List of Mathematical Symbols Rendered

List of mathematical symbols 1

List of mathematical symbolsThis is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML,which depends on appropriate fonts being installed, and in TeX, as an image.

This list is incomplete.

Symbols

Symbolin

HTML

Symbolin

TeX

Name Explanation Examples Read as

Category

= equalityis equal to; equalseverywhere

x= y means x and y represent the same thingor value.

2= 21+ 1= 2

inequalityis not equal to; does notequal

everywhere

x y means that x and y do not represent thesame thing or value.(The forms !=, /= or are generally used inprogramming languages where ease oftyping and use of ASCII text is preferred.)

2 + 2 5

strict inequality

is less than, is greater than

order theory

x< y means x is less than y.x> y means x is greater than y.

3< 45> 4

proper subgroup

is a proper subgroup of

group theory

H< G means H is a proper subgroup of G. 5Z< ZA3 < S3

(very) strict inequality

is much less than, is muchgreater than

order theory

x y means x is much less than y.x y means x is much greater than y.

0.003 1000000

asymptotic comparison

is of smaller order than, isof greater order than

analytic number theory

f g means the growth of f isasymptotically bounded by g.(This is I. M. Vinogradov's notation. Anothernotation is the Big O notation, which lookslike f= O(g).)

x ex

inequality

is less than or equal to, isgreater than or equal to

order theory

x y means x is less than or equal to y.x y means x is greater than or equal to y.(The forms = are generally used inprogramming languages where ease oftyping and use of ASCII text is preferred.)

34 and 5554 and 55

subgroup

is a subgroup of

group theory

H G means H is a subgroup of G. Z ZA3 S3

reduction

is reducible to

computational complexitytheory

A B means the problem A can be reducedto the problem B. Subscripts can be added tothe to indicate what kind of reduction.

If

then


Karp reductionis Karp reducible to; ispolynomial-time many-one

reducible to


L1 L2 means that the problem L1 is Karpreducible to L2.

[1]If L1 L2 and L2 P, then L1 P.

proportionalityis proportional to; varies aseverywhere

y x means that y = kx for some constant k. if y = 2x, then y x.

Karp reduction[2]

is Karp reducible to; ispolynomial-time many-one

reducible to


A B means the problem A can bepolynomially reduced to the problem B.

If L1 L2 and L2 P, then L1 P.

+ addition plus; addarithmetic

4 + 6 means the sum of 4 and 6. 2 + 7 = 9

disjoint union

the disjoint union of ... and...

set theory

A1 + A2 means the disjoint union of sets A1and A2.

A1 = {3, 4, 5, 6} A2 = {7, 8, 9, 10} A1 + A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)}

subtractionminus; take; subtractarithmetic

9 4 means the subtraction of 4 from 9. 8 3 = 5

negative sign

negative; minus; theopposite of

arithmetic

3 means the negative of the number 3. (5) = 5

set-theoretic complement

minus; without

set theory

AB means the set that contains all theelements of A that are not in B.( can also be used for set-theoreticcomplement as described below.)

{1,2,4}{1,3,4}= {2}


multiplicationtimes; multiplied byarithmetic

3 4 means the multiplication of 3 by 4.(The symbol * is generally used inprogramming languages, where ease oftyping and use of ASCII text is preferred.)

7 8 = 56

Cartesian product

the Cartesian product of ...and ...; the direct product of

... and ...

set theory

XY means the set of all ordered pairs withthe first element of each pair selected from Xand the second element selected from Y.

{1,2} {3,4} = {(1,3),(1,4),(2,3),(2,4)}

cross product

cross

linear algebra

u v means the cross product of vectors uand v

(1,2,5) (3,4,1) =(22, 16, 2)

group of units

the group of units of

ring theory

R consists of the set of units of the ring R,along with the operation of multiplication.This may also be written R* as describedbelow, or U(R).

multiplicationtimes; multiplied byarithmetic

3 4 means the multiplication of 3 by 4. 7 8 = 56

dot product

dot

linear algebra

u v means the dot product of vectors u andv

(1,2,5) (3,4,1) = 6

division (Obelus)

divided by; over

arithmetic

6 3 or 6 3 means the division of 6 by 3. 2 4 = 0.512 4 = 3

quotient group

mod

group theory

G/H means the quotient of group G moduloits subgroup H.

{0, a, 2a, b, b+a, b+2a}/{0, b}= {{0, b}, {a, b+a}, {2a, b+2a}}

quotient set

mod

set theory

A/~ means the set of all ~ equivalenceclasses in A.

If we define ~ by x~y xy , then/~= {x+n: n: x(0,1]}

plus-minusplus or minusarithmetic

6 3 means both 6 + 3 and 6 3. The equation x = 5 4, has two solutions, x = 7 and x = 3.

plus-minus

plus or minus

measurement

10 2 or equivalently 10 20% means therange from 10 2 to 10 + 2.

