Mathematical operators and symbols in Unicode: Difference between revisions

Revision as of 04:56, 4 July 2016

Latest revision as of 06:49, 10 June 2025

(66 intermediate revisions by 35 users not shown)

Line 1:

{{Short description|none}}

{{redirect|⊞|the key|Windows key}}

[[File:0009-cmls-2012_preview_ecran.jpg|thumb|300px|Blackboard at the Laurent Schwartz Center for Mathematics, [[École Polytechnique]]]]

{{SpecialChars}}

The [[Unicode]] Standard encodes almost all standard characters used in mathematics.<ref name="utr25">{{cite web | title = Unicode Technical Report #25: Unicode Support for Mathematics | publisher = The Unicode Consortium | date = 2 April 2012 | url = httphttps://www.unicode.org/reports/tr25/ | format = PDF | accessdateaccess-date = 14 August 2014}}</ref>

Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation.<ref name="utr25"/>

Mathematical [[OperatorOperation (mathematics)|Mathematical operators]] and [[List of mathematical symbols|symbols]] are in multiple [[Unicode block]]s. Some of these blocks are dedicated to, or primarily contain, mathematical characters while others are a mix of mathematical and non-mathematical characters. This article covers all Unicode characters with a derived property of "Math".<ref>{{cite web | title = Unicode Character Database: Derived Core Properties | publisher = The Unicode Consortium | date = 19 February 2014 | url = httphttps://www.unicode.org/Public/UCD/latest/ucd/DerivedCoreProperties.txt | accessdateaccess-date = 14 August 2014

}}</ref><ref>{{cite web | title = Unicode Technical Annex #44: Unicode Character Database | publisher = The Unicode Consortium | date = 25 September 2013 | url = httphttps://www.unicode.org/reports/tr44/ | format = PDF | accessdateaccess-date = 14 August 2014}}</ref>

<!-- Note that capitalization below intentionally uses Unicode block name capitalization instead of the normal Wikipedia sentence case for headings. -->

Line 12 ⟶ 11:

===Mathematical Operators block===

{{Main|Mathematical Operators|l1=Mathematical Operators (Unicode block)}}

The Mathematical Operators block (U+2200&ndash;U+22FF) contains characters for mathematical, logical, and set notation.

{{Unicode chart Mathematical Operators}}

Line 21 ⟶ 20:

{{Unicode chart Supplemental Mathematical Operators}}

===<span id="Mathematical Alphanumeric Symbols block" ></span>Mathematical Alphanumeric Symbols block===

{{Main|Mathematical Alphanumeric Symbols|l1=Mathematical Alphanumeric Symbols (Unicode block)}}

The Mathematical Alphanumeric Symbols block (U+1D400&ndash;U+1D7FF) contains Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The reserved code points (the "holes") in the alphabetic ranges areup filledto byU+1D551 previously definedduplicate characters in the Letter like Symbols block [[#Letterlike Symbols block|shownLetterlike belowSymbols block]]. In order, these are: {{not a typo|ℎ / ℬ ℰ ℱ ℋ ℐ ℒ ℳ ℛ / ℯ ℊ ℴ / ℭ ℌ ℑ ℜ ℨ / ℂ ℍ ℕ ℙ ℚ ℝ ℤ}}.

{{Unicode chart Mathematical Alphanumeric Symbols}}

===<span id="Letterlike Symbols block" ></span>Letterlike Symbols block===

{{Main|Letterlike Symbols|l1=Letterlike Symbols (Unicode block)}}

The Letterlike Symbols block (U+2100&ndash;U+214F) includes variables. Most alphabetic math symbols are in the Mathematical Alphanumeric Symbols block [[#Mathematical Alphanumeric Symbols block|shown above]].

