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

5 is equal to 2+3≠not equal signinequality5 ≠ 4

5 is not equal to 4≈approximately equalapproximation

*sin*(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 less than or equal to

*y*( )parenthesescalculate expression inside first 2 × (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 or -2±minus - plusboth minus and plus operations3 ∓ 5 = -2 or 8*asteriskmultiplication2 * 3 = 6×times signmultiplication2 × 3 = 6⋅multiplication dotmultiplication2 ⋅ 3 = 6÷division sign / obelusdivision6 ÷ 2 = 3/division slashdivision6 / 2 = 3—horizontal linedivision / fractionmodmoduloremainder calculation7 mod 2 = 1.perioddecimal point, decimal separator2.56 = 2+56/100

*a*powerexponent2

^{b}^{3 }= 8

*a^b*caretexponent2 ^ 3

^{ }= 8√

*a*square root

√*a⋅ *√*a = a*

^{3}√

*a*cube root

^{3}√

*a⋅*

^{3}

*√a ⋅*

^{3}

*√a = a*

^{3}√8 = 2

^{4}√

*a*fourth root

^{4}√

*a⋅*

^{4}

*√a ⋅*

^{4}

*√a ⋅*

^{4}

*√a =a*

^{4}√16 = ±2

^{n}√

*a*n-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

^{-7}pptper-trillion1ppt = 10

^{-12}10ppt × 30 = 3×10

^{-10}

*x*-

*y*|distancedistance between points x and y|

*x*-

*y*| = 5πpi constant

*π*= 3.141592654...

is the ratio between the circumference and diameter of a circle

*c*=

*π*⋅

*d*= 2⋅

*π*⋅

*r*radradiansradians angle unit360° = 2π rad

^{c}radiansradians angle unit360° = 2π

^{c}gradgradians / gonsgrads angle unit360° = 400 grad

^{g}gradians / gonsgrads angle unit360° = 400

^{g}

*x*x variableunknown value to findwhen 2

*x*= 4, then

*x*= 2≡equivalenceidentical to≜equal by definitionequal by definition:=equal by definitionequal by definition~approximately equalweak approximation11 ~ 10≈approximately equalapproximation

*sin*(0.01) ≈ 0.01∝proportional toproportional to

*y* ∝ *x* when *y* = *kx, k* constant

*x*⌋floor bracketsrounds number to lower integer⌊4.3⌋ = 4⌈

*x*⌉ceiling bracketsrounds number to upper integer⌈4.3⌉ = 5

*x*!exclamation markfactorial4! = 1*2*3*4 = 24|

*x*|vertical barsabsolute value| -5 | = 5

*f*(

*x*)function of xmaps values of x to f(x)

*f*(

*x*) = 3

*x*+5(

*f*∘

*g*)function composition(

*f*∘

*g*) (

*x*) =

*f*(

*g*(

*x*))

*f*(

*x*)=3

*x*,

*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*=

*t*

_{1 }-

*t*

_{0}∆discriminantΔ =

*b*

^{2}- 4

*ac*∑sigmasummation - sum of all values in range of series∑

*x*

_{i}= x_{1}

*+x*

_{2}

*+...+x*

_{n}∑∑sigmadouble summation∏capital piproduct - product of all values in range of series∏

*x*

_{i}=x_{1}

*∙x*

_{2}

*∙...∙x*

_{n}

*e*e constant/ Euler's number

*e*= 2.718281828...

*e*= lim (1+1/

*x*)

*,*

^{x}*x*→∞γEuler-Mascheroni constantγ = 0.5772156649...φgolden ratiogolden ratio constantπpi constant

*π*= 3.141592654...

is the ratio between the circumference and diameter of a circle

*c*=

*π*⋅

*d*= 2⋅

*π*⋅

*r*

*a*·

*b*×crossvector product

*a*×

*b*

*A*⊗

*B*tensor producttensor product of A and B

*A*⊗

*B*inner product[ ]bracketsmatrix of numbers( )parenthesesmatrix of numbers|

*A*|determinantdeterminant of matrix Adet(

*A*)determinantdeterminant of matrix A ||

*x*||double vertical barsnorm

*A*

^{T}transposematrix 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*

^{-1}inverse matrix

*A A*

^{-1}=

*I*rank(

*A*)matrix rankrank of matrix Arank(

*A*) = 3dim(

*U*)dimensiondimension of matrix Adim(

*U*) = 3

*P*(

*A*)probability functionprobability of event A

*P*(

*A*) = 0.5

*P*(

*A*⋂

*B*)probability of events intersectionprobability that of events A and B

*P*(

*A*⋂

*B*) = 0.5

*P*(

*A*⋃

*B*)probability of events unionprobability that of events A or B

*P*(

*A*⋃

*B*) = 0.5

*P*(

*A*|

*B*)conditional probability functionprobability of event A given event B occured

*P*(

*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*)

