows

3.1.3  Other

All other Unicode characters. Many of these may occur in mathematical texts, though less commonly as part of the mathematical expressions themselves.

3.2  Classification by Typographical Behavior

Math characters fall into a number of subcategories, such as operators, digits, delimiters, and identifiers (constants and variables). This section discusses some of the typographical characteristics of these subcategories. These characteristics and classifications are useful in the absence of overriding information. For example, there is at least one document that uses the letter P, in upright roman typestyle, as a relational operator. 

3.2.1  Alphabetic

In general italic Latin characters are used to represent single-character Latin variables. In contrast, mathematical function names like sin, cos, tan, tanh, etc., are represented by upright and usually serifed text to distinguish them from products of variables. Such names should then not use the math alphanumeric characters. The upright uppercase Greek letters are favored over the italic ones. In Europe, upright d, D, e, and i can be used today for the two differential, exponential, and imaginary unit functionalities, respectively. In common American mathematical practice, these quantities are represented by italic quantities. Products of italicized variables have slightly wider spacing than the letters in italicized words in ordinary text.

3.2.2  Operators

Operators fall into one or more categories. These include:

Table 3.1 Some 0perator Categories

Category	Notes
binary	some spacing around binary operators
unary	closer to modified character than binary operators
n-ary	often called "large" operators, take limits ordinarily above and below when displayed out-of-line or right top and bottom when displayed in-line
arithmetic	arithmetic includes binary and unary operators
logical	unary not and binary and, or, exclusive or in a host of guises
set-theoretic	inclusion, exclusion, in a variety of guises
relational	binary operators like less/greater than in many forms

As in arithmetic, operators have precedence, which streamlines the interpretation of operands and reduces the notational complexity of expressions. Operator precedence is commonly used for this purpose in computer programming languages, calculus, and algebra. Assigning consistent default precedence to the operators allows software to automate the transition from data input (or plain text) to fully marked up forms of mathematical data such as TeX or MATHML.

For example, in arithmetic, 3+1/2 = 3.5, not 2. Similarly the plain-text expression α + β/γ means

not   .

As in arithmetic, precedence can be overruled by explicit delimitation, so (α + β)/γ gives the latter.

3.2.3  Large Operators

These include n-ary operators like summation and integration. These may expand in size to fit their associated expressions. They generally also take limits. The placement of the limits on an operator is different when it is used in-line compared to its use in displayed formulae. For example versus

.

Specifying a particular layout for limit expressions is outside the scope of the Unicode Standard. 

For the integrals, European tradition prefers a more upright shape. This is not expressed as different character codes or variants, but left as a matter of font choice.

3.2.4  Digits

Digits include 0-9 in various styles. All digits of a particular style have the same width.

3.2.5  Delimiters

Delimiters include punctuation, opening/closing delimiters such as parentheses and brackets, braces, and fences. Opening and closing delimiters and fences may expand in size to fit their associated expressions. Some bracket expressions do not appear to be "logical" to readers unfamiliar with the notation, e.g., ]x,y[.

3.2.6  Fences

Fences are similar to opening and closing delimiters, but are not paired.

3.2.7  Combining Marks

Combining marks are used with mathematical alphabetic characters (see Section 2.6 Accented Characters), instead of precomposed characters. Use <U+0061, U+0308> for the second derivative of acceleration with respect to time, not the precomposed letter ä. On the other hand, precomposed characters are used for operators whenever they exist. Combining slash (solidus) or vertical overlays can be used to indicate negation for operators that do not have precomposed negated forms.

Where both long and short combining marks exist, use the long, e.g., use U+0338, not U+0337 combining short overlay and use U+20D2, not U+20D3 combining short vertical line overlay. The actual shape or position of a combining mark is a typesetting problem and not specified in plain text. When using combining marks, the composite characters have the same typesetting class as the base character.