The math subset of this block is U+2102, U+2107, U+210A&ndash;U+2113, U+2115, U+2118&ndash;U+2119211D, U+2124, U+2128&ndash;U+2129, U+212C&ndash;U+212D, U+212F&ndash;U+2131, U+2133, &ndash;U+21352138, U+213C&ndash;U+2149, and U+214B.<ref>See https://www.unicode.org/Public/UCD/latest/ucd/DerivedCoreProperties.txt</ref>

{{Unicode chart Letterlike Symbols}}

Line 49 ⟶ 48:

The Miscellaneous Technical block (U+2300&ndash;U+23FF) includes braces and operators.

The math subset of this block is U+2308&ndash;U+230B, U+2320-&ndash;U+2321, U+237C, U+239B-&ndash;U+23B5, 23B7, U+23D0, and U+23DC-&ndash;U+23E2.

{{Unicode chart Miscellaneous Technical}}

===Geometric Shapes block===

{{Main|Geometric Shapes|l1=Geometric Shapes (Unicode block)}}

The Geometric Shapes block (U+25A0&ndash;U+25FF) contains geometric shape symbols.

The math subset of this block is U+25A0&ndash;25A1, U+25AE&ndash;25B7, U+25BC&ndash;25C1, U+25C6&ndash;25C7, U+25CA&ndash;25CB, U+25CF&ndash;25D3, U+25E2, U+25E4, U+25E7&ndash;25EC, and U+25F8&ndash;25FF.

{{Unicode chart Geometric Shapes}}

===Miscellaneous Symbols and Arrows block===

{{Main|Miscellaneous Symbols and Arrows|l1=Miscellaneous Symbols and Arrows (Unicode block)}}

The Miscellaneous Symbols and Arrows block (U+2B00&ndash;U+2BFF Arrows) contains arrows and geometric shapes with various fills.

The math subset of this block is U+2B30&ndash;2B44 and U+2B47&ndash;2B4C.

{{Unicode chart Miscellaneous Symbols and Arrows}}

===Arrows block===

{{Main|Arrows (Unicode block)}}

The Arrows block (U+2190&ndash;U+21FF) contains line, curve, and semicircle arrows and arrow-like operators.

The math subset of this block is U+2190&ndash;U+21A7, U+21A9&ndash;U+21AE, U+21B0&ndash;U+21B1, U+21B6&ndash;U+21B7, U+21BC&ndash;U+21DB, U+21DD, U+21E4&ndash;U+21E5, U+21F4&ndash;U+21FF.<ref>More symbols are supported by TeX math packages, see e.g. [https://tug.ctan.org/fonts/xcharter-math/unimath-xcharter.pdf#page=47 Will Robertson, Symbols defined by unicode-math].</ref>

{{Unicode chart Arrows}}

Line 80 ⟶ 74:

The Supplemental Arrows-B block (U+2900&ndash;U+297F) contains arrows and arrow-like operators (arrow tails, crossing arrows, curved arrows, and harpoons).

{{Unicode chart Supplemental Arrows-B}}

===Miscellaneous Symbols and Arrows block===

{{Main|Miscellaneous Symbols and Arrows|l1=Miscellaneous Symbols and Arrows (Unicode block)}}

The Miscellaneous Symbols and Arrows block (U+2B00&ndash;U+2BFF Arrows) contains arrows and geometric shapes with various fills.

The math subset of this block is U+2B30&ndash;U+2B44, U+2B47&ndash;U+2B4C.<ref>The quadruple arrows U+2B45 and U+2B46 are supported by TeX math packages, per [https://tug.ctan.org/fonts/xcharter-math/unimath-xcharter.pdf#page=75 Will Robertson, Symbols defined by unicode-math].</ref>

{{Unicode chart Miscellaneous Symbols and Arrows}}

===Combining Diacritical Marks for Symbols block===

Line 94 ⟶ 95:

==Characters in other blocks==

Mathematical characters also appear in other blocks. Below is a list of these characters as of Unicode version 9{{Unicode version|version=16.0}}:

{|

Line 100 ⟶ 101:

| <!-- newspaper column 1 -->

*[[Basic Latin (Unicode block)|Basic Latin block]]

