数学符号表

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数学符号表

Mathematical Symbols

List of all mathematical symbols and signs - meaning and examples.

Basic math symbols
Geometry symbols
Algebra symbols
Probability & statistics symbols
Set theory symbols
Logic symbols
Calculus & analysis symbols
Number symbols
Greek symbols
Roman numerals

Basic math symbols

SymbolSymbol NameMeaning / definitionExample=equals signequality5 = 2+3≠not equal signinequality5 ≠ 4>strict inequalitygreater than5 > 4<strict inequalityless than4 < 5≥inequalitygreater than or equal to5 ≥ 4≤inequalityless than or equal to4 ≤ 5( )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 $\frac{6}{2}=3$ modmoduloremainder calculation7 mod 2 = 1.perioddecimal point, decimal separator2.56 = 2+56/100a^bpowerexponent2³= 8a^bcaretexponent2 ^ 3= 8√asquare root

√a · √a = a

√9 = ±3³√acube root³√a · ³√a · ³√a = a³√8 = 2⁴√afourth root⁴√a · ⁴√a · ⁴√a · ⁴√a = a⁴√16 = ±2ⁿ√an-th root (radical) for n=3, ⁿ√8 = 2%percent1% = 1/10010% × 30 = 3‰per-mille1‰ = 1/1000 = 0.1%10‰ × 30 = 0.3ppmper-million1ppm = 1/100000010ppm × 30 = 0.0003ppbper-billion1ppb = 1/100000000010ppb × 30 = 3×10^-7pptper-trillion1ppt = 10^-1210ppt × 30 = 3×10^-10

Geometry symbols

SymbolSymbol NameMeaning / definitionExample∠angleformed by two rays∠ABC = 30º

measured angle

ABC = 30º

spherical angle

AOB = 30º∟right angle= 90ºα = 90ººdegree1 turn = 360ºα = 60º´arcminute1º = 60´α = 60º59'´´arcsecond1´ = 60´´α = 60º59'59''

lineinfinite line ABline segmentline from point A to point B

rayline that start from point A

arcarc from point A to point B

= 60º|perpendicularperpendicular lines (90º angle)AC | BC||parallelparallel linesAB || CD≅congruent toequivalence of geometric shapes and size∆ABC ≅ ∆XYZ~similaritysame shapes, not same size∆ABC ~ ∆XYZΔtriangletriangle shapeΔABC ≅ ΔBCD|x-y|distancedistance between points x and y| x-y | = 5πpi constantπ = 3.141592654...

is the ratio between the circumference and diameter of a circle

c = π·d = 2·π·rradradiansradians angle unit360º = 2π radgradgradsgrads angle unit360º = 400 grad

Algebra symbols

SymbolSymbol NameMeaning / definitionExamplexx 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 to

f(x) ∝ g(x)

∞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 ⌊x⌋floor bracketsrounds number to lower integer⌊4.3⌋= 4⌈x⌉ceiling bracketsrounds number to upper integer⌈4.3⌉= 5x!exclamation markfactorial4! = 1*2*3*4 = 24| x |single vertical barabsolute value| -5 | = 5f (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 / difference∆t = t₁- t₀∆discriminantΔ = b² - 4ac ∑sigmasummation - sum of all values in range of series∑ x_i= x₁+x₂+...+x_n∑∑sigmadouble summation

∏capital piproduct - product of all values in range of series∏ x_i=x₁∙x₂∙...∙x_nee 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 × bA⊗Btensor producttensor product of A and BA ⊗ B $\langle x,y \rangle$ inner product [ ]bracketsmatrix of numbers ( )parenthesesmatrix of numbers | A |determinantdeterminant of matrix A det(A)determinantdeterminant of matrix A || x ||double vertical barsnorm A^Ttransposematrix transpose

