Math Symbols List: 200+ Symbols & Meanings

Math notation reference

Find a symbol. Read it in context.

Names, meanings, worked examples and LaTeX for 233 symbol entries, from arithmetic to calculus and proof writing. Some symbols appear more than once because their meanings change across subjects.

A mathematical symbol is shorthand for an operation, relationship or object. For example, ≤ means “less than or equal to,” ∈ means “is an element of,” and ∑ tells you to add indexed terms. Letters such as θ and μ represent quantities whose meanings must be defined in the problem.

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200+ mathematical symbols: meanings and examples

Choose a category to jump to its entries. A definition describes a common convention; follow your textbook when notation differs.

233 of 233 entries shown

Math symbols by subject, with plain-text examples and LaTeX
Symbol	Name	Category	Meaning and common use	LaTeX	Example
+	Plus / addition	Arithmetic	Adds quantities or indicates a positive quantity.	`+`	3 + 2 = 5
−	Minus / subtraction	Arithmetic	Subtracts a quantity or indicates a negative value.	`-`	8 − 3 = 5
±	Plus or minus	Arithmetic	Indicates two possible signs or a tolerance.	`\pm`	x = 4 ± 1
∓	Minus or plus	Arithmetic	Paired sign used with plus or minus expressions.	`\mp`	a ± b, c ∓ d
×	Multiplication sign	Arithmetic	Multiplies two quantities.	`\times`	4 × 6 = 24
·	Multiplication dot	Arithmetic	Multiplies quantities, often variables or vectors.	`\cdot`	a · b
÷	Division sign	Arithmetic	Divides one quantity by another.	`\div`	12 ÷ 3 = 4
/	Slash / division bar	Arithmetic	Represents division in linear text.	`/`	a / b
⁄	Fraction slash	Arithmetic	Slash separating an inline numerator and denominator; a stacked fraction uses \frac{a}{b}.	`\text{\textfractionsolidus}`	3⁄4
%	Percent	Arithmetic	Means per hundred.	`\%`	25% = 25/100
‰	Per mille	Arithmetic	Means per thousand.	`\text{\textperthousand}`	4‰ = 4/1000
√	Square root	Arithmetic	For a non-negative real x, the non-negative number whose square is x.	`\sqrt{x}`	√9 = 3
∛	Cube root	Arithmetic	For real x, the unique real number whose cube is x.	`\sqrt[3]{x}`	∛(−8) = −2
∜	Fourth root	Arithmetic	For a non-negative real x, the non-negative number whose fourth power is x.	`\sqrt[4]{x}`	∜16 = 2
^	Exponent marker	Arithmetic	Plain-text power marker; LaTeX uses ^ followed by an exponent.	`x^{2}`	x^2
!	Factorial	Arithmetic	For positive integer n, n! = 1 × 2 × … × n; by definition, 0! = 1.	`!`	5! = 120; 0! = 1
‖	Double vertical bar	Arithmetic	Paired bars commonly denote a norm; the parallel relation is written ∥.	`\\|`	‖v‖
\|	Vertical bar	Arithmetic	May mean absolute value, divides, or such that.	`\|`	\|x\|
⌊ ⌋	Floor brackets	Arithmetic	Greatest integer less than or equal to a value.	`\lfloor x \rfloor`	⌊3.8⌋ = 3
⌈ ⌉	Ceiling brackets	Arithmetic	Least integer greater than or equal to a value.	`\lceil x \rceil`	⌈3.2⌉ = 4
=	Equal to	Relations	States that two expressions have the same value.	`=`	2 + 3 = 5
≠	Not equal to	Relations	States that two expressions are not equal.	`\neq`	x ≠ 0
<	Less than	Relations	Left quantity is smaller than right quantity.	`<`	2 < 7
>	Greater than	Relations	Left quantity is larger than right quantity.	`>`	9 > 4
≤	Less than or equal to	Relations	Left quantity is smaller than or equal to right.	`\leq`	x ≤ 5
≥	Greater than or equal to	Relations	Left quantity is larger than or equal to right.	`\geq`	x ≥ 0
≪	Much less than	Relations	Left quantity is far smaller than right.	`\ll`	ε ≪ 1
≫	Much greater than	Relations	Left quantity is far larger than right.	`\gg`	n ≫ 1
≈	Approximately equal to	Relations	Values are equal to the stated or intended level of approximation.	`\approx`	π ≈ 3.14
≅	Congruent to / approximately equal	Relations	Often means congruent in geometry.	`\cong`	△ABC ≅ △DEF
≡	Identically equal / congruent	Relations	Identity or modular congruence by context.	`\equiv`	(x+1)^2 ≡ x²+2x+1
