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Mathematical Operators (Unicode block)
Mathematical Operators (Unicode block)
Mathematical Operators (Unicode block)
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Mathematical Operators
RangeU+2200..U+22FF
(256 code points)
PlaneBMP
ScriptsCommon
Symbol setsMathematical symbols
Logic and Set operators
Relation symbols
Assigned256 code points
Unused0 reserved code points
Unicode version history
1.0.0 (1991)242 (+242)
3.2 (2002)256 (+14)
Unicode documentation
Code chart ∣ Web page
Note: [1][2]
This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols.

Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation.

Notably absent are the plus sign (), greater than sign () and less than sign (), due to them already appearing in the Basic Latin Unicode block, and the plus-or-minus sign (), multiplication sign () and obelus (), due to them already appearing in the Latin-1 Supplement block, although a distinct minus sign () is included, semantically different from the Basic Latin hyphen-minus (-).

Block

[edit]
Mathematical Operators[1]
Official Unicode Consortium code chart (PDF)
  0 1 2 3 4 5 6 7 8 9 A B C D E F
U+220x ∀ ∁ ∂ ∃ ∄ ∅ ∆ ∇ ∈ ∉ ∊ ∋ ∌ ∍ ∎ ∏
U+221x ∐ ∑ − ∓ ∔ ∕ ∖ ∗ ∘ ∙ √ ∛ ∜ ∝ ∞ ∟
U+222x ∠ ∡ ∢ ∣ ∤ ∥ ∦ ∧ ∨ ∩ ∪ ∫ ∬ ∭ ∮ ∯
U+223x ∰ ∱ ∲ ∳ ∴ ∵ ∶ ∷ ∸ ∹ ∺ ∻ ∼ ∽ ∾ ∿
U+224x ≀ ≁ ≂ ≃ ≄ ≅ ≆ ≇ ≈ ≉ ≊ ≋ ≌ ≍ ≎ ≏
U+225x ≐ ≑ ≒ ≓ ≔ ≕ ≖ ≗ ≘ ≙ ≚ ≛ ≜ ≝ ≞ ≟
U+226x ≠ ≡ ≢ ≣ ≤ ≥ ≦ ≧ ≨ ≩ ≪ ≫ ≬ ≭ ≮ ≯
U+227x ≰ ≱ ≲ ≳ ≴ ≵ ≶ ≷ ≸ ≹ ≺ ≻ ≼ ≽ ≾ ≿
U+228x ⊀ ⊁ ⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉ ⊊ ⊋ ⊌ ⊍ ⊎ ⊏
U+229x ⊐ ⊑ ⊒ ⊓ ⊔ ⊕ ⊖ ⊗ ⊘ ⊙ ⊚ ⊛ ⊜ ⊝ ⊞ ⊟
U+22Ax ⊠ ⊡ ⊢ ⊣ ⊤ ⊥ ⊦ ⊧ ⊨ ⊩ ⊪ ⊫ ⊬ ⊭ ⊮ ⊯
U+22Bx ⊰ ⊱ ⊲ ⊳ ⊴ ⊵ ⊶ ⊷ ⊸ ⊹ ⊺ ⊻ ⊼ ⊽ ⊾ ⊿
U+22Cx ⋀ ⋁ ⋂ ⋃ ⋄ ⋅ ⋆ ⋇ ⋈ ⋉ ⋊ ⋋ ⋌ ⋍ ⋎ ⋏
U+22Dx ⋐ ⋑ ⋒ ⋓ ⋔ ⋕ ⋖ ⋗ ⋘ ⋙ ⋚ ⋛ ⋜ ⋝ ⋞ ⋟
U+22Ex ⋠ ⋡ ⋢ ⋣ ⋤ ⋥ ⋦ ⋧ ⋨ ⋩ ⋪ ⋫ ⋬ ⋭ ⋮ ⋯
U+22Fx ⋰ ⋱ ⋲ ⋳ ⋴ ⋵ ⋶ ⋷ ⋸ ⋹ ⋺ ⋻ ⋼ ⋽ ⋾ ⋿
Notes
1.^ As of Unicode version 17.0

