Mathematical symbols in Unicode

Fractions

½ ↉ ⅓ ⅔ ¼ ¾ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅐ ⅛ ⅜ ⅝ ⅞ ⅑ ⅒ ⅟ Fractions

I have ordered these by denominator then numerator. These are not consecutive within Unicode, the half and quarters appearing in different blocks. The oddity ↉ seems to have something to do with baseball scoring. Very little turns up in searches for this character, but Wikipedia links it to its "Baseball scorekeeping" page.

Super and subscripts

⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ superscripts

The 1, 2 and 3 superscripts are not consecutive with the other ones.

⁺ ⁻ ⁼ ⁽ ⁾ ⁱ ˡ ⁿ ˢ ˣ non-numerical superscripts

I've included things which are labelled as "modifier letter" in the Unicode description.

₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ numerical subscripts

₊ ₋ ₌ ₍ ₎ ₐ ₑ ₒ ₓ ₔ ₕ ₖ ₗ ₘ ₙ ₚ ₛ ₜ non-numerical subscripts

Some fonts make U+208F, not listed above, into a small n, but this is not actually in Unicode.

Signs

± (\pm) Plus or minus

∓ (\mp) Minus or plus

Sets of numbers in blackboard bold

Either blackboard bold or bold is used for sets of numbers. Blackboard bold was virtually unused until the 1990s.

ℕ natural numbers

ℤ integers

The use of Z is from German "Zahlen".

ℙ Primes

This is not as common as the other symbols.

ℚ rational

The Q stands for "quotient".

ℝ real

ℂ complex

Complex numbers

ℑ (\Im) Imaginary part

See notes under ℜ.

ℜ (\Re) Real part

"Fraktur" R/I are used by some authors to denote the real/imaginary part of a complex number, but I haven't found an example for the references yet, all the books I looked at used Im(z) and Re(z). In fact van der Waerden often uses ℜ for an example ring.

Sets

∃ (\exists) There exists

∄ (\nexists) There does not exist

∀ (\forall) For all (universal quantifier)

∅ (\O, \emptyset) Empty set [May be confused with ⌀]

∖ Set difference

∁ [Duplicate of C] (\complement) Complement

Various authors use either an apostrophe, a bar over the set's name, or a superscript c to indicate the complement of a set. The Wikipedia article gives a reference to use of a specially-shaped C to Bourbaki.

Membership

The first element here indicates that "A has the relation with B" and the second one represents "B has the relation with A".

∈ ∍ ∊ (\in) Element of

∉ ∌ (\notin) Not element of

⊂ ⊃ Subset of

⊆ ⊇ Subset or equal

⊄ ⊅ Not a subset of

Joins

∪ (\cup) Union

∩ (\cap) Intersection

Logical operators

⇒ Implies

⇔ If and only if (equivalence)

∧ Logical AND

∨ Logical OR

¬ Negation

⊤ Down tack

Used to mean "True" in logic.

⊥ Up tack

Used to mean "False" (upside down "true") in logic.

→ Conditional implication

Reasoning and statements

∴ Therefore

∵ Because

This symbol is not as common as ∴.

∎ End of proof

The use of a solid square to mark the end of a proof is common, but far from universal.

Constants

∞ Infinity

π Pi

This is the normal Greek letter rather than a special mathematical symbol.

ı 𝚤 undotted i

An undotted i is used to indicate an identity element in a group or an identity operator by some authors. There is no special mathematical Unicode symbol for this, it is part of the usual set of Latin characters, although there is a script form of it.

⋮ ⋯ ⋰ ⋱ Ellipsis

The mathematician's "et cetera".

Cardinals

ℵ [Duplicate of א] Aleph, cardinality symbol

(See for example Halmos page 101, Simmons §6 and §7.)

Operations and Relations

∑ [Duplicate of Σ] Summation

∏ [Duplicate of Π] Product

∆ [Duplicate of Δ] Increment

⟌ Long division

÷ (\div) division

× (\times) multiplication

∣ [Duplicate of |] Divides

For example a∣b means "a divides b", so 3∣6. See for example Hardy page 1.

∗ [Duplicate of *] Asterisk

∤ Does not divide

For example a∤b means "a does not divide b", so 3∤7. See for example Hardy page 1.

