Comprehensive List of Mathematical Symbols Comprehensive List of Mathematical Symbols Comprehensive List of Mathematical Symbols For the corresponding web guides, see Mathematical Symbols. Table of Contents 1 Constant ......................... 3 1.1 Key Mathematical Numbers ........... 3 1.2 Key Mathematical Sets ............. 4 1.3 Key Mathematical Infinities ........... 5 1.4 Other Key Mathematical Objects ........ 6 2 Variables ......................... 6 2.1 Variables for Numbers .............. 6 2.2 Variables in Geometry ............. 7 2.3 Variables in Calculus .............. 7 2.4 Variables in Linear Algebra ........... 8 2.5 Variables in Set Theory and Logic ....... 8 2.6 Variables in Probability and Statistics ...... 9 3 Delimiters ........................ 10 3.1 Common Delimiters ............... 10 3.2 Other Delimiters ................ 10 4 Operators ......................... 11 2 Table of Contents Comprehensive List of Mathematical Symbols 4.1 Common Operators ............... 11 4.2 Function-related Operators ........... 12 4.3 Elementary Functions .............. 12 4.4 Algebra-related Operators ............ 13 4.5 Geometry-related Operators .......... 14 4.6 Logic-related Operators ............. 15 4.7 Set-related Operators .............. 16 4.8 Vector-related Operators ............ 16 4.9 Matrix-related Operators ............ 17 4.10 Probability-related Operators .......... 18 4.11 Statistics-related Operators ........... 18 4.12 Key Probability Functions and Distributions .. 19 4.13 Calculus-related Operators ........... 20 5 Relational Symbols .................... 21 5.1 Equality-based Relational Symbols ....... 21 5.2 Comparison-based Relational Symbols ..... 21 5.3 Number-related Relational Symbols ...... 22 5.4 Geometry-related Relational Symbols ..... 22 5.5 Set-related Relational Symbols ......... 22 5.6 Logic-related Relational Symbols ........ 23 5.7 Probability-related Relational Symbols ..... 23 5.8 Calculus-related Relational Symbols ...... 24 6 Notational Symbols ................... 24 6.1 Common Notational Symbols .......... 24 6.2 Notational Symbols in Geometry and Trigonom- etry ........................ 25 6.3 Notational Symbols in Calculus ......... 25 6.4 Notational Symbols in Probability and Statistics 26 7 Additional Resources .................. 26 1 Constant 1.1 Key Mathematical Numbers 1.1 Key Mathematical Numbers 3 Comprehensive List of Mathematical Symbols Symbols LaTeX Code Example (Explanation) 0 $0$ 3 + 0 = 3 (Zero, additive identity) 1 $1$ 5 × 1 = 5 (One, multiplicative identity) √ √ √ 2 $\sqrt{2}$ ( 2 + 1)2 = 3 + 2 2 (Square root of 2) e $e$ ln(e2) = 2 (Euler’s constant) 2 π 1 1 ··· π $\pi$ = 2 + 2 + (Pi, Archimedes’ 6 1 2 constant) √ 1 + 5 φ $\varphi$ φ = (Phi, golden ratio) 2 i $i$ (1 + i)2 = 2i (Imaginary unit) 1.2 Key Mathematical Sets Symbols LaTeX Code Example (Explanation) ? $\varnothing$ |?