Variation sequences

[edit]

The Mathematical Operators block has sixteen variation sequences defined for standardized variants.[3][4] They use U+FE00 VARIATION SELECTOR-1 (VS01) to denote variant symbols (depending on the font):

Variation sequences
Base character Base +VS01 Description
U+2205 EMPTY SET ∅ ∅︀ zero with long diagonal stroke overlay form
U+2229 INTERSECTION ∩ ∩︀ with serifs
U+222A UNION ∪ ∪︀ with serifs
U+2268 LESS-THAN BUT NOT EQUAL TO ≨ ≨︀ with vertical stroke
U+2269 GREATER-THAN BUT NOT EQUAL TO ≩ ≩︀ with vertical stroke
U+2272 LESS-THAN OR EQUIVALENT TO ≲ ≲︀ following the slant of the lower leg
U+2273 GREATER-THAN OR EQUIVALENT TO ≳ ≳︀ following the slant of the lower leg
U+228A SUBSET OF WITH NOT EQUAL TO ⊊ ⊊︀ with stroke through bottom members
U+228B SUPERSET OF WITH NOT EQUAL TO ⊋ ⊋︀ with stroke through bottom members
U+2293 SQUARE CAP ⊓ ⊓︀ with serifs
U+2294 SQUARE CUP ⊔ ⊔︀ with serifs
U+2295 CIRCLED PLUS ⊕ ⊕︀ with white rim
U+2297 CIRCLED TIMES ⊗ ⊗︀ with white rim
U+229C CIRCLED EQUALS ⊜ ⊜︀ with equal sign touching the circle
U+22DA LESS-THAN EQUAL TO OR GREATER-THAN ⋚ ⋚︀ with slanted equal
U+22DB GREATER-THAN EQUAL TO OR LESS-THAN ⋛ ⋛︀ with slanted equal

History

[edit]

The following Unicode-related documents record the purpose and process of defining specific characters in the Mathematical Operators block:

Version Final code points[a] Count UTC ID L2 ID WG2 ID Document
1.0.0 U+2200..22F1 242 (to be determined)
UTC/1999-013 Karlsson, Kent (1999-05-27), Tildes and micro sign decompositions
L2/99-176R Moore, Lisa (1999-11-04), "Not Tilde", Minutes from the joint UTC/L2 meeting in Seattle, June 8-10, 1999
L2/00-115R2 Moore, Lisa (2000-08-08), "Motion 83-M21", Minutes Of UTC Meeting #83
L2/01-342 Suignard, Michel (2001-09-10), "T.9 B.1 List of combining characters/Variation selectors", Comments accompanying the US positive vote on the FPDAM 1 to ISO/IEC 10646-1:2001
L2/07-268 N3253 (pdf, doc) Umamaheswaran, V. S. (2007-07-26), "M50.7 (Math symbol glyph correction) [U+22C4]", Unconfirmed minutes of WG 2 meeting 50, Frankfurt-am-Main, Germany; 2007-04-24/27
L2/15-268 Beeton, Barbara; Freytag, Asmus; Iancu, Laurențiu; Sargent, Murray (2015-10-30), Proposal to Represent the Slashed Zero Variant of Empty Set
L2/15-254 Moore, Lisa (2015-11-16), "B.12.1.2 Proposal to Represent the Slashed Zero Variant of Empty Set", UTC #145 Minutes
L2/24-173 Pentzlin, Karl (2024-06-06), Proposal to encode a Middle Asterisk as referred to in the German standard DIN 2137 [Affects U+2217]
L2/24-166 Anderson, Deborah; Goregaokar, Manish; Kučera, Jan; Whistler, Ken; Pournader, Roozbeh; Constable, Peter (2024-07-18), "20. Middle Asterisk [Affects U+2217]", Recommendations to UTC #180 July 2024 on Script Proposals
L2/24-159 Constable, Peter (2024-07-29), "Section 20. Middle Asterisk", UTC #180 Minutes, Consider adding an annotation to U+2217 that it may be used to represent the telephony asterisk
3.2 U+22F2..22FF 14 L2/00-119[b] N2191R Whistler, Ken; Freytag, Asmus (2000-04-19), Encoding Additional Mathematical Symbols in Unicode
L2/00-234 N2203 (rtf, txt) Umamaheswaran, V. S. (2000-07-21), "8.18", Minutes from the SC2/WG2 meeting in Beijing, 2000-03-21 -- 24
L2/00-115R2 Moore, Lisa (2000-08-08), "Motion 83-M11", Minutes Of UTC Meeting #83
  1. ^ Proposed code points and characters names may differ from final code points and names
  2. ^ Refer to the history section of the Miscellaneous Mathematical Symbols-B block for additional math-related documents