If a = 100 1 mm, then a 99 mm and a 101 mm.

minus-plusminus or plusarithmetic

6 (3 5) means both 6 + (3 5) and 6 (3 + 5).

cos(x y) = cos(x) cos(y) sin(x) sin(y).


square rootthe (principal) square rootof

real numbers

means the nonnegative number whosesquare is .

complex square root

the (complex) square rootof

complex numbers

if is represented in polarcoordinates with , then

.

|| absolute value or modulusabsolute value of; modulusof

numbers

|x| means the distance along the real line (oracross the complex plane) between x andzero.

|3|= 3|5|= |5|= 5|i| = 1|3 + 4i|= 5

Euclidean norm orEuclidean length ormagnitude

Euclidean norm of

geometry

|x| means the (Euclidean) length of vector x. For x= (3,-4)

determinant

determinant of

matrix theory

|A| means the determinant of the matrix A

cardinality

cardinality of; size of; orderof

set theory

|X| means the cardinality of the set X.(# may be used instead as described below.)

|{3, 5, 7, 9}|= 4.

|||| normnorm of; length oflinear algebra

||x|| means the norm of the element x of anormed vector space.[3]

||x + y|| ||x|| + ||y||

nearest integer function

nearest integer to

numbers

||x|| means the nearest integer to x.(This may also be written [x], x, nint(x) orRound(x).)

||1|| = 1, ||1.6|| = 2, ||2.4|| = 2, ||3.49|| = 3

divisor, divides

divides

number theory

a|b means a divides b.ab means a does not divide b.(This symbol can be difficult to type, and itsnegation is rare, so a regular but slightlyshorter vertical bar | character can be used.)

Since 15 = 35, it is true that 3|15 and 5|15.

conditional probability

given

probability

P(A|B) means the probability of the event aoccurring given that b occurs.

if X is a uniformly random day of the year P(X is May 25 | X isin May) = 1/31

restriction

restriction of to ;restricted to

set theory

f|A means the function f restricted to the setA, that is, it is the function with domainAdom(f) that agrees with f.

The function f:RR defined by f(x)= x2 is not injective, butf|

R+ is injective.

such that

such that; so that

everywhere

| means such that, see ":" (describedbelow).

S = {(x,y) | 0 < y < f(x)}The set of (x,y) such that y is greater than 0 and less than f(x).


|| parallel is parallel togeometry

x||y means x is parallel to y. If l||m and mn then ln.

incomparability

is incomparable to

order theory

x||y means x is incomparable to y. {1,2}||{2,3} under set containment.

exact divisibility

exactly divides

number theory

pa||n means pa exactly divides n (i.e. pa

divides n but pa+1 does not).23||360.

# cardinalitycardinality of; size of; orderof

set theory

#X means the cardinality of the set X.(|| may be used instead as describedabove.)

#{4, 6, 8}= 3

connected sum

connected sum of; knotsum of; knot composition

of

topology, knot theory

A#B is the connected sum of the manifolds Aand B. If A and B are knots, then this denotesthe knot sum, which has a slightly strongercondition.

A#Sm is homeomorphic to A, for any manifold A, and the sphereSm.

aleph numberalephset theory

represents an infinite cardinality(specifically, the -th one, where is anordinal).

|| = 0, which is called aleph-null.

beth numberbethset theory

represents an infinite cardinality (similarto , but does not necessarily index all ofthe numbers indexed by . ).

cardinality of thecontinuumcardinality of the

continuum; cardinality ofthe real numbers; c;

set theory

The cardinality of is denoted by orby the symbol (a lowercase Fraktur letterC).

: such thatsuch that; so thateverywhere

: means such that, and is used in proofs andthe set-builder notation (described below).

n : n is even.

field extension

extends; over

field theory

K : F means the field K extends the field F.This may also be written as K F.

:

inner product of matrices

inner product of

linear algebra

A : B means the Frobenius inner product ofthe matrices A and B.The general inner product is denoted byu,v, u|v or (u|v), as described below.For spatial vectors, the dot product notation,xy is common. See also Bra-ket notation.

index of a subgroup

index of subgroup

group theory

The index of a subgroup H in a group G isthe "relative size" of H in G: equivalently,the number of "copies" (cosets) of H that fillup G


! factorial factorialcombinatorics

n! means the product 1 2 ... n. 4! = 1 2 3 4 = 24

logical negation

not

propositional logic

The statement !A is true if and only if A isfalse.A slash placed through another operator isthe same as "!" placed in front.(The symbol ! is primarily from computerscience. It is avoided in mathematical texts,where the notation A is preferred.)

!(!A)Axy !(x=y)

~ probability distributionhas distributionstatistics

X ~ D, means the random variable X has theprobability distribution D.

X ~ N(0,1), the standard normal distribution

row equivalence

is row equivalent to

matrix theory

A~B means that B can be generated by usinga series of elementary row operations on A

same order of magnitude

roughly similar; poorlyapproximates

approximation theory

m~n means the quantities m and n have thesame order of magnitude, or general size.(Note that ~ is used for an approximationthat is poor, otherwise use .)

2~589~ 100but 2 10

asymptotically equivalent

is asymptotically equivalentto

asymptotic analysis

f~g means .x~x+1

equivalence relation

are in the same equivalenceclass

everywhere

a~b means (and equivalently).

