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Please read: a personal appeal from Wikipedia founder Jimmy Wales Read now List of mathematical symbols From Wikipedia, the free encyclopedia Jump to: navigation, search This list is incomplete; you can help by expanding it. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2011) This is a listing of common symbols found within all branches of mathematics. Symbols are used in mathematical notation to express a formula or to replace a constant. It is important to recognize that a mathematical concept is independent of the symbol chosen to represent it when reading the list. The symbols below are usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics) but in some situations a different convention may be used. For example, the meaning of "≡" may represent congruence or a definition depending on context. Further, in mathematical logic, the concept of numerical equality is sometimes represented by "≡" instead of "=", with the latter taking the duty of representing equality of well-formed formulas. In short, convention rather than the symbol dictates the meaning. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in TEX, as an image. Contents [hide] 1 Symbols 2 Variations 3 See also 4 References 5 External links [edit] Symbols Name Sy Sy Read as mb mb ol ol Explanation Examples in in Category HT TE ML X equality is equal to; x = y means x and y represent the 2=2 = equals same thing or value. 1+1=2 everywhere x ≠ y means that x and y do not inequality represent the same thing or value. ≠ is not equal to; does not equal (The forms !=, /= or <> are generally used in programming 2+2≠5 languages where ease of typing and everywhere use of ASCII text is preferred.) strict inequality x < y means x is less than y. 3<4 is less than, 5>4 is greater than x > y means x is greater than y. < order theory proper subgroup > H < G means H is a proper subgroup 5Z < Z is a proper of G. A3 < S3 subgroup of group theory (very) strict x ≪ y means x is much less than y. ≪ inequality x ≫ y means x is much greater than 0.003 ≪ 1000000 is much less y. than, ≫ is much greater than order theory asymptotic comparison f ≪ g means the growth of f is is of smaller asymptotically bounded by g. order than, x ≪ ex is of greater (This is I. M. Vinogradov's notation. order than Another notation is the Big O notation, which looks like f = O(g).) analytic number theory x ≤ y means x is less than or equal to inequality y. is less than or x ≥ y means x is greater than or equal equal to, to y. 3 ≤ 4 and 5 ≤ 5 is greater than 5 ≥ 4 and 5 ≥ 5 or equal to (The forms <= and >= are generally used in programming languages order theory where ease of typing and use of ≤ subgroup ASCII text is preferred.) is a subgroup Z≤Z H ≤ G means H is a subgroup of G. ≥ of A3 ≤ S3 group theory reduction If A ≤ B means the problem A can be is reducible to reduced to the problem B. Subscripts can be added to the ≤ to indicate computational then what kind of reduction. complexity theory congruence 7k ≡ 28 (mod 2) is only true if k is an ≦ relation even integer. Assume that the problem requires k to be non- 10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10 ...is less than negative; the domain is defined as 0 ... is greater ≦ k ≦ ∞. ≧ than... modular arithmetic x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector vector y. inequality x ≧ y means that each component of ... is less than vector x is greater than or equal to or equal... is each corresponding component of greater than or vector y. equal... It is important to note that x ≦ y order theory remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true. Karp reduction is Karp reducible to; is polynomial- L ≺ L2 means that the problem L1 is If L1 ≺ L2 and L2 ∈ P, then L1 ≺ time many-one 1 reducible to Karp reducible to L2.[1] ∈ P. computational complexity theory proportionality is proportional y ∝ x means that y = kx for some to; if y = 2x, then y ∝ x. constant k. varies as ∝ everywhere Karp reduction[2] A ∝ B means the problem A can be If L1 ∝ L2 and L2 ∈ P, then L1 is Karp polynomially reduced to the problem ∈ P. reducible to; B. is polynomial- time many-one reducible to computational complexity theory addition plus; 4 + 6 means the sum of 4 and 6. 2+7=9 add arithmetic + disjoint union A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, the disjoint A1 + A2 means the disjoint union of 9, 10} ⇒ union of ... sets A1 and A2. A1 + A2 = {(3,1), (4,1), (5,1), and ... (6,1), (7,2), (8,2), (9,2), (10,2)} set theory subtraction minus; 9 − 4 means the subtraction of 4 take; 8−3=5 from 9. subtract arithmetic negative sign negative; −3 means the negative of the number − minus; the opposite of 3. −(−5) = 5 arithmetic set-theoretic complement A − B means the set that contains all the elements of A that are not in B. minus; {1,2,4} − {1,3,4} = {2} without (∖ can also be used for set-theoretic complement as described below.) set theory plus-minus The equation x = 5 ± √4, has ± plus or minus 6 ± 3 means both 6 + 3 and 6 − 3. two solutions, x = 7 and x = 3. arithmetic plus-minus 10 ± 2 or equivalently 10 ± 20% If a = 100 ± 1 mm, then a ≥ 99 plus or minus means the range from 10 − 2 to 10 + mm and a ≤ 101 mm. 2. measurement minus-plus 6 ± (3 ∓ 5) means both 6 + (3 − 5) cos(x ± y) = cos(x) cos(y) ∓ ∓ minus or plus and 6 − (3 + 5). sin(x) sin(y). arithmetic 3 × 4 means the multiplication of 3 multiplication by 4. times; (The symbol * is generally used in 7 × 8 = 56 multiplied by programming