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Mathematical Operators (Unicode block)

View on Grokipedia
from Grokipedia
The Mathematical Operators Unicode block is a standardized range of 256 characters spanning code points U+2200 through U+22FF, dedicated to encoding symbols essential for , including binary and n-ary operators, logical relations, geometric shapes, and other specialized glyphs primarily used in technical and scientific contexts. This block forms a core part of the Standard, version 17.0, where it supports precise rendering in applications such as mathematical systems, symbolic computation software, and digital documents requiring formal mathematical expressions. The encoding draws from established international standards, including ANSI Y10.20 for mathematical symbols, ISO 6862 for , and conventions from , ensuring compatibility with legacy mathematical publishing tools. Due to the polysemous nature of many mathematical symbols—which can represent different operations depending on context—the block adopts a predominantly shape-based encoding rather than strictly semantic assignments, allowing flexibility in interpretation while prioritizing visual consistency across fonts and systems. Notable disunifications distinguish mathematical variants from , such as the minus sign U+2212 − (distinct from the hyphen-minus U+002D -) to enable proper spacing and sizing in equations. Key categories within the block encompass:
  • Logical operators and quantifiers, like the universal quantifier ∀ U+2200 and existential quantifier ∃ U+2203, used in predicate logic.
  • Set and relation symbols, including membership ∈ U+2208, empty set ∅ U+2205, and subset ⊂ U+2282.
  • N-ary operators for summations and products, such as ∑ U+2211 (n-ary summation) and ∏ U+220F (n-ary product).
  • Geometric and arrow symbols, like the ring operator ∘ U+2218 and rightwards arrow → U+2192 (though arrows extend into adjacent blocks).
These characters integrate with broader mathematical support, including mechanisms for stretchy operators in markup languages like , enhancing and in .

Overview

Block Fundamentals

The Mathematical Operators Unicode block occupies the code point range U+2200 to U+22FF, spanning 256 consecutive positions in the Unicode standard. As of Unicode 17.0, released on September 9, 2025, all 256 code points in this block are fully assigned to characters, with no unassigned, reserved, or non-character positions remaining. The official block name is "Mathematical Operators", and none of its characters have a default emoji presentation; however, select symbols from this block can be rendered in emoji style via variation selectors or contextual mechanisms in supporting systems. For a complete visual and nominal enumeration of the characters, consult the official chart PDF and the character names list in the Unicode Character Database. The symbols in this block maintain compatibility with the mathematical typography conventions outlined in ISO/IEC TR 9573-13, ensuring alignment with established entity sets for .

