Basic Math Symbols (Algebra Geometry Statistics etc.)

Basic math symbols

SymbolSymbol NameMeaning / definitionExample
=equals signequality5 = 2+3
5 is equal to 2+3
not equal signinequality5 ≠ 4
5 is not equal to 4
approximately equalapproximationsin(0.01) ≈ 0.01,
x ≈ y means x is approximately equal to y
>strict inequalitygreater than5 > 4
5 is greater than 4
<strict inequalityless than4 < 5
4 is less than 5
inequalitygreater than or equal to5 ≥ 4,
x ≥ y means x is greater than or equal to y
inequalityless than or equal to4 ≤ 5,
x ≤ y means x is greater than or equal to y
( )parenthesescalculate expression inside first2 × (3+5) = 16
[ ]bracketscalculate expression inside first[(1+2)×(1+5)] = 18
+plus signaddition1 + 1 = 2
minus signsubtraction2 − 1 = 1
±plus – minusboth plus and minus operations3 ± 5 = 8 and -2
±minus – plusboth minus and plus operations3 ± 5 = -2 and 8
*asteriskmultiplication2 * 3 = 6
×times signmultiplication2 × 3 = 6
·multiplication dotmultiplication2 · 3 = 6
÷division sign / obelusdivision6 ÷ 2 = 3
/division slashdivision6 / 2 = 3
horizontal linedivision / fraction
modmoduloremainder calculation7 mod 2 = 1
.perioddecimal point, decimal separator2.56 = 2+56/100
abpowerexponent23 = 8
a^bcaretexponent2 ^ 3 = 8
asquare roota · a  = a√9 = ±3
3acube root3a · 3√a  · 3√a  = a3√8 = 2
4afourth root4a · 4√a  · 4√a  · 4√a  = a4√16 = ±2
nan-th root (radical)for n=3, n√8 = 2
%percent1% = 1/10010% × 30 = 3
per-mille1‰ = 1/1000 = 0.1%10‰ × 30 = 0.3
ppmper-million1ppm = 1/100000010ppm × 30 = 0.0003
ppbper-billion1ppb = 1/100000000010ppb × 30 = 3×10-7
pptper-trillion1ppt = 10-1210ppt × 30 = 3×10-10

Geometry symbols

SymbolSymbol NameMeaning / definitionExample
angleformed by two rays∠ABC = 30°
measured angleABC = 30°
spherical angleAOB = 30°
right angle= 90°α = 90°
°degree1 turn = 360°α = 60°
degdegree1 turn = 360degα = 60deg
primearcminute, 1° = 60′α = 60°59′
double primearcsecond, 1′ = 60″α = 60°59′59″
lineinfinite line
ABline segmentline from point A to point B
rayline that start from point A
arcarc from point A to point B = 60°
perpendicularperpendicular lines (90° angle)AC ⊥ BC
| |parallelparallel linesAB | | CD
congruent toequivalence of geometric shapes and size∆ABC≅ ∆XYZ
~similaritysame shapes, not same size∆ABC~ ∆XYZ
Δtriangletriangle shapeΔABC≅ ΔBCD
|xy|distancedistance between points x and yxy | = 5
πpi constantπ = 3.141592654…

is the ratio between the circumference and diameter of a circle

c = π·d = 2·π·r
radradiansradians angle unit360° = 2π rad
cradiansradians angle unit360° = 2π c
gradgradians / gonsgrads angle unit360° = 400 grad
ggradians / gonsgrads angle unit360° = 400 g