{|

**U+002B PLUS SIGN: {{Large|+}}

**| U+003C002B LESS-THAN SIGN:|| {{Large|&lt;+}} || PLUS SIGN

|-

**U+003D EQUALS SIGN: {{Large|1==}}

| U+002D || {{Large|-}} || HYPHEN-MINUS<ref>As per [https://www.unicode.org/Public/16.0.0/ucd/DerivedCoreProperties.txt Unicode 16.0.0], the ASCII hyphen-minus is not a mathematical symbol. To express the minus sign in math, {{unichar|2212|minus sign}} is used instead.</ref>

**U+003E GREATER-THAN SIGN: {{Large|&gt;}}

|-

**U+005E CIRCUMFLEX ACCENT: {{Large|^}}

**| U+007C003C VERTICAL LINE:|| {{Large|{{!}}&lt;}} || LESS-THAN SIGN

|-

**U+007E TILDE: {{Large|~}}

| U+003D || {{Large|1==}} || EQUALS SIGN

|-

| U+003E || {{Large|&gt;}} || GREATER-THAN SIGN

|-

| U+005E || {{Large|^}} || CIRCUMFLEX ACCENT

|-

| U+007C || {{Large|<nowiki>|</nowiki>}} || VERTICAL LINE

|-

| U+007E || {{Large|~}} || TILDE

|}

*[[Latin-1 Supplement (Unicode block)|Latin-1 Supplement block]]

{|

**U+00AC NOT SIGN: {{Large|¬}}

**| U+00B100AC PLUS-MINUS SIGN:|| {{Large|±¬}} || NOT SIGN

|-

**U+00D7 MULTIPLICATION SIGN: {{Large|×}}

| U+00B0 || {{Large|°}} || DEGREE SIGN<ref>As per [https://www.unicode.org/Public/16.0.0/ucd/DerivedCoreProperties.txt Unicode 16.0.0], the degree sign is not a mathematical symbol, since it is a measurement unit symbol rather than a math symbol. Consistently, TeX packages support many non-math symbols. But this article is designed to cover only Unicode characters with a derived property of "Math".</ref>

**U+00F7 DIVISION SIGN: {{Large|÷}}

|-

*[[Greek and Coptic|Greek and Coptic block]]:

**| U+03D000B1 GREEK BETA SYMBOL:|| {{Large|ϐ±}} || PLUS-MINUS SIGN

|-

**U+03D1 GREEK THETA SYMBOL: {{Large|ϑ}}

| U+00D7 || {{Large|×}} || MULTIPLICATION SIGN

**U+03D2 GREEK UPSILON WITH HOOK SYMBOL: {{Large|ϒ}}

|-

**U+03D5 GREEK PHI SYMBOL: {{Large|ϕ}}

**| U+03F000F7 GREEK KAPPA SYMBOL:|| {{Large|ϰ÷}} || DIVISION SIGN

|}

**U+03F1 GREEK RHO SYMBOL: {{Large|ϱ}}

*[[Greek and Coptic|Greek and Coptic block]]

**U+03F4 GREEK CAPITAL THETA SYMBOL: {{Large|ϴ}}

{|

**U+03F5 GREEK LUNATE EPSILON SYMBOL: {{Large|ϵ}}

| U+03D0 || {{Large|ϐ}} || GREEK BETA SYMBOL

**U+03F6 GREEK REVERSED LUNATE EPSILON SYMBOL: {{Large|϶}}

|-

| U+03D1 || {{Large|ϑ}} || GREEK THETA SYMBOL

|-

| U+03D2 || {{Large|ϒ}} || GREEK UPSILON WITH HOOK SYMBOL

|-

| U+03D5 || {{Large|ϕ}} || GREEK PHI SYMBOL

|-

| U+03F0 || {{Large|ϰ}} || GREEK KAPPA SYMBOL

|-

| U+03F1 || {{Large|ϱ}} || GREEK RHO SYMBOL

|-

| U+03F4 || {{Large|ϴ}} || GREEK CAPITAL THETA SYMBOL

|-

| U+03F5 || {{Large|ϵ}} || GREEK LUNATE EPSILON SYMBOL

|-

| U+03F6 || {{Large|϶}} || GREEK REVERSED LUNATE EPSILON SYMBOL

|}

*[[Arabic (Unicode block)|Arabic block]]