∝	Proportional to	Relations	One quantity varies in proportion to another.	`\propto`	y ∝ x
∼	Similar to / distributed as	Relations	Similarity or distribution relationship by context.	`\sim`	X ∼ N(0,1)
≃	Approximately equal / equivalent	Relations	Context-dependent approximation or equivalence; define its use. Asymptotic ratio equivalence is commonly written ∼.	`\simeq`	f(x) ≃ g(x)
≺	Precedes	Relations	Ordering relation, often in logic or algebra.	`\prec`	a ≺ b
≻	Succeeds	Relations	Reverse ordering relation.	`\succ`	b ≻ a
⊢	Provable from / syntactic consequence	Relations	A conclusion can be derived from assumptions using a specified proof system.	`\vdash`	P, P ⇒ Q ⊢ Q
⊨	Models / satisfies	Relations	A structure satisfies a formula, or assumptions semantically entail a conclusion.	`\models`	M ⊨ φ; P ⊨ Q
⊥	Perpendicular / contradiction	Relations	Perpendicular lines or false statement by context.	`\perp`	AB ⊥ CD
∥	Parallel	Relations	Lines or vectors are parallel.	`\parallel`	l₁ ∥ l₂
{ }	Set braces	Sets	Encloses elements of a set.	`\{ \}`	A = {1,2,3}
∅	Empty set	Sets	Set containing no elements.	`\varnothing`	A ∩ B = ∅
∈	Element of	Sets	Belongs to a set.	`\in`	3 ∈ ℕ
∉	Not an element of	Sets	Does not belong to a set.	`\notin`	−1 ∉ ℕ
∋	Contains as member	Sets	Reverse membership symbol.	`\ni`	A ∋ 3
⊂	Subset (convention varies)	Sets	Often a proper subset, but some authors allow equality; ⊊ unambiguously excludes equality.	`\subset`	{1} ⊂ {1,2}
⊆	Subset or equal to	Sets	Every element belongs to the other set.	`\subseteq`	A ⊆ B
⊃	Superset (convention varies)	Sets	Often a proper superset, but some authors allow equality; ⊋ unambiguously excludes equality.	`\supset`	{1,2} ⊃ {1}
⊇	Superset or equal to	Sets	Contains every element of another set.	`\supseteq`	B ⊇ A
∪	Union	Sets	Elements in at least one of the sets, including elements in both.	`\cup`	A ∪ B
∩	Intersection	Sets	Elements shared by both sets.	`\cap`	A ∩ B
∖	Set difference	Sets	Elements in first set but not second.	`\setminus`	A ∖ B
△	Symmetric difference	Sets	Elements in exactly one of two sets.	`\triangle`	A △ B
ᶜ	Complement	Sets	Elements outside a set relative to universe.	`A^{c}`	Aᶜ
℘	Power set	Sets	One notation for the set of all subsets of A; 𝒫(A) is another common notation.	`\wp`	℘(A)
×	Cartesian product	Sets	Ordered pairs from two sets.	`\times`	A × B
∣	Such that / divides	Sets	Defines a condition or divisibility relation.	`\mid`	{x ∈ ℝ ∣ x > 0}
ℕ	Natural numbers	Sets	Counting numbers; zero convention varies.	`\mathbb{N}`	n ∈ ℕ
ℤ	Integers	Sets	Whole positive, negative, and zero numbers.	`\mathbb{Z}`	−4 ∈ ℤ
ℚ	Rational numbers	Sets	Numbers a/b with integers a and b and b ≠ 0.	`\mathbb{Q}`	1/3 ∈ ℚ
ℝ	Real numbers	Sets	All rational and irrational real values.	`\mathbb{R}`	π ∈ ℝ
ℂ	Complex numbers	Sets	Numbers a + bi, where a and b are real and i² = −1.	`\mathbb{C}`	i ∈ ℂ
¬	Not	Logic	Negates a proposition.	`\neg`	¬P
∧	And	Logic	Both propositions must be true.	`\land`	P ∧ Q
∨	Or	Logic	At least one proposition is true.	`\lor`	P ∨ Q
⊕	Exclusive or	Logic	Exactly one proposition is true.	`\oplus`	P ⊕ Q
⇒	Implies	Logic	If first statement holds, second follows.	`\Rightarrow`	P ⇒ Q
⇐	Implied by	Logic	First statement follows from second.	`\Leftarrow`	P ⇐ Q
⇔	If and only if	Logic	Both statements imply each other.	`\Leftrightarrow`	P ⇔ Q
∀	For all	Logic	Universal quantifier.	`\forall`	∀x ∈ ℝ, x² ≥ 0
∃	There exists	Logic	Existential quantifier.	`\exists`	∃x ∈ ℝ: x² = 2
∄	There does not exist	Logic	Negated existential quantifier.	`\nexists`	∄x ∈ ℝ: x² = −1
∴	Therefore	Logic	Introduces a conclusion.	`\therefore`	x=2, ∴ x²=4
∵	Because	Logic	Introduces a reason.	`\because`	x²=4 ∵ x=2
⊤	True	Logic	Always true proposition.	`\top`	P = ⊤
⊥	False / contradiction	Logic	Always false proposition.	`\bot`	P = ⊥
⊼	NAND	Logic	Not both propositions are true.	`\barwedge`	P ⊼ Q = ¬(P ∧ Q)
