math_symbols.md

Mathematical Symbols

1. Basic Arithmetic Operators

Symbol	Meaning	Example
$+$	Addition	$2 + 3 = 5$
$-$	Subtraction	$7 - 4 = 3$
$\times$	Multiplication	$2 \times 3 = 6$
$\div$	Division	$6 \div 2 = 3$
$%$	Percentage	$25 % of 80 = 20$

2. Comparison Symbols

Symbol	Meaning	Example
$=$	Equal to	$5 = 5$
$\neq$	Not equal to	$4 \neq 5$
$<$	Less than	$3 < 5$
$>$	Greater than	$7 > 5$
$\leq$	Less than or equal to	$3 \leq 3$
$\geq$	Greater than or equal to	$5 \geq 5$
$\approx$	Approximately equal to	$π \approx 3.14$
$\equiv$	Identical to	$x \equiv x$
$\sim$	Similar to	$△ A B C \sim △ D E F$
$\propto$	Proportional to	$y \propto x$

3. Set Theory Symbols

Symbol	Meaning	Example
$\in$	Element of	$2 \in Z$
$\notin$	Not an element of	$π \notin Q$
$∋$	Contains as an element	$R ∋ π$
$\subset$	Proper subset	${1, 2} \subset {1, 2, 3}$
$\subseteq$	Subset	${1, 2} \subseteq {1, 2, 3}$
$\supset$	Proper superset	${1, 2, 3} \supset {1, 2}$
$\supseteq$	Superset	${1, 2, 3} \supseteq {1, 2}$
$\cup$	Union	$A \cup B$
$\cap$	Intersection	$A \cap B$
$∖$	Set difference	$A ∖ B$
$\emptyset$	Empty set	${x \in N : x < 0} = \emptyset$

4. Logic Symbols

Symbol	Meaning	Example
$\land$	And	$p \land q$
$\lor$	Or	$p \lor q$
$\neg$	Not	$\neg p$
$\Rightarrow$	Implies	$p \Rightarrow q$
$\Leftrightarrow$	If and only if	$p \Leftrightarrow q$
$\forall$	For all	$\forall x \in R$
$\exists$	There exists	$\exists x \in R$
$∴$	Therefore	$A = B, B = C, ∴ A = C$
$∵$	Because	$∵ A = B and B = C, ∴ A = C$

5. Geometry Symbols

Symbol	Meaning	Example
$∠$	Angle	$∠ A B C$
$⊥$	Perpendicular	$l_{1} ⊥ l_{2}$
$∥$	Parallel	$l_{1} ∥ l_{2}$
$≅$	Congruent	$△ A B C ≅ △ D E F$
$\sim$	Similar	$△ A B C \sim △ D E F$
$△$	Triangle	$△ A B C$
$◻$	Square	$◻ A B C D$
$\circ$	Degree	$90^{\circ}$
$π$	Pi	$π r^{2}$

6. Calculus Symbols

Symbol	Meaning	Example
$\int$	Integral	$\int_{a}^{b} f (x) d x$
$\oint$	Contour integral	$\oint_{C} f (z) d z$
$\partial$	Partial derivative	$\frac{\partial f}{\partial x}$
$\nabla$	Gradient	$\nabla f$
$lim$	Limit	$lim_{x \to \infty} f (x)$
$\sum$	Summation	$\sum_{i = 1}^{n} i$
$\prod$	Product	$\prod_{i = 1}^{n} i$
$\frac{d}{d x}$	Derivative	$\frac{d}{d x} f (x)$
$\iint$	Double integral	$\iint_{D} f (x, y) d x d y$

7. Algebra Symbols

Symbol	Meaning	Example
$\sqrt{}$	Square root	$\sqrt{4} = 2$
$\sqrt[n]{}$	nth root	$\sqrt[3]{8} = 2$
$‖ x ‖$	Absolute value	$‖ - 5 ‖ = 5$
$n!$	Factorial	$5! = 120$
$(\binom{n}{k})$	Binomial coefficient	$(\binom{5}{2}) = 10$
$\oplus$	Exclusive or	$1 \oplus 1 = 0$
$\otimes$	Kronecker product	$A \otimes B$

8. Number Theory Symbols

Symbol	Meaning	Example
$∣$	Divides	$3 ∣ 9$
$∤$	Does not divide	$3 ∤ 10$
$(\mod)$	Modulo operation	$7 (\mod 3) \equiv 1$
$gcd$	Greatest common divisor	$gcd (12, 18) = 6$
$lcm$	Least common multiple	$lcm (4, 6) = 12$
$\infty$	Infinity	$lim_{x \to 0} \frac{1}{x} = \infty$