1~5 mod 4

approximately equalis approximately equal toeverywhere

xy means x is approximately equal to y. 3.14159

isomorphism

is isomorphic to

group theory

GH means that group G is isomorphic(structurally identical) to group H.( can also be used for isomorphic, asdescribed below.)

Q/{1,1} V,where Q is the quaternion group and V is the Klein four-group.

wreath productwreath product of by group theory

AH means the wreath product of the groupA by the group H.This may also be written Awr H.

is isomorphic to the automorphism group of thecomplete bipartite graph on (n,n) vertices.

normal subgroup

is a normal subgroup of

group theory

NG means that N is a normal subgroup ofgroup G.

Z(G)G

ideal

is an ideal of

ring theory

IR means that I is an ideal of ring R. (2)Z

antijoin

the antijoin of

relational algebra

RS means the antijoin of the relations Rand S, the tuples in R for which there is not atuple in S that is equal on their commonattribute names.

R S = R - R S


semidirect product

the semidirect product of

group theory

N H is the semidirect product of N (anormal subgroup) and H (a subgroup), withrespect to . Also, if G= N H, then G issaid to split over N.( may also be written the other way round,as , or as .)

semijoin

the semijoin of

relational algebra

R S is the semijoin of the relations R and S,the set of all tuples in R for which there is atuple in S that is equal on their commonattribute names.

R S = a1,..,an(R S)

natural jointhe natural join ofrelational algebra

R S is the natural join of the relations Rand S, the set of all combinations of tuples inR and S that are equal on their commonattribute names.

thereforetherefore; so; henceeverywhere

Sometimes used in proofs before logicalconsequences.

All humans are mortal. Socrates is a human. Socrates ismortal.

becausebecause; sinceeverywhere

Sometimes used in proofs before reasoning. 3331 is prime it has no positive integer factors other than itselfand one.

end of proof

QED; tombstone; Halmossymbol

everywhere

Used to mark the end of a proof.(May also be written Q.E.D.)

material implication

implies; if thenpropositional logic,

Heyting algebra

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 havethe meaning for functions given below.)( may mean the same as ,[4] or it mayhave 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).

material equivalence

if and only if; iff

propositional logic

A B means A is true if B is true and A isfalse if B is false.

x+ 5= y+ 2 x+ 3= y

logical negation

not

propositional logic

The statement A is true if and only if A isfalse.A slash placed through another operator isthe same as "" placed in front.(The symbol ~ has many other uses, so orthe slash notation is preferred. Computerscientists will often use ! but this is avoidedin mathematical texts.)

(A) Axy (x= y)


logical conjunction or meetin a latticeand; min; meet

propositional logic, latticetheory

The statement A B is true if A and B areboth true; else it is false.For functions A(x) and B(x), A(x) B(x) isused to mean min(A(x), B(x)).

n< 4 n>2 n= 3 when n is a natural number.

wedge product

wedge product; exteriorproduct

linear algebra

u v means the wedge product of vectors uand v. This generalizes the cross product tohigher dimensions.(For vectors in R3, can also be used.)

exponentiation

(raised) to the power of

everywhere

a ^ b means a raised to the power of b(a ^ b is more commonly written ab. Thesymbol ^ is generally used in programminglanguages where ease of typing and use ofplain ASCII text is preferred.)

2^3 = 23 = 8

logical disjunction or joinin a latticeor; max; join

propositional logic, latticetheory

The statement A B is true if A or B (orboth) are true; if both are false, the statementis false.For functions A(x) and B(x), A(x) B(x) isused to mean max(A(x), B(x)).

n 4 n 2 n 3 when n is a natural number.

exclusive or

xor

propositional logic,Boolean algebra

The statement A B is true when either A orB, but not both, are true. A B means thesame.

(A) A is always true, A A is always false.

direct sum

direct sum of

abstract algebra

The direct sum is a special way ofcombining several objects into one generalobject.(The bun symbol , or the coproduct symbol, is used; is only for logic.)

Most commonly, for vector spaces U, V, and W, the followingconsequence is used:U = V W (U = V + W) (V W = {0})

universal quantificationfor all; for any; for eachpredicate logic

x: P(x) means P(x) is true for all x. n: n2 n.

existential quantificationthere exists; there is; thereare

predicate logic

x: P(x) means there is at least one x suchthat P(x) is true.

n: n is even.

! uniqueness quantificationthere exists exactly onepredicate logic

!x: P(x) means there is exactly one x suchthat P(x) is true.

!n: n+ 5= 2n.

=::=

:

definition

is defined as; is equal bydefinition to

everywhere

x:= y, y=: x or x y means x is defined tobe another name for y, under certainassumptions taken in context.(Some writers use to mean congruence).P: Q means P is defined to be logicallyequivalent to Q.


congruenceis congruent togeometry

ABC DEF means triangle ABC iscongruent to (has the same measurements as)triangle DEF.

isomorphic

is isomorphic to

abstract algebra

GH means that group G is isomorphic(structurally identical) to group H.( can also be used for isomorphic, asdescribed above.)

