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Contents 1 Common symbols 2 Advanced and less-frequently used math symbols 3 Note on symbol names 4 Abbreviated function names 5 See also 6 External links

[edit] Common symbols

This is a listing of common symbols found within all branches of the science of mathematics.

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
		Read as
		Category
=		equality	x = y means x and y represent the same thing or value.	1 + 1 = 2
		is equal to; equals
		everywhere
≠		inequation	x ≠ y means that x and y do not represent the same thing or value. (As ≠ can be hard to type, the more “keyboard friendly” forms !=, /= or <> may be seen. These are avoided in mathematical texts.)	2 + 2 ≠ 5
		is not equal to; does not equal
		everywhere
< > ≪ ≫		strict inequality	x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y.	3 < 4 5 > 4 0.003 ≪ 1000000
		is less than, is greater than, is much less than, is much greater than
		order theory
≤ ≥		inequality	x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. (As ≤ and ≥ can be hard to type, the more “keyboard friendly” forms <= and >= may be seen. These are avoided in mathematical texts.)	3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5
		is less than or equal to, is greater than or equal to
		order theory
∝		proportionality	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
		is proportional to; varies as
		everywhere
+		addition	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
		plus
		arithmetic
		disjoint union	A1 + A2 means the disjoint union of sets A1 and A2.	A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒ A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
		the disjoint union of ... and ...
		set theory
−		subtraction	9 − 4 means the subtraction of 4 from 9.	8 − 3 = 5
		minus
		arithmetic
		negative sign	−3 means the negative of the number 3.	−(−5) = 5
		negative; minus; the opposite of
		arithmetic
		set-theoretic complement	A − B means the set that contains all the elements of A that are not in B. (∖ can also be used for set-theoretic complement as described below.)	{1,2,4} − {1,3,4}  =  {2}
		minus; without
		set theory
×		multiplication	3 × 4 means the multiplication of 3 by 4.	7 × 8 = 56
		times
		arithmetic
		Cartesian product	X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.	{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
		the Cartesian product of ... and ...; the direct product of ... and ...
		set theory
		cross product	u × v means the cross product of vectors u and v	(1,2,5) × (3,4,−1) = (−22, 16, − 2)
		cross
		vector algebra
		group of units	R× consists of the set of units of the ring R, along with the operation of multiplication. This may also be written R* as described below, or U(R).
		the group of units of
		ring theory
·		multiplication	3 · 4 means the multiplication of 3 by 4.	7 · 8 = 56
		times
		arithmetic
		dot product	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
		dot
		vector algebra
÷ ⁄		division	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = .5 12 ⁄ 4 = 3
		divided by
		arithmetic
		quotient group	G / H means the quotient of group G modulo its subgroup H.	{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
		mod
		group theory
		quotient set	A/~ means the set of all ~ equivalence classes in A.	If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then ℝ/~ = {x + n : n ∈ ℤ : x ∈ (0,1]}
		mod
		set theory
±		plus-minus	6 ± 3 means both 6 + 3 and 6 − 3.	The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
		plus or minus
		arithmetic
		plus-minus	10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.	If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
		plus or minus
		measurement
∓		minus-plus	6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5).	cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
		minus or plus
		arithmetic
√		square root	means the positive number whose square is x.
		the principal square root of; square root
		real numbers
		complex square root	if is represented in polar coordinates with , then .
		the complex square root of …; square root
		complex numbers
|…|		absolute value or modulus	|x| means the distance along the real line (or across the complex plane) between x and zero.	|3| = 3 |–5| = |5| = 5 | i | = 1 | 3 + 4i | = 5
		absolute value (modulus) of
		numbers
		Euclidean distance	|x – y| means the Euclidean distance between x and y.	For x = (1,1), and y = (4,5), |x – y| = √([1–4]2 + [1–5]2) = 5
		Euclidean distance between; Euclidean norm of
		geometry
		determinant	|A| means the determinant of the matrix A
		determinant of
		matrix theory
		cardinality (AKA order)	|X| means the cardinality of the set X. (# or ♯ may be used instead as described below.)	|{3, 5, 7, 9}| = 4.
		cardinality of; size of
		set theory
||…||		norm	|| x || means the norm of the element x of a normed vector space.	|| x  + y || ≤  || x ||  +  || y ||
		norm of; length of
		linear algebra
		nearest integer function	||x|| means the nearest integer to x, with half-integers being rounded to even. (This may also be written [x], ⌊x⌉, nint(x) or Round(x).)	||1|| = 1, ||1.5|| = 2, ||−2.5|| = 2, ||3.49|| = 3