languages, where ease of typing and use of ASCII text is arithmetic preferred.) Cartesian product the Cartesian X×Y means the set of all ordered product of ... pairs with the first element of each {1,2} × {3,4} = and ...; pair selected from X and the second {(1,3),(1,4),(2,3),(2,4)} the direct element selected from Y. × product of ... and ... set theory cross product u × v means the cross product of (1,2,5) × (3,4,−1) = cross vectors u and v (−22, 16, − 2) linear algebra group of units R× consists of the set of units of the ring R, along with the operation of the group of multiplication. units of This may also be written R* as ring theory described below, or U(R). multiplication a * b means the product of a and b. times; (Multiplication can also be denoted 4 * 3 means the product of 4 * multiplied by with × or ⋅, or even simple and 3, or 12. juxtaposition. * is generally used arithmetic where ease of typing and use of ASCII text is preferred, such as programming languages.) convolution convolution; convolved f * g means the convolution of f and with g. . functional analysis complex conjugate z* means the complex conjugate of z. conjugate . ( can also be used for the complex conjugate of z, as described below.) numbers group of units R* consists of the set of units of the ring R, along with the operation of the group of multiplication. units of This may also be written R× as ring theory described above, or U(R). hyperreal numbers *R means the set of hyperreal the (set of) *N is the hypernatural numbers. Other sets can be used in hyperreals numbers. place of R. non-standard analysis Hodge dual *v means the Hodge dual of a vector If are the standard basis Hodge dual; v. If v is a k-vector within an n- Hodge star dimensional oriented inner product vectors of , space, then *v is an (n−k)-vector. linear algebra multiplication times; 3 · 4 means the multiplication of 3 7 · 8 = 56 multiplied by by 4. · arithmetic dot product u · v means the dot product of (1,2,5) · (3,4,−1) = 6 vectors u and v dot linear algebra placeholder A · means a placeholder for an argument of a function. Indicates the (silent) functional nature of an expression without assigning a specific symbol functional for an argument. analysis tensor product, tensor product of modules means the tensor product of {1, 2, 3, 4} ⊗ {1, 1, 2} = ⊗ V and U.[3] means the tensor product tensor product of modules V and U {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} of over the ring R. linear algebra division (Obelus) 2 ÷ 4 = 0.5 6 ÷ 3 or 6 ⁄ 3 means the division of 6 divided by; by 3. over 12 ⁄ 4 = 3 ÷ arithmetic quotient group {0, a, 2a, b, b+a, b+2a} / {0, G / H means the quotient of group G mod b} = {{0, b}, {a, b+a}, {2a, ⁄ group theory modulo its subgroup H. b+2a}} quotient set If we define ~ by x ~ y ⇔ x − A/~ means the set of all ~ y ∈ ℤ, then mod equivalence classes in A. ℝ/~ = { {x + n : n ∈ ℤ } : x ∈ [0,1) } set theory square root the (principal) means the nonnegative number square root of whose square is . √ real numbers complex if is represented in square root polar coordinates with , then the (complex) . square root of complex numbers mean overbar; (often read as “x bar”) is the mean … bar (average value of ). . statistics complex conjugate means the complex conjugate of z. conjugate . (z* can also be used for the conjugate of z, as described above.) complex numbers finite sequence, tuple means the finite sequence/tuple finite . x sequence, . tuple model theory algebraic closure The field of algebraic numbers is sometimes denoted as algebraic is the algebraic closure of the field because it is the algebraic closure of F. closure of the rational numbers . field theory topological closure is the topological closure of the set In the space of the real S. (topological) numbers, (the rational closure of This may also be denoted as cl(S) or numbers are dense in the real numbers). Cl(S). topology unit vector (pronounced "a hat") is the â hat normalized version of vector , having length 1. geometry absolute value; |3| = 3 modulus |x| means the distance along the real |–5| = |5| = 5 absolute value line (or across the complex plane) of; modulus of between x and zero. |i|=1 numbers | 3 + 4i | = 5 Euclidean norm or Euclidean length or magnitude |x| means the (Euclidean) length of For x = (3,-4) | Euclidean vector x. … norm of | geometry determinant |A| means the determinant of the determinant of matrix A matrix theory cardinality |X| means the cardinality of the set X. cardinality of; size of; |{3, 5, 7, 9}| = 4. (# may be used instead as described order of below.) set theory norm norm of; || x || means the norm of the element || x + y || ≤ || x || + || y || length of x of a normed vector space.