⌈ ⌉ ⌊ ⌋ (\lceil, \rceil, \lfloor, \rfloor) Ceiling and floor

⌈x⌉ means the smallest integer which is greater than or equal to x, and ⌊x⌋ means the largest integer which is less than or equal to x. These are called the "ceiling" and "floor" operators. This is a "new-fangled" usage based on computer programming languages. Traditionally [x] meant the integer part of a number, and this is what is used in Spivak page 72 for example.

⊕ (\oplus) Circled plus

This is used for exclusive OR (XOR) in logic, and direct sum in algebra.

⊗ (\otimes) direct product

∘ Ring operator

Used for function composition like g∘f.

∙ Bullet operator

Used for vector dot products, a∙b.

Roots

√ Square root

∛ Cube root

∜ Quadratic root, fourth root

Calculus

∫ Integral sign

∂ (\partial) partial

∇ Gradient

∮ Contour integral

Binary relations

≡ Identical to

This is often used for an equivalence relation.

≤ (\leq) Less than or equal to

≦ (\leqq) Less than or equal to

This means exactly the same thing as ≤. This version of the symbol is used in some texts, such as Ahlfors but others such as Rudin use ≤.

≥ (\geq) Greater than or equal to

≧ (\geqq) Greater than or equal to

This means exactly the same thing as ≥. See also the discussion under ≦.

≠ (\neq) Not equal to

≈ Approximately equal

∝ Proportional to

≌ (\cong) All equal to

In geometry this is used to signify congruency (see for example Moise p. 88), and in algebra it is used to indicate an isomorphism (see for example van der Waerden page 26).

Algebra

⊲ ⊳ Normal subgroup

This is used for various purposes other than just normal subgroups, for example an ideal of a ring.

⊴ ⊵ Normal subgroup of or equal to

⋪ ⋫ Not normal subgroup

⋬ ⋭ Neither normal subgroup nor equal

Geometry

∠ (\angle) Angle

∡ Measured angle

⦜ Right angle

∟ Right angle

∢ Spherical angle

⟂ (\perp) Perpendicular [May be confused with ⊥]

∥ (\|) Parallel

∦ Not parallel

⌀ Diameter [May be confused with ∅]

Functions

℘ Weierstrass elliptic function

See, for example, Ahlfors, pp. 272-277. I think this is the only function which has its own Unicode code point.

Polygons and other shapes

⬭ ⬬ ⬯ ⬮ Ellipses

⬠ ⬟ ⯂ ⭔ ⭓ Pentagons

The designers of Unicode may not have envisaged futuristic technology where it would become able to rotate shapes on a page without having a separate character for each 90 rotation.

⬡ ⬢ ⬣ Hexagons

⯃ ⯄ 🐙 Octagons

There are no heptagons (7-sided) or nonagons (9-sided).

Less commonly used symbols

Some symbols are much less commonly used. These are some more specialised meanings which I tracked down. Symbols which I couldn't find any examples of being used at all are in Mystery symbols below. I've resisted the temptation to put the meanings in scare quotes. I welcome additions to this list. Please email me at the address at the bottom of the page.

⊙ (\odot) Circled dot

This is used for the Hadamard product of matrices.

⊠ Squared times

This is used in functional analysis to indicate free multiplicative convolution.

⋈ Join

This is used in relational algebra, the study of databases.

⊌ Multiset

Although the concept of multisets seems to exist, I could find no examples of this symbol in use.

⟃ ⟄ Open subset

This symbol seems to exist and I assume that the meaning of A⟃B would be "A is an open subset of B", but I could find no examples of this symbol in use other than this discussion at tex.stackexchange.com

⋐ ⋑ Double subset

This is used to signify a compact embedding in topology.

⋔ (\pitchfork) Pitchfork

This is used to signify transversality in differential topology.

⊏ Square image of

According to Wiktionary, this is used to mean a variety of things by different authors.

⨛ ⨜ Integral with overbar/underbar

These symbols are used for example in Rudin chapter 6 for the upper and lower Riemann integrals. According to Wikipedia these are correctly named the Darboux integrals, although neither of the references given in the Wikipedia article, to either Rudin or Spivak, gives that name.

≺ ⊰ Precedes

This symbol and the succeeds symbol are used to indicate orderings in ways prescribed by a particular author.

≻ ⊱ Succeeds

See precedes above.

⊩ Forces

This is used in the field of forcing in set theory.

