Math Symbols

Symbol	Pronunciation	Short Description	Example
Grouping Symbols
$( )$	parentheses	Used to group expressions and indicate order of operations	$(b + c)$
$[ ]$	square brackets or brackets	Used for intervals, matrices, and function arguments	$[0, 1]$
$[a, b)$	interval from a to b	The interval from a to b, inclusive of a and exclusive of b	$[0, 1)$
$\{ \}$	curly braces or braces	Used for sets, piecewise functions, and grouping multiple elements	$\{1, 2, 3\}$
$⟨ ⟩$	angle brackets	Used for inner products, expected values, and Dirac notation	$\langle x, y \rangle$
$⌊ \ ⌋$	floor brackets	Floor function - rounds down to nearest integer	$\lfloor 3.7 \rfloor = 3$
$⌈ \ ⌉$	ceiling brackets	Ceiling function - rounds up to nearest integer	$\lceil 3.2 \rceil = 4$
$\lvert \ \rvert$	mod, modulo, absolute	Used for norms, determinants, and absolute values	$\lvert x \rvert$
Powers and Roots
$x^2$	x squared	Square of a number (raised to the power of 2)	$3^2 = 9$
$x^3$	x cubed	Cube of a number (raised to the power of 3)	$2^3 = 8$
$x^n$	x to the n or x to the nth power	Number raised to the power of n	$4^n$
$\sqrt{x}$	square root of x	Principal square root of a number	$\sqrt{16} = 4$
$\sqrt[3]{x}$	cube root of x	Root that when cubed gives the original number	$\sqrt[3]{8} = 2$
$\sqrt[n]{x}$	nth root of x	Root that when raised to the nth power gives the original number	$\sqrt[4]{16} = 2$
Calculus Symbols
$\frac{d}{dx}$	d over dx	First derivative	$\frac{d}{dx}(x^2) = 2x$
$\int$	integral	Indefinite integral	$\int x^2 dx = \frac{x^3}{3} + C$
$\int_a^b$	integral from a to b	Definite integral	$\int_0^1 x^2 dx = \frac{1}{3}$
$\oint$	contour integral	Closed path integral	$\oint_C \vec{F} \cdot d\vec{r}$
$\iint$	double integral	Double integral	$\iint_R f(x,y) dA$
$\lim$	limit	Limit of a function	$\lim_{x \to 0} \frac{\sin x}{x} = 1$
$\lim_{x \to a}$	limit as x approaches a	Limit as x approaches a	$\lim_{x \to 2} (x^2 - 4)/(x - 2) = 4$
$\Delta$	delta	Change in quantity	$\Delta x = x_2 - x_1$
$\infty$	infinity	Infinite value	$\lim_{x \to \infty} 1/x = 0$
$\sum_{i=1}^n$	summation from 1 to n	Summation notation	$\sum_{i=1}^n i = \frac{n(n+1)}{2}$
$\sum_{n=1}^{\infty} a_n$	sum of a sub n from n equals 1 to infinity	Infinite series	$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$
$\sum_{k=0}^n \binom{n}{k}$	sum from k equals zero to n of n choose k	Sum of binomial coefficients (equals $2^n$)	$\sum_{k=0}^n \binom{n}{k} = 2^n$
Algebra Symbols
$≡$	identical to or congruent to	Mathematical congruence or identity	$x² + 2x + 1 ≡ (x+1)²$
$aₙ$	a sub n	Nth term of sequence	$aₙ = 2n + 1$
$n \choose k$	n choose k	Binomial coefficient	$5 \choose 2 = 10$
$!$	factorial	Product of first n natural numbers	$5! = 120$
$ⁿPᵣ$	n P r	Permutation	$⁵P₂ = 20$
$ⁿCᵣ$	n C r	Combination	$⁵C₂ = 10$
$\log_b a$	log base b of a	Logarithm	$\log_2 8 = 3$
$ln x$	natural log of x	Natural logarithm (base e)	$ln e = 1$
$a:b$	a to b	Ratio of a to b	$a:b = 2:3$
$a:b::c:d$	a to b as c to d	Proportion	$2:4::3:6$
Probability Symbols
$P(A)$	probability of A	Probability of event A occurring	$P(\text{heads}) = 0.5$
$P(A \mid B)$	probability of A given B	Conditional probability of A occurring given B has occurred	$P(\text{rain} \mid \text{clouds}) = 0.8$
$P(A \cup B)$	probability of A union B or probability of A or B	Probability that either A or B (or both) occurs	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$P(A \cap B)$	probability of A intersection B or probability of A and B	Probability that both A and B occur	$P(A \cap B) = 0.2$