.

congruence relation... is congruent to ...modulo ...

modular arithmetic

a b (mod n) means a b is divisible by n 5 2 (mod 3)

{,} set bracketsthe set of set theory

{a,b,c} means the set consisting of a, b, andc.[5]

= {1, 2, 3, }

{:}{|}

set builder notation

the set of such thatset theory

{x: P(x)} means the set of all x for whichP(x) is true.[5] {x| P(x)} is the same as {x:P(x)}.

{n: n2


set-theoretic complementminus; withoutset theory

AB means the set that contains all thoseelements of A that are not in B.[6]

( can also be used for set-theoreticcomplement as described above.)

{1,2,3,4}{3,4,5,6}= {1,2}

function arrowfrom toset theory, type theory

f:X Y means the function f maps the set Xinto the set Y.

Let f:{0} be defined by f(x):= x2.

function arrowmaps toset theory

f:a b means the function f maps theelement a to the element b.

Let f:xx+1 (the successor function).

function compositioncomposed withset theory

fg is the function, such that (fg)(x) =f(g(x)).[7]

if f(x) := 2x, and g(x) := x + 3, then (fg)(x) = 2(x + 3).

N

natural numbers

N; the (set of) naturalnumbers

numbers

N means either {0, 1, 2, 3, ...} or {1, 2, 3,...}.The choice depends on the area ofmathematics being studied; e.g. numbertheorists prefer the latter; analysts, settheorists and computer scientists prefer theformer. To avoid confusion, always check anauthor's definition of N.Set theorists often use the notation (forleast infinite ordinal) to denote the set ofnatural numbers (including zero), along withthe standard ordering relation .

= {|a|: a } or = {|a|>0: a }

Z

integers

Z; the (set of) integers

numbers

means {..., 3, 2, 1, 0, 1, 2, 3, ...}. + or> means {1, 2, 3, ...}. * or means {0, 1,2, 3, ...}.

= {p,p: p {0} }

npZnZp

integers mod n

Zn; the (set of) integersmodulo n

numbers

n means {[0], [1], [2], ...[n1]} withaddition and multiplication modulo n.Note that any letter may be used instead of n,such as p. To avoid confusion with p-adicnumbers, use /p or /(p) instead.

3= {[0], [1], [2]}

p-adic integers

the (set of) p-adic integers

numbers

Note that any letter may be used instead of p,such as n or l.

P

projective space

P; the projective space, theprojective line, the

projective plane

topology

means a space with a point at infinity. ,

probability

the probability of

probability theory

(X) means the probability of the event Xoccurring.This may also be written as P(X), Pr(X), P[X]or Pr[X].

If a fair coin is flipped, (Heads)= (Tails)= 0.5.

Q

rational numbers

Q; the (set of) rationalnumbers; the rationals

numbers

means {p/q: p, q}. 3.14000...


R

real numbers

R; the (set of) real numbers;the reals

numbers

means the set of real numbers. (1)

C

complex numbers

C; the (set of) complexnumbers

numbers

means {a+bi: a,b}. i= (1)

H

quaternions or Hamiltonianquaternions

H; the (set of) quaternions

numbers

means {a+bi+cj+dk: a,b,c,d}.

O Big O notationbig-oh ofComputational complexity

theory

The Big O notation describes the limitingbehavior of a function, when the argumenttends towards a particular value or infinity.

If f(x) = 6x4 2x3 + 5 and g(x) = x4 , then

infinity infinitynumbers

is an element of the extended number linethat is greater than all real numbers; it oftenoccurs in limits.

floorfloor; greatest integer;entier

numbers

x means the floor of x, i.e. the largestinteger less than or equal to x.(This may also be written [x], floor(x) orint(x).)

4 = 4, 2.1 = 2, 2.9 = 2, 2.6 = 3

ceiling ceilingnumbers

x means the ceiling of x, i.e. the smallestinteger greater than or equal to x.(This may also be written ceil(x) orceiling(x).)

4 = 4, 2.1 = 3, 2.9 = 3, 2.6 = 2

nearest integer functionnearest integer tonumbers

x means the nearest integer to x.(This may also be written [x], ||x||, nint(x) orRound(x).)

2 = 2, 2.6 = 3, -3.4 = -3, 4.49 = 4

[:] degree of a field extensionthe degree offield theory

[K : F] means the degree of the extension K :F.

[(2) : ] = 2[ : ] = 2[ : ] =


[][,][,,]

equivalence class

the equivalence class of

abstract algebra

[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 theequivalence relation.

Let a~ b be true iff a b(mod5). Then [2]= {, 8, 3, 2, 7,}.

floor

floor; greatest integer;entier

numbers

[x] means the floor of x, i.e. the largestinteger less than or equal to x.(This may also be written x, floor(x) orint(x). Not to be confused with the nearestinteger function, as described below.)

[3] = 3, [3.5] = 3, [3.99] = 3, [3.7] = 4

nearest integer function

nearest integer to

numbers

[x] means the nearest integer to x.(This may also be written x, ||x||, nint(x) orRound(x). Not to be confused with the floorfunction, as described above.)