{|

**U+0606 ARABIC-INDIC CUBE ROOT: {{Large|؆}}

| U+0606 || {{Large|؆}} || ARABIC-INDIC CUBE ROOT

**U+0607 ARABIC-INDIC FOURTH ROOT: {{Large|؇}}

|-

**U+0608 ARABIC RAY: {{Large|؈}}

| U+0607 || {{Large|؇}} || ARABIC-INDIC FOURTH ROOT

*[[General Punctuation (Unicode block)|General Punctuation block]]:

|-

**U+2016 DOUBLE VERTICAL LINE: {{Large|‖}}

**| U+20320608 PRIME:|| {{Large|′؈}} || ARABIC RAY

|}

**U+2033 DOUBLE PRIME: {{Large|″}}

*[[General Punctuation (Unicode block)|General Punctuation block]]

**U+2034 TRIPLE PRIME: {{Large|‴}}

{|

**U+2040 CHARACTER TIE: {{Large|⁀}}

**| U+20442016 FRACTION SLASH:|| {{Large|⁄‖}} || DOUBLE VERTICAL LINE

|-

**U+2052 COMMERCIAL MINUS SIGN: {{Large|⁒}}

| U+2032 || {{Large|′}} || PRIME

**U+2061 FUNCTION APPLICATION (non-marking character)

|-

**U+2062 INVISIBLE TIMES (non-marking character)

| U+2033 || {{Large|″}} || DOUBLE PRIME

**U+2063 INVISIBLE SEPARATOR (non-marking character)

|-

**U+2064 INVISIBLE PLUS (non-marking character)

| U+2034 || {{Large|‴}} || TRIPLE PRIME

|-

| U+2040 || {{Large|⁀}} || CHARACTER TIE

|-

| U+2044 || {{Large|⁄}} || FRACTION SLASH

|-

| U+2052 || {{Large|⁒}} || COMMERCIAL MINUS SIGN

|-

| U+2061 || {{small|''note''}} || FUNCTION APPLICATION

|-

| U+2062 || {{small|''note''}} || INVISIBLE TIMES

|-

| U+2063 || {{small|''note''}} || INVISIBLE SEPARATOR

|-

| U+2064 || {{small|''note''}} || INVISIBLE PLUS

|}

''Note: non-marking character''

*[[Superscripts and Subscripts|Superscripts and Subscripts block]]

{|

| U+207A || {{Large|⁺}} || SUPERSCRIPT PLUS SIGN

|-

| U+207B || {{Large|⁻}} || SUPERSCRIPT MINUS

|-

| U+207C || {{Large|⁼}} || SUPERSCRIPT EQUALS SIGN

|-

| U+207D || {{Large|⁽}} || SUPERSCRIPT LEFT PARENTHESIS

|-

| U+207E || {{Large|⁾}} || SUPERSCRIPT RIGHT PARENTHESIS

|-

| U+208A || {{Large|₊}} || SUBSCRIPT PLUS SIGN

|-

| U+208B || {{Large|₋}} || SUBSCRIPT MINUS

|-

| U+208C || {{Large|₌}} || SUBSCRIPT EQUALS SIGN

|-

| U+208D || {{Large|₍}} || SUBSCRIPT LEFT PARENTHESIS

|-

| U+208E || {{Large|₎}} || SUBSCRIPT RIGHT PARENTHESIS

|}

| <!-- newspaper column 2 -->

*[[SuperscriptsMiscellaneous and SubscriptsSymbols|Superscripts andMiscellaneous SubscriptsSymbols block]]:

