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Technical
Reports| Technical
Reports |
| Version | 1.0 |
| Authors | Barbara Beeton (bnb@ams.org), Asmus Freytag (asmus@unicode.org), Murray Sargent III (murrays@microsoft.com) |
| Date | 2002-05-08 |
| This Version | http://www.unicode.org/unicode/reports/tr25/tr25-5.html |
| Previous Version | http://www.unicode.org/unicode/reports/tr25/tr25-4.html |
| Latest Version | http://www.unicode.org/unicode/reports/tr25 |
| Tracking Number | 5 |
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, computer algebra languages like OpenMath, internal representations of mathematics in systems like Mathematica and MathCAD, computer programs, and plain text. This technical report describes the Unicode mathematics character groups and gives some of their default math properties.
Status
This document has been approved by the Unicode Technical Committee for public review as a Draft Unicode Technical Report. Publication does not imply endorsement by the Unicode Consortium. This is a draft document which may be updated, replaced, or superseded by other documents at any time. This is not a stable document; it is inappropriate to cite this document as other than a work in progress.
Please send comments to the authors. A list of current Unicode Technical Reports is found on http://www.unicode.org/unicode/reports/. For more information about versions of the Unicode Standard, see http://www.unicode.org/unicode/standard/versions/.
The References provide related information that is useful in understanding this document. Please mail corrigenda and other comments to the author(s).
Appendix A: Mathematical Character
Classification
References
Modifications
This technical report starts with a discussion of the mathematics character repertoire incorporating the relevant block descriptions of the Unicode Standard [TUS]. Associated character properties are discussed next, including a number of properties that are not yet part of the Unicode Standard. Character classifications by usage, by typography, and by precedence are given. Some implementation guidelines for input methods and use of Unicode math characters in programming languages are presented next.
Unicode 3.2 provides a quite complete set of standard math characters to support publication of mathematics on and off the web. Specifically, Unicode 3.1 introduced 996 new alphanumeric symbols and Unicode 3.2 introduces 591 new symbols, in addition to the 340 math-specific symbols already encoded in Unicode 3.0, for a total of 1927 mathematical symbols. This repertoire is the result of input from many sources, notably from the STIX Project (Scientific and Technical Information Exchange) [STIX], a cooperation of mathematical 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 this work must remain incomplete; mathematicians and other scientists are continually inventing new mathematical symbols and the plan is to add them as they become 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 and the working group lobbied in favor of the inclusion of the new characters. In addition, the new characters lend themselves to a useful plain text encoding of mathematics (see Sec. 4) that is much more compact than MathML or TEX, the typesetting language and program designed by Donald Knuth [TeX].
The Mathematical Alphanumeric Symbols block (U+1D400 – U+1D7FF) contains a large extension of letterlike symbols used 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, 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.
Basic Set of Alphanumeric Characters. Mathematical notation uses a basic set of mathematical alphanumeric characters which consists of:
Only unaccented forms of the letters are used for mathematical notation, because general accents such as the acute accent would interfere with common mathematical diacritics. Examples of common mathematical diacritics that can interfere with general accents are the circumflex, macron, or the single or double dot above, the latter two of which are 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.
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.
Additional Characters. In addition to this basic set, mathematical notation also uses the four Hebrew-derived characters (U+2135 – U+2138). Occasional uses of other alphabetic and numeric characters are known. Examples include U+0428 cyrillic capital letter sha, U+306E hiragana letter no , and Eastern Arabic-Indic digits (U+06F0 – U+06F9). However, these characters are used in only the 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
|
Math Style |
Characters from Basic Set |
Location |
|
plain (upright, serifed) |
Latin, Greek and digits |
BMP |
|
bold |
Latin, Greek and digits |
Plane 1 |
|
italic |
Latin and Greek |
Plane 1* |
|
bold italic |
Latin and Greek |
Plane 1 |
|
script (calligraphic) |
Latin |
Plane 1* |
|
bold script (calligraphic) |
Latin |
Plane 1 |
|
Fraktur |
Latin |
Plane 1* |
|
bold Fraktur |
Latin |
Plane 1 |
|
double-struck |
Latin and digits |
Plane 1* |
|
sans-serif |
Latin and digits |
Plane 1 |
|
sans-serif bold |
Latin, Greek and digits |
Plane 1 |
|
sans-serif italic |
Latin |
Plane 1 |
|
sans-serif bold italic |
Latin and Greek |
Plane 1 |
|
monospace |
Latin and digits |
Plane 1 |
* 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 25 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.
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. 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, e.g. uppercase A and uppercase Alpha (A). Their use is normally avoided in mathematical notation unless no confusion is possible in a given context.