9. Matrix and Linear Algebra Symbols

Symbol	Meaning	Example
$det$	Determinant	$det (A)$
$tr$	Trace of a matrix	$tr (A)$
$A^{T}$	Matrix transpose	$(A^{T})_{i j} = A_{j i}$
$A^{- 1}$	Matrix inverse	$A A^{- 1} = I$
$\ker$	Kernel	$\ker (T)$
$Im$	Image	$Im (T)$
$rank$	Rank	$rank (A)$
$\dim$	Dimension	$\dim (V)$
$span$	Span	$span {v_{1}, v_{2}, \dots, v_{n}}$
$\cdot$	Dot product	$a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
$\times$	Cross product	$a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})$
$\otimes$	Kronecker product	$A \otimes B$
$0$	Zero vector	$0 = (0, 0, \dots, 0)$

10. Complex Number Symbols

Symbol	Meaning	Example
$i$	Imaginary unit	$i^{2} = - 1$
$z^{*}$	Complex conjugate	$(a + b i)^{*} = a - b i$
$‖ z ‖$	Modulus	$‖ 3 + 4 i ‖ = 5$
$\arg (z)$	Argument	$\arg (1 + i) = \frac{π}{4}$

11. Statistics Symbols

Symbol	Meaning	Example
$μ$	Population mean	$μ = \frac{1}{N} \sum_{i = 1}^{N} x_{i}$
$σ$	Standard deviation	$σ = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)^{2}}$
$σ^{2}$	Variance	$σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} (x_{i} - μ)^{2}$
$ρ$	Correlation coefficient	$- 1 \leq ρ \leq 1$
$χ^{2}$	Chi-squared	$χ^{2} = \sum \frac{(O - E)^{2}}{E}$

12. Special Constants

Symbol	Meaning	Example
$e$	Euler's number	$lim_{n \to \infty} {(1 + \frac{1}{n})}^{n} = e$
$π$	Pi	$π \approx 3.14159$
$i$	Imaginary unit	$i^{2} = - 1$
$γ$	Euler-Mascheroni constant	$γ \approx 0.57721$
$ϕ$	Golden ratio	$ϕ = \frac{1 + \sqrt{5}}{2} \approx 1.618$
$ζ (3)$	Apéry's constant	$ζ (3) = \sum_{n = 1}^{\infty} \frac{1}{n^{3}} \approx 1.20206$
$\ln (2)$	Natural logarithm of 2	$\ln (2) \approx 0.69315$
$δ$	Khinchin's constant	$δ \approx 2.68545$
$λ$	Lemniscate constant	$λ \approx 2.62205$
$K$	Catalan's constant	$K \approx 0.91596$
$M$	Meissel-Mertens constant	$M \approx 0.261497$

13. Other Symbols

Symbol	Meaning	Example
$\nabla$	Gradient, divergence, or curl	$\nabla f$
$◻$	End of proof	$∴ x = 2 ◻$
$\sim$	Distributed as	$X \sim N (μ, σ^{2})$
$≍$	Asymptotically equivalent	$f (x) ≍ g (x)$
$≪$	Much less than	$1 ≪ 1000$
$≫$	Much greater than	$1000 ≫ 1$
$\to$	Tends to	$x \to \infty$
$\mapsto$	Maps to	$f : x \mapsto x^{2}$

Greek Alphabet

Symbol (Lowercase)	Symbol (Uppercase)	Pronunciation	Meaning	Example
$α$	$A$	Alpha	Angle, coefficient	$α = 45^{\circ}$
$β$	$B$	Beta	Angle, coefficient	$β = 0.5$
$γ$	$Γ$	Gamma	Gamma function	$Γ (x)$
$δ$	$Δ$	Delta	Delta function, change	$Δ x = x_{2} - x_{1}$
$ϵ$	$E$	Epsilon	Small quantity, permittivity	$ϵ > 0$
$ζ$	$Z$	Zeta	Zeta function	$ζ (s)$
$η$	$H$	Eta	Efficiency, viscosity	$η = 0.8$
$θ$	$Θ$	Theta	Angle	$θ = 30^{\circ}$
$ι$	$ι$	Iota	Small quantity, index	$ι_{n}$
$κ$	$K$	Kappa	Curvature	$κ = \frac{1}{R}$
$λ$	$Λ$	Lambda	Eigenvalue, wavelength	$λ = 500 nm$
$μ$	$M$	Mu	Mean, magnetic moment	$μ = 10$
$ν$	$N$	Nu	Frequency	$ν = 60 Hz$
$ξ$	$Ξ$	Xi	Random variable	$ξ \sim N (0, 1)$
$o$	$O$	Omicron	Omicron	$o (n)$
$π$	$Π$	Pi	Pi, product	$π \approx 3.14$
$ρ$	$P$	Rho	Density, resistivity	$ρ = 1000 {kg/m}^{3}$
$σ$	$Σ$	Sigma	Standard deviation, sum	$σ = 2$
$τ$	$T$	Tau	Tau, torque	$τ = 5 Nm$
$υ$	$Υ$	Upsilon	Upsilon	$Υ (x)$
$ϕ$	$Φ$	Phi	Phi, potential	$ϕ = 10 V$
$χ$	$X$	Chi	Chi, chi-squared	$χ^{2}$
$ψ$	$Ψ$	Psi	Psi, wave function	$Ψ (x, t)$
$ω$	$Ω$	Omega	Omega, angular frequency	$ω = 2 π f$