{|

**U+207A SUPERSCRIPT PLUS SIGN: {{Large|⁺}}

**| U+207B2605 SUPERSCRIPT MINUS:|| {{Large|⁻★}} || BLACK STAR

|-

**U+207C SUPERSCRIPT EQUALS SIGN: {{Large|⁼}}

**| U+207D2606 SUPERSCRIPT LEFT PARENTHESIS:|| {{Large|⁽☆}} || WHITE STAR

|-

**U+207E SUPERSCRIPT RIGHT PARENTHESIS: {{Large|⁾}}

**| U+208A2640 SUBSCRIPT PLUS SIGN:|| {{Large|₊♀}} || FEMALE SIGN

|-

**U+208B SUBSCRIPT MINUS: {{Large|₋}}

**| U+208C2642 SUBSCRIPT EQUALS SIGN:|| {{Large|₌♂}} || MALE SIGN

|-

**U+208D SUBSCRIPT LEFT PARENTHESIS: {{Large|₍}}

| U+2660 || {{Large|♠}} || BLACK SPADE SUIT

**U+208E SUBSCRIPT RIGHT PARENTHESIS: {{Large|₎}}

|-

*[[Miscellaneous Symbols|Miscellaneous Symbols block]]:

**| U+26052661 BLACK STAR:|| {{Large|★♡}} || WHITE HEART SUIT

|-

**U+2606 WHITE STAR: {{Large|☆}}

**| U+26402662 FEMALE SIGN:|| {{Large|♀♢}} || WHITE DIAMOND SUIT

|-

**U+2642 MALE SIGN: {{Large|♂}}

**| U+26602663 BLACK SPADE SUIT:|| {{Large|♠♣}} || BLACK CLUB SUIT

|-

**U+2661 WHITE HEART SUIT: {{Large|♡}}

**| U+2662266D WHITE DIAMOND SUIT:|| {{Large|♢♭}} || MUSIC FLAT SIGN

|-

**U+2663 BLACK CLUB SUIT: {{Large|♣}}

**| U+266D266E MUSIC FLAT SIGN:|| {{Large|♭♮}} || MUSIC NATURAL SIGN

|-

**U+266E MUSIC NATURAL SIGN: {{Large|♮}}

**| U+266F MUSIC SHARP SIGN:|| {{Large|♯}} || MUSIC SHARP SIGN

|}

*[[Alphabetic Presentation Forms|Alphabetic Presentation Forms block]]:

*[[Alphabetic Presentation Forms|Alphabetic Presentation Forms block]]

**U+FB29 HEBREW LETTER ALTERNATIVE PLUS SIGN: {{Large|﬩}}

{|

*[[Small Form Variants|Small Form Variants block]]:

| U+FB29 || {{Large|﬩}} || HEBREW LETTER ALTERNATIVE PLUS SIGN

**U+FE61 SMALL ASTERISK: {{Large|﹡}}

|}

**U+FE62 SMALL PLUS SIGN: {{Large|﹢}}

*[[Small Form Variants|Small Form Variants block]]

**U+FE63 SMALL HYPHEN-MINUS: {{Large|﹣}}

{|

**U+FE64 SMALL LESS-THAN SIGN: {{Large|﹤}}

**| U+FE65FE61 SMALL GREATER-THAN SIGN:|| {{Large|﹥﹡}} || SMALL ASTERISK

|-

**U+FE66 SMALL EQUALS SIGN: {{Large|﹦}}

**| U+FE68FE62 SMALL REVERSE SOLIDUS:|| {{Large|﹨﹢}} || SMALL PLUS SIGN

|-

*[[Halfwidth and fullwidth forms|Halfwidth and Fullwidth Forms block]]:

**| U+FF0BFE63 FULLWIDTH PLUS SIGN:|| {{Large|＋﹣}} || SMALL HYPHEN-MINUS

|-

**U+FF1C FULLWIDTH LESS-THAN SIGN: {{Large|＜}}

**| U+FF1DFE64 FULLWIDTH EQUALS SIGN:|| {{Large|＝﹤}} || SMALL LESS-THAN SIGN

|-

**U+FF1E FULLWIDTH GREATER-THAN SIGN: {{Large|＞}}

| U+FE65 || {{Large|﹥}} || SMALL GREATER-THAN SIGN

**U+FF3C FULLWIDTH REVERSE SOLIDUS: {{Large|＼}}

|-

**U+FF3E FULLWIDTH CIRCUMFLEX ACCENT: {{Large|＾}}

**| U+FF5CFE66 FULLWIDTH VERTICAL LINE:|| {{Large|｜﹦}} || SMALL EQUALS SIGN

|-

**U+FF5E FULLWIDTH TILDE: {{Large|～}}

**| U+FFE2FE68 FULLWIDTH NOT SIGN:|| {{Large|￢﹨}} || SMALL REVERSE SOLIDUS

|}

**U+FFE9 HALFWIDTH LEFTWARDS ARROW: {{Large|￩}}

*[[Halfwidth and fullwidth forms|Halfwidth and Fullwidth Forms block]]

**U+FFEA HALFWIDTH UPWARDS ARROW: {{Large|￪}}

{|

**U+FFEB HALFWIDTH RIGHTWARDS ARROW: {{Large|￫}}

| U+FF0B || {{Large|＋}} || FULLWIDTH PLUS SIGN

**U+FFEC HALFWIDTH DOWNWARDS ARROW: {{Large|￬}}

|-

| U+FF1C || {{Large|＜}} || FULLWIDTH LESS-THAN SIGN

|-

| U+FF1D || {{Large|＝}} || FULLWIDTH EQUALS SIGN

|-

| U+FF1E || {{Large|＞}} || FULLWIDTH GREATER-THAN SIGN

|-

| U+FF3C || {{Large|＼}} || FULLWIDTH REVERSE SOLIDUS

|-

| U+FF3E || {{Large|＾}} || FULLWIDTH CIRCUMFLEX ACCENT

|-

| U+FF5C || {{Large|｜}} || FULLWIDTH VERTICAL LINE

|-

| U+FF5E || {{Large|～}} || FULLWIDTH TILDE

|-

| U+FFE2 || {{Large|￢}} || FULLWIDTH NOT SIGN

|-

| U+FFE9 || {{Large|￩}} || HALFWIDTH LEFTWARDS ARROW

|-

| U+FFEA || {{Large|￪}} || HALFWIDTH UPWARDS ARROW

|-

| U+FFEB || {{Large|￫}} || HALFWIDTH RIGHTWARDS ARROW

|-

| U+FFEC || {{Large|￬}} || HALFWIDTH DOWNWARDS ARROW

|}

*[[Garay (Unicode block)|Garay block]]

{|

| U+10D8E || {{Large|𐶎}} || GARAY PLUS SIGN

|-

| U+10D8F || {{Large|𐶏}} || GARAY MINUS SIGN

|}

|}

== See also ==

* [[ListGlossary of mathematical symbols]]

* [[List of logic symbols]]

* [[Greek letters used in mathematics, science, and engineering]]

* [[List of letters used in mathematics and science]]

* [[List of mathematical uses of Latin letters]]

* [[Latin letters used in mathematics]]

* [[Unicode subscripts and superscripts]]

* [[Unicode symbols]]

* [[CJK Compatibility|CJK Compatibility Unicode symbols]] includes symbols for [[International System of Units|SI]] units

* [[Order of magnitude|Units for order of magnitude]] shows position of [[International System of Units|SI]] units

== References ==

{{reflist}}

== External links ==

*{{cite book | title = Mathematical Markup Language (MathML) W3C Recommendation | publisher = W3C | version = 3.0 | edition = 2nd | date = 10 April 2014 | url = http://www.w3.org/TR/MathML/}}

*Images of glyphs in [http://www.w3.org/TR/MathML2/chapter6.html#chars.16x16-tables section 6.3.3] of the {{cite book | title = Mathematical Markup Language (MathML) W3C Recommendation | publisher = W3C

Line 206 ⟶ 306:

{{Unicode navigation}}

{{Mathematical symbols notation language}}

[[Category:Unicode blocks|Mathematical operators*]]

[[Category:Mathematical symbols]]