| = 0 (Empty set) N $\mathbb{N}$ ∀x, y ∈ N, x + y ∈ N (Set of natural numbers) Z $\mathbb{Z}$ N ⊆ Z (Set of integers) 4 1.2 Key Mathematical Sets Comprehensive List of Mathematical Symbols Z+ $\mathbb{Z}_+$ 3 ∈ Z+ (Set of positive integers) √ Q $\mathbb{Q}$ 2 ∈/ Q (Set of rational numbers) R $\mathbb{R}$ ∀x ∈ R (x2 ≥ 0) (Set of real numbers) R+ $\mathbb{R}_+$ ∀x, y ∈ R+ (xy ∈ R+) (Set of positive real numbers) C $\mathbb{C}$ ∃z ∈ C (z2 + 1 = 0) (Set of complex numbers) Zn $\mathbb{Z}_n$ In the world of Z2, (Set of integer modulo 1 + 1 = 0. n) R3 $\mathbb{R}^3$ (5, 1, 2) ∈ R3 (Three-dimensional Euclidean space) 1.3 Key Mathematical Infinities Symbols LaTeX Code Example (Explanation) ℵ0 $\aleph_0$ ℵ0 + 5 = ℵ0 (Cardinality of natural numbers) ℵ c $\mathfrak{c}$ c = 2 0 (Cardinality of real numbers) ω $\omega$ ∀n ∈ N (n < ω) (Smallest infinite ordinal number) 1.3 Key Mathematical Infinities 5 Comprehensive List of Mathematical Symbols 1.4 Other Key Mathematical Objects Symbols LaTeX Code Example (Explanation) 0 $\mathbf{0}$ ∀v ∈ V, v + 0 = v (Zero vector) e $e$ e ◦ e = e (Identity element of a group) I $I$ AI = IA = A (Identity matrix) Z C $C$ 1 dx = x + C (Constant of integration) > $\top$ For each proposition P , (Tautology) P ∧ > ≡ P . ⊥ $\bot$ For each proposition P , (Contradiction) P ∧ ¬P ≡ ⊥. Z $Z$ Z ∼ N(0, 1) (Standard normal distribution) 2 Variables 2.1 Variables for Numbers Symbols LaTeX Code Example (Explanation) m, n, p, q $m$, $n$, $p$, $q$ m + n − q = 1 (Integers and natural numbers) 6 2.1 Variables for Numbers Comprehensive List of Mathematical Symbols a, b, c $a$, $b$, $c$ ax + by = 0 (Coefficients for functions and equations) x, y, z $x$, $y$, $z$ If 2x + 5 = 3, then (Unknowns in x = −1. functions and equations) ∆ $\Delta$ ∆ = b2 − 4ac for (Discriminant) quadratic polynomials X10 i, j, k $i$, $j$, $k$ i = 55 (Index variables) i=1 t $t$ At t = 5, the velocity (Time variable) is v(5) = 32. z $z$ zz¯ = |z|2 (Complex numbers) 2.2 Variables in Geometry Symbols LaTeX Code Example (Explanation) P , Q, R, S $P$, $Q$, $R$, $S$ PQ ⊥ QR (Vertices) ℓ $\ell$ ℓ1 k ℓ2 (Lines) α, β, γ, θ $\alpha$, $\beta$, α + β + θ = 180◦ (Angles) $\gamma$, $\theta$ 2.3 Variables in Calculus Symbols LaTeX Code Example (Explanation) 2.3 Variables in Calculus 7 Comprehensive List of Mathematical Symbols f(x), g(x, y), h(z) $f(x)$, $g(x,y)$, f(2) = g(3, 1) + 5 (Functions) $h(z)$ 3 an, bn, cn $a_n$, $b_n$, an = (Sequences) $c_n$ n + 2 eh − e0 h, ∆x $h$, $\Delta x$ lim = 1 → (Limiting variables in h 0 h derivatives) δ, ε $\delta$, For all ε > 0, there is a (Small quantities in $\varepsilon$ δ > 0 such that |x| < δ proofs involving implies that |2x| < ε. limits) F (x), G(x) $F(x)$, $G(x)$ F 0(x) = f(x) (Antiderivatives) 2.4 Variables in Linear Algebra Symbols LaTeX Code Example (Explanation) u, v, w $\mathbf{u}$, 3u + 4v = w (Vectors) $\mathbf{v}$, $\mathbf{w}$ A, B, C $A$, $B$, $C$ AX = B (Matrices) λ $\lambda$ Av = λv (Eigenvalues) 2.5 Variables in Set Theory and Logic Symbols LaTeX Code Example (Explanation) A, B, C $A$, $B$, $C$ A ⊆ B ∪ C (Sets) 8 2.5 Variables in Set Theory and Logic Comprehensive List of Mathematical Symbols a, b, c $a$, $b$, $c$ a ∈ A (Elements) P , Q, R $P$, $Q$, $R$ P ∨ ¬P ≡ > (Propositions) 2.6 Variables in Probability and Statistics Symbols LaTeX Code Example (Explanation) X, Y , Z $X$, $Y$, $Z$ E(X + Y ) = (Random variables) E(X) + E(Y ) µ $\mu$ H0 : µ = 5 (Population means) σ $\sigma$ σ1 = σ2 (Population standard deviations) s $s$ s =6 σ (Sample standard deviations) n $n$ For n ≥ 30, use the (Sample sizes) normal distribution. ρ $\rho$ Ha : ρ < 0 (Population correlations) r $r$ If r = 0.75, then (Sample correlations) r2 = 0.5625. π $\pi$ π = 0.5 (Population proportions) X p $p$ p = (Sample proportions) n 2.6 Variables in Probability and Statistics 9 Comprehensive List of Mathematical Symbols 3 Delimiters 3.1 Common Delimiters Symbols LaTeX Code Example (Explanation) . $.