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 is unified with the pre-1949 symbol for the German currency unit Mark and U+2113 is unified with the common non-SI symbol for the liter [SI]. 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 per recommendation by the American Mathematical Society. For consistency, other script characters in the Letterlike Symbols block are now shown in the same typestyle.
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 was prepared in consultation with the American Mathematical Society and leading mathematical publishers, and shows much simpler forms that are derived from the forms written on a blackboard. However, both serifed and non-serifed forms can be used in mathematical texts, and inline fonts are found in works published by certain publishers. There is no intention to support such stylistic preference via character encoding, therefore only one set of double struck mathematical alphanumeric symbols have been encoded.
With Unicode 3.0 and the concurrent second edition of ISO/IEC 10646-1, the reference glyphs for U+03C6 GREEK LETTER SMALL PHI and U+03D5 GREEK PHI SYMBOL were swapped. In ordinary Greek text, the character U+03C6 is used exclusively, although this characters has considerably 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).
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 reversed assignment of representative glyphs in versions of the Unicode Standard prior to Unicode 3.0 had 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 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.
In Unicode 3.2 the reference glyphs for 2278 neither less-than nor greater-than and 2279 neither greater-than nor less-than are changed from using a vertical cancellation to using a slanted cancellation. This change was made in order to match their the long standing canonical decompositions for these characters, which use 0338 Combining long solidus overlay. Irrespective of this change to the reference glyphs, the symmetric forms using the vertical stroke are acceptable glyph variants. Using 2278 or 2279 followed by FE00 Variation Selector-1 (VS1) will request these upright variants explicitly, as will using 2275 or 2276 followed by 20D2 combining long vertical line overlay.
Unless fonts are created with the intention to add support for both forms (via VS1 for the upright forms) there is no need to revise the glyphs for 2287 and 2279 in existing fonts: the glyphic range implied by using the base character alone encompasses both shapes.
Mathematical characters can be located by looking in the blocks that contain such characters 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 |
Characters |
|
Basic Latin |
U+0021–U+007E |
Variables, operators, digits* |
|
Greek |
U+0370–U+03FF |
Variables* |
|
General Punctuation |
U+2000–U+206F |
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 |
|
Mathematical Alphanumeric Symbols |
U+1D400–U+1D7FF |
Variables and digits |
|
Other blocks |
… |
Characters for occasional use |
*This block contains non-mathematical characters as well.
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 greek letter capital omega is the canonical equivalent of U+2126 and its use is preferred. Latin letters duplicated include 212A kelvin sign and U+212B 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 adds two new mathematical angle bracket characters (U+27E8 and U+27E9) that are unequivocally intended for mathematical use.
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 there are multiple ways of expressing the character: as precomposed or as a sequence of base character and combining mark. It would be nice to have a single way to represent any given character, since this 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, the normalization forms presented there are not 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. This rule concurs with Normalization Form C (NFC) used on the web. If a negated operator is needed that does not have a precomposed form, the character U+0338 combining long solidus overlay can be used to indicate negation.
On the other hand, for accented alphabetic characters used as variables, ideally only decomposed sequences are used since there are no precomposed math alphanumerical symbols.
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. Note that this approach does not concur with Normalization Form C 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 such letters when used in mathematical expressions and use a single algorithm to place embellishments.
The Unicode blocks U+2200 – U+22FF and U+2A00 – U+2AFF 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.
Semantics. Mathematical operators often have more than one meaning 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 14, Code Charts in The Unicode Standard, Version 3.0 [TUS] 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 2.17 "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. Because no semantic distinction applies, 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. U+22EE – U+22F1 are a set of ellipses used in matrix notation.
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 TEX and their use with XML documents is discouraged, see Unicode Technical Report #20, "Unicode in XML and other Markup Languages" [UXML].
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, Chapter 14, Code 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.
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 show this kind of size relation.
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. [ED: TBD: summary picture]
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 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 reference glyphs of existing characters shown in the charts. Mathematical fonts may need to deviate in positioning of these triangles.
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, ♥, ♠, etc. can be found as operators as well as subscripts.