[4] || linear algebra … nearest integer function ||x|| means the nearest integer to x. || nearest integer ||1|| = 1, ||1.6|| = 2, ||−2.4|| = −2, (This may also be written [x], ⌊x⌉, ||3.49|| = 3 to nint(x) or Round(x).) numbers divisor, a|b means a divides b. ∣ divides a∤b means a does not divide b. Since 15 = 3×5, it is true that 3|15 and 5|15. divides (This symbol can be difficult to type, and its negation is rare, so a regular ∤ number theory but slightly shorter vertical bar | character can be used.) conditional probability P(A|B) means the probability of the if X is a uniformly random day event a occurring given that b of the year P(X is May 25 | X given occurs. is in May) = 1/31 probability restriction f| means the function f restricted to restriction of A The function f : R → R the set A, that is, it is the function … to …; defined by f(x) = x2 is not with domain A ∩ dom(f) that agrees restricted to injective, but f|R+ is injective. with f. set theory such that S = {(x,y) | 0 < y < f(x)} such that; | means “such that”, see ":" The set of (x,y) such that y is so that (described below). greater than 0 and less than f(x). everywhere parallel is parallel to x || y means x is parallel to y. If l || m and m ⊥ n then l ⊥ n. geometry incomparabilit y is {1,2} || {2,3} under set x || y means x is incomparable to y. || incomparable to containment. order theory exact divisibility pa || n means pa exactly divides n 23 || 360. exactly divides (i.e. pa divides n but pa+1 does not). number theory cardinality #X means the cardinality of the set # cardinality of; X. #{4, 6, 8} = 3 size of; (|…| may be used instead as order of described above.) set theory connected sum connected sum A#B is the connected sum of the of; manifolds A and B. If A and B are A#Sm is homeomorphic to A, knot sum of; knots, then this denotes the knot for any manifold A, and the knot sum, which has a slightly stronger sphere Sm. composition of condition. topology, knot theory primorial n# is product of all prime numbers 12# = 2 × 3 × 5 × 7 × 11 = primorial less than or equal to n. 2310 number theory aleph number ℵα represents an infinite cardinality |ℕ| = ℵ0, which is called aleph- ℵ aleph (specifically, the α-th one, where α is an ordinal). null. set theory beth number ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not ℶ beth necessarily index all of the numbers indexed by ℵ. ). set theory cardinality of the continuum cardinality of the continuum; The cardinality of is denoted by c; or by the symbol (a lowercase cardinality of Fraktur letter C). the real numbers set theory such that : means “such that”, and is used in : such that; so that proofs and the set-builder notation (described below). ∃ n ∈ ℕ: n is even. everywhere field extension K : F means the field K extends the extends; field F. ℝ:ℚ over This may also be written as K ≥ F. field theory A : B means the Frobenius inner inner product product of the matrices A and B. of matrices The general inner product is denoted inner product by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as of described below. For spatial vectors, the dot product notation, x·y is linear algebra common. See also Bra-ket notation. index of a subgroup The index of a subgroup H in a group G is the "relative size" of H in index of G: equivalently, the number of subgroup "copies" (cosets) of H that fill up G group theory factorial factorial n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24 combinatorics The statement !A is true if and only if A is false. logical ! negation A slash placed through another operator is the same as "!" placed in !(!A) ⇔ A not front. x ≠ y ⇔ !(x = y) propositional (The symbol ! is primarily from logic computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.) probability distribution X ~ D, means the random variable X X ~ N(0,1), the standard ~ has distribution has the probability distribution D. normal distribution statistics row equivalence A~B means that B can be generated is row by using a series of elementary row equivalent to operations on A matrix theory same order of magnitude m ~ n means the quantities m and n have the same order of magnitude, or 2 ~ 5 roughly general size. similar; 8 × 9 ~ 100 poorly (Note that ~ is used for an approximates approximation that is poor, but π2 ≈ 10 otherwise use ≈ .) approximation theory asymptotically equivalent is asymptotically x ~ x+1 equivalent to f ~ g means . asymptotic analysis equivalence relation are in the same a ~ b means (and 1 ~ 5 mod 4 equivalence equivalently ). class everywhere approximately equal x ≈ y means x is approximately equal to y. is π ≈ 3.14159 approximately This may also be written ≃, ≅, ~, ♎ ≈ equal to (Libra Symbol), or ≒. everywhere isomorphism G ≈ H means that group G is Q / {1, −1} ≈ V, isomorphic (structurally identical) to where Q is the quaternion is isomorphic group H. group and V is the Klein four- to group. (≅ can also be used for isomorphic, group theory as described below.) wreath product A ≀ H means the wreath product of is isomorphic to the ≀ wreath product the group A by the group H. of … by … automorphism group of the complete bipartite graph on This may also be written A wr H. (n,n) vertices. group theory normal subgroup N ◅ G means that N is a normal is a normal Z(G) ◅ G subgroup of subgroup of group G. group theory ◅ ideal I ◅ R means that I is an ideal of ring is an ideal of (2) ◅ Z R. ▻ ring theory antijoin R ▻ S means the antijoin of the the antijoin of relations R and S, the tuples in R for which there is not a tuple in S that is R S = R - R S equal on their common attribute relational names. algebra N ⋊φ H is the semidirect product of semidirect N (a normal subgroup) and H (a product subgroup), with respect to φ. Also, if G = N ⋊φ H, then G is said to split the semidirect ⋉ over N. product of (⋊ may also be written the other way group theory round, as ⋉, or as ×.) ⋊ semijoin R ⋉ S is the semijoin of the relations R and S, the set of all tuples in R for the semijoin of which there is a tuple in S that is R S= a1,..,an(R S) equal on their common attribute relational names. algebra natural join R ⋈ S is the natural join of the ⋈ relations R and S, the set of all the natural join combinations of tuples in R and S of that are equal on their common attribute names. relational algebra therefore therefore; All