## 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

Symbol Symbol Name Meaning / definition Example = equals sign equality 5 = 2+3 ≠ not equal sign inequality 5 ≠ 4 > strict inequality greater than 5 > 4 < strict inequality less than 4 < 5 ≥ inequality greater than or equal to 5 ≥ 4 ≤ inequality less than or equal to 4 ≤ 5 ( ) parentheses calculate expression inside first 2 × (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 + plus sign addition 1 + 1 = 2 − minus sign subtraction 2 − 1 = 1 ± plus - minus both plus and minus operations 3 ± 5 = 8 and -2 ∓ minus - plus both minus and plus operations 3 ∓ 5 = -2 and 8 * asterisk multiplication 2 * 3 = 6 × times sign multiplication 2 × 3 = 6 ∙ multiplication dot multiplication 2 ∙ 3 = 6 ÷ division sign / obelus division 6 ÷ 2 = 3 / division slash division 6 / 2 = 3 – horizontal line division / fraction mod modulo remainder calculation 7 mod 2 = 1 . period decimal point, decimal separator 2.56 = 2+56/100 a^{b}power exponent 2 ^{3 }= 8a^bcaret exponent 2 ^ 3 ^{ }= 8√ asquare root √

a ·√a = a√9 = ±3 ^{ 3}√acube root ^{ 3}√a ·^{ 3}√a ·^{ 3}√a = a^{ 3}√8 = 2^{ 4}√afourth root ^{ 4}√a ·^{ 4}√a ·^{ 4}√a ·^{ 4}√a = a^{ 4}√16 = ±2^{n}√an-th root (radical) for n=3,^{n}√8 = 2% percent 1% = 1/100 10% × 30 = 3 ‰ per-mille 1‰ = 1/1000 = 0.1% 10‰ × 30 = 0.3 ppm per-million 1ppm = 1/1000000 10ppm × 30 = 0.0003 ppb per-billion 1ppb = 1/1000000000 10ppb × 30 = 3×10 ^{-7}ppt per-trillion 1ppt = 10 ^{-12}10ppt × 30 = 3×10 ^{-10}## Geometry symbols

Symbol Symbol Name Meaning / definition Example ∠ angle formed by two rays ∠ABC = 30º measured angle ABC = 30º spherical angle AOB = 30º ∟ right angle = 90º α = 90º º degree 1 turn = 360º α = 60º ´ arcminute 1º = 60´ α = 60º59' ´´ arcsecond 1´ = 60´´ α = 60º59'59'' line infinite line AB line segment line from point A to point B ray line that start from point A arc arc from point A to point B = 60º |perpendicular perpendicular lines (90º angle) AC |BC|| parallel parallel lines AB || CD ≅ congruent to equivalence of geometric shapes and size ∆ABC ≅ ∆XYZ ~ similarity same shapes, not same size ∆ABC ~ ∆XYZ Δ triangle triangle shape ΔABC ≅ ΔBCD | x-y|distance distance between points x and y | x-y| = 5π pi constant π= 3.141592654...is the ratio between the circumference and diameter of a circle

c=π·d= 2·π·rrad radians radians angle unit 360º = 2π rad grad grads grads angle unit 360º = 400 grad ## Algebra symbols

Symbol Symbol Name Meaning / definition Example xx variable unknown value to find when 2 x= 4, thenx= 2≡ equivalence identical to ≜ equal by definition equal by definition := equal by definition equal by definition ~ approximately equal weak approximation 11 ~ 10 ≈ approximately equal approximation sin(0.01) ≈ 0.01∝ proportional to proportional to

f(x) ∝g(x)∞ lemniscate infinity symbol ≪ much less than much less than 1 ≪ 1000000 ≫ much greater than much greater than 1000000 ≫ 1 ( ) parentheses calculate expression inside first 2 * (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 { } braces set ⌊ x⌋floor brackets rounds number to lower integer ⌊4.3⌋= 4 ⌈ x⌉ceiling brackets rounds number to upper integer ⌈4.3⌉= 5 x!exclamation mark factorial 4! = 1*2*3*4 = 24 | x|single vertical bar absolute value | -5 | = 5 f(x)function of x maps 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]∆ delta change / difference ∆ t=t_{1 }-t_{0}∆ discriminant Δ = b^{2}- 4ac∑ sigma summation - sum of all values in range of series ∑ x_{i}= x_{1}+x_{2}+...+x_{n}∑∑ sigma double summation ∏ capital pi product - product of all values in range of series ∏ x_{i}=x_{1}∙x_{2}∙...∙x_{n}ee constant / Euler's number e= 2.718281828...e= lim (1+1/x),^{x}x→∞γ Euler-Mascheroni constant γ = 0.527721566... φ golden ratio golden ratio constant π pi constant π= 3.141592654...is the ratio between the circumference and diameter of a circle

c=π·d= 2·π·r## Linear Algebra Symbols

Symbol Symbol Name Meaning / definition Example ∙ dot scalar product a∙b× cross vector product a×bA⊗Btensor product tensor product of A and B A⊗Binner product [ ] brackets matrix of numbers ( ) parentheses matrix of numbers | A|determinant determinant of matrix A det( A)determinant determinant of matrix A || x||double vertical bars norm A^{ T}transpose matrix transpose (