Purpose and Scope

The Mathematical Operators Unicode block (U+2200–U+22FF) serves as a foundational collection for encoding specialized symbols essential to advanced mathematical, logical, and set-theoretic notation in environments. Its primary purpose is to enable consistent representation of operators, relations, geometric shapes, and delimiters that are predominantly used in scientific and technical contexts, ensuring portability across digital documents, markup languages, and applications without reliance on proprietary fonts or formatting. This block addresses the need for symbols that convey precise mathematical semantics, such as integrals, quantifiers, and set relations, which are integral to formal expressions in fields like , , and logic. Notably, the block excludes basic arithmetic operators—such as the plus sign (U+002B), (U+002D), (U+00D7), and (U+00F7)—which are instead allocated to the Basic Latin (U+0000–U+007F) and (U+0080–U+00FF) blocks to maintain compatibility with legacy encodings like ASCII and ISO 8859-1. The focus here is on more specialized characters, like the minus sign (U+2212) disunified from the for distinct spacing and semantic roles in equations, or (U+2205), highlighting a deliberate to prioritize mathematical precision over general reuse. This exclusion prevents overlap with everyday text processing while supporting advanced notation that requires specific rendering behaviors. In usage contexts, the block underpins typesetting systems in mathematical modes, including / for document preparation and for web-based rendering, allowing seamless integration into structured markup for expressions like logical implications or geometric constructs. Positioned within the Basic Multilingual Plane (BMP, Plane 0), it offers broad compatibility with early implementations, though it complements symbols in the Supplementary Multilingual Plane, such as those in the block (U+1D400–U+1D7FF). For extended notation, it overlaps conceptually with the block (U+2A00–U+2AFF), which provides additional variants, but the core Mathematical Operators block retains its role in encoding legacy and essential symbols. The block's stability underscores its maturity, with no additions or modifications since Unicode 15.0, as confirmed in versions 16.0 and 17.0, reflecting a deliberate choice to preserve existing encodings for reliable long-term use in mathematical software and standards. This lack of change addresses gaps in earlier coverage by emphasizing enduring support for core notation without introducing disruptions to established implementations.

Character Categories

Logical and Quantifier Symbols

The Logical and Quantifier Symbols in the Mathematical Operators Unicode block encompass characters including those from U+2200 to U+2228, which represent essential notations for predicate and propositional logic, including quantifiers and connectives used to formalize mathematical statements and proofs. These symbols facilitate precise expression of logical relationships, such as universality and existence, and are integral to fields like set theory, formal logic, and theoretical computer science. All characters in this group are classified under the Unicode category "Sm" (Symbol, Math), indicating their role as mathematical operators without canonical decompositions except where specified. Central to this category are the quantifiers: ∀ (U+2200, FOR ALL) denotes universal quantification, asserting that a property holds for every element in a domain, as in the predicate logic notation ∀x P(x)\forall x \, P(x), where P(x)P(x) is true for all xx. The existential quantifier ∃ (U+2203, THERE EXISTS) indicates that there is at least one element satisfying the property, exemplified by ∃x P(x)\exists x \, P(x), meaning P(x)P(x) holds for some xx. Its negated form ∄ (U+2204, THERE DOES NOT EXIST) combines with a compatibility decomposition to ∃ overlaid with a negation slash (U+0338), expressing that no such element exists. Additionally, ∅ (U+2205, EMPTY SET) symbolizes the set containing no elements, often used in foundational logic to denote vacuous truths, with no canonical decomposition but a variation sequence available for a stroked variant. The nabla ∇ (U+2207, NABLA) serves as a differential operator, particularly for the gradient or divergence in vector analysis, though it appears in logical contexts for Laplacian operators in proof derivations. Logical connectives in this range include ∧ (U+2227, LOGICAL AND), representing conjunction of propositions, where P∧QP \wedge Q is true only if both PP and QQ hold, and ∨ (U+2228, LOGICAL OR), denoting disjunction, true if at least one of PP or QQ is true. These connectives, standardized in ISO 80000-2 as symbols for "and" and "or" in mathematical expressions, enable the construction of complex logical statements essential for proving and . A practical example is the notation ∃x∈S\exists x \in S, which asserts the of an element xx within a set SS, commonly employed in proofs to establish non-emptiness or . These symbols originated from established mathematical conventions and were incorporated into Unicode through harmonization with ISO standards, including ISO 9573-13 for entity sets in publishing and ISO 80000-2 for , ensuring consistent rendering in digital mathematical documents. In mathematical proofs, quantifiers like ∀ and ∃ provide structural precedence, binding variables to domains before applying connectives such as ∧ and ∨ to build rigorous arguments. The ∅ further supports logical foundations by representing the absence of quantification targets.