humans are mortal. ∴ so; hence Sometimes used in proofs before logical consequences. Socrates is a human. ∴ Socrates is mortal. everywhere because 3331 is prime ∵ it has no ∵ because; since Sometimes used in proofs before reasoning. positive integer factors other than itself and one. everywhere ■ end of proof QED; Used to mark the end of a proof. tombstone; □ Halmos symbol (May also be written Q.E.D.) everywhere ∎ It is the generalisation of the Laplace D'Alembertian operator in the sense that it is the differential operator which is ▮ non-Euclidean invariant under the isometry group Laplacian of the underlying space and it reduces to the Laplace operator if vector calculus restricted to time independent ‣ functions. A ⇒ B means if A is true then B is ⇒ material implication also true; if A is false then nothing is said about B. implies; x = 2 ⇒ x2 = 4 is true, but x2 = (→ may mean the same as ⇒, or it → if … then may have the meaning for functions 4 ⇒ x = 2 is in general false (since x could be −2). given below.) propositional logic, Heyting (⊃ may mean the same as ⇒,[5] or it ⊃ algebra may have the meaning for superset given below.) material ⇔ equivalence if and only if; A ⇔ B means A is true if B is true x+5=y+2⇔x+3=y iff and A is false if B is false. ↔ propositional logic The statement ¬A is true if and only if A is false. logical A slash placed through another ¬ negation operator is the same as "¬" placed in front. ¬(¬A) ⇔ A not x ≠ y ⇔ ¬(x = y) ˜ propositional logic (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.) logical conjunction or meet in a lattice The statement A ∧ B is true if A and B are both true; else it is false. and; n < 4 ∧ n >2 ⇔ n = 3 when n min; For functions A(x) and B(x), A(x) ∧ is a natural number. meet B(x) is used to mean min(A(x), B(x)). propositional logic, lattice theory ∧ wedge product u ∧ v means the wedge product of wedge any multivectors u and v. In three product; dimensional Euclidean space the exterior wedge product and the cross product product of two vectors are each other's Hodge dual. exterior algebra exponentiation a ^ b means a raised to the power of b 2^3 = 23 = 8 … (raised) to the power of (a ^ b is more commonly written ab. … The symbol ^ is generally used in programming languages where ease everywhere of typing and use of plain ASCII text is preferred.) logical disjunction or join in a The statement A ∨ B is true if A or B lattice (or both) are true; if both are false, the statement is false. n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n ∨ or; max; For functions A(x) and B(x), A(x) ∨ is a natural number. join B(x) is used to mean max(A(x), B(x)). propositional logic, lattice theory exclusive or xor The statement A ⊕ B is true when (¬A) ⊕ A is always true, A ⊕ either A or B, but not both, are true. A is always false. propositional A ⊻ B means the same. ⊕ logic, Boolean algebra The direct sum is a special way of direct sum ⊻ direct sum of combining several objects into one general object. Most commonly, for vector spaces U, V, and W, the following consequence is used: (The bun symbol ⊕, or the U = V ⊕ W ⇔ (U = V + W) ∧ abstract coproduct symbol ∐, is used; ⊻ is (V ∩ W = {0}) algebra only for logic.) universal quantification ∀ for all; for any; ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. for each predicate logic existential quantification ∃ x: P(x) means there is at least one ∃ there exists; x such that P(x) is true. ∃ n ∈ ℕ: n is even. there is; there are predicate logic uniqueness quantification ∃! there exists exactly one ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. predicate logic =: := ≡ x := y, y =: x or x ≡ y means x is defined to be another name for y, definition under certain assumptions taken in context. : is defined as; is equal by ⇔ definition to (Some writers use ≡ to mean congruence). everywhere P :⇔ Q means P is defined to be ≜ logically equivalent to Q. ≝ ≐ congruence △ABC ≅ △DEF means triangle is congruent to ABC is congruent to (has the same ≅ geometry measurements as) triangle DEF. isomorphic G ≅ H means that group G is isomorphic (structurally identical) to . is isomorphic group H. to (≈ can also be used for isomorphic, abstract as described above.) algebra congruence relation ... is congruent a ≡ b (mod n) means a − b is ≡ to ... modulo ... divisible by n 5 ≡ 2 (mod 3) modular arithmetic set brackets {, the set of … {a,b,c} means the set consisting of ℕ = { 1, 2, 3, …} a, b, and c.[6] } set theory {: } set builder notation {| {x : P(x)} means the set of all x for [6] the set of … which P(x) is true. {x | P(x)} is the {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} } such that same as {x : P(x)}. set theory {; } ∅ empty set ∅ means the set with no elements.[6] the empty set {n ∈ ℕ : 1 < n2 < 4} = ∅ { } means the same. { set theory } a ∈ S means a is an element of the (1/2)−1 ∈ ℕ ∈ set set S;[6] a ∉ S means a is not an membership element of S.[6] 2−1 ∉ ℕ ∉ is an element of; is not an element of everywhere, set theory (subset) A ⊆ B means every element ⊆ subset of A is also an element of B.[7] (A ∩ B) ⊆ A (proper subset) A ⊂ B means A ⊆ B is a subset of ℕ⊂ℚ but A ≠ B. ⊂ set theory (Some writers use the symbol ⊂ as if ℚ⊂ℝ it were the same as ⊆.) A ⊇ B means every element of B is ⊇ superset also an element of A. (A ∪ B) ⊇ B is a superset of A ⊃ B means A ⊇ B but A ≠ B. ℝ⊃ℚ ⊃ set theory (Some writers use the symbol ⊃ as if it were the same as ⊇.) set-theoretic union A ∪ B means the set of those ∪ the union of … or …; elements which are either in A, or in A ⊆ B ⇔ (A ∪ B) = B B, or in both.