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 2.3 Use of Symbol Pieces
|
|
2-row |
3-row |
5-row |
|
Summation |
23B2, 23B3 |
|
|
|
Integral |
2320, 2321 |
2320, 23AE, 2321 |
2320, 3×23AE, 2321 |
|
Left Parenthesis |
239B, 239D |
239B, 239D |
239B, 3×239C, 239D |
|
Right Parenthesis |
239E, 23A0 |
239E, 239F, 23A0 |
239E, 3×239F, 23A0 |
|
Left Bracket |
23A1, 23A3 |
23A1, 23A2, 23A4 |
23A1, 3×23A2, 23A3 |
|
Right Bracket |
23A4, 23A6 |
23A4, 23A5, 23A6 |
23A4, 3×23A5, 23A6 |
|
Left Brace |
23B0, 23B1 |
23A7, 23A8, 2389 |
23A7, 23AA, 23A8, 23AA, 2389 |
|
Right Brace |
23B1, 23B0 |
23AB, 23AC, 23AD |
23AB, 23AA, 23AC, 23AA, 23AD |
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.
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 aij means ai, 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 typeset a small space between the variables.
Similarly an expression like mc2 implies that the mass m multiplies the square of the speed c. To represent the implied multiplication in mc2, one inserts a non-displaying U+2061invisible times between the m and the c. A related case is the use of U+2062 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 fi j(cos(ab)), which means the same as fi,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.
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.
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 variants of negated mathematical symbols that can be realized via composition, by using U+20D2 COMBINING LOng VERTICAL LINE OVERLAY for negation instead of the slanted U+0338 combining LONG solidus overlay. This contrasts to the use of U+FE00 VARIATION SELECTOR-1 for those symbols for which only a partial vertical stroke is used, and for which the use of U+20D2 would not give the intended result. The part of the description in SMALL CAPS is the character name of the corresponding standard character, with the part in lower case indicating the variation in appearance.
Table 2.4 Negated relations using vertical line overlay
| Std Symbol | Alternate Symbol | Description | ||
 |
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 |
 |
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 following table lists negated forms of mathematical relations that can only be encoded by using U+0338 COMBINING LONG SOLIDUS OVERLAY or U+20D2 COMBINING LONG VERTICAL LINE OVERLAY. 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 2.5 Using vertical line or solidus overlay
| Glyph / Sequence | Description | |
 |
220A,0338 | negated SMALL ELEMENT OF |
 |
220A,20D2 | negated SMALL ELEMENT OF with vertical stroke |
 |
220D,0338 | negated SMALL CONTAINS AS MEMBER |
 |
220D,20D2 | negated SMALL CONTAINS AS MEMBER with vertical stroke |
 |
2242,0338 | negated MINUS TILDE |
 |
2242,20D2 | negated MINUS TILDE with vertical stroke |
 |
2263,0338 | negated STRICTLY EQUIVALENT TO |
 |
2263,20D2 | negated STRICTLY EQUIVALENT TO with vertical stroke |
 |
2266,0338 | negated LESS-THAN OVER EQUAL TO |
 |
2266,20D2 | negated LESS-THAN OVER EQUAL TO with vertical stroke |
 |
2267,0338 | negated GREATER-THAN OVER EQUAL TO |
 |
2267,20D2 | negated GREATER-THAN OVER EQUAL TO with vertical stroke |
 |
22F7,0338 | negated ELEMENT OF WITH OVERBAR |
 |
22F7,20D2 | negated ELEMENT OF WITH OVERBAR with vertical stroke |
 |
22FE,0338 | negated SMALL CONTAINS WITH OVERBAR |
 |
22FE,20D2 | negated SMALL CONTAINS WITH OVERBAR EQUALS with vertical stroke |
 |
2A6C,20D2 | negated SIMILAR MINUS SIMILAR |
 |
2A6C,0338 | negated SIMILAR MINUS SIMILAR with vertical stroke |
 |
2A70,0338 | negated APPROXIMATELY EQUAL OR EQUAL TO |
 |
2A70,20D2 | negated APPROXIMATELY EQUAL OR EQUAL TO with vertical stroke |
 |
2A7D,0338 | negated LESS-THAN OR SLANTED EQUAL TO |
 |
2A7D,20D2 | negated LESS-THAN OR SLANTED EQUAL TO with vertical stroke |
 |
2A7E,0338 | negated GREATER-THAN OR SLANTED EQUAL TO |
 |
2A7E,20D2 | negated GREATER-THAN OR SLANTED EQUAL TO with vertical stroke |
 |
2A95,0338 | negated SLANTED EQUAL TO OR LESS-THAN |
 |
2A95,20D2 | negated SLANTED EQUAL TO OR LESS-THAN with vertical stroke |
 |
2A96,0338 | negated SLANTED EQUAL TO OR GREATER-THAN |
 |
2A96,20D2 | negated SLANTED EQUAL TO OR GREATER-THAN with vertical stroke |
 |
2A99,0338 | negated DOUBLE-LINE EQUAL TO OR LESS-THAN |
 |
2A99,20D2 | negated DOUBLE-LINE EQUAL TO OR LESS-THAN with vertical stroke |
 |
2A9A,0338 | negated DOUBLE-LINE EQUAL TO OR GREATER-THAN |
 |
2A9A,20D2 | negated DOUBLE-LINE EQUAL TO OR GREATER-THAN with vertical stroke |
 |
2AC5,0338 | negated SUBSET OF ABOVE EQUALS SIGN |
 |
2AC5,20D2 | negated SUBSET OF ABOVE EQUALS SIGN with vertical stroke |
 |
2AC6,0338 | negated SUPERSET OF ABOVE EQUALS SIGN |
 |
2AC6,20D2 | negated SUPERSET OF ABOVE EQUALS SIGN with vertical stroke |
The variation selector VS1 is used to represent well-defined variants of particular math symbols. The variations include: different slope of 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 Variation Selector 1 (VS1 or U+FE00) to mathematical operator base characters. To select one of the predefined variations, follow the base character with the variation selector. Only the valid, recognized combinations are listed in the table of standardized variants. All combinations not listed here are unspecified and are reserved for future standardization; no conformant process may interpret them as standardized variants. For more information, see Section 13.7, Variation Selectors, in Unicode 3.2 [U3.2].