4  Implementation Guidelines

4.1  Use of Normalization with Mathematical Text

If Normalization Form C is applied to mathematical text, some accents or overlays used with BMP alphabetic characters may be composed with their base character, even though for mathematical text the decomposed forms would have been preferred. Parsers should allow for this. Normalization forms KC or KD remove the distinction between different mathematical alphabets. These forms cannot be used with mathematical texts. For more details on Normalization see Unicode Standard Annex #15, Unicode Normalization Forms [Normalization] and the discussion in Section 2.6 Accented Characters.

4.2  Input of Mathematical and Other Unicode Characters

In view of the large number of characters used in mathematics, it is useful to give a brief and informal discussion of possible approaches for input methods. While the ASCII math symbols are easy to find, e.g., + - / * [ ] ( ) { }, they often need to be used as themselves.

Post-entry correction. From a syntactical point of view, U+2212 minus sign is certainly preferable to the U+002D hyphen-minus in the ASCII range and U+2032 prime is preferable to U+0027 apostrophe, but users may locate the ASCII characters more easily. Similarly it is easier to type ASCII letters than italic letters, but when used as mathematical variables, such letters are traditionally italicized in print. Accordingly a user might want to make italic the default alphabet in a math context, reserving the right to overrule this default when necessary. Other post-entry enhancements include automatic-ligature and left-right quote substitutions, which can be done automatically by some word processors. Suffice it to say that intelligent input algorithms can dramatically simplify the entry of mathematical symbols.

Input method editors.  Many systems support interfaces for a user-selectable Input Method Editor (IME). While the functionality envisaged is based on the needs of East Asian language input, the task of selecting one of over thousand mathematical symbols at input time can be solved with a similar approach and could make use of the existing interfaces.

Math keyboards. A special math shift facility for keyboard entry could bring up proper math symbols. The values chosen can be displayed on an on-screen keyboard. For example, the left Alt key could access the most common mathematical characters and Greek letters, the right Alt key could access italic characters plus a variety of arrows, and the right Ctrl key could access script characters and other mathematical symbols. On systems that support it, the numeric keypad offers locations for a variety of symbols, such as sub/superscript digits using the left Alt key. Left Alt CapsLock could lock into the left-Alt symbol set, etc. This approach yields what one might call a "sticky" shift. Other possibilities involve the NumLock and ScrollLock keys in combinations with the left/right Ctrl/Alt keys. Pretty soon one realizes that this approach rapidly approaches literally billions of combinations, that is, several orders of magnitude more than Unicode can handle!

Macros. The auto-correct and keyboard macro features of some word processing systems provide other ways of entering mathematical characters for people familiar with TeX. For example, typing \alpha inserts α if the appropriate auto-correct entry is present. This approach is noticeably faster than using menus.

Hexadecimal input. A handy hex-to-Unicode entry method works with recent Microsoft text software (similar approaches are available on other systems) to insert Unicode characters in general and math characters in particular. Basically one types a character’s hexadecimal code (in ASCII), making corrections as need be, and then types Alt+x. The hexadecimal code is replaced by the corresponding Unicode character. The Alt+x can be a toggle, that is, type it once to convert a hex code to a character and type it again to convert the character back to a hex code. If the hex code is preceded by one or more hexadecimal digits, one needs to "select" the code so that the preceding hexadecimal characters aren’t included in the code. The code can range up to the value 0x10FFFF, which is the highest character in the 17 planes of Unicode.

Pull-down menus. Pull-down menus are a popular method for handling large character sets, but they are slow. A better approach is the symbol box, which is an array of symbols either chosen by the user or displaying the characters in a font. Symbols in symbol boxes can be dragged and dropped onto key combinations on an on-screen keyboard, or directly into applications. On-screen keyboards and symbol boxes are valuable for entry of mathematical expressions and of Unicode text in general.

4.3  Use of Math Characters in Computer Programs

It can be very useful to have typical mathematical symbols available in computer programs.  To realize the full potential of supporting mathematical symbols as part of identifiers,  a development environment should display the desired characters in both edit and debug windows. A preprocessor can translate MathML, for example, into C++, but it will not be able to make the debug windows use the math-oriented characters since the language cannot handle the underlying Unicode characters. Java has made an important step in this direction by allowing Unicode in identifiers. The mathematical alphanumeric symbols make this approach quite powerful for the user with relatively little effort for compilers.

