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Table of mathematical symbols
From Wikipedia, the free encyclopedia
Basic mathematical symbols
Symbol
Name
Explanation ExamplesRead as
Category
=equality
x= ymeansxand yrepresent the same thing orvalue.
1 + 1 = 2is equal to;
equals
everywhere
!=
inequationx ymeans thatxand ydo not represent thesame thing or value.
(The symbols != and are primarily fromcomputer science. They are avoided inmathematical texts.)
1 2is not equal to;does not equal
everywhere
strictinequality
x< ymeansxis less than y.
x> ymeansxis greater than y.
xymeansxis much less than y.
xymeansxis much greater than y.
3 < 45 > 4
0.003 1000000
is less than, isgreater than,
is much lessthan, is muchgreater than
order theory
=
inequalityx ymeansxis less than or equal to y.
x ymeansxis greater than or equal to y.
(The symbols = are primarily fromcomputer science. They are avoided inmathematical texts.)
3 4 and 5 55 4 and 5 5
is less than orequal to, is
greater than or
equal to
order theory
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everywhere
+
addition
4 + 6 means the sum of 4 and 6. 2 + 7 = 9plus
arithmetic
disjoint union
A1 +A2 means the disjoint union of sets A1 andA2.
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 disjointunion of ... and
...set theory
subtraction
9 4 means the subtraction of 4 from 9. 8 3 = 5minus
arithmetic
negative sign
3 means the negative of the number 3. (5) = 5negative;minus; theopposite of
arithmetic
set-theoreticcomplementA B means the set that contains all theelements ofA that are not in B.
can also be used for set-theoretic complementas 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 = 56times
arithmetic
Cartesianproduct
XYmeans the set of all ordered pairs with thefirst element of each pair selected from X and thesecond element selected from Y.
{1,2} {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the Cartesian
product of ...and ...; thedirect productof ... 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
multiplication
3 4 means the multiplication of 3 by 4. 7 8 = 56times
arithmeticdot product
u v means the dot product of vectors u and v (1,2,5) (3,4,1) = 6dot
vector algebra
division
6 3 or 6 3 means the division of 6 by 3.2 4 = .5
12 4 = 3divided by
arithmetic
quotient group G / Hmeans the quotient of group G modulo itssubgroup H.
{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b},{a, b+a}, {2a, b+2a}}mod
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group theory
quotient setA/~ means the set of all ~ equivalence classes inA.
If we define ~ by x ~ y x y ,then/~ = {{x+ n : n} : x (0,1]}
mod
set theory
plus-minus
6 3 means both 6 + 3 and 6 - 3.The equationx= 5 4, has twosolutions,x= 7 andx= 3.
plus or minus
arithmetic
plus-minus10 2 or equivalently 10 20% means the rangefrom 10 2 to 10 + 2.
Ifa = 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 principalsquare root of;
square root
real numbers
complexsquare root
if is represented in polarcoordinates with , then
.
the complexsquare root
of
square root
complexnumbers
||
absolute valueor modulus
|x| means the distance along the real line (oracross the complex plane) betweenxand zero.
|3| = 3
|5| = |5| = 5
| i| = 1
| 3 + 4i| = 5
absolute value(modulus) of
numbers
Euclideandistance
|x y| means the Euclidean distance between xand y.
Forx = (1,1), and y = (4,5),
|x y| = ([14]2 + [15]2) = 5
Euclideandistancebetween;Euclidean
norm ofGeometry
Determinant
|A| means the determinant of the matrix Adeterminant of
Matrix theory
Cardinality
|X| means the cardinality of the setX. |{3, 5, 7, 9}| = 4.cardinality of
Set theory
divides A single vertical bar is used to denote divisibility. Since 15 = 35, it is true that 3|15 and
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a|b means a divides b. 5|15.divides
NumberTheory
Conditionalprobability
A single vertical bar is used to describe theprobability of an event given another eventhappening.P(A|B) means a given b.
IfP(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4Given
Probability
!factorial
n! is the product 1 2 ... n. 4! = 1 2 3 4 = 24factorial
combinatorics
Ttranspose
Swap rows for columns Aij = (AT)ji
transpose
matrixoperations
~
probabilitydistribution
X ~ D, means the random variableXhas theprobability distribution D.