Algebra symbols

SymbolSymbol NameMeaning / definitionExample
xx variableunknown value to findwhen 2x = 4, then x = 2
equivalenceidentical to
equal by definitionequal by definition
:=equal by definitionequal by definition
~approximately equalweak approximation11 ~ 10
approximately equalapproximationsin(0.01) ≈ 0.01
proportional toproportional toy ∝ x when y = kx, k constant
lemniscateinfinity symbol
much less thanmuch less than1 ≪ 1000000
much greater thanmuch greater than1000000 ≫ 1
( )parenthesescalculate expression inside first2 * (3+5) = 16
[ ]bracketscalculate expression inside first[(1+2)*(1+5)] = 18
{ }bracesset
xfloor bracketsrounds number to lower integer⌊4.3⌋ = 4
xceiling bracketsrounds number to upper integer⌈4.3⌉ = 5
x!exclamation markfactorial4! = 1*2*3*4 = 24
x |single vertical barabsolute value| -5 | = 5
f (x)function of xmaps values of x to f(x)f (x) = 3x+5
(f ∘ g)function composition(f ∘ g) (x) = f (g(x))f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)
(a,b)open interval(a,b) = {x | a < x < b}x∈ (2,6)
[a,b]closed interval[a,b] = {x | a ≤ x ≤ b}x ∈ [2,6]
deltachange / differencet = t t0
discriminantΔ = b2 – 4ac
sigmasummation – sum of all values in range of series xi= x1+x2+…+xn
∑∑sigmadouble summation
capital piproduct – product of all values in range of series xi=x1∙x2∙…∙xn
ee constant / Euler’s numbere = 2.718281828…e = lim (1+1/x)x , x→∞
γEuler-Mascheroni  constantγ = 0.527721566…
φgolden ratiogolden ratio constant
πpi constantπ = 3.141592654…

is the ratio between the circumference and diameter of a circle

c = π·d = 2·π·r

Linear Algebra Symbols

SymbolSymbol NameMeaning / definitionExample
·dotscalar producta · b
×crossvector producta × b
ABtensor producttensor product of A and BA ⊗ B
inner product
[ ]bracketsmatrix of numbers
( )parenthesesmatrix of numbers
A |determinantdeterminant of matrix A
det(A)determinantdeterminant of matrix A
|| x ||double vertical barsnorm
ATtransposematrix transpose(AT)ij = (A)ji
AHermitian matrixmatrix conjugate transpose(A)ij = (A)ji
A*Hermitian matrixmatrix conjugate transpose(A*)ij = (A)ji
A -1inverse matrixA A-1 = I
rank(A)matrix rankrank of matrix Arank(A) = 3
dim(U)dimensiondimension of matrix Arank(U) = 3

Probability and statistics symbols

SymbolSymbol NameMeaning / definitionExample
P(A)probability functionprobability of event AP(A) = 0.5
P(A ∩ B)probability of events intersectionprobability that of events A and BP(AB) = 0.5
P(A ∪ B)probability of events unionprobability that of events A or BP(AB) = 0.5
P(A | B)conditional probability functionprobability of event A given event B occuredP(A | B) = 0.3
f (x)probability density function (pdf)P(a  x  b) = ∫ f (x) dx
F(x)cumulative distribution function (cdf)F(x) = P(X x)
μpopulation meanmean of population valuesμ = 10
E(X)expectation valueexpected value of random variable XE(X) = 10
E(X | Y)conditional expectationexpected value of random variable X given YE(X | Y=2) = 5
var(X)variancevariance of random variable Xvar(X) = 4
σ2variancevariance of population valuesσ= 4
std(X)standard deviationstandard deviation of random variable Xstd(X) = 2
σXstandard deviationstandard deviation value of random variable XσX  = 2
medianmiddle value of random variable x
cov(X,Y)covariancecovariance of random variables X and Ycov(X,Y) = 4
corr(X,Y)correlationcorrelation of random variables X and Ycorr(X,Y) = 0.6
ρX,Ycorrelationcorrelation of random variables X and YρX,Y = 0.6
summationsummation – sum of all values in range of series
∑∑double summationdouble summation
Momodevalue that occurs most frequently in population
MRmid-rangeMR = (xmax+xmin)/2
Mdsample medianhalf the population is below this value
Q1lower / first quartile25% of population are below this value
Q2median / second quartile50% of population are below this value = median of samples
Q3upper / third quartile75% of population are below this value
xsample meanaverage / arithmetic meanx = (2+5+9) / 3 = 5.333
s 2sample variancepopulation samples variance estimators 2 = 4
ssample standard deviationpopulation samples standard deviation estimators = 2
zxstandard scorezx = (xx) / sx
X ~distribution of Xdistribution of random variable XX ~ N(0,3)
N(μ,σ2)normal distributiongaussian distributionX ~ N(0,3)
U(a,b)uniform distributionequal probability in range a,bX ~ U(0,3)
exp(λ)exponential distributionf (x) = λeλx , x≥0
gamma(c, λ)gamma distributionf (x) = λ c xc-1eλx / Γ(c), x≥0
χ 2(k)chi-square distributionf (x) = xk/2-1ex/2 / ( 2k/2 Γ(k/2) )
F (k1, k2)F distribution
Bin(n,p)binomial distributionf (k) = nCk pk(1-p)n-k
Poisson(λ)Poisson distributionf (k) = λkeλ / k!
Geom(p)geometric distributionf (k) =  p(1-p) k
HG(N,K,n)hyper-geometric distribution
Bern(p)Bernoulli distribution