		nearest integer to
		numbers
∣ ∤		divisor, divides	a|b means a divides b. As with inequality, this symbol can be difficult to type and its negation is rare, in which case a regular but slightly shorter vertical bar "|" character is used.	Since 15 = 3×5, it is true that 3|15 and 5|15.
		divides
		number theory
		conditional probability	P(A|B) means the probability of the event a occurring given that b occurs.	If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4
		given
		probability
		restriction	f|A means the function f restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f.	The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective.
		restriction of … to …; restricted to
		set theory
||		parallel	x || y means x is parallel to y.	If l || m and m ⊥ n then l ⊥ n. In physics this is also used to express .
		is parallel to
		geometry, physics
		incomparability	x || y means x is incomparable to y.	{1,2} || {2,3} under set containment.
		is incomparable to
		order theory
		exact divisibility	pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not).	23 || 360.
		exactly divides
		number theory
# ♯		cardinality (AKA order)	#X means the cardinality of the set X. (|…| may be used instead as described above.)	#{4, 6, 8} = 3
		cardinality of; size of
		set theory
:		such that	: means “such that”, and is used in proofs and the set-builder notation (described below).	∃ n ∈ ℕ: n is even.
		such that; so that
		everywhere
		field extension	K : F means the field K extends the field F. This may also be written as K ≥ F.	ℝ : ℚ
		extends; over
		field theory
		inner product of matrices	A : B means the inner product of the matrices A and B. The general inner product is denoted by 〈u, v〉, 〈u | v〉 or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common. See also Bra-ket notation.
		inner product of
		linear algebra
!		factorial	n! means the product 1 × 2 × ... × n.	4! = 1 × 2 × 3 × 4 = 24
		factorial
		combinatorics
		logical negation	The statement !A is true if and only if A is false. A slash placed through another operator is the same as "!" placed in front. (The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.)	!(!A) ⇔ A  x ≠ y  ⇔  !(x = y)
		not
		propositional logic
~		probability distribution	X ~ D, means the random variable X has the probability distribution D.	X ~ N(0,1), the standard normal distribution
		has distribution
		statistics
		row equivalence	A~B means that B can be generated by using a series of elementary row operations on A
		is row equivalent to
		matrix theory
		same order of magnitude	m ~ n means the quantities m and n have the same order of magnitude, or general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈ .)	2 ~ 5 8 × 9 ~ 100 but π2 ≈ 10
		roughly similar; poorly approximates
		approximation theory
		asymptotically equivalent	f ~ g means .	x ~ x+1
		is asymptotically equivalent to
		asymptotic analysis
		equivalence relation	a ~ b means (and equivalently ).	1 ~ 5 mod 4
		are in the same equivalence class
		everywhere
≈		approximately equal	x ≈ y means x is approximately equal to y.	π ≈ 3.14159
		is approximately equal to
		everywhere
		isomorphism	G ≈ H means that group G is isomorphic (structurally identical) to group H. (≅ can also be used for isomorphic, as described below.)	Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein four-group.
		is isomorphic to
		group theory
≀		wreath product	A ≀ H means the wreath product of the group A by the group H. This may also be written A wr H.	is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.
		wreath product of … by …
		group theory
◅		normal subgroup	N ◅ G means that N is a normal subgroup of group G.	Z(G) ◅ G
		is a normal subgroup of
		group theory
		ideal	I ◅ R means that I is an ideal of ring R.	(2) ◅ Z
		is an ideal of
		ring theory
∴		therefore	Sometimes used in proofs before logical consequences.	All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
		therefore; so; hence
		everywhere
∵		because	Sometimes used in proofs before reasoning.	3331 is prime ∵ it has no positive integer factors other than itself and one.
		because; since
		everywhere
⇒ → ⊃		material implication	A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. (→ may mean the same as ⇒, or it may have the meaning for functions given below.) (⊃ may mean the same as ⇒, or it may have the meaning for superset given below.)	x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
		implies; if … then
		propositional logic, Heyting algebra
⇔ ↔		material equivalence	A ⇔ B means A is true if B is true and A is false if B is false.	x + 5 = y +2  ⇔  x + 3 = y
		if and only if; iff
		propositional logic
¬ ˜		logical negation	The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use ! but this is avoided in mathematical texts.)	¬(¬A) ⇔ A x ≠ y  ⇔  ¬(x =  y)
		not
		propositional logic
∧		logical conjunction or meet in a lattice	The statement A ∧ B is true if A and B are both true; else it is false. For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).	n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
		and; min; meet
		propositional logic, lattice theory