$P(A')$	probability of A complement	Probability that A does not occur	$P(A') = 1 - P(A)$
$E[X]$	expected value of X	Average value of random variable X	$E[X] = \sum x_i p_i$
$\bar{x}$	x bar	The average of x (sample mean)	$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$
$f(x)$	f of x	Probability density function	$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
$F(x)$	F of x	Cumulative distribution function	$F(x) = P(X \leq x)$
$∴$	therefore	Logical conclusion	$x = 2, ∴ x² = 4$
$∵$	because	Reason or cause	$x² = 4 ∵ x = 2$
$⇒$	implies	Logical implication	$x > 0 ⇒ x² > 0$
$⇐$	implied by	Reverse implication	$x² > 0 ⇐ x > 0$
$⇔$	if and only if	Biconditional	$x = 1 ⇔ x² = 1$
$\partial$	partial	Partial derivative	$\frac{\partial f}{\partial x}$
Linear Algebra
$\vec{v}$	vector v	A vector in a vector space	$\vec{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$
$\mathbf{A} \odot \mathbf{B}$	A Hadamard B or element-wise product	Hadamard (element-wise) product	$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \odot \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 5 & 12 \\ 21 & 32 \end{bmatrix}$
Set Theory Symbols
$∀$	for all	Universal quantifier	$∀x ∈ ℝ, x² ≥ 0$
$\in$	belongs to or element of	Indicates membership of an element in a set	$x \in A$ means x is an element of set A
$\notin$	does not belong to or not element of	Indicates non-membership of an element in a set	$x \notin A$ means x is not an element of set A
$\subseteq$	subset of or contained in	Set A is a subset of set B (includes equality)	$A \subseteq B$ means every element of A is in B
$\subset$	proper subset of	Set A is a proper subset of set B (strict inclusion)	$A \subset B$ means A is subset of B but A ≠ B
$\supseteq$	superset of or contains	Set A is a superset of set B (includes equality)	$A \supseteq B$ means every element of B is in A
$\supset$	proper superset of	Set A is a proper superset of set B (strict inclusion)	$A \supset B$ means A is superset of B but A ≠ B
$\cup$	union or cup	Set of elements in either set	$A \cup B = \{x \mid x \in A \lor x \in B\}$
$\cap$	intersection or cap	Set of elements common to both sets	$A \cap B = \{x \mid x \in A \land x \in B\}$
$\setminus$	set minus or without	Set difference (elements in first set but not second)	$A \setminus B = \{x \mid x \in A \land x \notin B\}$
$\triangle$	symmetric difference	Set of elements in exactly one of the two sets	$A \triangle B = (A \setminus B) \cup (B \setminus A)$
$\emptyset$	empty set or null set	Set with no elements	$\emptyset = \{\}$
$\vert A \vert$	cardinality of A or size of A	Number of elements in set A	$\vert A \vert = n$ means A has n elements
$\mathbb{N}$	natural numbers	Set of non-negative integers	$\mathbb{N} = \{0, 1, 2, 3, \ldots\}$
$\mathbb{Z}$	integers	Set of all integers	$\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$
$\mathbb{Q}$	rational numbers	Set of numbers that can be expressed as fractions	$\mathbb{Q} = \{\frac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0\}$
$\mathbb{R}$	real numbers	Set of all real numbers	$\mathbb{R}$ includes all rational and irrational numbers
$\mathbb{C}$	complex numbers	Set of numbers of the form a + bi	$\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$
$\mathbb{A}$	algebraic numbers	Complex numbers that are roots of polynomials	$\sqrt{2}, i \in \mathbb{A}$ but $\pi \notin \mathbb{A}$
$\mathbb{I}$	irrational numbers	Real numbers that are not rational	$\pi, e, \sqrt{2} \in \mathbb{I}$