[2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4

Iverson bracket

1 if true, 0 otherwise

propositional logic

[S] maps a true statement S to 1 and a falsestatement S to 0.

[0=5]=0, [7>0]=1, [2{2,3,4}]=1, [5{2,3,4}]=0

image

image of under everywhere

f[X] means { f(x): x X }, the image of thefunction f under the set X dom(f).(This may also be written as f(X) if there isno risk of confusing the image of f under Xwith the function application f of X. Anothernotation is Imf, the image of f under itsdomain.)

closed interval

closed interval

order theory

. 0 and 1/2 are in the interval [0,1].

commutator

the commutator of

group theory, ring theory

[g,h] = g1h1gh (or ghg1h1), if g, h G(a group).[a,b]= ab ba, if a, b R (a ring orcommutative algebra).

xy = x[x,y] (group theory).[AB,C] = A[B,C]+ [A,C]B (ring theory).

triple scalar product

the triple scalar product of

vector calculus

[a,b,c]= a b c, the scalar product ofab with c.

[a,b,c]= [b,c,a]= [c,a,b].


()( , )

function application

of

set theory

f(x) means the value of the function f at theelement x.

If f(x):= x2, then f(3)= 32= 9.

image

image of under everywhere

f(X) means { f(x): x X }, the image of thefunction f under the set X dom(f).(This may also be written as f[X] if there is arisk of confusing the image of f under X withthe function application f of X. Anothernotation is Imf, the image of f under itsdomain.)

combinations

(from) n choose r

combinatorics

means the number of combinations of

r elements drawn from a set of n elements.(This may also be written as nCr.)

precedence grouping

parentheses

everywhere

Perform the operations inside theparentheses first.

(8/4)/2= 2/2= 1, but 8/(4/2)= 8/2= 4.

tuple

tuple; n-tuple; orderedpair/triple/etc; row vector;

sequence

everywhere

An ordered list (or sequence, or horizontalvector, or row vector) of values. (Note thatthe notation (a,b) is ambiguous: it could bean ordered pair or an open interval. Settheorists and computer scientists often useangle brackets instead of parentheses.)

(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).

highest common factor

highest common factor;greatest common divisor;

hcf; gcd

number theory

(a, b) means the highest common factor of aand b.(This may also be written hcf(a, b) or gcd(a,b).)

(3, 7) = 1 (they are coprime); (15, 25) = 5.

(,)],[

open interval

open interval

order theory

. (Notethat the notation (a,b) is ambiguous: it couldbe an ordered pair or an open interval. Thenotation ]a,b[ can be used instead.)

4 is not in the interval (4, 18). (0, +) equals the set of positivereal numbers.

(,]],]

left-open interval

half-open interval;left-open interval

order theory

. (1, 7] and (, 1]

[,)[,[

right-open interval

half-open interval;right-open interval

order theory

. [4, 18) and [1, +)


,

inner product

inner product of

linear algebra

u,v means the inner product of u and v,where u and v are members of an innerproduct space.Note that the notation u, v may beambiguous: it could mean the inner productor the linear span.There are many variants of the notation,such as u|v and (u|v), which aredescribed below. For spatial vectors, the dotproduct notation, xy is common. Formatrices, the colon notation A:B may beused. As and can be hard to type, themore keyboard friendly forms < and > aresometimes seen. These are avoided inmathematical texts.

The standard inner product between two vectors x=(2,3) andy=(1,5) is:x,y=21+35= 13

average

average of

statistics

let S be a subset of N for example, represents the average of all the element inS.

for a time series :g(t) (t = 1, 2,...) we can define the structurefunctions Sq( ):

linear span

(linear) span of;linear hull of

linear algebra

S means the span of S V. That is, it is theintersection of all subspaces of V whichcontain S.u1,u2,is shorthand for {u1,u2,}.Note that the notation u,v may beambiguous: it could mean the inner productor the linear span.The span of S may also be written as Sp(S).

.

subgroup generated by a set

the subgroup generated by

group theory

means the smallest subgroup of G(where S G, a group) containing everyelement of S.

is shorthand for .

In S3, and.

tuple

tuple; n-tuple; orderedpair/triple/etc; row vector;

sequence

everywhere

An ordered list (or sequence, or horizontalvector, or row vector) of values. (Thenotation (a,b) is often used as well.)

is an ordered pair (or 2-tuple). is an orderedtriple (or 3-tuple).

is the empty tuple (or 0-tuple).

|(|)

inner product

inner product of

linear algebra

u|v means the inner product of u and v,where u and v are members of an innerproduct space.[8] (u|v) means the same.Another variant of the notation is u,vwhich is described above. For spatialvectors, the dot product notation, xy iscommon. For matrices, the colon notationA:B may be used. As and can be hard totype, the more keyboard friendly forms are sometimes seen. These areavoided in mathematical texts.

| ket vectorthe ket ; the vector Dirac notation

| means the vector with label , which isin a Hilbert space.