⊽	NOR	Logic	Neither proposition is true.	`\mathbin{\overline{\vee}}`	P ⊽ Q = ¬(P ∨ Q)
⊣	Reverse turnstile	Logic	Reverses ⊢ when explicitly defined that way; notation depends on the text.	`\dashv`	Q ⊣ P means P ⊢ Q
∎	End of proof	Logic	Marks the completion of a proof.	`\blacksquare`	QED ∎
∠	Angle	Geometry	Angle formed by two rays.	`\angle`	∠ABC
△	Triangle	Geometry	Triangle shape or triangle notation.	`\triangle`	△ABC
□	Square	Geometry	Square figure or proof end by context.	`\square`	□ABCD
○	Circle	Geometry	Circle-shaped symbol; some geometry texts use it to name a circle.	`\bigcirc`	○O
⊙	Circle with centre	Geometry	Circle named by its centre.	`\odot`	⊙O
⌒	Arc	Geometry	Marks an arc between endpoints; typeset the arc accent above the endpoint letters.	`\overset{\frown}{AB}`	arc AB
≅	Congruent	Geometry	Figures have same shape and size.	`\cong`	△ABC ≅ △DEF
∼	Similar	Geometry	Figures have same shape, proportional size.	`\sim`	△ABC ∼ △DEF
∥	Parallel	Geometry	Distinct coplanar lines that do not intersect.	`\parallel`	AB ∥ CD
⊥	Perpendicular	Geometry	Lines meet at right angle.	`\perp`	AB ⊥ CD
°	Degree	Geometry	Angle unit equal to 1/360 of a full turn.	`90^\circ`	90°
′	Prime	Geometry	An arcminute is 1/60 of a degree; primes also denote derivatives in calculus.	`30^{\prime}`	60′ = 1°
″	Double prime	Geometry	An arcsecond is 1/60 of an arcminute; double primes can denote second derivatives.	`45^{\prime\prime}`	60″ = 1′
≜	Defined as	Geometry	States a definition in some texts.	`\triangleq`	r ≜ 5
→	Ray / directed segment notation	Geometry	An arrow above AB denotes ray AB or directed segment AB, according to context.	`\overrightarrow{AB}`	ray AB: starts at A and passes through B
↔	Line through points	Geometry	A double-headed arrow above AB denotes the line through distinct points A and B.	`\overleftrightarrow{AB}`	line AB: extends in both directions
↦	Maps to	Geometry	Maps an input to output.	`\mapsto`	x ↦ x²
v⃗	Vector arrow accent	Geometry	An arrow above a letter marks a vector; the Unicode combining arrow follows the letter.	`\vec{v}`	v⃗
⊥̸	Not perpendicular	Geometry	Lines are not perpendicular.	`\not\perp`	l₁ ⊥̸ l₂
∦	Not parallel	Geometry	Lines are not parallel.	`\nparallel`	l₁ ∦ l₂
∫	Integral	Calculus	Integral notation: an indefinite integral denotes antiderivatives; a definite integral gives signed accumulation.	`\int`	∫ x dx = x²/2 + C
∬	Double integral	Calculus	Integrates over a two-dimensional region.	`\iint`	∬f dA
∭	Triple integral	Calculus	Integrates over a three-dimensional region.	`\iiint`	∭f dV
∮	Closed contour / line integral	Calculus	Integrates along a closed curve; contour integrals in general need not be closed.	`\oint`	∮C f(z) dz
∂	Partial derivative	Calculus	Differentiation with respect to one variable, with all other independent variables held fixed.	`\partial`	∂(x²y)/∂x = 2xy
d	Differential	Calculus	Differential notation; in an integral, dx identifies the variable of integration.	`d`	∫ f(x) dx
∇	Nabla / del	Calculus	Gradient, divergence, or curl operator.	`\nabla`	∇f
Δ	Finite difference	Calculus	A finite change, often Δy = y₂ − y₁; written with capital Greek delta.	`\Delta`	Δy/Δx
lim	Limit	Calculus	Introduces the value approached in a specified limiting process, when it exists.	`\lim`	lim (x→0) (sin x)/x = 1, with x in radians
∞	Infinity	Calculus	Represents unboundedness in limits; ∞ is not an ordinary real number.	`\infty`	x → ∞
′	First derivative	Calculus	Derivative of a function.	`f'(x)`	f′(x)
″	Second derivative	Calculus	Second derivative of a function.	`f''(x)`	f″(x)
o	Little o	Calculus	f = o(g) means f/g → 0 in the stated limit, where the ratio is defined.	`o(g(x))`	x² = o(x) as x → 0
O	Big O	Calculus	f = O(g) means \|f\| is bounded by a fixed multiple of \|g\| in the stated limit or domain.	`O(g(x))`	3n² + n = O(n²) as n → ∞
∼	Asymptotic equivalence	Calculus	f ∼ g means f/g → 1 in the stated limit, where the ratio is defined.	`\sim`	sin x ∼ x as x → 0, with x in radians