⊸ Multimap

This is used in linear logic. Wikipedia says it is sometimes called "lollipop" due to its shape.

ℶ Second transfinite symbol, Beth numbers.

ℷ Gimel symbol, third transfinite symbol

ℸ Dalet symbol, fourth transfinite symbol

∐ (\coprod) N-ary coproduct

This is used to mean coproduct in category theory.

⨧ Plus sign with subscript two

This is used to denote addition modulo 2, so 1⨧1=0. Burton gives \(+_n\) and \(\cdot_n\) for addition and multiplication modulo \(n\). Wikipedia directs this symbol to a page about "Nimbers", but the symbol itself is not used anywhere on the current version of that page.

≲ ≳ ≴ ≵ Greater/less than or (not) equivalent to

Web searches turn these up in some references in astronomy where the use seems to be "approximately greater than or equal to" rather than "equivalent to".

∱ ⨑ ∲ ∳ Clockwise and anticlockwise contour integrals

The meaning of these symbols seems obvious, but they don't seem to be in use anywhere on the web. The difference between ∱ and ∲, and ⨑ and ∳, is not clear.

≣ Strictly equivalent to

According to this discussion at maths stackexchange, this symbol is used as a substitute for ≡ when that is already being used to indicate congruence.

Mystery symbols

The purpose of some of the "mathematical symbols" in Unicode is unknown to me. Sometimes the word corresponding to the symbol shows up in searches, but the symbols themselves don't have any examples so it's not clear how they are used, or even whether they are ever used. Please feel free to email if you have better information.

ℇ Euler constant

Unicode describes this as EULER CONSTANT, but, as every schoolboy knows, mathematicians use the Greek letter γ for the quantity usually known as Euler's constant (see for example Hardy page 59), the difference between the harmonic series and the logarithm. Google searches suggest it is a phonetic symbol. Wikipedia claims "It is unknown which constant this is supposed to be" with a reference to a mailing list post by one Kenneth Whistler from 2002 who says that originally this was just called "Euler's" without the word "constant". It may be something like the ghost characters in Japanese, things which were accidentally added to a coding standard that never actually existed.

⬲ ⟴ Left/right arrow with circled plus

Nothing turns up in searches for these characters.

∻ Homothety

Although this term exists, I could not find an example of how this symbol is used. Oddly enough it is called "\kernelcontraction" in the Latex package unicode-math.

∺ Geometric proportion

I could not find any examples of how this symbol is used. Web searches only turn up the usual machine-generated garbage.

⊚ Circled ring operator

Neither Wiktionary nor Wikipedia has any information on this character, with the Wikipedia link actually circularly going back to itself.

⋒ ⋓ double intersection / union

I could not track down a meaning for these.

⊜ Circled equals

Nothing shows up in searches except for the usual machine-generated guff, and neither Wikipedia nor Wiktionary has anything on this.

⊶ ⊷ Original of / image of

These are listed in Wiktionary, but I haven't discovered what they are used for. Wiktionary links to a page about "original" for ⊶, but this page has no mathematical definition of the word. There is no better information about ⊷. The consecutive symbol, ⊸, however, does appear to be used in linear logic.

⊹ Hermitian conjugate matrix

Apparently this symbol means "Hermitian conjugate matrix", but I could not find any examples of its use.

⋹ Element of with two horizontal strokes

I found no examples of use.

⋇ Division times

An unholy combination of × and ÷. Something like alcohol-free beer perhaps, first you brew the alcohol in and then you take it out again? This Maths stackexchange discussion suggests that it is used in statistics and group theory, but note the comment under the top answer where it is pointed out that the suggested reference does not actually use this silly-looking symbol.

∔ Dot plus

Wiktionary claims this is used for the Cartesian product of two sets, but gives no reference or examples, and there is a request for clarification next to that entry.

⊛ Circled asterisk

Web searches turn up nothing useful for this.

⨉ [Duplicate of ×] n-ary times operator

This seems to be just a duplicate of × which comes out larger. The symbol is not used anywhere on Wikipedia and is redirected to the page for the multiplication sign itself.

⨊ ⨋ ⨌ ⨍ ⨎ ⨏ ⨐ ⨑ ⨒ ⨓ ⨔ ⨕ ⨖ ⨗ ⨘ ⨙ ⨚ Integral and summation signs

I haven't looked these up yet.