A qubit's state can be represented as |0+ |1, where and are complex numbers s.t. ||2+ ||2= 1.

| bra vectorthe bra ; the dual of Dirac notation

| means the dual of the vector |, a linearfunctional which maps a ket | onto theinner product |.


summationsum over from to of

arithmetic

means a1+ a2+ + an. = 12+ 22+ 32+ 42

= 1+ 4+ 9+ 16= 30

productproduct over from to of

arithmetic

means a1a2an. = (1+2)(2+2)(3+2)(4+2)

= 3 4 5 6= 360

Cartesian product

the Cartesian product of;the direct product of

set theory

means the set of all (n+1)-tuples

(y0, , yn).

coproductcoproduct over from to of

category theory

A general construction which subsumes thedisjoint union of sets and of topologicalspaces, the free product of groups, and thedirect sum of modules and vector spaces.The coproduct of a family of objects isessentially the "least specific" object towhich each object in the family admits amorphism.

derivative prime;derivative of

calculus

f(x) means the derivative of the function f atthe point x, i.e., the slope of the tangent to fat x.(The single-quote character ' is sometimesused instead, especially in ASCII text.)

If f(x):=x2, then f(x)=2x

derivative dot;time derivative of

calculus

means the derivative of x with respect totime. That is .

If x(t):=t2, then .

indefinite integral orantiderivativeindefinite integral ofthe antiderivative of

calculus

f(x)dx means a function whose derivativeis f.

x2dx= x3/3 + C

definite integral

integral from to of with respect to

calculus

abf(x)dx means the signed area between

the x-axis and the graph of the function fbetween x= a and x= b.

abx2dx= b3/3 a3/3;

line integral

line/path/curve integral of along

calculus

Cfds means the integral of f along thecurve C, , where r is

a parametrization of C.(If the curve is closed, the symbol may beused instead, as described below.)


Contour integral or closedline integralcontour integral of

calculus

Similar to the integral, but used to denote asingle integration over a closed curve orloop. It is sometimes used in physics textsinvolving equations regarding Gauss's Law,and while these formulas involve a closedsurface integral, the representations describeonly the first integration of the volume overthe enclosing surface. Instances where thelatter requires simultaneous doubleintegration, the symbol would be moreappropriate. A third related symbol is theclosed volume integral, denoted by thesymbol . The contour integral can alsofrequently be found with a subscript capitalletter C, C, denoting that a closed loopintegral is, in fact, around a contour C, orsometimes dually appropriately, a circle C.In representations of Gauss's Law, asubscript capital S, S, is used to denote thatthe integration is over a closed surface.

If C is a Jordan curve about 0, then .

gradientdel, nabla, gradient ofvector calculus

f (x1, , xn) is the vector of partialderivatives (f / x1, , f / xn).

If f (x,y,z) := 3xy + z, then f=(3y, 3x, 2z)

divergence

del dot, divergence of

vector calculus

If , then .

curl

curl of

vector calculus

If , then .

partial derivativepartial, dcalculus

f/xi means the partial derivative of f withrespect to xi, where f is a function on (x1, ,xn).

If f(x,y) := x2y, then f/x = 2xy

boundary

boundary of

topology

M means the boundary of M {x : ||x|| 2} = {x : ||x|| = 2}

degree of a polynomial

degree of

algebra

f means the degree of the polynomial f.(This may also be written deg f.)

(x2 1) = 2

delta delta; change incalculus

x means a (non-infinitesimal) change in x.(If the change becomes infinitesimal, andeven d are used instead. Not to be confusedwith the symmetric difference, written ,above.)

is the gradient of a straight line

Laplacian

Laplace operator

vector calculus

The Laplace operator is a second orderdifferential operator in n-dimensionalEuclidean space

If is a twice-differentiable real-valued function, then theLaplacian of is defined by


Dirac delta functionDirac delta ofhyperfunction

(x)

Kronecker delta

Kronecker delta of

hyperfunction

ij

projectionProjection ofrelational algebra

restricts to theattribute set.

Pi

3.1415926_ or 227Used in various formulas involving circles; is equivalent to the amount of area a circlewould take up in a square of equal widthwith an area of 4 square units, roughly3.14/4. It is also the ratio of thecircumference to the diameter of a circle.

A=R2=314.16R=10

selectionSelection ofrelational algebra

The selection selects all thosetuples in for which holds between the

and the attribute. The selectionselects all those tuples in for

which holds between the attribute andthe value .


perpendicularis perpendicular togeometry

xy means x is perpendicular to y; or moregenerally x is orthogonal to y.

If lm and mn in the plane then l||n.

orthogonal complement

orthogonal/perpendicularcomplement of; perp

linear algebra

W means the orthogonal complement of W(where W is a subspace of the inner productspace V), the set of all vectors in Vorthogonal to every vector in W.

Within , .

coprime

is coprime to

number theory

xy means x has no factor greater than 1 incommon with y.

34 55.

independent

is independent of

probability

AB means A is an event whose probabilityis independent of event B.