X ~ N(0,1), the standard normal
distributionhas
distributionstatistics
Rowequivalence
A~B means that B can be generated by using aseries of elementary row operations onA
is rowequivalent to
Matrix theory
same order ofmagnitude
m ~ n means the quantities m and n have thesame order of magnitude, or general size.
(Note that~ is used for an approximation that ispoor, otherwise use .)
2 ~ 5
8 9 ~ 100
but 2 10
roughly similar
poorlyapproximates
Approximationtheory
asymptoticallyequivalent
f~ gmeans . x ~ x+1is
asymptoticallyequivalent to
Asymptoticanalysis
Equivalencerelation
a ~ b means (and equivalently ). 1 ~ 5 mod 4are in the
sameequivalence
class
everywhere
approximatelyequal x ymeansxis approximately equal to y. 3.14159
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isapproximately
equal to
everywhere
isomorphism
G Hmeans that group G is isomorphic to groupH.
Q / {1, 1} V,where Q is the quaternion group and Visthe Klein four-group.
is isomorphicto
group theory
normalsubgroup
NG means that Nis a normal subgroup ofgroup G.
Z(G) Gis a normalsubgroup of
group theory
ideal
IRmeans that Iis an ideal of ring R. (2) Zis an ideal of
ring theory
materialimplication AB means ifA is true then B is also true; ifA
is false then nothing is said about B.
may mean the same as , or it may have themeaning for functions given below.
may mean the same as , or it may have themeaning for superset given below.
x= 2 x2 = 4 is true, butx2 = 4 x= 2 is in general false (since xcould be2).
implies; if then
propositionallogic, Heyting
algebra
materialequivalence
AB meansA is true ifB is true andA is falseifB is false.
x+ 5 = y+2 x+ 3 = yif and only if;
iff
propositionallogic
logicalnegation
The statement A is true if and only ifA is false.
A slash placed through another operator is thesame as "" placed in front.
(The symbol~ has many other uses, so or theslash notation is preferred.)
(A) Ax y (x= y)
not
propositionallogic
logicalconjunction ormeet in a
lattice
The statementAB is true ifA and B are bothtrue; else it is false.
For functionsA(x) and B(x),A(x) B(x) is used tomean min(A(x), B(x)).
(Old notation) uv means the cross product ofvectors u and v.
n < 4 n >2 n = 3 when n is anatural number.and; min
propositionallogic, lattice
theory
logicaldisjunction orjoin in a lattice
The statementAB is true ifA orB (or both) aretrue; if both are false, the statement is false.
For functionsA(x) and B(x),A(x) B(x) is used to
n 4 n 2 n 3 when n is anatural number.
or; max
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mean max(A(x), B(x)).propositionallogic, lattice
theory
exclusive or
The statementAB is true when either A or B,but not both, are true.AB means the same.
(A) A is always true,AA is alwaysfalse.
xor
propositionallogic, Boolean
algebra
direct sumThe direct sum is a special way of combiningseveral modules into one general module (thesymbol is used, is only for logic).
Most commonly, for vector spaces U, V,and W, the following consequence isused:U= VW (U= V+ W) (V W={0})
direct sum of
Abstractalgebra
universalquantification
x: P(x) means P(x) is true for allx. n: n2 n.for all; for any;for each
predicate logic
existentialquantification x: P(x) means there is at least one xsuch that
P(x) is true.n: n is even.there exists
predicate logic
!
uniquenessquantification
!x: P(x) means there is exactly onexsuch thatP(x) is true.
! n: n + 5 = 2n.there existsexactly one
predicate logic
:=
:
definition x:= yorx ymeansxis defined to be another
name fory
(Some writers use to mean congruence).
P:Q means Pis defined to be logicallyequivalent to Q.
coshx:= (1/2)(exp(x)+exp(-x))is defined as
everywhere
congruence
ABC DEF means triangle ABC is congruentto (has the same measurements as) triangle DEF.
is congruent to
geometry
congruence
relation
a b (mod n) means a b is divisible by n 5 11 (mod 3)... is congruentto ... modulo ...
modulararithmetic
{ , }set brackets
{a,b,c} means the set consisting ofa, b, and c. = { 1, 2, 3, }the set of
set theory
{ : } set buildernotation{x: P(x)} means the set of allxfor which P(x) istrue. {x| P(x)} is the same as {x: P(x)}. {n : n
2 < 20} = { 1, 2, 3, 4}
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{ | }the set of
such that
set theory
{ }
empty set
means the set with no elements. { } means thesame. {n : 1 < n
2 < 4} = the empty set
set theory
setmembership
aS means a is an element of the set S; aSmeans a is not an element ofS.