→	Approaches	Calculus	Variable tends toward a value.	`\to`	x → 0
↗	Increases to	Calculus	Indicates increasing convergence to a limit; it conveys more than a one-sided approach.	`\nearrow`	1 − 1/n ↗ 1 as n → ∞
↘	Decreases to	Calculus	Indicates decreasing convergence to a limit; it conveys more than a one-sided approach.	`\searrow`	1 + 1/n ↘ 1 as n → ∞
∑	Summation	Calculus	Adds terms in a sequence.	`\sum`	∑ᵢ₌₁ⁿ xᵢ
∏	Product	Calculus	Multiplies terms in a sequence.	`\prod`	∏ᵢ₌₁ⁿ xᵢ
∐	Coproduct	Calculus	Disjoint union or categorical coproduct.	`\coprod`	∐ Aᵢ
v⃗	Vector	Linear Algebra	A vector is an element of a vector space; geometric vectors can represent magnitude and direction.	`\vec{v}`	v⃗ = (2,3)
‖ ‖	Vector norm	Linear Algebra	Length or norm of a vector.	`\\|v\\|`	‖v‖
·	Dot product	Linear Algebra	Scalar product of two vectors.	`\cdot`	u · v
×	Cross product	Linear Algebra	Vector product in three dimensions.	`\times`	u × v
⊗	Tensor product	Linear Algebra	Tensor or Kronecker product by context.	`\otimes`	A ⊗ B
⊕	Direct sum	Linear Algebra	Combines vector spaces or modules.	`\oplus`	V ⊕ W
Aᵀ	Transpose	Linear Algebra	Rows and columns of a matrix swapped.	`A^T`	Aᵀ
A⁻¹	Inverse matrix	Linear Algebra	For an invertible square matrix A, A⁻¹ satisfies AA⁻¹ = A⁻¹A = I.	`A^{-1}`	AA⁻¹ = A⁻¹A = I
det	Determinant	Linear Algebra	Scalar associated with a square matrix.	`\det`	det(A)
tr	Trace	Linear Algebra	Sum of the main diagonal entries of a square matrix.	`\operatorname{tr}`	tr(A)
rank	Matrix rank	Linear Algebra	Dimension of column or row space.	`\operatorname{rank}`	rank(A)
λ	Eigenvalue	Linear Algebra	A scalar λ for which Av = λv has a nonzero vector solution v.	`\lambda`	Av = λv, v ≠ 0
I	Identity matrix	Linear Algebra	Square matrix with ones on the main diagonal and zeros elsewhere.	`I`	AI = A
0	Zero matrix	Linear Algebra	Matrix with every entry equal to zero.	`0`	A + 0 = A
⟨ ⟩	Inner product	Linear Algebra	Generalised dot product.	`\langle u,v \rangle`	⟨u,v⟩
μ	Population mean	Statistics	Mean of an entire population.	`\mu`	μ = E[X]
x̄	Sample mean	Statistics	Average of observed sample values.	`\bar{x}`	x̄ = 12
σ	Population standard deviation	Statistics	The non-negative square root of population variance.	`\sigma`	σ = √(σ²)
s	Sample standard deviation	Statistics	Usually the square root of sample variance computed with denominator n − 1, for n > 1.	`s`	s = √(s²)
σ²	Population variance	Statistics	Population mean of squared deviations from the population mean.	`\sigma^2`	σ² = E[(X − μ)²]
s²	Sample variance	Statistics	Usually the sum of squared deviations from x̄ divided by n − 1, for n > 1.	`s^2`	s² = ∑ᵢ(xᵢ − x̄)²/(n − 1)
ρ	Population correlation	Statistics	Population correlation coefficient.	`\rho`	ρ = 0.7
r	Sample correlation	Statistics	Sample correlation coefficient.	`r`	r = −0.3
p̂	Sample proportion	Statistics	Estimated proportion from a sample.	`\hat{p}`	p̂ = 0.52
p	p-value / probability	Statistics	A p-value is a null-model probability of a result at least as extreme as observed; p can also denote a probability parameter.	`p`	p = 0.03
P(A)	Probability of event A	Statistics	Probability assigned to event A, a value from 0 to 1.	`P(A)`	P(A) = 0.4
P(A\|B)	Conditional probability	Statistics	Probability of A conditional on B; for events with P(B) > 0, P(A∣B) = P(A∩B)/P(B).	`P(A\mid B)`	P(A∣B) = P(A∩B)/P(B)
E[X]	Expected value	Statistics	Probability-weighted mean of a random variable, when the expectation exists.	`\mathbb{E}[X]`	E[X] = μ
Var(X)	Variance	Statistics	Expected squared distance from mean.	`\operatorname{Var}(X)`	Var(X) = σ²
Cov(X,Y)	Covariance	Statistics	Expected product of centered variables, when defined: E[(X − E[X])(Y − E[Y])].	`\operatorname{Cov}(X,Y)`	Cov(X,Y) = E[(X − E[X])(Y − E[Y])]
χ²	Chi-square	Statistics	Test statistic or distribution.	`\chi^2`	χ² = 5.2