If AB, then P(A|B) = P(A).

bottom element

the bottom element

lattice theory

means the smallest element of a lattice. x: x=

bottom type

the bottom type; bot

type theory

means the bottom type (a.k.a. the zero typeor empty type); bottom is the subtype ofevery type in the type system.

types T,


* multiplicationtimes; multiplied byarithmetic

a*b means the product of a and b.(Multiplication can also be denoted with or , or even simple juxtaposition. * isgenerally used where ease of typing and useof ASCII text is preferred, such asprogramming languages.)

4 * 3 means the product of 4 and 3, or 12.

convolution

convolution, convolvedwith

functional analysis

f*g means the convolution of f and g..

complex conjugate

conjugate

complex numbers

z* means the complex conjugate of z.( can also be used for the conjugate of z,as described below.)

.

group of units

the group of units of

ring theory

R* consists of the set of units of the ring R,along with the operation of multiplication.This may also be written R as describedabove, or U(R).

hyperreal numbers

the (set of) hyperreals

non-standard analysis

*R means the set of hyperreal numbers.Other sets can be used in place of R.

*N is the hypernatural numbers.

Hodge dual

Hodge dual, Hodge star

linear algebra

*v means the Hodge dual of a vector v. If v isa k-vector within an n-dimensional orientedinner product space, then *v is an(nk)-vector.

If are the standard basis vectors of ,

o Hadamard productentrywise productlinear algebra

For two matrices (or vectors) of the samedimensions the Hadamardproduct is a matrix of the same dimensions

with elements given by. This is

often used in matrix based programmingsuch as MATLAB where the operation isdone by A.*B

x mean overbar, barstatistics

(often read as x bar) is the mean(average value of ).

.

complex conjugate

conjugate

complex numbers

means the complex conjugate of z.(z* can also be used for the conjugate of z,as described above.)

.

algebraic closure

algebraic closure of

field theory

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 .

topological closure

(topological) closure of

topology

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 numbersare dense in the real numbers).


VariationsIn mathematics written in Arabic, some symbols may be reversed to make right-to-left(boustrophedon) writing andreading easier. [11]

References[1] Rnyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN963-9132-16-0[2] Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Boston: Course Technology, p.822,

ISBN0-534-42057-5[3] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press,

p.66, ISBN0-521-63503-9, OCLC43641333[4] Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material Implication", Introduction to Logic (8th ed.),

New York: Macmillan, pp.268269, ISBN0023250356, LCCN89-37742[5] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p.3, ISBN0-412-60610-0[6] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p.4, ISBN0-412-60610-0[7] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p.5, ISBN0-412-60610-0[8] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press,

p.62, ISBN0-521-63503-9, OCLC43641333[9] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press,

pp.6970, ISBN0-521-63503-9, OCLC43641333[10] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press,

pp.7172, ISBN0-521-63503-9, OCLC43641333[11] M. Benatia, A. Lazrik, and K. Sami, " Arabic mathematical symbols in Unicode (http:/ / www. ucam. ac. ma/ fssm/ rydarab/ doc/ expose/

unicodeme. pdf)", 27th Internationalization and Unicode Conference, 2005.

External links The complete set of mathematics Unicode characters (http:/ / krestavilis. com/ math. php) Jeff Miller: Earliest Uses of Various Mathematical Symbols (http:/ / jeff560. tripod. com/ mathsym. html) Numericana: Scientific Symbols and Icons (http:/ / www. numericana. com/ answer/ symbol. htm) TCAEP - Institute of Physics (http:/ / www. tcaep. co. uk/ science/ symbols/ maths. htm) GIF and PNG Images for Math Symbols (http:/ / us. metamath. org/ symbols/ symbols. html) Mathematical Symbols in Unicode (http:/ / tlt. psu. edu/ suggestions/ international/ bylanguage/ math.

html#browsers) Using Greek and special characters from Symbol font in HTML (http:/ / www. alanwood. net/ demos/ symbol.

html) Unicode Math Symbols (http:/ / www. vex. net/ ~trebla/ symbols/ select. html) - a quick form for using unicode

math symbols. DeTeXify handwritten symbol recognition (http:/ / detexify. kirelabs. org/ classify. html) doodle a symbol in

the box, and the program will tell you what its name isSome Unicode charts of mathematical operators: Index of Unicode symbols (http:/ / www. unicode. org/ charts/ #symbols) Range 2100 214F: Letterlike Symbols (http:/ / www. unicode. org/ charts/ PDF/ U2100. pdf) Range 2190 21FF: Arrows (http:/ / www. unicode. org/ charts/ PDF/ U2190. pdf) Range 2200 22FF: Unicode Mathematical Operators (http:/ / www. unicode. org/ charts/ PDF/ U2200. pdf)Some Unicode cross-references: Short list of commonly used LaTeX symbols (http:/ / www. artofproblemsolving. com/ Wiki/ index. php/

LaTeX:Symbols) and Comprehensive LaTeX Symbol List (http:/ / mirrors. med. harvard. edu/ ctan/ info/symbols/ comprehensive/ )

MathML Characters (http:/ / www. robinlionheart. com/ stds/ html4/ entities-mathml) - sorts out Unicode, HTMLand MathML/TeX names on one page