		wedge product	u ∧ v means the wedge product of vectors u and v. This generalizes the cross product to higher dimensions. (For vectors in R3, × can also be used.)
		wedge product; exterior product
		linear algebra
∨		logical disjunction or join in a lattice	The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).	n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
		or; max; join
		propositional logic, lattice theory
⊕ ⊻		exclusive or	The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.	(¬A) ⊕ A is always true, A ⊕ A is always false.
		xor
		propositional logic, Boolean algebra
		direct sum	The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).	Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0})
		direct sum of
		abstract algebra
∀		universal quantification	∀ x: P(x) means P(x) is true for all x.	∀ n ∈ ℕ: n2 ≥ n.
		for all; for any; for each
		predicate logic
∃		existential quantification	∃ x: P(x) means there is at least one x such that P(x) is true.	∃ n ∈ ℕ: n is even.
		there exists; there is; there are
		predicate logic
∃!		uniqueness quantification	∃! x: P(x) means there is exactly one x such that P(x) is true.	∃! n ∈ ℕ: n + 5 = 2n.
		there exists exactly one
		predicate logic
:= ≡ :⇔ ≜ ≝ ≐		definition	x := y or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context. (Some writers use ≡ to mean congruence). P :⇔ Q means P is defined to be logically equivalent to Q.
		is defined as; equal by definition
		everywhere
≅		congruence	△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
		is congruent to
		geometry
		isomorphic	G ≅ H means that group G is isomorphic (structurally identical) to group H. (≈ can also be used for isomorphic, as described above.)	.
		is isomorphic to
		abstract algebra
≡		congruence relation	a ≡ b (mod n) means a − b is divisible by n	5 ≡ 11 (mod 3)
		... is congruent to ... modulo ...
		modular arithmetic
{ , }		set brackets	{a,b,c} means the set consisting of a, b, and c.	ℕ = { 1, 2, 3, …}
		the set of …
		set theory
{ : } { | }		set builder notation	{x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}.	{n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4}
		the set of … such that
		set theory
∅ { }		empty set	∅ means the set with no elements. { } means the same.	{n ∈ ℕ : 1 < n2 < 4} = ∅
		the empty set
		set theory
∈ ∉		set membership	a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.	(1/2)−1 ∈ ℕ 2−1 ∉ ℕ
		is an element of; is not an element of
		everywhere, set theory
⊆ ⊂		subset	(subset) A ⊆ B means every element of A is also an element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.)	(A ∩ B) ⊆ A ℕ ⊂ ℚ ℚ ⊂ ℝ
		is a subset of
		set theory
⊇ ⊃		superset	A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.)	(A ∪ B) ⊇ B ℝ ⊃ ℚ
		is a superset of
		set theory
∪		set-theoretic union	A ∪ B means the set of those elements which are either in A, or in B, or in both.	A ⊆ B  ⇔  (A ∪ B) = B
		the union of … or …; union
		set theory
∩		set-theoretic intersection	A ∩ B means the set that contains all those elements that A and B have in common.	{x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
		intersected with; intersect
		set theory
∆		symmetric difference	A ∆ B means the set of elements in exactly one of A or B.	{1,5,6,8} ∆ {2,5,8} = {1,2,6}
		symmetric difference
		set theory
∖		set-theoretic complement	A ∖ B means the set that contains all those elements of A that are not in B. (− can also be used for set-theoretic complement as described above.)	{1,2,3,4} ∖ {3,4,5,6} = {1,2}
		minus; without
		set theory
→		function arrow	f: X → Y means the function f maps the set X into the set Y.	Let f: ℤ → ℕ∪{0} be defined by f(x) := x2.
		from … to
		set theory, type theory
↦		function arrow	f: a ↦ b means the function f maps the element a to the element b.	Let f: x ↦ x+1 (the successor function).
		maps to
		set theory
∘		function composition	fog is the function, such that (fog)(x) = f(g(x)).	if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
		composed with
		set theory
ℕ N		natural numbers	N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}. The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author's definition of N. Set theorists often use the notation ω to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.	ℕ = {|a| : a ∈ ℤ}
		N; the (set of) natural numbers
		numbers
ℤ Z		integers	ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}. ℤ+ or ℤ> means {1, 2, 3, ...} . ℤ≥ means {0, 1, 2, 3, ...} .	ℤ = {p, −p : p ∈ ℕ ∪ {0}​}
		Z; the (set of) integers
		numbers
ℤn ℤp Zn Zp		integers mod n	ℤn means {[0], [1], [2], ...[n−1]} with addition and multiplication modulo n. Note that any letter may be used instead of n, such as p. To avoid confusion with p-adic numbers, use ℤ/pℤ or ℤ/(p) instead.	ℤ3 = {[0], [1], [2]}
		Zn; the (set of) integers modulo n
		numbers
		p-adic integers	Note that any letter may be used instead of p, such as n or l.
		the (set of) p-adic integers
		numbers
ℚ Q		rational numbers	ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.	3.14000... ∈ ℚ π ∉ ℚ
		Q; the (set of) rational numbers; the rationals
		numbers
ℝ R		real numbers	ℝ means the set of real numbers.	π ∈ ℝ √(−1) ∉ ℝ
		R; the (set of) real numbers; the reals
		numbers