z	z-score	Statistics	For σ > 0, the number of population standard deviations x lies above or below μ.	`z`	z = (x − μ)/σ
t	t statistic	Statistics	Student t test statistic.	`t`	t = −2.1
H₀	Null hypothesis	Statistics	Hypothesis specifying the null model or parameter claim assessed by a test.	`H_0`	H₀: μ = 0
H₁	Alternative hypothesis	Statistics	Competing claim in hypothesis test.	`H_1`	H₁: μ ≠ 0
α	Alpha	Greek Letters	Angle, significance level, or parameter.	`\alpha`	α = 0.05
β	Beta	Greek Letters	Coefficient, angle, or type II error rate.	`\beta`	β₁x
γ	Gamma	Greek Letters	Lowercase gamma can denote an angle, parameter, or Euler–Mascheroni constant; Γ denotes the gamma function.	`\gamma`	γ = 30°
δ	Lowercase delta	Greek Letters	Small change or parameter.	`\delta`	δx
ε	Epsilon	Greek Letters	Small positive quantity or error.	`\varepsilon`	ε > 0
ζ	Zeta	Greek Letters	Zeta function or parameter.	`\zeta`	ζ(s)
η	Eta	Greek Letters	Efficiency or parameter.	`\eta`	η = 0.9
θ	Theta	Greek Letters	Angle or unknown parameter.	`\theta`	sin θ
ι	Iota	Greek Letters	Indexing map or inclusion map.	`\iota`	ι:A→B
κ	Kappa	Greek Letters	Curvature or constant.	`\kappa`	κ(s)
λ	Lambda	Greek Letters	Eigenvalue, wavelength, or rate.	`\lambda`	λ = 2
μ	Mu	Greek Letters	Population mean or parameter.	`\mu`	μ = 10
ν	Nu	Greek Letters	Frequency or degrees of freedom.	`\nu`	ν = n−1
ξ	Xi	Greek Letters	Random variable or coordinate.	`\xi`	ξ ∈ ℝ
ο	Omicron	Greek Letters	Greek omicron; traditional LaTeX often substitutes Latin o, while unicode-math supports \omicron.	`o`	ο
π	Pi	Greek Letters	Circle constant, or another quantity such as the prime-counting function; the product operator is ∏.	`\pi`	π ≈ 3.14159
ρ	Rho	Greek Letters	Density or correlation coefficient.	`\rho`	ρ = m/V
σ	Sigma	Greek Letters	Often standard deviation, a permutation, or a divisor-sum function; summation uses ∑.	`\sigma`	σ = 2
τ	Tau	Greek Letters	Time constant or torque.	`\tau`	τ = rF sin θ
υ	Upsilon	Greek Letters	Greek letter, rarely used in elementary math.	`\upsilon`	υ
φ	Phi	Greek Letters	Angle, function, or golden ratio notation.	`\varphi`	φ = (1 + √5)/2 ≈ 1.618
χ	Chi	Greek Letters	Chi-square distribution or variable.	`\chi`	χ²
ψ	Psi	Greek Letters	Wave function or variable.	`\psi`	ψ(x)
ω	Omega	Greek Letters	Often angular frequency; in set theory, ω denotes the first infinite ordinal.	`\omega`	ω = 2πf
Γ	Capital gamma	Greek Letters	Gamma function or matrix.	`\Gamma`	Γ(n)
Δ	Capital delta	Greek Letters	Finite change or discriminant.	`\Delta`	Δx
Θ	Capital theta	Greek Letters	An asymptotically tight bound, with both upper and lower bounds up to positive constants.	`\Theta`	3n² + n = Θ(n²) as n → ∞
Λ	Capital lambda	Greek Letters	Greek capital lambda, often a matrix or parameter; logical and is ∧.	`\Lambda`	Λ = diag(λ₁, λ₂)
Ξ	Capital xi	Greek Letters	Random variable or special function.	`\Xi`	Ξ
Π	Capital pi	Greek Letters	Greek capital pi; may label a quantity. The distinct product operator is ∏, written \prod.	`\Pi`	Π = ∏ᵢ aᵢ
Σ	Capital sigma	Greek Letters	Greek capital sigma, often a covariance matrix; the distinct summation operator is ∑, written \sum.	`\Sigma`	Σ = I
Φ	Capital phi	Greek Letters	Commonly the cumulative distribution function of the standard normal distribution.	`\Phi`	Φ(0) = 0.5
Ψ	Capital psi	Greek Letters	Function or state vector.	`\Psi`	Ψ
Ω	Capital omega	Greek Letters	Sample space or asymptotic bound.	`\Omega`	Ω
∣	Divides	Number Theory	First integer divides second exactly.	`\mid`	3 ∣ 12
∤	Does not divide	Number Theory	First integer does not divide second.	`\nmid`	5 ∤ 12
≡	Congruent modulo	Number Theory	For integers a, b and positive integer m, a ≡ b (mod m) means m divides a − b.	`\equiv`	17 ≡ 5 (mod 12)
mod	Modulo	Number Theory	For integer a and positive integer m, a mod m is the remainder from 0 through m − 1.	`\bmod`	17 mod 12 = 5
gcd	Greatest common divisor	Number Theory	Largest positive common divisor.	`\gcd`	gcd(12,18)=6