Set Theory and Relation Symbols

The Mathematical Operators Unicode block includes a dedicated range from U+2208 to U+223F encompassing symbols essential for expressing concepts in and binary relations, such as membership, set operations, and logical inferences. These characters facilitate precise notation in mathematical texts, enabling representations of discrete structures like sets and their interactions without reliance on external formatting. Unlike logical connectives, which emphasize propositional relationships, these symbols prioritize set-theoretic constructions and relational assertions central to foundational . Key symbols in this range include U+2208 ∈ (element of), used to denote that an element belongs to a set, and its negation U+2209 ∉ (not an element of), indicating exclusion from a set. Set operations are represented by U+2229 ∩ (intersection), which signifies the common elements between two sets, and U+222A ∪ (union), denoting the combined elements of the sets involved. The set minus operation appears as U+2216 ∖, subtracting one set's elements from another. In , U+2223 ∣ (divides) expresses divisibility, where a∣ba \mid b means aa divides bb evenly. The inverse of element of is U+220B ∋ (contains as member), often used for readability in certain notations. Logical reasoning symbols include U+2234 ∴ (therefore), marking conclusions, and U+2235 ∵ (because), indicating premises. Additionally, U+2237 ∝ (proportional to) conveys direct proportionality between quantities, as in geometric or algebraic contexts. These symbols function primarily as binary relational operators, supporting set-theoretic applications like defining subsets or proving properties via membership. Unicode assigns these characters the bidirectional class "ON" (Other Neutral), ensuring they remain neutral in processing and do not initiate levels, which is crucial for mathematical expressions embedded in multilingual documents. Certain symbols, such as U+220B ∋, possess mirrored variants for right-to-left rendering; for instance, ∋ mirrors to resemble a reversed form in Arabic-script contexts to maintain visual logic. This is governed by the Unicode Bidirectional Algorithm, preventing distortion in mixed-directionality layouts. Representative examples illustrate their utility: the set of natural numbers can be described as {x∣x∈N}\{x \mid x \in \mathbb{N}\}, where ∈\in asserts membership, while the even naturals might be N∖{x∣x≢0(mod2)}\mathbb{N} \setminus \{x \mid x \not\equiv 0 \pmod{2}\}, employing set minus and negation. In proofs, a statement might conclude with ∴P\therefore P follows from QQ, using ∴ to link premises logically. Proportionality appears in relations like f(x)∝x2f(x) \propto x^2, highlighting scaling behaviors in functions. These notations underscore the symbols' role in compactly conveying complex set relations. This range addresses gaps in early Unicode by categorizing symbols specifically for set-theoretic and relational uses, with no new characters added since Unicode 3.2 in 2002, reflecting the stability of core mathematical notation in the standard.

Arithmetic and Operator Symbols

The Arithmetic and Operator Symbols subsection of the Mathematical Operators Unicode block (U+2200–U+22FF) encompasses characters primarily used for denoting basic arithmetic operations, aggregations like sums and products, roots, and special notations such as infinity. These symbols facilitate the representation of computational and quantitative mathematical expressions, distinguishing them from relational or logical constructs in other categories. All characters here belong to the General Category "Sm" (Symbol, Math), indicating their role as mathematical operators that can stretch or vary in rendering based on context, as outlined in Unicode Technical Report #25. Key symbols include U+220F ∏ N-ARY PRODUCT, which denotes the product of a sequence of terms, analogous to repeated ; for instance, the product over all prime numbers pp is written as ∏pp\prod_p p in contexts like the Euler product for the . Similarly, U+2211 ∑ N-ARY SUMMATION represents the of terms, such as the for the sum of the first nn positive integers: ∑i=1ni=n(n+1)2,\sum_{i=1}^n i = \frac{n(n+1)}{2}, where the symbol stretches vertically to encompass limits and indices in extended notations. These n-ary operators are stretchable in mathematical systems, allowing them to grow with the number of terms. Root symbols provide notation for extracting roots: U+221A √ SQUARE ROOT indicates the principal square root, as in x2=∣x∣\sqrt{x^2} = |x|
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