[7] union set theory set-theoretic intersection A ∩ B means the set that contains all ∩ intersected with; those elements that A and B have in {x ∈ ℝ : x2 = 1} ∩ ℕ = {1} common.[7] intersect set theory A ∆ B means the set of elements in symmetric exactly one of A or B. ∆ difference (Not to be confused with delta, Δ, {1,5,6,8} ∆ {2,5,8} = {1,2,6} symmetric described below.) difference set theory set-theoretic A ∖ B means the set that contains all complement those elements of A that are not in ∖ minus; without B.[7] {1,2,3,4} ∖ {3,4,5,6} = {1,2} (− can also be used for set-theoretic complement as described above.) set theory function arrow from … to f: X → Y means the function f maps Let f: ℤ → ℕ∪{0} be defined → the set X into the set Y. by f(x) := x2. set theory, type theory function arrow f: a ↦ b means the function f maps Let f: x ↦ x+1 (the successor ↦ maps to the element a to the element b. function). set theory function composition ∘ composed with f∘g is the function, such that (f∘g)(x) if f(x) := 2x, and g(x) := x + 3, = f(g(x)).[8] then (f∘g)(x) = 2(x + 3). set theory For two matrices (or vectors) of the same dimensions Hadamard the Hadamard product is a matrix of product the same dimensions with elements o entrywise product given by . linear algebra This is often used in matrix based programming such as MATLAB where the operation is done by A.*B ℕ natural numbers N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}. ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > N; The choice depends on the area of 0: a ∈ ℤ} the (set of) mathematics being studied; e.g. N natural number theorists prefer the latter; numbers analysts, set theorists and computer scientists prefer the former. To avoid numbers confusion, always check an author's definition of N. Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤. integers ℤ Z; ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}. the (set of) ℤ = {p, −p : p ∈ ℕ ∪ {0}} integers ℤ or ℤ means {1, 2, 3, ...} . ℤ or + > * Z ℤ≥ means {0, 1, 2, 3, ...} . numbers integers mod n ℤn means {[0], [1], [2], ...[n−1]} ℤn with addition and multiplication Zn; modulo n. the (set of) ℤ3 = {[0], [1], [2]} ℤp integers Note that any letter may be used modulo n instead of n, such as p. To avoid confusion with p-adic numbers, use numbers ℤ/pℤ or ℤ/(p) instead. p-adic integers Zn the (set of) p- adic integers Note that any letter may be used instead of p, such as n or l. Zp numbers projective space P; ℙ the projective space; ℙ means a space with a point at the projective infinity. , line; the projective P plane topology probability ℙ(X) means the probability of the If a fair coin is flipped, event X occurring. ℙ(Heads) = ℙ(Tails) = 0.5. the probability of This may also be written as P(X), Pr(X), P[X] or Pr[X]. probability theory rational numbers ℚ Q; 3.14000... ∈ ℚ the (set of) ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. rational π∉ℚ numbers; Q the rationals numbers real numbers ℝ R; π∈ℝ the (set of) ℝ means the set of real numbers. real numbers; √(−1) ∉ ℝ the reals R numbers complex numbers ℂ C; the (set of) ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ complex C numbers numbers quaternions or Hamiltonian ℍ quaternions ℍ means {a + b i + c j + d k : a,b,c,d H; ∈ ℝ}. the (set of) H quaternions numbers Big O notation The Big O notation describes the If f(x) = 6x4 − 2x3 + 5 and g(x) limiting behavior of a function, = x4 , then O big-oh of when the argument tends towards a particular value or infinity. Computational complexity theory infinity ∞ is an element of the extended ∞ infinity number line that is greater than all real numbers; it often occurs in limits. numbers floor ⌊ floor; ⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x. ⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, … greatest integer; (This may also be written [x], ⌊−2.6⌋ = −3 ⌋ entier floor(x) or int(x).) numbers ⌈ ceiling ⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, … ceiling to x. ⌈−2.6⌉ = −2 ⌉ numbers (This may also be written ceil(x) or ceiling(x).) nearest integer ⌊ function ⌊x⌉ means the nearest integer to x. ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, … nearest integer to (This may also be written [x], ||x||, ⌊4.49⌉ = 4 ⌉ numbers nint(x) or Round(x).) degree of a [ℚ(√2) : ℚ] = 2 field extension [: [K : F] means the degree of the [ℂ : ℝ] = 2 the degree of extension K : F. ] [ℝ : ℚ] = ∞ field theory equivalence [] class [a] means the equivalence class of a, Let a ~ b be true iff a ≡ b (mod i.e. {x : x ~ a}, where ~ is an the 5). equivalence relation. equivalence [, class of [a]R means the same, but with R as Then [2] = {…, −8, −3, 2, 7, …}. the equivalence relation. ] abstract algebra floor [x] means the floor of x, i.e. the largest integer less than or equal to x. floor; [, greatest (This may also be written ⌊x⌋, [3] = 3, [3.5] = 3, [3.99] = 3, integer; [−3.7] = −4 ,] entier floor(x) or int(x). Not to be confused with the nearest integer function, as described below.) numbers nearest integer [x] means the nearest integer to x. function (This may also be written ⌊x⌉, ||x||, [2] = 2, [2.6] = 3, [-3.4] = -3, nearest integer nint(x) or Round(x). Not to be [4.49] = 4 to confused with the floor function, as described above.) numbers Iverson bracket 1 if true, 0 [S] maps a true statement S to 1 and [0=5]=0, [7>0]=1, [2 ∈ otherwise a false statement S to 0. {2,3,4}]=1, [5 ∈ {2,3,4}]=0 propositional logic f[X] means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ image dom(f). image of … (This may also be written as f(X) if under … there is no risk of confusing the image of f under X with the function everywhere application f of X. Another notation is Im f, the image of f under its domain.) closed interval 0 and 1/2 are in the interval closed interval . [0,1]. order theory commutator [g, h] = g−1h−1gh (or ghg−1h−1), if g, xy = x[x, y] (group theory). the h ∈ G (a group). commutator of [AB, C] = A[B, C] + [A, C]B [a, b] = ab − ba, if a, b ∈ R (a ring or (ring theory). group theory, commutative algebra). ring theory triple scalar product the triple [a, b, c] = a × b · c, the scalar [a, b, c] = [b, c, a] = [c, a, b]. scalar product product of a × b with c. of vector calculus function application f(x) means the value of the function f If f(x) := x2, then f(3) = 32 = 9. of at the element x. set theory f(X) means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ image dom(f). image of … (This may also be written as f[X] if under … there is a risk of confusing the image of f under X with the function everywhere application f of X. Another notation is Im f, the image of f under its () domain.) combinations means the number of (from) n (, choose r combinations of r elements drawn from a set of n elements. ) combinatorics (This may also be written as nC .) r precedence grouping Perform the operations inside the (8/4)/2 = 2/2 = 1, but 8/(4/2) = parentheses parentheses first. 8/2 = 4. everywhere tuple An ordered list (or sequence, or (a, b) is an ordered pair (or 2- horizontal vector, or row vector) of tuple). tuple; n-tuple; values. ordered (a, b, c) is an ordered triple (or pair/triple/etc; (Note that the notation (a,b) is 3-tuple). row vector; ambiguous: it could be an ordered sequence pair or an open interval. Set ( ) is the empty tuple (or 0- theorists and computer scientists tuple). everywhere often use angle brackets ⟨ ⟩ instead of parentheses.) highest common factor highest (a, b) means the highest common common factor of a and b. (3, 7) = 1 (they are coprime); factor; (15, 25) = 5. greatest (This may also be written hcf(a, b) common or gcd(a, b).) divisor; hcf; gcd number theory (, ) open interval . 4 is not in the interval (4, 18). open interval (Note that the notation (a,b) is (0, +∞) equals the set of ambiguous: it could be an ordered positive real numbers. ], order theory pair or an open interval. The notation ]a,b[ can be used instead.) [ multichoose (( multichoose means n multichoose k. )) combinatorics (, left-open interval ] half-open interval; . (−1, 7] and (−∞, −1] left-open ], interval ] order theory right-open [, interval . [4, 18) and [1, +∞) ) half-open interval; right-open interval [, order theory [ ⟨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. inner product The standard inner product There are many variants of the inner product between two vectors x = (2, 3) notation, such as ⟨u | v⟩ and (u | v), of and y = (−1, 5) is: which are described below. For ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13 spatial vectors, the dot product linear algebra notation, x·y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more “keyboard friendly” forms < and > are ⟨⟩ sometimes seen. These are avoided in mathematical texts. for a time series :g(t) (t = 1, average 2,...) ⟨,⟩ average of let S be a subset of N for example, represents the average of all the we can define the structure element in S. functions Sq( ): statistics ⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. linear span ⟨u1, u2, …⟩is shorthand for ⟨{u1, u2, …}⟩. (linear) span of; linear hull of Note that the notation ⟨u, v⟩ may be . ambiguous: it could mean the inner linear algebra product or the linear span. The span of S may also be written as Sp(S). subgroup generated by a means the smallest subgroup of In S , set G (where S ⊆ G, a group) containing 3 every element of S. and the subgroup generated by is shorthand for . . group theory tuple is an ordered pair (or 2- An ordered list (or sequence, or tuple). tuple; n-tuple; horizontal vector, or row vector) of ordered values. is an ordered triple (or pair/triple/etc; row vector; 3-tuple). (The notation (a,b) is often used as sequence well.) is the empty tuple (or 0- everywhere tuple). ⟨u | v⟩ means the inner product of u and v, where u and v are members of an inner product space.[9] (u | v) means the same. ⟨|⟩ inner product Another variant of the notation is ⟨u, v⟩ which is described above. For inner product spatial vectors, the dot product of notation, x·y is common. For (|) linear algebra matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more “keyboard friendly” forms < and > are sometimes seen. These are avoided in mathematical texts. ket vector A qubit's state can be the ket …; |φ⟩ means the vector with label φ, represented as α|0⟩+ β|1⟩, |⟩ the vector … which is in a Hilbert space. where α and β are complex numbers s.t. |α|2 + |β|2 = 1. Dirac notation bra vector ⟨φ| means the dual of the vector |φ⟩, ⟨| the bra …; the dual of … a linear functional which maps a ket |ψ⟩ onto the inner product ⟨φ|ψ⟩. Dirac notation summation sum over … ∑ from … to … of means a1 + a2 + … + an. = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 arithmetic product product over = … from … to (1+2)(2+2)(3+2)(4+2) … of means a1a2···an. = 3 × 4 × 5 × 6 = 360 arithmetic ∏ Cartesian product the Cartesian means the set of all (n+1)- product of; tuples the direct product of (y0, …, yn). set theory A general construction which coproduct subsumes the disjoint