The advantages of using Unicode characters directly in computer program identifiers are at least threefold:

1. Many formulas in document files can be programmed simply by copying them into a program file and inserting appropriate multiplication dots. This dramatically reduces coding time and errors.
2. The use of the same notation in programs and the associated journal articles and books leads to an unprecedented level of self-documentation.
3. In addition to providing useful tools for the present, these proposed initial steps ease the way towards the ultimate goal of teaching computers to understand and use arbitrary mathematical expressions.

For more information on identifiers and syntax characters see Section 5.15 Identifiers in [U4.0] as well as Annex 7, Programming Language Identifiers in [Normalization]. A Unicode Technical Report on Identifier and Syntax Patterns [Identifier] is in preparation.

4.4 Recognizing Mathematical Expressions

It is possible to use a number of heuristics for identifying mathematical expressions and treating them accordingly, for example to tag expressions input as plain text with a rich-text math style. Such heuristics are not foolproof, but they lead to the most popular choices. Ultimately the approach could be used in post-entry correction . The user could then override cases that were tagged incorrectly. A math style would connect in a straightforward way to appropriate MathML tags.

 The basic idea is that math characters identify themselves as such and potentially identify their surrounding characters as math characters as well. For example, the fraction (U+2044) and ASCII slashes, symbols in the range U+2200 through U+22FF, the symbol combining marks (U+20D0..U+20FF), and in general, Unicode characters with the math property, identify the characters immediately surrounding them as parts of mathematical expressions. See also Section 3.1.1 Strongly Mathematical Characters.

If Latin letter mathematical variables are already given in one of the math alphabets, they are considered parts of math expressions. If they are not, one can still have some recognition heuristics as well as the opportunity to italicize appropriate variables. Specifically ASCII letter pairs surrounded by whitespace are often mathematical expressions, and as such should be converted to using math italics. If a letter pair fails to appear in a list of common English and European two-letter words, it is treated as a mathematical expression and converted to italics. Many Unicode characters are not mathematical in nature and suggest that their neighbors are not parts of mathematical expressions.

Strings of characters containing no white space but containing one or more unambiguous mathematical characters are generally treated as mathematical expressions. Certain two-, three-, and four-letter words inside such expressions should not use italics. These include trigonometric function names like sin and cos, as well as ln, cosh, etc. Words or abbreviations that are often used as subscripts, also should not be italicized, even when they clearly appear inside mathematical expressions.

4.5 Some Examples of Mathematical Notation

This section gives some additional, but still relatively straightforward examples of mathematical notation for the benefit of readers not familiar with it. The simple built-up fraction

appears in inline text as (abc)/d; similarly the inline text (a+c)/d could appear as

 

in display. For the ratio

,

an inline format is α₂³/ (β₂³ + δ₂³). In equations such as

 

the size of the integral or bracket scales with the size of the enclosed text. This example also shows the positioning of multiple sub- and superscripts as well as the positioning of limit expressions on the integral.

Punctuation following math in display is commonly on the local baseline or centerline.

5 Mathematical classification

The data file [Data] provides a classification of characters by primary usage in mathematical notation. The classes used in this file are defined as follows:

Table A-1 Classes of Mathematical Characters

Class	Name	Comments
N	Normal	This includes all the digits and symbols requiring only one form
A	Alphabetic
B	Binary
C	Close	Paired with opening delimiter
D	Diacritic
F	Fence	Unpaired delimiter
O	Open	Paired with closing delimiter
L	Large	N-Ary or Large operator, often takes limits
P	Punctuation
R	Relation	Includes arrows

The same file also contains mappings to standard entity sets commonly used for SGML and MathML documents.