A^{T})= (_{ij}A)_{ji}A^{ †}Hermitian matrix matrix conjugate transpose (

A^{†})= (_{ij}A)_{ji}A^{ *}Hermitian matrix matrix conjugate transpose (

A^{*})= (_{ij}A)_{ji}A^{ -1}inverse matrix A A^{-1}=Irank( A)matrix rank rank of matrix A rank(

A) = 3dim( U)dimension dimension of matrix A rank(

U) = 3## Probability and statistics symbols

Symbol Symbol Name Meaning / definition Example P(A)probability function probability of event A P(A) = 0.5P(A∩B)probability of events intersection probability that of events A and B P(A∩B) = 0.5P(A∪B)probability of events union probability that of events A or B P(A∪B) = 0.5P(A|B)conditional probability function probability of event A given event B occured P(A | B) = 0.3f(x)probability density function (pdf) P(a≤x≤b) =∫ f(x)dxF(x)cumulative distribution function (cdf) F(x) =P(X≤x)μpopulation mean mean of population values μ= 10E(X)expectation value expected value of random variable X E(X) = 10E(X | Y)conditional expectation expected value of random variable X given Y E(X | Y=2) = 5var(X)variance variance of random variable X var(X) = 4σ ^{2}variance variance of population values σ = 4^{2 }std(X)standard deviation standard deviation of random variable X std(X) = 2σ _{X}standard deviation standard deviation value of random variable X σ =_{X}_{ }2median middle value of random variable x cov(X,Y)covariance covariance of random variables X and Y cov(X,Y) = 4corr(X,Y)correlation correlation of random variables X and Y corr(X,Y) = 0.6ρ_{X,Y}correlation correlation of random variables X and Y ρ_{X,Y}= 0.6∑ summation summation - sum of all values in range of series ∑∑ double summation double summation Momode value that occurs most frequently in population MRmid-range

MR= (x+_{max}x)/2_{min}Mdsample median half the population is below this value Q _{1}lower / first quartile 25% of population are below this value Q _{2}median / second quartile 50% of population are below this value = median of samples Q _{3}upper / third quartile 75% of population are below this value xsample mean average / arithmetic mean x= (2+5+9) / 3 = 5.333s_{ }^{2}sample variance population samples variance estimator s^{ }^{ 2}= 4ssample standard deviation population samples standard deviation estimator s= 2z_{x}standard score

z= (_{x}x-x) /s_{x}X~distribution of X distribution of random variable X X~N(0,3)N(μ,σ^{2})normal distribution gaussian distribution X~N(0,3)U(a,b)uniform distribution equal probability in range a,b X~U(0,3)exp(λ)exponential distribution f(x)= λe^{-λx},x≥0gamma(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 ## Combinatorics Symbols

Symbol Symbol Name Meaning / definition Example n!factorial n! = 1·2·3·...·n5! = 1·2·3·4·5 = 120 _{n}P_{k}permutation _{5}P_{3}=5! / (5-3)! = 60_{n}C_{k}

combination _{5}C_{3}=5!/[3!(5-3)!]=10## Set theory symbols

Symbol Symbol Name Meaning / definition Example { } set a collection of elements A = {3,7,9,14},

B = {9,14,28}A ∩ B intersection objects that belong to set A and set B A ∩ B = {9,14} A ∪ B union objects that belong to set A or set B A ∪ B = {3,7,9,14,28} A ⊆ B subset subset has fewer elements or equal to the set {9,14,28} ⊆ {9,14,28} A ⊂ B proper subset / strict subset subset has fewer elements than the set {9,14} ⊂ {9,14,28} A ⊄ B not subset left set not a subset of right set {9,66} ⊄ {9,14,28} A ⊇ B superset set A has more elements or equal to the set B {9,14,28} ⊇ {9,14,28} A ⊃ B proper superset / strict superset set A has more elements than set B {9,14,28} ⊃ {9,14} A ⊅ B not superset set A is not a superset of set B {9,14,28} ⊅ {9,66} 2 ^{A}power set all subsets of A power set all subsets of A A = B equality both sets have the same members A={3,9,14},

B={3,9,14},

A=BA ^{c}complement all the objects that do not belong to set A A \ B relative complement objects that belong to A and not to B A = {3,9,14},

B = {1,2,3},

A-B = {9,14}A - B relative complement objects that belong to A and not to B A = {3,9,14},

B = {1,2,3},

A-B = {9,14}A ∆ B symmetric difference objects that belong to A or B but not to their intersection A = {3,9,14},

B = {1,2,3},

A ∆ B = {1,2,9,14}A ⊖ B symmetric difference objects that belong to A or B but not to their intersection A = {3,9,14},