union of sets and of topological spaces, the free coproduct over product of groups, and the direct ∐ … from … to … of sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" category object to which each object in the theory family admits a morphism. Δx means a (non-infinitesimal) delta change in x. delta; is the gradient of a straight (If the change becomes infinitesimal, change in line δ and even d are used instead. Not to be confused with the symmetric Δ calculus difference, written ∆, above.) Laplacian If ƒ is a twice-differentiable The Laplace operator is a second real-valued function, then the Laplace order differential operator in n- Laplacian of ƒ is defined by operator dimensional Euclidean space vector calculus Dirac delta function δ(x) Dirac delta of hyperfunction Kronecker delta Kronecker δij δ delta of hyperfunction Functional derivative Functional derivative of Differential operators partial derivative ∂f/∂xi means the partial derivative of partial; f with respect to xi, where f is a If f(x,y) := x2y, then ∂f/∂x = 2xy d function on (x1, …, xn). calculus boundary ∂ boundary of ∂M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} topology degree of a polynomial ∂f means the degree of the polynomial f. ∂(x2 − 1) = 2 degree of (This may also be written deg f.) algebra gradient ∇f (x1, …, xn) is the vector of partial If f (x,y,z) := 3xy + z², then ∇f = ∇ del; derivatives (∂f / ∂x1, …, ∂f / ∂xn). (3y, 3x, 2z) nabla; gradient of vector calculus divergence del dot; If divergence of , then . vector calculus curl If curl of , then . vector calculus f ′(x) means the derivative of the derivative function f at the point x, i.e., the slope of the tangent to f at x. … prime; ′ derivative of (The single-quote character ' is If f(x) := x2, then f ′(x) = 2x sometimes used instead, especially in calculus ASCII text.) derivative means the derivative of x with … dot; • respect to time. That is time derivative If x(t) := t2, then . of . calculus indefinite integral or antiderivative indefinite ∫ f(x) dx means a function whose integral of ∫x2 dx = x3/3 + C derivative is f. the ∫ antiderivative of calculus definite ∫ab f(x) dx means the signed area integral between the x-axis and the graph of ∫ b x2 dx = b3/3 − a3/3; the function f between x = a and x = a integral from b. … to … of … with respect to calculus line integral ∫C f ds means the integral of f along the curve C, , line/ path/ where r is a parametrization of C. curve/ integral of… along… (If the curve is closed, the symbol ∮ may be used instead, as described calculus below.) 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 Contour the enclosing surface. Instances integral; where the latter requires closed line simultaneous double integration, the integral symbol ∯ would be more If C is a Jordan curve about 0, ∮ contour appropriate. A third related symbol is the closed volume integral, then . integral of denoted by the symbol ∰. calculus 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. projection π Projection of restricts to the attribute set. relational algebra Pi Used in various formulas involving circles; π is equivalent to the amount pi; of area a circle would take up in a 3.1415926; square of equal width with an area of A=πR2=314.16→R=10 ≈22÷7 4 square units, roughly 3.14/4. It is also the ratio of the circumference to mathematical the diameter of a circle. constant The selection selects all selection those tuples in for which holds between the and the attribute. The σ Selection of selection selects all those relational tuples in for which holds algebra between the attribute and the value . cover {1, 8} <• {1, 3, 8} among the is covered by x <• y means that x is covered by y. subsets of {1, 2, …, 10} <: ordered by containment. order theory subtype <· is a subtype of T1 <: T2 means that T1 is a subtype of If S <: T and T <: U then S <: T2. U (transitivity). type theory conjugate transpose conjugate A† means the transpose of the transpose; complex conjugate of A.[10] † adjoint; If A = (aij) then A† = (aji). Hermitian This may also be written A*T, AT*, adjoint/conjug A*, AT or AT. ate/transpose matrix operations transpose AT means A, but with its rows T transpose swapped for columns. If A = (aij) then AT = (aji). matrix This may also be written A', At or Atr. operations top element the top ⊤ means the largest element of a ∀x : x ∨ ⊤ = ⊤ element lattice. ⊤ lattice theory top type ⊤ means the top or universal type; the top type; every type in the type system of ∀ types T, T <: ⊤ top interest is a subtype of top. type theory perpendicular is x ⊥ y means x is perpendicular to y; If l ⊥ m and m ⊥ n in the plane, perpendicular or more generally x is orthogonal to then l || n. to y. geometry orthogonal complement W⊥ means the orthogonal orthogonal/ complement of W (where W is a perpendicular subspace of the inner product space Within , . complement V), the set of all vectors in V of; orthogonal to every vector in W. perp ⊥ linear algebra coprime x ⊥ y means x has no factor greater is coprime to 34 ⊥ 55. than 1 in common with y. number theory independent A ⊥ B means A is an event whose is independent probability is independent of event If A ⊥ B, then P(A|B) = P(A). of B. probability bottom element ⊥ means the smallest element of a ∀x : x ∧ ⊥ = ⊥ lattice. the bottom element lattice theory bottom type ⊥ means the bottom type (a.k.a. the the bottom zero type or empty type); bottom is type; ∀ types T, ⊥ <: T the subtype of every type in the type bot system. type theory comparability