References

[Charts]	The online code charts can be found at http://www.unicode.org/charts/ An index to characters names with links to the corresponding chart is found at http://www.unicode.org/charts/charindex.html
[Data]	Classification of math characters by usage: MathClass-6.txt
[EAW]	Unicode Standard Annex #11, East Asian Width. http://www.unicode.org/unicode/reports/tr11 For a definition of East Asian Width
[FAQ]	Unicode Frequently Asked Questions http://www.unicode.org/unicode/faq/ For answers to common questions on technical issues.
[Feedback]	To report errors or submit suggestions please use http://www.unicode.org/reporting.html
[Glossary]	Unicode Glossary http://www.unicode.org/glossary/ For explanations of terminology used in this and other documents.
[Identifier]	Unicode Technical Report #31: Identifier and Syntax Patterns (in preparation) http://www.unicode.org/reports/tr31/
[LaTeX]	LaTeX: A Document Preparation System, User's Guide & Reference Manual, 2nd edition, by Leslie Lamport (Addison-Wesley, 1994; ISBN 1-201-52983-1)
[Math]	Math Property in the Unicode Character Database. For more information see http://www.unicode.org/Public/UNIDATA/UCD.html#Math
[MathML]	Mathematical Markup Language (MathML™)  Version 2.0 . (W3C Recommendation, revision of 21 February 2001) Editors: David Carlisle, Patrick Ion, Robert Miner and Nico Poppelier. http://www.w3.org/TR/MathML2/ Slated to go into a second edition at the time of this writing.
[Meystre]	P. Meystre and M. Sargent III (1991), Elements of Quantum Optics, Springer-Verlag
[Normalization]	Unicode Standard Annex #15: Unicode Normalization Forms http://www.unicode.org/unicode/reports/tr15/
[OpenMath]	The OpenMath Standard, 1.0, see: http://www.openmath.org/cocoon/openmath/standard/index.html
[PropMod]	Unicode Technical Report #23: The Unicode Character Property Model, http://www.unicode.org/reports/tr23/
[Reports]	Unicode Technical Reports http://www.unicode.org/unicode/reports/ For information on the status and development process for technical reports, and for a list of technical reports.
[SI]	International System of Units (SI) - Le Système International d'Unités. The metric system of weights and measures based on the meter, kilogram, second and ampere, Kelvin and candela. For background information see http://physics.nist.gov/cuu/Units/index.html.
[StdVar]	For the formal list of Standardized Variants in the Unicode Character Database, see: http://www.unicode.org/Public/UNIDATA/StandardizedVariants.html (with glyphs) or http://www.unicode.org/Public/UNIDATA/StandardizedVariants.txt
[STIX]	STIX Project Home Page: http://www.ams.org/STIX/
[TeX]	Donald E. Knuth,The TE Xbook, (Reading, Massachusetts: Addison-Wesley 1984) The TE Xbook is the manual for Donald Knuth's TE X composition system. Appendix G describes the somewhat idiosyncratic mechanism used by TE X to accomplish the composition of mathematical notation; it is based on the principles laid out in [Chaundy, Wick, Swanson], as well as on examination of a large number of published samples that demonstrated Knuth's style preferences. Donald E. Knuth, TE X, the Program, Volume B of Computers & Typesetting, (Reading, Massachusetts: Addison-Wesley 1986) See also http://www.ams.org/tex/publications.html
[U3.0]	The Unicode Standard, Version 3.0, (Reading, MA, Addison-Wesley, 2000. ISBN 0-201-61633-5)) or online as http://www.unicode.org/unicode/uni2book/u2.html
[U3.1]	Unicode Standard Annex #27: Unicode 3.1 http://www.unicode.org/unicode/reports/tr27/
[U3.2]	Unicode Standard Annex #28: Unicode 3.2 http://www.unicode.org/unicode/reports/tr28/
[U4.0]	The Unicode Standard, Version 4.0, (Reading, MA, Addison-Wesley, 2003. ISBN 0-321-18578-1) or online as http://www.unicode.org/versions/Unicode4.0.0/
[U4.0.1]	Unicode 4.0.1, http://www.unicode.org/versions/Unicode4.0.1/
[UCD]	Unicode Character Database. http://www.unicode.org/Public/UNIDATA/UCD.html For and overview of the Unicode Character Database and a list of its associated files
[Unicode]	The latest version of the Unicode Standard can be found at http://www.unicode.org/versions/latest
[UXML]	Unicode Technical Report #20: Unicode in XML and other Markup Languages http://www.unicode.org/unicode/reports/tr20/
[Versions]	Versions of the Unicode Standard http://www.unicode.org/unicode/standard/versions/ For details on the precise contents of each version of the Unicode Standard, and how to cite them.
[XML]	Tim Bray, Jean Paoli, C. M. Sperberg-McQueen, Eve Maler, Eds., Extensible Markup Language (XML) 1.0 (Second Edition), W3C Recommendation 6-October-2000, <http://www.w3.org/TR/REC-xml>