(A^T)_ij = (A)_ji

A^†Hermitian 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 A

rank(A) = 3

dim(U)dimensiondimension of matrix A

rank(U) = 3

Probability and statistics symbols

SymbolSymbol NameMeaning / definitionExampleP(A)probability functionprobability of event AP(A) = 0.5P(A ∩ B)probability of events intersectionprobability that of events A and BP(A∩B) = 0.5P(A ∪ B)probability of events unionprobability that of events A or BP(A∪B) = 0.5P(A | B)conditional probability functionprobability of event A given event B occuredP(A | B) = 0.3f (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 = 10E(X)expectation valueexpected value of random variable XE(X) = 10E(X | Y)conditional expectationexpected value of random variable X given YE(X | Y=2) = 5var(X)variancevariance of random variable Xvar(X) = 4σ²variancevariance of population valuesσ²= 4std(X)standard deviationstandard deviation of random variable Xstd(X) = 2σ_Xstandard deviationstandard deviation value of random variable Xσ_X= 2

medianmiddle value of random variable x

cov(X,Y)covariancecovariance of random variables X and Ycov(X,Y) = 4corr(X,Y)correlationcorrelation of random variables X and Ycorr(X,Y) = 0.6ρ_X,Ycorrelationcorrelation of random variables X and Yρ_X,Y = 0.6∑summationsummation - sum of all values in range of series

∑∑double summationdouble summation

Momodevalue that occurs most frequently in population MRmid-range

MR = (x_max+x_min)/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 = 5.333s²sample variancepopulation samples variance estimators² = 4ssample standard deviationpopulation samples standard deviation estimators = 2z_xstandard score

z_x = (x-x) / s_x

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,b X ~ U(0,3)exp(λ)exponential distributionf (x) = λe^-λx , x≥0 gamma(c, λ)gamma distribution

f (x) = λ c x^c-1e^-λx / Γ(c), x≥0

χ²(k)chi-square distribution

f (x) = x^k^/2-1e^-x/2 / ( 2^k/2Γ(k/2) )

F (k₁, k₂)F distribution Bin(n,p)binomial distribution

f (k) = _nC_k p^k(1-p)^n-k

Poisson(λ)Poisson distribution

f (k) = λ^ke^-λ / k!

Geom(p)geometric distribution

f (k) = p(1-p)^k

HG(N,K,n)hyper-geometric distribution Bern(p)Bernoulli distribution

Combinatorics Symbols

SymbolSymbol NameMeaning / definitionExamplen!factorialn! = 1·2·3·...·n5! = 1·2·3·4·5 = 120_nP_kpermutation $_{n}P_{k}=\frac{n!}{(n-k)!}$ ₅P₃ = 5! / (5-3)! = 60_nC_k

combination $_{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!}$ ₅C₃ = 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}2^Apower setall subsets of A $\mathcal{P}(A)$ power setall subsets of A A = Bequalityboth sets have the same membersA={3,9,14},
B={3,9,14},
A=BA^ccomplementall 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 ∈ Ax∉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} $\mathbb{U}$ universal setset of all possible values $\mathbb{N}$ ₀natural numbers / whole numbers set (with zero) $\mathbb{N}$ ₀ = {0,1,2,3,4,...}0 ∈ $\mathbb{N}$ ₀ $\mathbb{N}$ ₁natural numbers / whole numbers set (without zero) $\mathbb{N}$ ₁ = {1,2,3,4,5,...}6 ∈ $\mathbb{N}$ ₁ $\mathbb{Z}$ integer numbers set $\mathbb{Z}$ = {...-3,-2,-1,0,1,2,3,...}-6 ∈ $\mathbb{Z}$ $\mathbb{Q}$ rational numbers set $\mathbb{Q}$ = {x | x=a/b, a,b∈ $\mathbb{Z}$ }2/6 ∈ $\mathbb{Q}$ $\mathbb{R}$ real numbers set $\mathbb{R}$ = {x | -∞ < x <∞}6.343434 ∈ $\mathbb{R}$ $\mathbb{C}$ complex numbers set $\mathbb{C}$ = {z | z=a+bi, -∞<a<∞, -∞<b<∞}6+2i ∈ $\mathbb{C}$