ℂ C		complex numbers	ℂ means {a + b i : a,b ∈ ℝ}.	i = √(−1) ∈ ℂ
		C; the (set of) complex numbers
		numbers
𝕂 K		real or complex numbers	K means both R and C: a statement containing K is true if either R or C is substituted for the K.
		K
		linear algebra
∞		infinity	∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.
		infinity
		numbers
⌊…⌋		floor	⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written [x], floor(x) or int(x).)	⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3
		floor; greatest integer; entier
		numbers
⌈…⌉		ceiling	⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x. (This may also be written ceil(x) or ceiling(x).)	⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2
		ceiling
		numbers
⌊…⌉		nearest integer function	⌊x⌉ means the nearest integer to x, with half-integers being rounded to even. (This may also be written [x], ||x||, nint(x) or Round(x).)	⌊2⌉ = 2, ⌊2.5⌉ = 2, ⌊3.5⌉ = 3, ⌊4.5⌉ = 4, ⌊7.2⌉ = 7, ⌊8.9⌉ = 9
		nearest integer to
		numbers
[ : ]		degree of a field extension	[K : F] means the degree of the extension K : F.	[ℚ(√2) : ℚ] = 2 [ℂ : ℝ] = 2 [ℝ : ℚ] = ∞
		the degree of
		field theory
[ ] [ , ] [ , , ]		equivalence class	[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation. [a]R means the same, but with R as the equivalence relation.	Let a ~ b be true iff a ≡ b (mod 5). Then [2] = {…, −8, −3, 2, 7, …}.
		the equivalence class of
		abstract algebra
		floor	[x] means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written ⌊x⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.)	[3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4
		floor; greatest integer; entier
		numbers
		nearest integer function	[x] means the nearest integer to x, with half-integers being rounded to even. (This may also be written ⌊x⌉, ||x||, nint(x) or Round(x). Not to be confused with the floor function, as described above.)	[2] = 2, [2.5] = 2, [3.5] = 4, [4.5] = 4
		nearest integer to
		numbers
		closed interval	.	[0,1]
		closed interval
		order theory
		commutator	[g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group). [a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra).	xy = x[x, y] (group theory). [AB, C] = A[B, C] + [A, C]B (ring theory).
		the commutator of
		group theory, ring theory
		triple scalar product	[a, b, c] = a × b · c, the scalar product of a × b with c.	[a, b, c] = [b, c, a] = [c, a, b].
		the triple scalar product of
		vector calculus
( ) ( , )		function application	f(x) means the value of the function f at the element x.	If f(x) := x2, then f(3) = 32 = 9.
		of
		set theory
		precedence grouping	Perform the operations inside the parentheses first.	(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
		parentheses
		everywhere
		tuple	An ordered list (or sequence, or horizontal vector, or row vector) of values. (Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval.)	(a, b) is an ordered pair (or 2-tuple). (a, b, c) is an ordered triple (or 3-tuple). ( ) is the empty tuple (or 0-tuple).
		tuple; n-tuple; ordered pair/triple/etc; row vector
		everywhere
		highest common factor	(a, b) means the highest common factor of a and b. (This may also be written hcf(a, b) or gcd(a, b).)	(3, 7) = 1 (they are coprime); (15, 25) = 5.
		highest common factor; greatest common divisor; hcf; gcd
		number theory
( , ) ] , [		open interval	. (Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.)	(4,18)
		open interval
		order theory
( , ] ] , ]		left-open interval	.	(−1, 7] and (−∞, −1]
		half-open interval; left-open interval
		order theory
[ , ) [ , [		right-open interval	.	[4, 18) and [1, +∞)
		half-open interval; right-open interval
		order theory
〈〉 <> 〈,〉 <,>		inner product	〈u,v〉 means the inner product of u and v, where u and v are members of an inner product space. Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the linear span. There are many variants of the notation, such as 〈u | v〉 and (u | v), which are described below. The less-than and greater-than symbols are primarily from computer science; they are avoided in mathematical texts. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used.	The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2 × −1 + 3 × 5 = 13
		inner product of
		linear algebra
		linear span	〈S〉 means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. 〈u1, u2, …〉is shorthand for 〈{u1, u2, …}〉. Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the linear span. The span of S may also be written as Sp(S).	.
		(linear) span of; linear hull of
		linear algebra
		subgroup generated by a set	〈S〉 means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S. 〈g1, g2, …〉is shorthand for 〈{g1, g2, …}〉.	In S3, 〈(1 2)〉 = {id, (1 2)} and 〈(1 2 3)〉 = {id, (1 2 3), (1 3 2)}.
		the subgroup generated by
		group theory
〈|〉 <|> (|)		inner product	〈u | v〉 means the inner product of u and v, where u and v are members of an inner product space. (u | v) means the same. Another variant of the notation is 〈u, v〉 which is described above. The less-than and greater-than symbols are primarily from computer science; they are avoided in mathematical texts. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used.