is comparable x ⊥ y means that x is comparable to {e, π} ⊥ {1, 2, e, 3, π} under to y. set containment. order theory entailment A ⊧ B means the sentence A entails ⊧ entails the sentence B, that is in every model in which A is true, B is also A ⊧ A ∨ ¬A true. model theory inference infers; is derived from x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A. propositional ⊢ logic, predicate logic partition is a partition p ⊢ n means that p is a partition of n. (4,3,1,1) ⊢ 9, of . number theory vertical ellipsis Denotes that certain constants and terms are missing out (i.e. for vertical clarity) and that only the important ellipsis terms are being listed. everywhere [edit] Variations In mathematics written in Arabic, some symbols may be reversed to make right-to-left writing and reading easier. [11] [edit] See also Greek letters used in mathematics, science, and engineering ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology) List of mathematical abbreviations Mathematical alphanumeric symbols Mathematical notation Notation in probability and statistics Physical constants Latin letters used in mathematics Table of logic symbols Table of mathematical symbols by introduction date Unicode mathematical operators [edit] References 1. ^ Rónyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0 2. ^ Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Boston: Course Technology, p. 822, ISBN 0-534-42057-5 3. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 71–72, ISBN 0-521-63503-9, OCLC 43641333 4. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 66, ISBN 0-521-63503-9, OCLC 43641333 5. ^ Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material Implication", Introduction to Logic (8th ed.), New York: Macmillan, pp. 268–269, ISBN 0-02- 325035-6, LCCN 8937742 6. ^ a b c d e Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0- 412-60610-0 7. ^ a b c d Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412- 60610-0 8. ^ Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412- 60610-0 9. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 62, ISBN 0-521-63503-9, OCLC 43641333 10. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 69–70, ISBN 0-521-63503-9, OCLC 43641333 11. ^ M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode", 27th Internationalization and Unicode Conference, 2005. [edit] External links The complete set of mathematics Unicode characters Jeff Miller: Earliest Uses of Various Mathematical Symbols Numericana: Scientific Symbols and Icons 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 Unicode Math Symbols - a quick form for using unicode math symbols. DeTeXify handwritten symbol recognition — doodle a symbol in the box, and the program will tell you what its name is Some Unicode charts of mathematical operators: Index of Unicode symbols Range 2100–214F: Unicode Letterlike Symbols Range 2190–21FF: Unicode Arrows Range 2200–22FF: Unicode Mathematical Operators Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B Range 2A00–2AFF: Unicode Supplementary Mathematical Operators Some Unicode cross-references: 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 Retrieved from "http://en.wikipedia.org/w/index.php?title=List_of_mathematical_symbols&oldid=516661834" View page ratings Rate this page What's this? Trustworthy Objective Complete Well-written I am highly knowledgeable about this topic (optional) I have a relevant college/university degree It is part of my profession It is a deep personal passion The source of my knowledge is not listed here I would like to help improve Wikipedia, send me an e-mail (optional) We will send you a confirmation e-mail. We will not share your e-mail address with outside parties as per our feedback privacy statement. Submit ratings Saved successfully Your ratings have not been submitted yet Your ratings have expired Please reevaluate this page and submit new ratings. An error has occurred. Please try again later. Thanks! Your ratings have been saved. Do you want to create an account? An account will help you track your edits, get involved in discussions, and be a part of the community. Create an accountorLog inMaybe later Thanks! Your ratings have been saved. Did you know that you can edit this page? Edit this pageMaybe later Categories: Mathematical notation Mathematics-related lists Mathematical symbols Mathematical tables Mathematical logic Lists of symbols Hidden categories: Incomplete lists from March 2010 Articles needing additional references from November 2011 All articles needing additional references Personal tools Create account Log in Namespaces Article Talk Variants Views Read Edit View history Actions Search Special:Search Navigation Main page Contents Featured content Current events Random article Donate to Wikipedia Wikipedia Shop Interaction Help About Wikipedia Community portal Recent changes Contact Wikipedia Toolbox What links here Related changes Upload file Special pages Permanent link Cite this page Rate this page Print/export Create a book Download as PDF Printable version Languages አማርኛ ال عرب ية বাাংলা Български Bosanski Català Česky Cymraeg Deutsch Español Euskara Français 한국어 हिन्दी Bahasa Indonesia Italiano עברית Қазақша Lumbaart Magyar മലയാളം Nederlands 日本語 orsk (bokm l) Norsk (nynorsk) Polski Português Română Русский Simple English Slovenčina Slovenščina Soomaaliga Basa Sunda Suomi Svenska Türkçe Українська اردو Tiếng Việt Võro 中文 This page was last modified on 20 October 2012 at 17:20. 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