B = {1,2,3},

A ⊖ B = {1,2,9,14}a∈Aelement of set membership A={3,9,14}, 3 ∈ A x∉Anot element of no set membership A={3,9,14}, 1 ∉ A ( a,b)ordered pair collection of 2 elements A×B cartesian product set of all ordered pairs from A and B |A| cardinality the number of elements of set A A={3,9,14}, |A|=3 #A cardinality the number of elements of set A A={3,9,14}, #A=3 aleph-null infinite cardinality of natural numbers set aleph-one cardinality of countable ordinal numbers set Ø empty set Ø = { } C = {Ø} universal set set 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∈## Logic symbols

Symbol Symbol Name Meaning / definition Example ·and and x·y^ caret / circumflex and x^y& ampersand and x&y+ plus or x+y∨ reversed caret or x∨y| vertical line or x|yx'single quote not - negation x'xbar not - negation x ¬ not not - negation ¬ x! exclamation mark not - negation ! x⊕ circled plus / oplus exclusive or - xor x⊕y~ tilde negation ~ x⇒ implies ⇔ equivalent if and only if (iff) ↔ equivalent if and only if (iff) ∀ for all ∃ there exists ∄ there does not exists ∴ therefore ∵ because / since ## Calculus & analysis symbols

Symbol Symbol Name Meaning / definition Example limit limit value of a function εepsilon represents a very small number, near zero ε→0ee constant / Euler's number e= 2.718281828...e= lim (1+1/x),^{x}x→∞y'derivative derivative - Leibniz's notation (3 x^{3})' = 9x^{2}y''second derivative derivative of derivative (3 x^{3})'' = 18xy^{(n)}nth derivative n times derivation (3 x^{3})^{(3)}= 18derivative derivative - Lagrange's notation d(3x^{3})/dx= 9x^{2}second derivative derivative of derivative d^{2}(3x^{3})/dx^{2}= 18xnth derivative n times derivation time derivative derivative by time - Newton notation time second derivative derivative of derivative partial derivative ∂( x^{2}+y^{2})/∂x= 2x∫ integral opposite to derivation ∬ double integral integration of function of 2 variables ∭ triple integral integration 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 unit i≡ √-1z= 3 + 2iz*complex conjugate z=a+bi→z*=a-biz*= 3 + 2izcomplex conjugate z=a+bi→z=a-biz= 3 + 2i∇ nabla / del gradient / divergence operator ∇ f(x,y,z)vector unit vector x*yconvolution y(t) =x(t) *h(t)Laplace transform F(s) = {f(t)}Fourier transform X(ω) = {f(t)}δdelta function ∞ lemniscate infinity symbol ## Numeral symbols

Name European Roman Hindu Arabic Hebrew zero 0 ٠ one 1 I ١ א two 2 II ٢ ב three 3 III ٣ ג four 4 IV ٤ ד five 5 V ٥ ה six 6 VI ٦ ו seven 7 VII ٧ ז eight 8 VIII ٨ ח nine 9 IX ٩ ט ten 10 X ١٠ י eleven 11 XI ١١ יא twelve 12 XII ١٢ יב thirteen 13 XIII ١٣ יג fourteen 14 XIV ١٤ יד fifteen 15 XV ١٥ טו sixteen 16 XVI ١٦ טז seventeen 17 XVII ١٧ יז eighteen 18 XVIII ١٨ יח nineteen 19 XIX ١٩ יט twenty 20 XX ٢٠ כ thirty 30 XXX ٣٠ ל fourty 40 XL ٤٠ מ fifty 50 L ٥٠ נ sixty 60 LX ٦٠ ס seventy 70 LXX ٧٠ ע eighty 80 LXXX ٨٠ פ ninety 90 XC ٩٠ צ one hundred 100 C ١٠٠ ק

## Greek alphabet letters

Greek Symbol Greek Letter Name English Equivalent Pronunciation Upper Case Lower Case Α α Alpha a al-fa Β β Beta b be-ta Γ γ Gamma g ga-ma Δ δ Delta d del-ta Ε ε Epsilon e ep-si-lon Ζ ζ Zeta z ze-ta Η η Eta h eh-ta Θ θ Theta th te-ta Ι ι Iota i io-ta Κ κ Kappa k ka-pa Λ λ Lambda l lam-da Μ μ Mu m m-yoo Ν ν Nu n noo Ξ ξ Xi x x-ee Ο ο Omicron o o-mee-c-ron Π π Pi p pa-yee Ρ ρ Rho r row Σ σ Sigma s sig-ma Τ τ Tau t ta-oo Υ υ Upsilon u oo-psi-lon Φ φ Phi ph f-ee Χ χ Chi ch kh-ee Ψ ψ Psi ps p-see Ω ω Omega o o-me-ga ## Roman numerals

Number Roman numeral 0 not defined 1 I 2 II 3 III 4 IV 5 V 6 VI 7 VII 8 VIII 9 IX 10 X 11 XI 12 XII 13 XIII 14 XIV 15 XV 16 XVI 17 XVII 18 XVIII 19 XIX 20 XX 30 XXX 40 XL 50 L 60 LX 70 LXX 80 LXXX 90 XC 100 C 200 CC 300 CCC 400 CD 500 D 600 DC 700 DCC 800 DCCC 900 CM 1000 M 5000 V 10000 X 50000 L 100000 C 500000 D 1000000 M

## See also