lcm	Least common multiple	Number Theory	Smallest positive common multiple.	`\operatorname{lcm}`	lcm(4,6)=12
≔	Defined as	Number Theory	Introduces a definition; \coloneqq is supplied by mathtools.	`\coloneqq`	f(n) ≔ n² + 1
φ	Euler totient	Number Theory	For positive integer n, counts the integers k with 1 ≤ k ≤ n and gcd(k,n) = 1.	`\varphi`	φ(9) = 6
μ	Möbius function	Number Theory	μ(1) = 1; μ(n) = 0 if a prime square divides n; otherwise μ(n) = (−1)^k for k distinct prime factors.	`\mu`	μ(6) = 1; μ(12) = 0
⌊ ⌋	Greatest-integer function	Number Theory	Rounds down to nearest integer.	`\lfloor x \rfloor`	⌊−1.2⌋=−2
∑	Summation	Discrete Math	Adds indexed values.	`\sum`	∑ᵢ aᵢ
∏	Product	Discrete Math	Multiplies indexed values.	`\prod`	∏ᵢ aᵢ
!	Factorial	Discrete Math	For non-negative integer n, counts orderings of n distinct objects; n! = 1 × … × n and 0! = 1.	`!`	6! = 720
nPᵣ	Permutation count	Discrete Math	Number of ordered selections of r distinct items from n distinct items without repetition, for 0 ≤ r ≤ n.	`{}_nP_r`	₅P₂ = 5!/(5−2)! = 20
nCᵣ	Combination count	Discrete Math	Number of unordered selections of r items from n distinct items without repetition; also written as a binomial coefficient.	`{}_nC_r`	₅C₂ = 5!/(2!3!) = 10
∑	Sigma notation	Discrete Math	Compact notation for repeated addition.	`\sum`	∑ᵢ₌₁ⁿ i
∏	Pi notation	Discrete Math	Compact notation for repeated multiplication.	`\prod`	∏ᵢ₌₁ⁿ i
↔	Biconditional	Discrete Math	Logical if-and-only-if statement.	`\leftrightarrow`	P ↔ Q
⇒	Implication	Discrete Math	One condition leads to another.	`\Rightarrow`	P ⇒ Q
∴	Therefore	Discrete Math	Conclusion in a proof or argument.	`\therefore`	P and P ⇒ Q; ∴ Q
∘	Composition	Abstract Algebra	Combines functions or operations.	`\circ`	f ∘ g
≅	Isomorphic to	Abstract Algebra	Structures have same algebraic form.	`\cong`	G ≅ H
≇	Not isomorphic	Abstract Algebra	Structures are not isomorphic.	`\ncong`	G ≇ H
◁	Normal subgroup	Abstract Algebra	H is a normal subgroup of G; whether ◁ also requires H ≠ G depends on the author.	`\triangleleft`	H ◁ G
⋉	Semidirect product (left symbol)	Abstract Algebra	Semidirect product notation; the group action and factor convention must be specified.	`\ltimes`	H ⋉ N
⋊	Semidirect product (right symbol)	Abstract Algebra	Commonly N ⋊ H denotes a semidirect product in which H acts on the normal factor N.	`\rtimes`	N ⋊φ H, where φ: H → Aut(N)
Aut(G)	Automorphism group	Abstract Algebra	All automorphisms of group G.	`\operatorname{Aut}(G)`	Aut(G)
ker	Kernel	Abstract Algebra	Elements mapped to identity or zero.	`\ker`	ker(T)
im	Image	Abstract Algebra	Outputs reached by a map.	`\operatorname{im}`	im(T)
⟨S⟩	Generated subgroup	Abstract Algebra	Smallest subgroup containing S.	`\langle S \rangle`	⟨a⟩
ℵ	Aleph	Special Symbols	Indexed alephs denote infinite cardinal numbers; ℵ₀ is the size of a countably infinite set.	`\aleph`	\|ℕ\| = ℵ₀
ℏ	Reduced Planck constant	Special Symbols	Physics constant used in quantum formulas.	`\hbar`	ℏ
∠	Angle symbol	Special Symbols	Marks an angle in geometry.	`\angle`	∠XYZ
∡	Measured angle	Special Symbols	Marks a measured angle.	`\measuredangle`	∡ABC
∟	Right angle	Special Symbols	A 90° angle; the exact LaTeX command \rightangle requires a package such as unicode-math.	`\rightangle`	∠ABC = 90°
□	End of proof	Special Symbols	Marks end of proof in many texts.	`\square`	□
…	Ellipsis	Special Symbols	Indicates continuation in a sequence.	`\ldots`	1,2,3,…
⋯	Centered ellipsis	Special Symbols	Indicates continuation in products or sums.	`\cdots`	a₁+⋯+aₙ
⋮	Vertical ellipsis	Special Symbols	Indicates continuation down a matrix.	`\vdots`	⋮
⋱	Diagonal ellipsis	Special Symbols	Indicates diagonal continuation in matrix.	`\ddots`	⋱
∎	Tombstone / QED	Special Symbols	Marks end of mathematical proof.	`\blacksquare`	Proof ∎
⊤	Top	Special Symbols	True proposition or top element.	`\top`	x = ⊤
⊥	Bottom	Special Symbols	False proposition or bottom element.	`\bot`	x = ⊥