Additional References

The following four books are entirely about the composition of mathematics
[Chaundy]	T.W. Chaundy, P.R. Barrett and Charles Batey, The Printing of Mathematics, (London: Oxford University Press 1954, third impression, 1965) [out of print]
[Wick]	Karel Wick, Rules for Type-setting Mathematics, (Prague: Publishing House of the Czechoslovak Academy of Sciences 1965) [out of print]
[Swanson]	Ellen Swanson, Mathematics into Type, (Providence, RI: American Mathematical Society, 1971, revised 1979, updated 1999 by Arlene O'Sean and Antoinette Schleyer) The original edition is based on "traditional" composition (Monotype and "cold type", that is Varityper and Selectric Composer); the 1979 edition adds material for computer composition, and the 1999 edition mostly assumes TEX or a comparably advanced system.
[Byrd]	Mathematics in Type, (Richmond, VA: The William Byrd Press 1954) [out of print]
The following books contain material on mathematical composition, but it is not the principal topic covered
[Maple]	The Maple Press Company Style Book, (York, PA: 1931) (reprinted 1942) Contains sections on fractions; mathematical signs; simple equations; alignment of equations; braces, brackets and parentheses; integrals, sigmas and infinities; hyphens, dashes and minus signs; superiors and inferiors; ... [out of print]
[Manual]	A Manual of Style, Twelfth Edition, Revised (Chicago: The University of Chicago Press 1969) A chapter "Mathematics in Type" was produced using the Penta (computer) system.

Acknowledgements

Patrick Ion graciously reviewed the text of this report and suggested many improvements. Rick McGowan redrew many of the figures. Magda Danish managed the collection of glyph images for the tables of negated operators.

Modifications

Changes from Tracking Number 5

Rewrote the Overview. Brought table 2.7 into alignment with the standardized variant listing in the Unicode Character Database: 2278 and 2279 have been moved to table 2.5. 2225 was removed from table 2.7 since there is now a new character 2AFD and the variation is no longer needed. Added Table 2.3. Added Section 2.15. Removed section 3.3. Renumbered the appendix to become Section 5. Moved the actual classification of characters into a separate data file. Updated references to the Unicode Standard to Unicode 4.0 where appropriate. Improved the layout of tables 2.5, 2.6 and 2.7. Many minor spelling, wording and formatting fixes throughout. Updated status and conformance section. Completed the classification in sections 3.1.1 and 3.1.2.  Changed header and improved visual layout of the data file. (AF)

Changes from Tracking Number 4

Added section 2.16. Added section 3.3. Removed section 5 on plain text math. Added Appendix A. Added a few typographical samples. (AF)

Changes from Tracking Number 3

Fixed some CSS issues.

Changes from Tracking Number 2

Changed many special symbols to NCRs. Fixed an HTML glitch affecting table formatting and fixed contents of Table 2.4. A number of additional typographical mistakes and inconsistencies in the original proposed draft have been corrected. Merged duplicated text in section 2.7 and made additional revisions to further align the text with Unicode 3.2. Minor wording changes for clarity or consistency throughout.  (bnb/AF).

Changes from Tracking Number 1

A large number of minor, but annoying typographical and HTML mistakes in the original proposed draft have been corrected. This includes the occasional mistaken character name or code point. Additional entries were made to the references section and new bookmarks and internal links have been added to refer to them from the text.  Other minor improvements to the text and formatting have been carried out. Added section 2.10 and revised the first paragraph of section 2 to bring the text inline with Unicode 3.2 (bnb/AF)

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