# Basic Math Symbols (Algebra Geometry Statistics etc.)

## Basic mathematical symbols

 Symbol Symbol Name Meaning / definition Example = equals sign equality 5 = 2+35 is equal to 2+3 ≠ not equal sign inequality 5 ≠ 45 is not equal to 4 ≈ approximately equal approximation sin(0.01) ≈ 0.01,x ≈ y means x is approximately equal to y > strict inequality greater than 5 > 45 is greater than 4 < strict inequality less than 4 < 54 is less than 5 ≥ inequality greater than or equal to 5 ≥ 4,x ≥ y means x is greater than or equal to y ≤ inequality less than or equal to 4 ≤ 5,x ≤ y means x is greater than or equal to y ( ) 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 ab power exponent 23 = 8 a^b caret exponent 2 ^ 3 = 8 √a square root √a · √a  = a √9 = ±3 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 % 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

## Algebra symbols

 Symbol Symbol Name Meaning / definition Example x x variable unknown value to find when 2x = 4, then x = 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 y ∝ x when y = kx, k constant ∞ 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 = t1 – t0 ∆ discriminant Δ = b2 – 4ac ∑ sigma summation – sum of all values in range of series ∑ xi= x1+x2+…+xn ∑∑ sigma double summation ∏ capital pi product – product of all values in range of series ∏ xi=x1∙x2∙…∙xn e e 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 × b A⊗B tensor product tensor product of A and B A ⊗ B inner 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 AT transpose matrix transpose (AT)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 = I rank(A) matrix rank rank of matrix A rank(A) = 3 dim(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) = 0.5 P(A ∩ B) probability of events intersection probability that of events A and B P(A∩B) = 0.5 P(A ∪ B) probability of events union probability that of events A or B P(A∪B) = 0.5 P(A | B) conditional probability function probability 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 mean mean of population values μ = 10 E(X) expectation value expected value of random variable X E(X) = 10 E(X | Y) conditional expectation expected value of random variable X given Y E(X | Y=2) = 5 var(X) variance variance of random variable X var(X) = 4 σ2 variance variance of population values σ2 = 4 std(X) standard deviation standard deviation of random variable X std(X) = 2 σX standard deviation standard deviation value of random variable X σX  = 2 median middle value of random variable x cov(X,Y) covariance covariance of random variables X and Y cov(X,Y) = 4 corr(X,Y) correlation correlation of random variables X and Y corr(X,Y) = 0.6 ρX,Y correlation correlation of random variables X and Y ρX,Y = 0.6 ∑ summation summation – sum of all values in range of series ∑∑ double summation double summation Mo mode value that occurs most frequently in population MR mid-range MR = (xmax+xmin)/2 Md sample median half the population is below this value Q1 lower / first quartile 25% of population are below this value Q2 median / second quartile 50% of population are below this value = median of samples Q3 upper / third quartile 75% of population are below this value x sample mean average / arithmetic mean x = (2+5+9) / 3 = 5.333 s 2 sample variance population samples variance estimator s 2 = 4 s sample standard deviation population samples standard deviation estimator s = 2 zx standard score zx = (x–x) / sx X ~ distribution of X distribution of random variable X X ~ N(0,3) N(μ,σ2) 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≥0 gamma(c, λ) gamma distribution f (x) = λ c xc-1e–λx / Γ(c), x≥0 χ 2(k) chi-square distribution f (x) = xk/2-1e–x/2 / ( 2k/2 Γ(k/2) ) F (k1, k2) F distribution Bin(n,p) binomial distribution f (k) = nCk pk(1-p)n-k Poisson(λ) Poisson distribution f (k) = λke–λ / k! Geom(p) geometric distribution f (k) =  p(1-p) k HG(N,K,n) hyper-geometric distribution Bern(p) Bernoulli distribution

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