LaTeX note: examples are source code for math mode, not text to paste into every calculator. Some commands require packages; see typing guidance. Copy copies the visible Unicode symbol or notation, which may contain more than one character.

How to read symbols with more than one meaning

First identify the subject, then read the expression on both sides of the symbol. The same mark can have several legitimate meanings:

Vertical bars: | |

|−5| = 5: absolute value, or distance from zero
|{2, 4, 6}| = 3: the number of elements in a finite set
3 ∣ 12: 3 divides 12 exactly
P(A ∣ B): probability of A given B
|A|: may mean a determinant when A is a square matrix

Similar-looking relationships

= states exact equality: 1/2 = 0.5
≈ indicates approximation: π ≈ 3.14159
≡ may mark an identity or modular congruence: 17 ≡ 5 (mod 12)
≅ can mean congruent figures or isomorphic structures
∼ may mean similarity, a distribution, or asymptotic equivalence

Greek letters are not fixed operations. Lowercase σ often denotes standard deviation, uppercase Σ is a letter, and the summation operator ∑ means “add these terms.” Likewise, lowercase π is commonly the circle constant, while ∏ is the product operator.

Arithmetic and comparison notation

The order of operations decides how signs combine. Multiplication and division have the same priority and are read from left to right; addition and subtraction also share a priority. Parentheses can change the grouping.

8 − 3 × 2 = 8 − 6 = 2
(8 − 3) × 2 = 5 × 2 = 10

x < 5 excludes 5; x ≤ 5 includes it. On a number line, an open endpoint excludes a boundary and a filled endpoint includes it. For real numbers, √9 = 3 is the non-negative square root, but solving x² = 9 gives two answers, x = ±3.

Floor and ceiling behave differently for negatives: ⌊−2.3⌋ = −3 and ⌈−2.3⌉ = −2. Floor goes to the greatest integer at or below the value, rather than simply removing the decimal part.

Continue with multiplication, fractions and exponents when you need the calculation methods behind the symbols.

Algebra, functions and equations

An equation asks when two expressions agree. 2x + 3 = 11 is true when x = 4. An identity such as (x + 1)² ≡ x² + 2x + 1 is true for every real x.

In f: A → B, A is the domain and B is the codomain. In x ↦ x², the arrow describes the rule applied to each input. Thus, if f(x) = x², then f(3) = 9. The notation f(x) is a function value, not automatically f multiplied by x.

Inverse is not the same as reciprocal

If f(x) = x + 3, its inverse function is f⁻¹(x) = x − 3. Its reciprocal is 1/f(x) = 1/(x + 3), defined when x ≠ −3. A one-to-one function has an inverse on its image; an inverse from the full codomain requires a bijection.

For practice reading and simplifying algebra, see linear expressions.

Set membership, subsets and logic

Let A = {1, 2, 3} and B = {3, 4}. Then 2 ∈ A says that 2 is an element. In contrast, {2} ⊆ A says that every element of the set {2} belongs to A.

For A equals {1, 2, 3}, 2 is an element of A, while the set {2} is a subset of A. — ∈ asks whether an object belongs to a set. ⊆ asks whether every element of one set belongs to another.

A ∪ B = {1, 2, 3, 4}: union keeps elements in either set, including both

A ∩ B = {3}: intersection keeps shared elements

A ∖ B = {1, 2}: difference removes elements that are in B

A △ B = {1, 2, 4}: symmetric difference keeps elements in exactly one set

A ⊆ A is always true. A proper subset is a subset that is not equal to the larger set. The symbol ⊂ has conflicting textbook conventions; use ⊊ when you need to make “proper subset” explicit. Also check whether ℕ includes 0 in your course.

∀x ∈ ℝ, x² ≥ 0 reads “for every real x, x squared is non-negative.” ∃x ∈ ℝ such that x² = 2 says at least one such number exists; it does not say the number is unique.

P ⇒ Q means “if P, then Q.” Its converse need not hold: x = 2 implies x² = 4, but x² = 4 also permits x = −2. P ⇔ Q asserts both directions. In ordinary mathematical logic, P ∨ Q includes the possibility that both are true.

Geometry and trigonometry symbols

∠ABC names the angle with vertex B. △ABC names a triangle. AB ∥ CD states that the indicated lines are parallel; AB ⊥ CD states they meet at a right angle. Congruent triangles, written △ABC ≅ △DEF, have corresponding equal sides and angles. Similar triangles, written with ∼, have equal corresponding angles and proportional corresponding sides.

For an acute angle θ in a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. A degree sign matters: sin(30°) = 1/2, while an unmarked input may be interpreted in radians.

Use the trigonometric calculator with the correct angle mode. The Pythagorean theorem connects exponent and equality notation: a² + b² = c² for a right triangle whose hypotenuse is c.

Calculus: derivatives, integrals and indexed sums

f′(x) and df/dx denote derivatives with respect to x. If f(x) = x², then f′(x) = 2x. For a multivariable function, ∂f/∂x differentiates with respect to x while holding the other independent variables fixed.

