Basic mathematical symbols
| Symbol | Symbol Name | Meaning / definition | Example |
| = | equals sign | equality | 5 = 2+3 5 is equal to 2+3 |
| ≠ | not equal sign | inequality | 5 ≠ 4 5 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 > 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 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 |
Geometry symbols
| Symbol | Symbol Name | Meaning / definition | Example |
| ∠ | angle | formed by two rays | ∠ABC = 30° |
| measured angle | ABC = 30° | ||
| 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 | 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 |
| |x–y| | distance | distance between points x and y | | x–y | = 5 |
| π | pi constant | π = 3.141592654…
is the ratio between the circumference and diameter of a circle | c = π·d = 2·π·r |
| rad | radians | radians angle unit | 360° = 2π rad |
| c | radians | radians angle 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 | 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 |