Using a variation selector allows users and font designers to make a distinction between alternate glyphs 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 used as symbols. See Section 2.31, "Reference Glyphs for Greek phi".
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.
In extremis, the Unicode Standard considers the variation selector somewhat optional. Processes or fonts that cannot support it should yield acceptable results by ignoring the variation selector.
Table 2.6 Variants of Mathematical Symbols using VS1
|
2268 + VS1 |
LESS-THAN BUT NOT EQUAL TO - with vertical stroke |
|
2269 + VS1 |
GREATER-THAN BUT NOT EQUAL TO - with vertical stroke |
|
22DA + VS1 |
LESS-THAN slanted EQUAL TO OR GREATER-THAN |
|
22DB + VS1 |
GREATER-THAN slanted EQUAL TO OR LESS-THAN |
|
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 |
|
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 |
|
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 |
|
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 |
|
2A3B + VS1 |
INTERIOR PRODUCT - tall variant with narrow foot |
|
2A3C + VS1 |
RIGHTHAND INTERIOR PRODUCT - tall variant with narrow foot |
| 2278 + VS1 | NEITHER LESS-THAN NOR GREATER-THAN with vertical stroke (*) |
| 2279 + VS1 | NEITHER GREATER-THAN NOR LESS-THAN with vertical stroke (*) |
|
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 |
|
2225 + VS1 |
Slanted PARALLEL TO |
|
2225 + VS1 + 20E5 |
Slanted PARALLEL TO with reverse slash |
|
2229 + VS1 |
INTERSECTION with serifs |
|
222A + VS1 |
UNION with serifs |
|
2293 + VS1 |
SQUARE CAP with serifs |
|
2294 + VS1 |
SQUARE CUP with serifs |
* The reference glyphs shown in the code charts [Charts] have been revised to show the slanted forms - this matches their existing decomposition using U+0338 combining long solidus overlay (see section 2.32 for more information).
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 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 Area (U+E000 – U+F8FF) can be used for such nonstandard symbols. It is 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.
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, <md> for one or more digits comprising a number, and <mi> for identifier. TeX [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. For example, 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 property is useful in heuristics that seek to identify mathematical expressions in plain text.
[ED: This classification is a work in progress.]
Strong mathematical characters are all characters that are primarily used for mathematical notation. This includes all characters with the math property [Sec. 4.9 of The Unicode Standard] [ED: Check that this is true after extension of the properties to the new characters.] with the following exceptions:
002D HYPHEN-MINUS
and the following additions [ED: any?]
These characters often 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.
All other Unicode characters. Many of these may occur in mathematical texts, though often not as part of the mathematical expressions themselves.
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 as a relational operator.
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 serifed text to distinguish them from products of variables. Such names should not use the math alphanumeric characters. The upright uppercase Greek are favored over the italic ones. In Europe, upright d, D, e, and i are used for the two differential, exponential, and imaginary part 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.
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/below when displayed out-of-line and right top/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 |
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.
Digits include 0-9 in various styles. All digits of a particular style have the same width.
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[.
Fences are similar to opening and closing delimiters, but are not paired.
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 and use U+20D2, not U+20D3. 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.
Operator precedence reduces the notational complexity of expressions and is commonly used for this purpose in computer programming languages, calculus, and algebra. Assigning consistent default precedence to the operators allows software to autmoate the transition from data input (or plain text) to fully marked up forms of mathematical data such as TeX or MATHML.
