Basic math symbols Symbol Name Meaning / definition Example = equality 5 = 2+3 5 is equal to 2+3 ≠ not equal sign inequality 5 ≠ 4 5 is not equal to 4 ≈ equal approximation sin(0.01) ≈ 0.01, x ≈ y means x is approximately equal to y > strict inequality greater than 5 > 4 5 is greater than 4 < strict inequality less than 4 < 5 4 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 less than or equal to y ( ) calculate expression inside first 2 × (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)×(1+5)] = 18 + addition 1 + 1 = 2 − subtraction 2 − 1 = 1 ± plus - minus both plus and minus operations 3 ± 5 = 8 and -2 s l m
± minus - plus both minus and plus operations 3 ∓ 5 = -2 and 8 * multiplication 2 * 3 = 6 × multiplication 2 × 3 = 6 ⋅ multiplication 2 ⋅ 3 = 6 ÷ division 6 ÷ 2 = 3 / division 6 / 2 = 3 — division / fraction modulo remainder calculation 7 mod 2 = 1 . period decimal point, decimal separator 2.56 = 2+56/100 power exponent 23 = 8 caret exponent 2 ^ 3 = 8 square root √a √a = a √9 = ±3 b d o
cube root 3√a 3√a 3√a = a 3√8 = 2 fourth root 4√a 4√a 4√a = a 4√16 = ±2 n-th root (radical) for n=3, n√8 = 2 % 1% = 1/100 10% × 30 = 3 ‰ 1‰ = 1/1000 = 0.1% 10‰ × 30 = 0.3 1ppm = 1/1000000 10ppm × 30 = 0.0003 per-billion 1ppb = 1/1000000000 10ppb × 30 = 3×10-7 per-trillion 1ppt = 10-12 10ppt × 30 = 3×10-10 Geometry symbols Symbol Symbol Name Meaning / definition Example ∠ angle formed by two rays ∠ABC = 30° measured angle ABC = 30° a n p
spherical angle AOB = 30° ∟ right angle = 90° α = 90° ° degree 1 turn = 360° α = 60° deg degree 1 turn = 360deg α = 60deg ′ prime arcminute, 1° = 60′ α = 60°59′ ″ double prime arcsecond, 1′ = 60″ α = 60°59′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 = 60° ⊥ perpendicular lines (90° 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 distance distance between points x and y | x-y | = 5 π pi constant π = 3.141592654... c = π⋅d = 2⋅π⋅r r
is the ratio between the circumference and diameter of a circle rad radians radians angle unit 360° = 2π rad c radians radians unit 360° = 2π c grad gradians / gons grads angle unit 360° = 400 grad g gradians / gons grads angle unit 360° = 400 g Algebra symbols 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 ~ weak approximation 11 ~ 10 ≈ equal approximation sin(0.01) ≈ 0.01 ∝ proportional to proportional to y ∝ x when y = kx, kconstant ∞ ≪ 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 4! = 1*2*3*4 = 24 | x | single vertical bar absolute value | -5 | = 5 f 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 ⇒(f ∘ g)(x)=3(x-1) open interval (a,b) = {x | < x < b} x∈ (2,6) closed interval [a,b] = {x | a ≤ x ≤ b} x ∈ [2,6] x)
∆ change / difference ∆t = t1 - t0 ∆ discriminant Δ = b2 - 4ac ∑ 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 / Euler's number e = 2.718281828... e = lim (1+1/x)x , x→∞ γ γ = 0.5772156649... φ golden ratio golden ratio constant π pi constant π = 3.141592654... is the 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 · × cross vector product a × b A⊗B tensor product tensor product of A and B A ⊗ B inner product
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≤ μ population mean mean of population values μ = 10 E(X) expected value of random variable X E(X) = 10 E(X | Y) conditional expectation expected of random variable X given Y E(X | Y=2) = 5 var(X) variance of random variable X var(X) = 4 σ2 variance of population values σ2 = 4 std(X) standard deviation of random variable X std(X) = 2 σX standard deviation value of random variable X σX = 2 median middle of random variable x covariance covariance of random variables X and Y cov(X,Y) = 4 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 that occurs most frequently in population MR 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 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 ~ of X distribution of random variable X X ~ N(0,3) N(μ,σ2) 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
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 Combinatorics Symbols Symbol Symbol Name Meaning / definition Example n! n! = 1⋅2⋅3⋅...⋅n 5! = 1⋅2⋅3⋅4⋅5 = 120 nPk permutation 5P3 = 5! / (5-3)! = 60 nCk combination 5C3 = 5!/[3!(5-3)!]=10 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 | y x' single not - negation x' x bar 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