(1/2)1
21
is an elementof; is not anelement of
everywhere,set theory
subset(subset)AB means every element ofA is alsoelement ofB.
(proper subset)AB meansAB butA B.
(Some writers use the symbolas if it were thesame as.)
(A B) A
is a subset of
set theory
superset AB means every element ofB is also elementofA.
AB meansAB butA B.
(Some writers use the symbolas if it were thesame as.)
(AB) B
is a supersetof
set theory
set-theoretic
union
(exclusive)AB means the set that contains all
the elements fromA, or all the elements from B,but not both."A orB, but not both."
(inclusive)AB means the set that contains allthe elements fromA, or all the elements from B, orall the elements from bothA and B."A orB or both".
AB (AB) = B (inclusive)the union of
or
union
set theory
set-theoreticintersection
A B means the set that contains all thoseelements thatA and B have in common. {x :x
2 = 1} = {1}intersectedwith; intersect
set theorysymmetricdifference
means the set of elements in exactly oneofA orB.
{1,5,6,8} {2,5,8} = {1,2,6}symmetricdifference
set theory
set-theoreticcomplement
AB means the set that contains all thoseelements ofA that are not in B.
can also be used for set-theoretic complementas described above.
{1,2,3,4} {3,4,5,6} = {1,2}minus; without
set theory
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( )
functionapplication f(x) means the value of the function fat the
elementx. Iff(x) :=x2, then f(3) = 32 = 9.of
set theory
precedencegrouping Perform the operations inside the parentheses
first.(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
parentheses
everywhere
f:XY
function arrow
f:X Ymeans the function fmaps the setXintothe set Y. Let f: be defined by f(x) :=x
2.from to
settheory,type
theory
o
functioncomposition
fogis the function, such that (fog)(x) = f(g(x)).iff(x) := 2x, and g(x) :=x+ 3, then(fog)(x) = 2(x+ 3).composed with
set theory
N
naturalnumbersN means { 1, 2, 3, ...}, but see the article onnatural numbers for a different convention.
= {|a| : a, a 0}N
numbers
Z
integers
means {..., 3, 2, 1, 0, 1, 2, 3, ...} and +means {1, 2, 3, ...} = .
= {p, -p :p} {0}Z
numbers
Q
rationalnumbers
means {p/q :p, q}.3.14000...
Q
numbers
R
real numbers
means the set of real numbers.
(1) R
numbers
C
complex
numbers means {a + bi: a,b}. i= (1)
C
numbers
arbitraryconstant
Ccan be any number, most likely unknown;usually occurs when calculating antiderivatives.
iff(x) = 6x + 4x, then F(x) = 2x + 2x +C, where F'(x) = f(x)
C
integralcalculus
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K
real orcomplexnumbers
K means the statement holds substituting K forRand also forC.
because
and
.
K
linear algebra
infinity is an element of the extended number line that
is greater than all real numbers; it often occurs inlimits.
infinity
numbers
||||
norm
||x|| is the norm of the elementxof a normedvector space.
||x + y|| ||x|| + || y||norm of
length 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 a1a2an.= (1+2)(2+2)(3+2)(4+2)
= 3 4 5 6 = 360
product over from to
of
arithmetic
Cartesianproduct
means the set of all (n+1)-tuples
(y0, , yn).
the Cartesianproduct of; thedirect product
of
set theory
coproduct
coproductover from to of
categorytheory
derivative f(x) is the derivative of the function fat the pointx, i.e., the slope of the tangent to fatx.
The dot notation indicates a time derivative. That
is .
Iff(x) :=x2, then f(x) = 2x prime
derivative of
calculus
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indefiniteintegral orantiderivative
f(x) dxmeans a function whose derivative is f. x2 dx=x3/3 + C
indefiniteintegral of
theantiderivative
of
calculus
definiteintegral
abf(x) dxmeans the signed area between the
x-axis and the graph of the function fbetweenx=a andx= b.
ab x2 dx= b3/3 - a3/3;
integral from to of
with respect to
calculus
contourintegral orclosed lineintegral
Similar to the integral, but used to denote a singleintegration over a closed curve or loop. It issometimes used in physics texts involvingequations regarding Gauss's Law, and while
these formulas involve a closed surface integral,the representations describe only the firstintegration of the volume over the enclosingsurface. Instances where the latter requiressimultaneous double integration, the symbol would be more appropriate. A third related symbolis the closed volume integral, denoted by thesymbol .