*μ*population meanmean of population values

*μ*= 10

*E*(

*X*)expectation valueexpected value of random variable X

*E*(

*X*) = 10

*E*(

*X | Y*)conditional expectationexpected value of random variable X given Y

*E*(

*X | Y=2*) = 5

*var*(

*X*)variancevariance of random variable X

*var*(

*X*) = 4σ

*variancevariance of population valuesσ*

^{2}*= 4*

^{2 }*std*(

*X*)standard deviationstandard deviation of random variable X

*std*(

*X*) = 2σ

*standard deviationstandard deviation value of random variable Xσ*

_{X}*=*

_{X}_{ }*2medianmiddle value of random variable x*

*cov*(

*X*,

*Y*)covariancecovariance of random variables X and Y

*cov*(

*X,Y*) = 4

*corr*(

*X*,

*Y*)correlationcorrelation of random variables X and Y

*corr*(

*X,Y*) = 0.6

*ρ*

_{X,Y}correlationcorrelation of random variables X and Y

*ρ*

_{X,Y}= 0.6∑summationsummation - sum of all values in range of series∑∑double summationdouble summation

*Mo*modevalue that occurs most frequently in population

*MR*mid-range

*MR*= (

*x*+

_{max}*x*)/2

_{min}*Md*sample medianhalf the population is below this valueQ

_{1}lower / first quartile25% of population are below this valueQ

_{2}median / second quartile50% of population are below this value = median of samplesQ

_{3}upper / third quartile75% of population are below this value

*x*sample meanaverage / arithmetic mean

*x*= (2+5+9) / 3 = 5.333

*s*

_{ }^{2}sample variancepopulation samples variance estimator

*s*

^{ }^{2}= 4

*s*sample standard deviation population samples standard deviation estimator

*s*= 2

*z*standard score

_{x}*z*= (

_{x}*x*-x)/

*s*

_{x}*X*~distribution of Xdistribution of random variable X

*X*~

*N*(0,3)

*N*(

*μ*,

*σ*

^{2})normal distributiongaussian distribution

*X*~

*N*(0,3)

*U*(

*a*,

*b*)uniform distributionequal probability in range a,b

*X*~

*U*(0,3)

*exp*(λ)exponential distribution

*f*(

*x*)

*= λe*

^{-λx},

*x*≥0

*gamma*(

*c*, λ)gamma distribution

*f*(

*x*)

*= λ c x*

^{c-1}

*e*

^{-λx}/ Γ(

*c*),

*x*≥0χ

^{ 2}(

*k*)chi-square distribution

*f*(

*x*)

*= x*

^{k}^{/2-1}

*e*

^{-x/2}/ ( 2

^{k/2 }Γ(

*k*/2) )

*F*(

*k*

_{1}

*, k*

_{2})F distribution

*Bin*(

*n*,

*p*)binomial distribution

*f*(

*k*)

*=*(1

_{n}C_{k}p^{k}*-p*)

^{n-k}*Poisson*(λ)Poisson distribution

*f*(

*k*)

*= λ*

^{k}e^{-λ}/

*k*!

*Geom*(

*p*)geometric distribution

*f*(

*k*)

*= p*(1

*-p*)

^{ k}*HG*(

*N*,

*K*,

*n*)hyper-geometric distribution

*Bern*(

*p*)Bernoulli distribution

*n*!factorial

*n*! = 1⋅2⋅3⋅...⋅

*n*5! = 1⋅2⋅3⋅4⋅5 = 120

*permutation*

_{n}P_{k}_{5}

*P*

_{3}

*=*5! / (5-3)! = 60

*combination*

_{n}C_{k}_{5}

*C*

_{3}

*=*5!/[3!(5-3)!]=10

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 ⊆ BsubsetA is a subset of B. set A is included in set B.{9,14,28} ⊆ {9,14,28}A ⊂ Bproper subset / strict subsetA is a subset of B, but A is not equal to B.{9,14} ⊂ {9,14,28}A ⊄ Bnot subsetset A is not a subset of set B{9,66} ⊄ {9,14,28}A ⊇ BsupersetA is a superset of B. set A includes set B{9,14,28} ⊇ {9,14,28}A ⊃ Bproper superset / strict supersetA is a superset of B, but B is not equal to A.{9,14,28} ⊃ {9,14}A ⊅ Bnot supersetset A is not a superset of set B{9,14,28} ⊅ {9,66}2