$ 25.9703 (Decimal separator) : $:$ 1 : 4 : 9 = 3 : 12 : 27 (Ratio indicator) , $,$ (3, 5, 12) (Object separator) (), [], {} $()$, $[]$, $\{ \}$ (a + b) × c (Order-of-operation indicators) (), [] $()$, $[]$ 3 ∈/ (3, 4], 4 ∈ (3, 4] (Interval indicators) 3.2 Other Delimiters Symbols LaTeX Code Example (Explanation) a 1 4 (), [], x y , $()$, $[]$, b $\begin{pmatrix} x 3 6 (Vector/matrix & y \end{pmatrix}$, indicators) $\begin{bmatrix} a \\ b \end{bmatrix}$ {} $\{ \}$ {π, e, i} (Set builder) |, : $\mid, :$ {x ∈ R | x2 − 2 = 0} (“Such that” markers) 10 3.2 Other Delimiters Comprehensive List of Mathematical Symbols ||, kk $| |, \| \|$ k(3, 4)k = 5 (Metric-related operators) f(x) x ≥ a 1 x ≥ 0 $\begin{cases} f(x) f(x) = g(x) x < a & x \ge a \\ g(x) & 0 x < 0 (Piecewise-function x < a \end{cases}$ marker) hi $\langle \rangle$ hka, bi = kha, bi (Inner product operator) de $\lceil \rceil$ d2.476e = 3 (Ceiling operator) bc $\lfloor \rfloor$ bπc = 3 (Floor operator) 4 Operators 4.1 Common Operators Symbols LaTeX Code Example (Explanation) x + y $x+y$ 2a + 3a = 5a (Sum) x − y $x-y$ 11 − 5 = 6 (Difference) −x $-x$ −3 + 3 = 0 (Additive inverse) x × y, x · y, xy $x \times y$, (m + 1)n = mn + n (Product) $x \cdot y$, $xy$ x ÷ y, x/y $x \div y$, $x/y$ 152 ÷ 3 = 50.6 (Quotient) 4.1 Common Operators 11 Comprehensive List of Mathematical Symbols x 53 + 5 53 5 $\displaystyle = + y \frac{x}{y}$ 6 6 6 (Fraction) xy $x^y$ 34 = 81 (Power) √ −b ± ∆ x ± y $x \pm y$ (Plus and minus) 2a √ √ x $\sqrt{x}$ 2 ≈ 1.414 (Positive square root) |x| $|x|$ |x − 3| < 5 (Absolute value) . x x% $x \%$ x% = (Percent) 100 4.2 Function-related Operators Symbols LaTeX Code Example (Explanation) dom f $\operatorname{dom}f$ If g(x) = ln x, then (Domain) dom(g) = R. ran f $\operatorname{ran}f$ If h(y) = sin y, then (Range) ran(h) = [−1.1]. f(x) $f(x)$ g(5) = g(4) + 3 (Image of an element) f(X) $f(X)$ f(A∩B) ⊆ f(A)∩f(B) (Image of a set) f ◦ g $f \circ g$ If g(3) = 5 and f(5) = (Composite 8, then (f ◦ g)(3) = 8. function) 4.3 Elementary Functions 12 4.3 Elementary Functions Comprehensive List of Mathematical Symbols Symbols LaTeX Code Example (Explanation) n 0 knx + ··· + k0x $k_n x^n + \cdots The polynomial (Polynomial) + k_0x^0$ x3 + 2x2 + 3 has a root in (−3, −2). ex, exp x $e^x$, $\exp x$ ex+y = ex · ey (Natural exponential function) bx $b^x$ 2x > x2 for large x. (General exponential function) ln x $\ln x$ ln(x2) = 2 ln x (Natural logarithmic function) log x $\log x$ log 10000 = 4 (Common logarithmic function) ln x logb x $\log_b x$ log2 x = (General logarithmic ln 2 function) sin x $\sin x$ sin π = 0 (Sine function) √ π 2 cos x $\cos x$ cos = (Cosine function) 4 2 sin x tan x $\tan x$ tan x = (Tangent function) cos x 4.4 Algebra-related Operators Symbols LaTeX Code Example (Explanation) gcd(x, y) $\gcd (x,y)$ gcd(35, 14) = 7 (Greatest common factor) 4.4 Algebra-related Operators 13 Comprehensive List of Mathematical Symbols bxc $\lfloor x \rfloor$ b3.6c = 3 (Floor operator) dxe $\lceil x \rceil$ dπe = 4 (Ceiling operator) min(A) $\min (A)$ If min(A) = 3, then (Minimum) min(A + 5) = 8.