Operands in subscripts, superscripts, fractions, roots, boxes, etc. are defined in part in terms of operators and operator precedence. While such notions are very familiar to mathematically oriented people, some of the symbols that are defined here as operators might surprise one at first. Most notably, the character SPACE is an important operator when interpreting mathematical text encoded in plain text.
Table A.1 A list of common operators ordered by precedence
| Operators By Precedence |
|
FF CR \ |
|
( [ { |
|
) ] } | |
|
Space " . , = - + LF Tab |
|
/ * × · • · ½ |
|
n √ |
|
∫ ∑ ∏ |
|
↓ ↑ |
Here Tab = U+0009, LF = U+000A, FF = U+000C, and CR = U+000D.
As in arithmetic, operators have precedence, which streamlines the interpretation of operands. The operators are grouped above in order of increasing precedence, with equal precedence values on the same line. 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.
The following gives a list of the syntax for a variety of mathematical constructs.
| exp1/exp2 | Create a built-up fraction with numerator exp1and denominator exp2. Numerator and denominator expressions are terminated by operators such as / *] ) ↑↓ · and blank (can be overruled by enclosing in parentheses). The "/" is given by U+2044. |
| ↑exp1 | Superscript expression exp1. The
superscripts 0 - 9 + - ( ) exist as Unicode symbols.
Sub/superscripts expressions are terminated by / * ] ) ↑↓
· and blank. Sub/superscript operators associate right to left. |
| ↓exp1 | Subscript expression exp1. The subscripts 0 - 9 + - ( ) exist as Unicode symbols. |
| [exp1] | Surround exp1 with built-up brackets. Similarly for { } and ( ). |
| [exp1]↑exp2 | Surround exp1 with built-up brackets
followed by superscripted exp2 (moved up high enough). Similarly for { } and ( ). |
| √exp1 | Square root of exp1. |
| · | Small raised dot that is not intended to print. It is used to terminate an operand, such as in a subscript, superscript, numerator, or denominator, when other operators cannot be used for this purpose. Similar raised dots like • and · also terminate operands, but they are intended to print. |
| ∑↓exp1↑exp2 | Summation from exp1 to exp2. ↓exp1 and ↑exp2 are optional. |
| ∏↑exp1↑exp2 | Product from exp1to exp2. |
| ∫↑exp1↑exp2 | Integral from exp1 to exp2. |
| exp1½exp2 | Align exp1 over exp2 (like fraction without bar). Useful for building up matrices as a set of columns. |
Diacritics are handled using Unicode combining marks (U+0300 – U+036F, U+20D0 – U+20FF). Note that many more operators can be added to fill out the capabilities of the approach in representing mathematical expressions in Unicode plain (or almost plain) text.
If Normalization Form C is applied to mathematical text, some accents or overlays used with BMP alphabetic characters may be incorrectly composed with their base character. 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".
In view of the large number of characters used in mathematics, it is useful to give some discussion of input methods. The ASCII math symbols are easy to find, e.g., + - / * [ ] ( ) { }, but often need to be used as themselves.
Post-entry correction. From a syntax point of view, the official Unicode minus sign (U+2212) is certainly preferable to the ASCII hyphen-minus (U+002D) and the prime (U+2032) is preferable to the ASCII apostrophe (U+0027), 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.
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. 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 autocorrect 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 autocorrect 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 the on-screen keyboard(s), or directly into applications. On-screen keyboards and symbol boxes are valuable for entry of mathematical expressions and of Unicode text in general.
Unicode plain-text mathematics. One use of the plain-text format is as a math input method, both for search text and for general editing.
It can be very useful to have typical mathematical symbols available in computer programs (see Section A.3 "Using Plain-Text Mathematics in Programming Languages" for a more detailed discussion). A key point is that the compiler 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 unless it can handle the underlying Unicode characters. Java has made an important step in this direction by allowing Unicode variable names. The mathematical alphanumeric symbols allow this approach to go further with relatively little effort for compilers.
The advantages of using the Unicode plain text in computer programs 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 should help one figure out how to accomplish the ultimate goal of teaching computers to understand and use arbitrary 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 mathematics property, identify the characters immediately surrounding them as parts of math expressions.
If English 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.
[This section is still preliminary]
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, similar the inline text (a+c)/d appears as
.
For the ratio
,
the inline format is reads 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.
The classes used in this appendix are
| Class | Name | Comments |
| N | Numeric | This includes all the digits, but a lot of symbols |
| 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 following listing provides an early draft of the classification. [Please ignore the non-standard notation in the first column, format content and presentation of this listing will change in future versions].
uniq xref C entity set description
+0021 P excl ISONUM exclamation mark +0021 N fact factor