The contour integral can also frequently be foundwith a subscript capital letterC, C, denoting thata closed loop integral is, in fact, around a contourC, or sometimes dually appropriately, a circle C.
In representations of Gauss's Law, a subscriptcapital S, S, is used to denote that theintegration is over a closed surface.
contourintegral of
calculus
gradient
f(x1, , xn) is the vector of partial derivatives(f/ x1, , f/ xn).
Iff(x,y,z) := 3xy+ z, then f= (3y, 3x,2z)
del, nabla,gradient of
vectorcalculus
divergence
If , then
.
del dot,
divergence ofvector
calculus
curl
If , then
.
curl of
vectorcalculus
partialdifferentialWith f(x1, , xn), f/xi is the derivative offwith
Iff(x,y) := x2y, then f/x = 2xy
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respect to xi, with all other variables keptconstant.
partial, d
calculus
boundary
Mmeans the boundary ofM {x : ||x|| 2} = {x : ||x|| = 2}boundary of
topology
Dirac deltafunction
(x)Dirac delta of
hyperfunction
Kroneckerdelta
ijKroneckerdelta of
hyperfunction
perpendicular
xymeansxis perpendicular to y; or moregenerallyxis orthogonal to y.
Iflm and mn in the plane thenl|| n.
isperpendicular
togeometry
coprime
xymeansxhas no factor in common with y. 34 55.is coprime to
number theory
bottomelement
x= meansxis the smallest element. x:x = the bottomelement
lattice theory
comparability
xymeans thatxis comparable to y.{e, } {1, 2, e, 3, } under setcontainment.
is comparableto
Order theory
||
parallel
x|| ymeansxis parallel to y.
Ifl|| m and mn then ln. In physicsthis is also used to expressis parallel to
geometry,physics
incomparability
x|| ymeansxis incomparable to y. {1,2} || {2,3} under set containment.
is
incomparableto
order theory
exactdivisibility
pf|| n meanspfexactly divides n. 23 || 360.exactly divides
number theory
entailment
AB means the sentenceA entails the sentenceB, that is in every model in whichA is true, B isalso true.
AA Aentails
model theory
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inference
xymeans yis derivable fromx. A B B A
infers or isderived from
propositionallogic,
predicate logic
,
( | )
< , >
:
inner product
x,y means the inner product ofxand yasdefined in an inner product space.
For spatial vectors, the dot product notation,xyis common.For matrices, the colon notation may be used.
The standard inner product between twovectorsx= (2, 3) and y= (1, 5) is:x, y = 2 1 + 3 5 = 13
A:B = AijBij
i,j
inner productof
linear algebra
tensorproduct,tensor productof modules
means the tensor product ofVand U.means the tensor product of modules
Vand Uover the ring R.
{1, 2, 3, 4} {1, 1, 2} ={{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor productof
linear algebra
*
convolution
f* gmeans the convolution offand g.
convolution,convoluted
with
functionalanalysis
x
mean
(often read as "x bar") is the mean (averagevalue ofxi).
.overbar, bar
statistics
complexconjugate
is the complex conjugate ofz.conjugatecomplexnumbers
delta equal to means equal by definition. When is used,equality is not true generally, but rather equality istrue under certain assumptions that are taken incontext. Some writers prefer . .
equal bydefinition
everywhere
See also
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Greek letters used in mathematicsISO 31-11Mathematical alphanumeric symbolsMathematical notationNotation in probability
Physical constantsRoman letters used in mathematicsTable of logic symbolsUnicode Mathematical OperatorsWikipedia:Mathematical symbols
External links
Jeff Miller: Earliest Uses of Various Mathematical Symbols (http://members.aol.com/jeff570/mathsym.html)
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)
Retrieved from "http://en.wikipedia.org/wiki/Table_of_mathematical_symbols"Categories: Mathematical notation | Mathematics-related lists | Mathematical symbols | Mathematical tables
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