Logic symbols

SymbolSymbol NameMeaning / definitionExample·andandx · y^caret / circumflexandx ^ y&ampersandandx & y+plusorx + y∨reversed caretorx ∨ y|vertical lineorx | yx'single quotenot - negationx'xbarnot - negationx¬notnot - negation¬ x!exclamation marknot - negation! x⊕circled plus / oplusexclusive or - xorx ⊕ y~tildenegation~ x⇒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 $\lim_{x\to x0}f(x)$ limitlimit value of a function εepsilonrepresents a very small number, near zeroε → 0ee constant / Euler's numbere = 2.718281828...e = lim (1+1/x)^x ,x→∞y 'derivativederivative - Lagrange's notation(3x³)' = 9x²y ''second derivativederivative of derivative(3x³)'' = 18xy⁽ⁿ⁾nth derivativen times derivation(3x³)⁽³⁾ = 18 $\frac{dy}{dx}$ derivativederivative - Leibniz's notationd(3x³)/dx = 9x² $\frac{d^2y}{dx^2}$ second derivativederivative of derivatived²(3x³)/dx² = 18x $\frac{d^ny}{dx^n}$ nth derivativen times derivation $\dot{y}$ time derivativederivative by time - Newton's notation

time second derivativederivative of derivative D_xyderivativederivative - Euler's notation D_x²ysecond derivativederivative of derivative $\frac{\partial f(x,y)}{\partial x}$ partial derivative ∂(x²+y²)/∂x = 2x∫integralopposite to derivation ∬double integralintegration of function of 2 variables ∭triple integralintegration of function of 3 variables ∮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 + 2iz*complex conjugatez = a+bi → z*=a-biz* = 3 + 2izcomplex conjugatez = a+bi → z = a-biz = 3 + 2i∇nabla / delgradient / divergence operator∇f (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

Numeral symbols

NameEuropeanRomanHindu ArabicHebrewzero0 ٠ one1I١אtwo2II٢בthree3III٣גfour4IV٤דfive5V٥הsix6VI٦וseven7VII٧זeight8VIII٨חnine9IX٩טten10X١٠יeleven11XI١١יאtwelve12XII١٢יבthirteen13XIII١٣יגfourteen14XIV١٤ידfifteen15XV١٥טוsixteen16XVI١٦טזseventeen17XVII١٧יזeighteen18XVIII١٨יחnineteen19XIX١٩יטtwenty20XX٢٠כthirty30XXX٣٠לfourty40XL٤٠מfifty50L٥٠נsixty60LX٦٠סseventy70LXX٧٠עeighty80LXXX٨٠פninety90XC٩٠צone hundred100C١٠٠ק

Greek alphabet letters

Greek SymbolGreek Letter NameEnglish EquivalentPronunciationUpper CaseLower CaseΑαAlphaaal-faΒβBetabbe-taΓγGammagga-maΔδDeltaddel-taΕεEpsiloneep-si-lonΖζZetazze-taΗηEtaheh-taΘθThetathte-taΙιIotaiio-taΚκKappakka-paΛλLambdallam-daΜμMumm-yooΝνNunnooΞξXixx-eeΟοOmicronoo-mee-c-ronΠπPippa-yeeΡρRhorrowΣσSigmassig-maΤτTautta-ooΥυUpsilonuoo-psi-lonΦφPhiphf-eeΧχChichkh-eeΨψPsipsp-seeΩωOmegaoo-me-ga

Roman numerals

NumberRoman numeral0not defined1I2II3III4IV5V6VI7VII8VIII9IX10X11XI12XII13XIII14XIV15XV16XVI17XVII18XVIII19XIX20XX30XXX40XL50L60LX70LXX80LXXX90XC100C200CC300CCC400CD500D600DC700DCC800DCCC900CM1000M5000V10000X50000L100000C500000D1000000M

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