Combinatorics Symbols

SymbolSymbol NameMeaning / definitionExample
n!factorialn! = 1·2·3·…·n5! = 1·2·3·4·5 = 120
nPkpermutation5P3 = 5! / (5-3)! = 60
nCk

 

combination5C3 = 5!/[3!(5-3)!]=10

Set theory symbols

SymbolSymbol NameMeaning / definitionExample
{ }seta collection of elementsA = {3,7,9,14},
B = {9,14,28}
A ∩ Bintersectionobjects that belong to set A and set BA ∩ B = {9,14}
A ∪ Bunionobjects that belong to set A or set BA ∪ B = {3,7,9,14,28}
A ⊆ Bsubsetsubset has fewer elements or equal to the set{9,14,28} ⊆ {9,14,28}
A ⊂ Bproper subset / strict subsetsubset has fewer elements than the set{9,14} ⊂ {9,14,28}
A ⊄ Bnot subsetleft set not a subset of right set{9,66} ⊄ {9,14,28}
A ⊇ Bsupersetset A has more elements or equal to the set B{9,14,28} ⊇ {9,14,28}
A ⊃ Bproper superset / strict supersetset A has more elements than set B{9,14,28} ⊃ {9,14}
A ⊅ Bnot supersetset A is not a superset of set B{9,14,28} ⊅ {9,66}
2Apower setall subsets of A
power setall subsets of A
A = Bequalityboth sets have the same membersA={3,9,14},
B={3,9,14},
A=B
Accomplementall the objects that do not belong to set A
A \ Brelative complementobjects that belong to A and not to BA = {3,9,14},
B = {1,2,3},
A-B = {9,14}
A – Brelative complementobjects that belong to A and not to BA = {3,9,14},
B = {1,2,3},
A-B = {9,14}
A ∆ Bsymmetric differenceobjects that belong to A or B but not to their intersectionA = {3,9,14},
B = {1,2,3},
A ∆ B = {1,2,9,14}
A ⊖ Bsymmetric differenceobjects that belong to A or B but not to their intersectionA = {3,9,14},
B = {1,2,3},
A ⊖ B = {1,2,9,14}
a∈Aelement ofset membership A={3,9,14}, 3 ∈ A
x∉Anot element ofno set membershipA={3,9,14}, 1 ∉ A
(a,b)ordered paircollection of 2 elements
A×Bcartesian productset of all ordered pairs from A and B
|A|cardinalitythe number of elements of set AA={3,9,14}, |A|=3
#Acardinalitythe number of elements of set AA={3,9,14}, #A=3
aleph-nullinfinite cardinality of natural numbers set
aleph-onecardinality of countable ordinal numbers set
Øempty setØ = { }C = {Ø}
universal setset of all possible values
0natural numbers / whole numbers  set (with zero)0 = {0,1,2,3,4,…}0 ∈  0
1natural numbers / whole numbers  set (without zero)1 = {1,2,3,4,5,…}6 ∈  1
integer numbers set = {…-3,-2,-1,0,1,2,3,…}-6 ∈
rational numbers set = {x | x=a/ba,b∈ }2/6 ∈
real numbers set = {x | -∞ < x <∞}6.343434∈
complex numbers set = {z | z=a+bi, -∞<a<∞,      -∞<b<∞}6+2i ∈