An indefinite integral denotes a family of antiderivatives: ∫ 2x dx = x² + C. A definite integral has bounds: ∫₀¹ 2x dx = 1. It gives signed accumulation; it is not always the unsigned geometric area when the function is negative.

∑_i=1⁴ i = 1 + 2 + 3 + 4 = 10
∏_i=1⁴ i = 1 × 2 × 3 × 4 = 24

In these indexed expressions, i is the index, 1 is the starting value and 4 is the final value. ∑ adds the terms; ∏ multiplies them. The infinity symbol ∞ is not an ordinary real number. A limit such as x → ∞ describes behavior as x grows without bound.

Big-O notation gives a bound relative to a comparison function, does not specify an exact growth rate or automatically a tight bound. For example, 3n + 2 = O(n) as n → ∞. See the math formula reference to connect notation with further formulas.

Probability and statistics notation

μ and σ commonly represent a population mean and population standard deviation. x̄ and s commonly represent their sample counterparts. For the sample 2, 4, 6, the mean is x̄ = (2 + 4 + 6)/3 = 4.

P(A ∣ B) is a conditional probability. For events with P(B) > 0, P(A ∣ B) = P(A ∩ B)/P(B). For example, on a fair six-sided die, let A mean “roll a 6” and B mean “roll an even number.” Then P(A) = 1/6 but P(A ∣ B) = 1/3.

X ∼ N(μ, σ²) commonly states that X has a normal distribution with mean μ and variance σ². Check the parameter convention in your source, because software and texts can parameterize distributions differently.

A p-value is not the probability that H₀ is true

Under the specified null model and its assumptions, a p-value measures how probable a test statistic at least as extreme as the observed one would be, using the test’s definition of “extreme.” It is not the probability that the null hypothesis is true or the probability that results happened “by chance.”

Use the statistics and probability formula sheet for formulas that use these quantities.

Typing math symbols and using LaTeX

Unicode is useful for text: copy π, ≤ or ∈ directly. A combining accent, such as the hat in p̂, is part of a multi-character sequence. Font support can differ across devices. Copying a symbol does not make a calculator understand its meaning.

LaTeX describes mathematical structure. In math mode, type \frac{a}{b} for a fraction, \sqrt{x} for a square root, \sum_{i=1}^{n} x_i for a sum, and A \subseteq B for a subset relation. Commands are case-sensitive: \sigma, \Sigma and \sum have different roles.

The table assumes standard LaTeX with amsmath and amssymb where needed, for example \operatorname, \mathbb, \nexists and \varnothing. For unusual glyphs, the table may give a clear equivalent expression rather than promise a command that every renderer supports. The command \coloneqq needs mathtools; \textfractionsolidus and \textperthousand need text-symbol support (such as textcomp) within \text. \rightangle requires a supporting package, such as unicode-math with a Unicode math engine. Some online equation editors provide only a subset of LaTeX.

Equation editors are helpful for fractions, limits, matrices and stacked indices. Keep the minus sign − distinct from a hyphen -, the letter x distinct from multiplication ×, and epsilon ε distinct from membership ∈. Always check the final display after pasting.

Frequently asked questions about math symbols

What are the most common mathematical symbols?

The basic operations are +, −, × and ÷. Common relationships include =, ≠, <, >, ≤ and ≥. You will also meet √ for square roots, π for the circle constant, ∈ for set membership, ∑ for summation and ∫ for integration.

What is the difference between = and ≡?

= states equality. ≡ can mark an identity or modular congruence. For instance, (x + 1)² ≡ x² + 2x + 1 is an identity, and 17 ≡ 5 (mod 12) means that 17 and 5 differ by a multiple of 12.

What does ∈ mean in math?

It means “is an element of.” If A = {1, 2, 3}, then 2 ∈ A and 4 ∉ A. Membership compares an object with a set; a subset relation compares two sets.

What is the difference between ⊂ and ⊆?

A ⊆ B permits A = B. Some authors use A ⊂ B for a proper subset, while others allow equality. A ⊊ B explicitly means A is a subset of B and A ≠ B. Follow the convention stated in your textbook.

What does the upside-down A symbol mean?

∀ means “for every” or “for all.” The statement ∀x ∈ ℝ, x² ≥ 0 says that the square of every real number is non-negative.

What does the backwards E symbol mean?

∃ means “there exists.” It asserts at least one example. The related notation ∃! means that exactly one example exists.

What does the vertical bar mean in mathematics?

Context decides: |x| can mean absolute value, |A| set cardinality or a determinant, a ∣ b divisibility, and P(A ∣ B) conditional probability. A bar in set-builder notation often reads “such that.”

How do I type the pi symbol?

Copy π from the reference, or type \pi in a LaTeX math field. Uppercase \Pi produces Π; \prod produces the product operator ∏.

Is this page a calculator?

It is a notation reference. Once you understand an expression, use the scientific calculator to evaluate supported calculations.

Sources & References

Definitions and examples on this page are written for this reference. These sources support the underlying notation, subject conventions and LaTeX guidance. Check your course’s notation when conventions vary.