Technical Reports Unicode® Technical Report #25 UNICODE SUPPORT FOR MATHEMATICS Authors Barbara Beeton ([email protected]), Asmus Freytag ([email protected]), Murray Sargent III ([email protected]) Date 2015-07-31 This Version http://www.unicode.org/reports/tr25/tr25-14.pdf Previous Version http://www.unicode.org/reports/tr25/tr25-13.pdf Latest Version http://www.unicode.org/reports/tr25/ Source Document http://www.unicode.org/reports/tr25/tr25-14.docx Data http://www.unicode.org/Public/math/revision-14/ Revision 14 Summary The Unicode Standard includes virtually all standard characters used in mathematics. This set supports a wide variety of math usage on computers, including in document presentation lan- guages like TEX, in math markup languages like MathML and OpenMath, in internal represen- tations of mathematics for applications like Mathematica, Maple, and MathCAD, in computer programs, and in plain text. This technical report describes the Unicode support for mathemat- ics and gives some of the imputed default math properties for Unicode characters. Status A Unicode Technical Report (UTR) contains informative material. Conformance to the 푈푛푖푐표푑푒 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]. Re- lated information that is useful in understanding this document is found in the References. For the latest version of the Unicode Standard see [Unicode]. For a list of current Unicode Technical Reports see [Reports]. For more information about versions of the Unicode Standard, see [Ver- sions]. Contents UNICODE SUPPORT FOR MATHEMATICS .................................................................................................................. 1 Unicode Technical Report #25 1 Unicode Support for Mathematics 1. OVERVIEW ............................................................................................................................................................................................... 3 2. MATHEMATICAL CHARACTER REPERTOIRE ......................................................................................................................... 4 2.1 Mathematical Alphanumeric Symbols Block ........................................................................................................... 4 2.2 Mathematical Alphabets ......................................................................................................................................... 5 2.3 Fonts Used for Mathematical Alphabets ................................................................................................................. 8 2.3.1 Representative Glyphs for Greek Phi ............................................................................................................. 10 2.3.2 Representative Glyphs for U+2278 and U+2279 ............................................................................................ 10 2.4 Locating Mathematical Characters ........................................................................................................................ 11 2.5 Duplicated Characters ........................................................................................................................................... 11 2.6 Accented Characters .............................................................................................................................................. 12 2.7 Operators ............................................................................................................................................................... 13 2.8 Superscripts and Subscripts ................................................................................................................................... 15 2.9 Arrows ................................................................................................................................................................... 15 2.10 Delimiters ............................................................................................................................................................ 16 2.11 Geometrical Shapes ............................................................................................................................................. 18 2.12 Other Symbols ..................................................................................................................................................... 22 2.13 Symbol Pieces ...................................................................................................................................................... 22 2.14 Invisible Operators ............................................................................................................................................... 23 2.15 Fraction Slash and Other Diagonals ..................................................................................................................... 24 2.16 Other Characters ................................................................................................................................................. 25 2.17 Negations ............................................................................................................................................................. 26 2.18 Variation Selector ................................................................................................................................................ 28 2.19 Novel Symbols not yet in Unicode ....................................................................................................................... 29 3. MATHEMATICAL CHARACTER PROPERTIES ........................................................................................................................ 29 3.1 Classification by Degree of Mathematical Usage ........................................................................................... 30 3.1.1 Strongly Mathematical Characters ................................................................................................................. 30 3.1.2 Weakly Mathematical Characters .................................................................................................................. 31 3.1.3 Other .............................................................................................................................................................. 32 3.2 Classification by Typographical Behavior ....................................................................................................... 32 3.2.1 Alphabetic ...................................................................................................................................................... 32 3.2.2 Operators ....................................................................................................................................................... 33 3.2.3 Large Operators .............................................................................................................................................. 33 3.2.4 Digits ............................................................................................................................................................... 34 3.2.5 Delimiters ....................................................................................................................................................... 34 3.2.6 Fences ............................................................................................................................................................. 34 3.2.7 Combining Marks ........................................................................................................................................... 34 4. IMPLEMENTATION GUIDELINES ................................................................................................................................................ 35 4.1 Use of Normalization with Mathematical Text ...................................................................................................... 35 4.2 Bidirectional Layout of Mathematical Text ........................................................................................................... 36 4.3 Input of Mathematical and Other Unicode Characters ......................................................................................... 37 4.4 Use of Math Characters in Computer Programs .................................................................................................... 38 4.5 Recognizing Mathematical Expressions ................................................................................................................. 39 4.6 Some Examples of Mathematical Notation ........................................................................................................... 40 5. DATA FILES ........................................................................................................................................................................................... 41 5.1 Mathematical Classification ........................................................................................................................... 41 5.2 Mapping to other Standards .......................................................................................................................... 42 6. SECURITY CONSIDERATIONS ......................................................................................................................................................