Logic symbols

SymbolSymbol NameMeaning / definitionExample
·andandx · y
^caret / circumflexandx ^ y
&ampersandandx & y
+plusorx + y
reversed caretorx ∨ y
|vertical lineorx | y
xsingle quotenot – negationx
xbarnot – negationx
¬notnot – negation¬ x
!exclamation marknot – negationx
circled plus / oplusexclusive or – xorx ⊕ y
~tildenegationx
implies
equivalentif and only if (iff)
equivalentif and only if (iff)
for all
there exists
there does not exists
therefore
because / since

Calculus & analysis symbols

SymbolSymbol NameMeaning / definitionExample
limitlimit value of a function
εepsilonrepresents a very small number, near zeroε  0
ee constant / Euler’s numbere = 2.718281828…e = lim (1+1/x)x ,x→∞
y ‘derivativederivative – Lagrange’s notation(3x3)’ = 9x2
y ”second derivativederivative of derivative(3x3)” = 18x
y(n)nth derivativen times derivation(3x3)(3) = 18
derivativederivative – Leibniz’s notationd(3x3)/dx = 9x2
second derivativederivative of derivatived2(3x3)/dx2 = 18x
nth derivativen times derivation
time derivativederivative by time – Newton’s notation
time second derivativederivative of derivative
Dx yderivativederivative – Euler’s notation
Dx2ysecond derivativederivative of derivative
partial derivative∂(x2+y2)/∂x = 2x
integralopposite to derivation∫ f(x)dx
∫∫double integralintegration of function of 2 variables∫∫ f(x,y)dxdy
∫∫∫triple integralintegration of function of 3 variables∫∫∫ f(x,y,z)dxdydz
closed contour / line integral
closed surface integral
closed volume integral
[a,b]closed interval[a,b] = {x | a  x  b}
(a,b)open interval(a,b) = {x | a < x < b}
iimaginary uniti ≡ √-1z = 3 + 2i
z*complex conjugatez = a+bi → z*=abiz* = 3 – 2i
zcomplex conjugatez = a+bi → z = abiz = 3 – 2i
nabla / delgradient / divergence operatorf (x,y,z)
vector
unit vector
x * yconvolutiony(t) = x(t) * h(t)
Laplace transformF(s) = {f (t)}
Fourier transformX(ω) =  {f (t)}
δdelta function
lemniscateinfinity symbol

Greek alphabet letters

Greek SymbolGreek Letter NameEnglish EquivalentPronunciation
Upper CaseLower Case
ΑαAlphaaal-fa
ΒβBetabbe-ta
ΓγGammagga-ma
ΔδDeltaddel-ta
ΕεEpsiloneep-si-lon
ΖζZetazze-ta
ΗηEtaheh-ta
ΘθThetathte-ta
ΙιIotaiio-ta
ΚκKappakka-pa
ΛλLambdallam-da
ΜμMumm-yoo
ΝνNunnoo
ΞξXixx-ee
ΟοOmicronoo-mee-c-ron
ΠπPippa-yee
ΡρRhorrow
ΣσSigmassig-ma
ΤτTautta-oo
ΥυUpsilonuoo-psi-lon
ΦφPhiphf-ee
ΧχChichkh-ee
ΨψPsipsp-see
ΩωOmegaoo-me-ga