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转 http://www.rapidtables.com/math/symbols/Basic_Math_Symbols.htm
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+56/100abpowerexponent23 = 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 = 3×10-10Geometry symbols
SymbolSymbol NameMeaning / definitionExample∠angleformed by two rays∠ABC = 30°
measured angle ABC = 30°
spherical angle AOB = 30°∟right angle= 90°α = 90°°degree1 turn = 360°α = 60°´arcminute1° = 60´α = 60°59'´´arcsecond1´ = 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 ~∆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 unit360° = 2π radgradgradsgrads angle unit360° = 400 gradAlgebra 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 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 ⌊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∑ xi= x1+x2+...+xn∑∑sigmadouble summation
∏capital piproduct - product of all values in range of series∏ xi=x1∙x2∙...∙xnee 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·rLinear Algebra Symbols
SymbolSymbol NameMeaning / definitionExample∙dotscalar producta ∙ b×crossvector producta × bA⊗Btensor 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 A Ttransposematrix transpose(AT)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 Arank(A) = 3
dim(U)dimensiondimension of matrix Arank(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) = 2σXstandard 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) = 4corr(X,Y)correlationcorrelation of random variables X and Ycorr(X,Y) = 0.6ρX,Ycorrelationcorrelation of random variables X and YρX,Y = 0.6∑summationsummation - sum of all values in range of series
∑∑double summationdouble summationMomodevalue 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 = 5.333s 2sample variancepopulation samples variance estimators 2 = 4ssample standard deviationpopulation samples standard deviation estimators = 2zxstandard scorezx = (x-x) / 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,b X ~ U(0,3)exp(λ)exponential distributionf (x) = λe-λx , x≥0 gamma(c, λ)gamma distributionf (x) = λ c xc-1e-λx / Γ(c), x≥0
χ 2(k)chi-square distributionf (x) = xk/2-1e-x/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) = λke-λ / 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 / definitionExamplen!factorialn! = 1·2·3·...·n5! = 1·2·3·4·5 = 120nPkpermutation
5P3 = 5! / (5-3)! = 60nCk
combination
5C3 = 5!/[3!(5-3)!]=10Set 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=BAccomplementall 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}
universal setset of all possible values
0natural numbers / whole numbers set (with zero)0 = {0,1,2,3,4,...}0 ∈ 01natural 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/b, a,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&ersandandx & 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
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
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 Dx2 ysecond derivativederivative of derivative
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 one1I١אtwo2II٢בthree3III٣גfour4IV٤דfive5V٥הsix6VI٦וseven7VII٧זeight8VIII٨חnine9IX٩טten10X١٠יeleven11XI١١יאtwelve12XII١٢יבthirteen13XIII١٣יגfourteen14XIV١٤ידfifteen15XV١٥טוsixteen16XVI١٦טזseventeen17XVII١٧יזeighteen18XVIII١٨יחnineteen19XIX١٩יטtwenty20XX٢٠כthirty30XXX٣٠לforty40XL٤٠מfifty50L٥٠נsixty60LX٦٠סseventy70LXX٧٠עeighty80LXXX٨٠פninety90XC٩٠צone hundred100C١٠٠קGreek 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-gaRoman numerals
NumberRoman numeral0not defined1I2II3III4IV5V6VI7VII8VIII9IX10X11XI12XII13XIII14XIV15XV16XVI17XVII18XVIII19XIX20XX30XXX40XL50L60LX70LXX80LXXX90XC100C200CC300CCC400CD500D600DC700DCC800DCCC900CM1000M5000V10000X50000L100000C500000D1000000M |
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