		inner product of
		linear algebra
∑		summation	means a1 + a2 + … + an.	= 12 + 22 + 32 + 42  = 1 + 4 + 9 + 16 = 30
		sum over … from … to … of
		arithmetic
∏		product	means a1a2···an.	= (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
		product over … from … to … of
		arithmetic
		Cartesian product	means the set of all (n+1)-tuples (y0, …, yn).
		the Cartesian product of; the direct product of
		set theory
∐		coproduct	A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.
		coproduct over … from … to … of
		category theory
′ •		derivative	f ′(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. The dot notation indicates a time derivative. That is .	If f(x) := x2, then f ′(x) = 2x
		… prime derivative of
		calculus
∫		indefinite integral or antiderivative	∫ f(x) dx means a function whose derivative is f.	∫x2 dx = x3/3 + C
		indefinite integral of the antiderivative of
		calculus
		definite integral	∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.	∫ab x2 dx = b3/3 − a3/3;
		integral from … to … of … with respect to
		calculus
∮		contour integral or closed line integral	Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰. The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface.	If C is a Jordan curve about 0, then .
		contour integral of
		calculus
∇		gradient	∇f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn).	If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
		del, nabla, gradient of
		vector calculus
		divergence		If , then .
		del dot, divergence of
		vector calculus
		curl		If , then .
		curl of
		vector calculus
∂		partial derivative	∂f/∂xi means the partial derivative of f with respect to xi, where f is a function on (x1, …, xn).	If f(x,y) := x2y, then ∂f/∂x = 2xy
		partial, d
		calculus
		boundary	∂M means the boundary of M	∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
		boundary of
		topology
		degree of a polynomial	∂f means the degree of the polynomial f. (This may also be written deg f.)	∂(x2 − 1) = 2
		degree of
		algebra
δ		Dirac delta function		δ(x)
		Dirac delta of
		hyperfunction
		Kronecker delta		δij
		Kronecker delta of
		hyperfunction
<: <·		cover	x <• y means that x is covered by y.	{1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
		is covered by
		order theory
		subtype	T1 <: T2 means that T1 is a subtype of T2.	If S <: T and T <: U then S <: U (transitivity).
		is a subtype of
		type theory
T		transpose	AT means A, but with its rows swapped for columns. This may also be written At or Atr.	If A = (aij) then AT = (aji).
		transpose
		matrix operations
⊤		top element	⊤ means the largest element of a lattice.	∀x : x ∨ ⊤ = ⊤
		the top element
		lattice theory
		top type	⊤ means the top or universal type; every type in the type system of interest is a subtype of top.	∀ types T, T <: ⊤
		the top type; top
		type theory
⊥		perpendicular	x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.	If l ⊥ m and m ⊥ n in the plane then l || n.
		is perpendicular to
		geometry
		orthogonal complement	W⊥ means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W.	Within , .
		orthogonal/perpendicular complement of; perp
		linear algebra
		coprime	x ⊥ y means x has no factor in common with y.	34  ⊥  55.
		is coprime to
		number theory
		bottom element	⊥ means the smallest element of a lattice.	∀x : x ∧ ⊥ = ⊥
		the bottom element
		lattice theory
		bottom type	⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system.	∀ types T, ⊥ <: T
		the bottom type; bot
		type theory
		comparability	x ⊥ y means that x is comparable to y.	{e, π} ⊥ {1, 2, e, 3, π} under set containment.
		is comparable to
		order theory
⊧		entailment	A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true.	A ⊧ A ∨ ¬A
		entails
		model theory
⊢		inference	x ⊢ y means y is derivable from x.	A → B ⊢ ¬B → ¬A.
		infers; is derived from
		propositional logic, predicate logic
⊗		tensor product, tensor product of modules	means the tensor product of V and U. means the tensor product of modules V and U over the ring R.	{1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
		tensor product of
		linear algebra
*		convolution	f * g means the convolution of f and g.	.
		convolution, convolved with
		functional analysis
		complex conjugate	z* means the complex conjugate of z. ( can also be used for the conjugate of z, as described below.)	.
		conjugate
		complex numbers
		group of units	R* consists of the set of units of the ring R, along with the operation of multiplication. This may also be written R× as described above, or U(R).
		the group of units of
		ring theory
x		mean	(often read as “x bar”) is the mean (average value of xi).	.
		overbar, … bar
		statistics
		complex conjugate	means the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.)	.
		conjugate
		complex numbers
		algebraic closure	is the algebraic closure of the field F.	The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers .
		algebraic closure of
		field theory
		topological closure	is the topological closure of the set S. This may also be denoted as cl(S) or Cl(S).	In the space of the real numbers, (the rational numbers are dense in the real numbers).
		(topological) closure of
		topology

