mathematical symbol

Basic symbols[edit]

Symbol in HTML	Symbol in TeX	Name	Explanation	Examples
		Read as
		Category
+	${\displaystyle +\!\,}$	addition plus; add arithmetic	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
		disjoint union the disjoint union of ... and ... set theory	A1 + A2 means the disjoint union of sets A1 and A2.	A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒ A1 + A2 = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)}
−	${\displaystyle -\!\,}$	subtraction minus; take; subtract arithmetic	36 − 11 means the subtraction of 11 from 36.	36 − 11 = 25
		negative sign negative; minus; the opposite of arithmetic	−3 means the additive inverse of the number 3.	−(−5) = 5
		set-theoretic complement minus; without set theory	A − B means the set that contains all the elements of A that are not in B. (∖ can also be used for set-theoretic complement as described below.)	{1, 2, 4} − {1, 3, 4} = {2}
±	${\displaystyle \pm }$ \pm	plus-minus plus or minus arithmetic	6 ± 3 means both 6 + 3 and 6 − 3.	The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. Note: {{sqrt|4}} was used to get √4.
		plus-minus plus or minus measurement	10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.	If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
∓	${\displaystyle \mp \!\,}$ \mp	minus-plus minus or plus arithmetic	6 ± (3 ∓ 5) means 6 + (3 − 5) and 6 − (3 + 5).	cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
× ⋅ ·	${\displaystyle \times }$ ${\displaystyle \cdot }$	multiplication times; multiplied by arithmetic	3 × 4 or 3 ⋅ 4 means the multiplication of 3 by 4.	7 ⋅ 8 = 56
		dot product scalar product dot linear algebra vector algebra	u ⋅ v means the dot product of vectors u and v	(1, 2, 5) ⋅ (3, 4, −1) = 6
		cross product vector product cross linear algebra vector algebra	u × v means the cross product of vectors u and v	(1, 2, 5) × (3, 4, −1) = i j k 1 2 5 3 4 −1 = (−22, 16, −2)
		placeholder (silent) functional analysis	A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument.	| · |
÷ ⁄	${\displaystyle \div \!\,}$ ${\displaystyle /\!\,}$	division (Obelus) divided by; over arithmetic	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = 0.5 12 ⁄ 4 = 3
		quotient group mod group theory	G / H means the quotient of group G modulo its subgroup H.	{0, a, 2a, b, b + a, b + 2a} / {0, b} = {{0, b}, {a, b + a}, {2a, b + 2a}}
		quotient set mod set theory	A/~ means the set of all ~ equivalence classes in A.	If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then ℝ/~ = {x + n : n ∈ ℤ, x ∈ [0,1)}.
√	${\displaystyle \surd }$ ${\displaystyle {\sqrt {\ }}}$	square root the (principal) square root of real numbers	√x means the nonnegative number whose square is x.	√4 = 2
		complex square root the (complex) square root of complex numbers	If z = r exp(iφ) is represented in polar coordinates with −π < φ ≤ π, then √z = √r exp(iφ/2).	√−1 = i
∑	${\displaystyle \sum }$	summation sum over ... from ... to ... of arithmetic	${\displaystyle \sum _{k=1}^{n}{a_{k}}}$ means ${\displaystyle a_{1}+a_{2}+\cdots +a_{n}}$.	${\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30}$
∫	${\displaystyle \int }$ \int	indefinite integral or antiderivative indefinite integral of - OR - the antiderivative of calculus	∫ f(x) dx means a function whose derivative is f.	${\displaystyle \int x^{2}dx={\frac {x^{3}}{3}}+C}$
		definite integral integral from ... to ... of ... with respect to calculus	∫b a f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.	∫b a x2 dx = b3 − a3/3
		line integral line/ path/ curve/ integral of ... along ... calculus	∫ C f ds means the integral of f along the curve C, ∫b a f(r(t)) | r'(t) | dt, where r is a parametrization of C. (If the curve is closed, the symbol ∮ may be used instead, as described below.)
∮	${\displaystyle \oint \!\,}$ \oint	Contour integral; closed line integral contour integral of calculus	Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰ . The contour integral can also frequently be found with a subscript capital letter C, ∮ C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮ S, is used to denote that the integration is over a closed surface.	If C is a Jordan curve about 0, then ∮ C 1/z dz = 2πi.
∴	${\displaystyle \therefore \!\,}$ \therefore	therefore therefore; so; hence everywhere	Sometimes used in proofs before logical consequences.	All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
∵	${\displaystyle \because \!\,}$ \because	because because; since everywhere	Sometimes used in proofs before reasoning.	11 is prime ∵ it has no positive integer factors other than itself and one.
!	${\displaystyle !\!\,}$	factorial factorial combinatorics	n! means the product 1 × 2 × ... × n.	${\displaystyle 4!=1\times 2\times 3\times 4=24}$
		logical negation not propositional logic	The statement !A is true if and only if A is false. A slash placed through another operator is the same as "!" placed in front. (The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.)	!(!A) ⇔ A x ≠ y ⇔  !(x = y)
¬ ˜	${\displaystyle \neg \!\,}$ ${\displaystyle \sim \!\,}$ \neg	logical negation not propositional logic	The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use ! but this is avoided in mathematical texts.)	¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y)
∝	${\displaystyle \propto \!\,}$ \propto	proportionality is proportional to; varies as everywhere	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x.
∞	${\displaystyle \infty \!\,}$ \infty	infinity infinity numbers	∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.	${\displaystyle \lim _{x\to 0}{\frac {1}{|x|}}=\infty }$
■ □ ∎ ▮ ‣	${\displaystyle \blacksquare \!\,}$ ${\displaystyle \Box \!\,}$ ${\displaystyle \blacktriangleright \!\,}$ \blacksquare, \Box, \blacktriangleright	end of proof QED; tombstone; Halmos finality symbol everywhere	Used to mark the end of a proof. (May also be written Q.E.D.)	(1) a + 0 := a   (def.) (2) a + succ(b) := succ(a + b)   (def.) Proposition. 3 + 2 = 5. Proof. 3 + 2 = 3 + succ(1)   (definition of succ) 3 + succ(1) = succ(3 + 1)   (2) succ(3 + 1) = succ(3 + succ(0))   (definition of succ) succ(3 + succ(0)) = succ(succ(3 + 0))   (2) succ(succ(3 + 0)) = succ(succ(3))   (1) succ(succ(3)) = succ(4) = 5   (definition of succ) ▮