^{A}power setall subsets of Apower setall subsets of AA = Bequalityboth sets have the same membersA={3,9,14},

B={3,9,14},

A=BA

^{c}complementall the objects that do not belong to set AA \ 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 totheir 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 of,

belongs toset membershipA={3,9,14}, 3 ∈ A

*x*∉Anot element ofno set membershipA={3,9,14}, 1 ∉ A(

*a*,

*b*)ordered paircollection of 2 elementsA×Bcartesian productset of all ordered pairs from A and BA×B = {(

*a*,

*b*)|

*a*∈A ,

*b*∈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|vertical barsuch thatA={x|3<x<14}aleph-nullinfinite cardinality of natural numbers setaleph-onecardinality of countable ordinal numbers setØempty setØ = { }C = {Ø}universal setset of all possible values

_{0}natural numbers / whole numbers set (with zero)

_{0}= {0,1,2,3,4,...}0 ∈

_{0}

_{1}natural 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+2

*i*∈

**⋅**andand

*x*

**⋅**

*y*^caret / circumflexand

*x*^

*y*&ersandand

*x*&

*y*+plusor

*x*+

*y*∨reversed caretor

*x*∨

*y*|vertical lineor

*x*|

*y*

*x*'single quotenot - negation

*x*'

*x*barnot - negationx¬notnot - negation¬

*x*!exclamation marknot - negation!

*x*⊕circled plus / oplusexclusive or - xor

*x*⊕

*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

*ε*epsilonrepresents a very small number, near zero

*ε*→

*0*

*e*e constant / Euler's number

*e*= 2.718281828...

*e*= lim (1+1/

*x*)

*,*

^{x}*x*→∞

*y*'derivativederivative - Lagrange's notation(3

*x*

^{3})' = 9

*x*

^{2}

*y*''second derivativederivative of derivative(3

*x*

^{3})'' = 18

*x*

*y*

^{(n)}nth derivativen times derivation(3

*x*

^{3})

^{(3)}= 18derivativederivative - Leibniz's notation

*d*(3

*x*

^{3})/

*dx*= 9

*x*

^{2}second derivativederivative of derivative

*d*

^{2}(3

*x*

^{3})/

*dx*

^{2}= 18

*x*nth derivativen times derivationtime derivativederivative by time - Newton's notationtime second derivativederivative of derivative

*D*derivativederivative - Euler's notation

_{x }y*D*

_{x}^{2}

*y*second derivativederivative of derivativepartial derivative∂(

*x*

^{2}+

*y*

^{2})/∂

*x*= 2

*x*∫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*}

*i*imaginary unit

*i*≡ √-1

*z*= 3 + 2

*i*

*z**complex conjugate

*z*=

*a*+

*bi*→

*z**=

*a*-

*bi*

*z**= 3 - 2

*i*

*z*complex conjugate

*z*=

*a*+

*bi*→

*z*=

*a*-

*bi*

*z*= 3 - 2

*i*Re(

*z*)real part of a complex number

*z*=

*a*+

*bi*→ Re(

*z*)=

*a*Re(3 - 2

*i*) = 3Im(

*z*)imaginary part of a complex number

*z*=

*a*+

*bi*→ Im(

*z*)=

*b*Im(3 - 2

*i*) = -2|

*z*|absolute value/magnitude of a complex number|

*z*| = |

*a*+

*bi*| = √(

*a*

^{2}+

*b*

^{2})|3 - 2

*i*| = √13arg(

*z*)argument of a complex numberThe angle of the radius in the complex planearg(3 + 2

*i*) = 33.7°∇nabla / delgradient / divergence operator∇

*f*(

*x*,

*y*,

*z*)vectorunit vector

*x**

*y*convolution

*y*(

*t*) =

*x*(

*t*) *

*h*(

*t*)Laplace transform

*F*(

*s*) = {

*f*(

*t*)}Fourier transform

*X*(

*ω*) = {

*f*(

*t*)}

*δ*delta function∞lemniscateinfinity symbol