[edit] Advanced and less-frequently used math symbols

Here are some other math symbols in a compact list, preceded by their Unicode values

• x2135: ℵ: Hebrew letter aleph, the sign for cardinality

• x2210 ∐: direct sum, but in other areas of math, it might mean the coproduct

• x22c9 ⋉: semidirect product

[edit] Note on symbol names

One symbol can have up to four different names in standard web publishing. For example, the Unicode symbol known as "Subset of or equal to" can be expressed as:

• Unicode: ⊆ rendered as ⊆ or as the raw Unicode character ⊆ in the wikitext
• HTML: ⊆ rendered as ⊆
• TeX (LaTex, MathML): \subseteq rendered as
• Postscript (Adobe, PDF) name: reflexsubset. The Postscript name is of interest if you are viewing a raw PDF 1.3 or earlier version files or other low-level issue. See Symbol (typeface) is some Postscript names that one will find if one examines PDF files. (see PostScript Language Reference Manual, pp. 256-257, "Symbol Font").

Some useful cross-references via Unicode are:

• Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List
• MathML Characters - sorts out Unicode, HTML and MathML/TeX names on one page
• Unicode values and MathML names
• Unicode values and Postscript names from the source code for Ghostscript

[edit] Abbreviated function names

There are a typical set of abbreviations used for functions whose names.

Operators borrowed from linear algebra include:

• char = characteristic
• det = determinant
• dim = dimension (vector space)
• Card = cardinality (AKA order)

and from category theory:

• Ker = kernel (category theory)
• Im = Image (mathematics)
• Tr or trace = field trace

and from other areas of abstract algebra:

• = , → See Exact functor, Topological half-exact functor
• δ = Delta functor
• End = Endomorphism ring
• Ext = Ext functor (extension of modules)
• Frob = Frobenius endomorphism
• Hom = Hom functor
• Spec = spectrum of a ring
• Sym = symmetric group
• Sym2 = symmetric square
• Tan = tangent space
• Tor = Tor functor (torsion)

• general and special linear groups are referred to with the acronyms SL, GL, PSL and PGL.

[edit] See also

• Greek letters used in mathematics
• ISO 31-11
• Mathematical alphanumeric symbols
• Mathematical notation
• Notation in probability and statistics
• Physical constants
• Roman letters used in mathematics
• Table of logic symbols
• Unicode Mathematical Operators
• Wikipedia:Mathematical symbols
• Help:Advanced editing#Special characters
• Help:Displaying a formula

[edit] External links

• Jeff Miller: Earliest Uses of Various Mathematical Symbols
• TCAEP - Institute of Physics
• GIF and PNG Images for Math Symbols
• Mathematical Symbols in Unicode
• Using Greek and special characters from Symbol font in HTML