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10 janeiro 2014

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+56/100
a bpowerexponent23 = 8
a^bcaretexponent2 ^ 3 = 8
asquare root
a ·  = a
= ±3
3acube root38 = 2
4aforth root416 = ±2
nan-th root (radical)for n=3, n8 = 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-trillion1ppb = 10-1210ppb × 30 = 3×10-10

Geometry symbols

SymbolSymbol NameMeaning / definitionExample
angleformed by two rays
ABC = 30º
measured angleABC = 30º
spherical angleAOB = 30º
right angle= 90ºα = 90º
ºdegree1 turn = 360ºα = 60º
´arcminute1º = 60´α = 60º59'
´´arcsecond1´ = 60´´α = 60º59'59''
ABlineline from point A to point B
rayline that start from point A
|perpendicularperpendicular lines (90ºangle)AC | BC
||parallelparallel linesAB || CD
congruent toequivalence of geometric shapes and size∆ABC ≅ ∆XYZ
~similaritysame shapes, not same size∆ABC ∆XYZ
Δtriangletriangle shapeΔABC ≅ ΔBCD
| x-y |distancedistance between points x and yx-y | = 5
πpi constantπ = 3.141592654...
is the ratio between the circumference and diameter of a circle
c = π·d = 2·π·r
radradiansradians angle unit360º = 2π rad
gradgradsgrads angle unit360º = 400 grad

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 to
f(x g(x)
lemniscateinfinity symbol
much less thanmuch less than≪ 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 integer4.34
xceiling bracketsrounds number to upper integer4.35
x!exclamation markfactorial4! = 1*2*3*4 = 24
x |single vertical barabsolute value| -5 | = 5
(x)function of xmaps values of x to f(x)(x) = 3x+5
(g)function composition
(g) (x) = (g(x))
(x)=3xg(x)=x-1 (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

Linear Algebra Symbols

SymbolSymbol NameMeaning / definitionExample
dotscalar product b
×crossvector product× b
ABtensor producttensor product of A and BA  B
\langle x,y \rangleinner 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 A
rank(A) = 3
dim(U)dimensiondimension of matrix A
rank(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
(x)probability density function (pdf)P( x  b) = ∫ f (x) dx
F(x)cumulative distribution function (cdf)F(x) = P( 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) = 3
ρX,Ycorrelationcorrelation of random variables X and YρX,Y = 3
summationsummation - sum of all values in range of series
∑∑double summationdouble summation
Momodevalue that occurs most frequently in population
MRmid-range
MR = (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 score
zx = (x-x) / sx
~distribution of Xdistribution of random variable X~ N(0,3)
N(μ,σ2)normal distributiongaussian distribution~ N(0,3)
U(a,b)uniform distributionequal probability in range a,b ~ U(0,3)
exp(λ)exponential distribution(x) = λe-λx , x≥0
gamma(c, λ)gamma distribution
(x) = λ c xc-1e-λx / Γ(c),x≥0
χ 2(k)chi-square distribution
(x) = xk/2-1e-x/2 / ( 2k/2Γ(k/2) )
(k1, k2)F distribution
Bin(n,p)binomial distribution
(k) = nCk pk(1-p)n-k
Poisson(λ)Poisson distribution
(k) = λke-λ / k!
Geom(p)geometric distribution
(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
nPkpermutation_{n}P_{k}=\frac{n!}{(n-k)!}5P3 = 5! / (5-3)! = 60
nCk

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}
 Bintersectionobjects that belong to set A and set B∩ B = {9,14}
 Bunionobjects that belong to set A or set B∪ B = {3,7,9,14,28}
 Bsubsetsubset has less elements or equal to the set{9,14,28} ⊆ {9,14,28}
 Bproper subset / strict subsetsubset has less elements than the set{9,14} ⊂ {9,14,28}
 Bnot subsetleft set not a subset of right set{9,66} ⊄ {9,14,28}
 Bsupersetset A has more elements or equal to the set B{9,14,28} ⊇ {9,14,28}
 Bproper superset / strict supersetset A has more elements than set B{9,14,28} ⊃ {9,14}
 Bnot supersetset A is not a superset of set B{9,14,28} ⊅ {9,66}
2Apower setall subsets of A
Ƥ (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}
 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}
aAelement ofset membershipA={3,9,14}, 3 ∈ A
xAnot 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
אalephinfinite cardinality
Øempty setØ = { }C = {Ø}
Uuniversal setset of all possible values
0natural numbers / whole numbers  set (with zero)0 = {0,1,2,3,4,...}∈ ℕ0
1natural numbers / whole numbers  set (without zero)1 = {1,2,3,4,5,...}∈ ℕ1
integer numbers setℤ = {...-3,-2,-1,0,1,2,3,...}-6 ∈ ℤ
rational numbers setℚ = {| x=a/ba,b∈ℕ}2/6 ∈ ℚ
real numbers setℝ = {x | -∞ < x <∞}6.343434 ∈ ℝ
complex numbers setℂ = {| z=a+bi, -∞<a<∞,      -∞<b<∞}6+2i ∈ ℂ