Unicode values and MathML names (http:/ / www. w3. org/ TR/ REC-MathML/ chap6/ bycodes. html) Unicode values and Postscript names (http:/ / svn. ghostscript. com/ ghostscript/ branches/ gs-db/ Resource/

Decoding/ Unicode) from the source code for Ghostscript

Article Sources and Contributors 22

Article Sources and ContributorsList of mathematical symbols Source: http://en.wikipedia.org/w/index.php?oldid=441263636 Contributors: 12jbooher, ABCD, AK Auto, Acroterion, Agent Foxtrot, Alan Liefting, Alex43223,Alison22, Alksentrs, Alpharigel, Ancheta Wis, AndrewHowse, Anomalocaris, Anonymous Dissident, ArnoldReinhold, Ashleycocks, AugPi, Avraham, AxelBoldt, BAxelrod, Bart133,Belovedfreak, BenFrantzDale, BenKovitz, Berteun, BiT, Bkell, Bkkbrad, Blokkendoos, Bonus Onus, Boud, Bryan Derksen, Btipling, Bwholm, CBM, CRGreathouse, Calrfa Wn, Camembert,CanisRufus, Capitalist, Charles Matthews, Church of emacs, ColinHelvensteijn, Computer97, Corti, Courcelles, DA3N, DARTH SIDIOUS 2, DRLB, Daniel Brockman, Dave R Barton, DavidShay, David spector, DavidHouse, Deagle AP, Decltype, Dicklyon, Dominus, DonkeyKong64, Dysprosia, EagleFan, Eclecticology, Efnar, Elano, Epbr123, Erik Postma, Estel, Fixblor, Flinx,Fredrik, Furrykef, G716, Giftlite, Gowdasathish, Gregbard, Gremagor, Greswik, Gurch, H2g2bob, Hbent, Hekerui, Hoot, Hu12, Hult041956, HumbleGod, IMacWin95, Iceera88, IdLoveOne,Ideyal, Imaginationac, Innotata, InverseHypercube, Itub, Jadony, Jan1nad, JanGB, Jaranda, Jbalint, Jbergquist, Jezmck, Jim.belk, Jkmaloo, Joc, JohnyDog, Jokes Free4Me, Josh Parris, Joshdick,[email protected], Jshadias, Julian Mendez, Justin W Smith, KGasso, Karol Langner, Kauffner, Kevinb, KlaudiuMihaila, Knowandgive, Kraftlos, Kuru, Lagelspeil, LakeHMM, Lambiam, LeszekJaczuk, Letdinosaursdie, Lfiguero, Linas, LittleDan, Lohray, Loren.wilton, LutzL, MFNickster, MZMcBride, MagicalPhats, Makeemlighter, Makuabob, Markus Kuhn, MathMartin, Mathaxiom,Maurice Carbonaro, Maxcyber10, Mckee, Melchoir, Mets501, Mfhall, MiNombreDeGuerra, Michael Hardy, Michael miceli, Michiel Helvensteijn, Mikael Hggstrm, Mikay, Mikez,Mindmatrix, Mmortal03, Momojeng, Monedula, MovGP0, Mygerardromance, Myncknm, Mysdaao, N3rd4i, NJA, Navigatr85, Nerd42, Nikola Smolenski, Nilkanthvns, Nima Baghaei,NocNokNeo, Noisy, Nosferattr, NuclearWarfare, Nutiketaiel, OlEnglish, Oleg Alexandrov, OliverTwist, Orz, P0mbal, Pak21, Paolo.dL, Pasixxxx, Patrick, Paul August, PaulTanenbaum,Pfoifry, Phil Boswell, Pizza1512, Pooryorick, PrimeHunter, Pschemp, Psiphiorg, Psource, Psy guy, Puellanivis, Qmark42, QoppaGamma, Quief, Qwertyus, R.e.b., RDBury, RNLion, Rade Kutil,Random user 8384993, Redacteur, Renata3, Rich Farmbrough, Rjwilmsi, RobHar, Robinh, Ronhjones, Ryulong, SMP, Salix alba, Sam Derbyshire, Sam Korn, Sango123, Scientific29, Scott776,Secretlondon, ShelfSkewed, Simonleyton, Skal, Sl, Sligocki, Smmurphy, Special+Utilizator+$, Spoon!, Srleffler, Stevertigo, Strange but untrue, Sunborn, Super-c-sharp, Sverdrup, Tanthanyes,Tauwasser, TedPavlic, Tekhnofiend, Teo64x, Thallinger, Thehotelambush, Thezulu, Thr4wn, Tim Starling, Timothy Clemans, Tizio, Tkuvho, Toby Bartels, Tom Lougheed, Tom harrison,Tresiden, Triwbe, Trovatore, Truthkeeper88, Tumble, Tyomitch, Ulf Karlsson, Vanish2, Voyajer, Wavelength, WhisperToMe, Wigie, WikHead, Wikipelli, Wile E. Heresiarch, WillowW, Writeron wiki, Xantolus, YahoKa, Ybenharim, Yonideworst, Yunesj, Zero0000, Zundark, , 390 anonymous edits

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