Logic symbols

SymbolSymbol NameMeaning / definitionExample
·andand
x · y
^caret / circumflexand
x ^ y
&ampersandand
x & y
+plusor
x + y
reversed caretor
x  y
|vertical lineor
x | y
x'single quotenot - negation
x'
xbarnot - negation
x
¬notnot - negation
¬ x
!exclamation marknot - negation
x
circled plus / oplusexclusive or - xor
x  y
~tildenegation
x
implies
equivalentif and only if
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 zero
ε  0
ee constant / Euler's numbere = 2.718281828...e = lim (1+1/x)x , x→∞
'derivativederivative - Leibniz's notation(3x3)' = 9x2
''second derivativederivative of derivative(3x3)'' = 18x
y(n)nth derivativen times derivation(3x3)(3) = 18
\frac{dy}{dx}derivativederivative - Lagrange'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 notation
time second 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] = {| a  x  b}
(a,b)open interval(a,b) = {| a < x < b}
iimaginary uniti ≡ √-1z = 3 + 2i
z*complex conjugate= a+bi → z*=a-biz* = 3 + 2i
zcomplex conjugate= a+bi → = a-biz = 3 + 2i
nabla / delgradient / divergence operator(x,y,z)
vector
unit vector
* yconvolutiony(t) = x(t) * h(t)
Laplace transformF(s) = {(t)}
Fourier transformX(ω) = {(t)}
δdelta function

Numeral symbols

NameEuropeanRomanHindu ArabicHebrew
zero0٠
one1I١א
two2II٢ב
three3III٣ג
four4IV٤ד
five5V٥ה
six6VI٦ו
seven7VII٧ז
eight8VIII٨ח
nine9IX٩ט
ten10X١٠י
eleven11XI١١יא
twelve12XII١٢יב
thirteen13XIII١٣יג
fourteen14XIV١٤יד
fifteen15XV١٥טו
sixteen16XVI١٦טז
seventeen17XVII١٧יז
eighteen18XVIII١٨יח
nineteen19XIX١٩יט
twenty20XX٢٠כ
thirty30XXX٣٠ל
fourty40XL٤٠מ
fifty50L٥٠נ
sixty60LX٦٠ס
seventy70LXX٧٠ע
eighty80LXXX٨٠פ
ninety90XC٩٠צ
one hundred100C١٠٠ק

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

Roman numerals

NumberRoman numeral
1I
2II
3III
4IV
5V
6VI
7VII
8VIII
9IX
10X
11XI
12XII
13XIII
14XIV
15XV
16XVI
17XVII
18XVIII
19XIX
20XX
30XXX
40XL
50L
60LX
70LXX
80LXXX
90XC
100C
200CC
300CCC
400CD
500D
600DC
700DCC
800DCCC
900CM
1000M
5000V
10000X
50000L
100000C
500000D
1000000M


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Segue alguns símbolos, caso necessitem utilizá-los:
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α β γ δ ∆ λ μ Ω ο ρ φ χ ψ ξ ε η θ π ∂ ∑ ∏ ℮ אօ ∞ ℝ ℕ ℚ ℤ Ø f◦g
½ ¼ ¾ ½ ⅓ ⅔ ⅛ ⅜ ⅝ ⅞ ² ³ ¹ º ª ₁ ₂ ₃ ₄ ≈ ≠ ≡ ∀ ∃ ⇒ ⇔ → ↔
∈∋∧ ∨ ⊂ ⊃ ∩ ∪ − + × ± ∓ ÷ √ ∛ ∜ ⊿∟ ∠→ ↑ ↓ ↕ ← ≤ ≥
outros
√ ∇ ∂ ∑ ∏ ∫ ≠ ≤ ≥ ∼ ≈ ≅ ≡ ∝ ⇒ ⇔ ∈ ∉ ⊂ ⊃ ⊆ ⊇ \ ∩ ∪ ∧ ∨ ∀ ∃ ℜ ℑ

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