Math Symbols List

List of all mathematical symbols and signs - meaning and examples.

Basic math symbols

Symbol	Symbol Name	Meaning / definition	Example
=	equals sign	equality	5 = 2+3 5 is equal to 2+3
≠	not equal sign	inequality	5 ≠ 4 5 is not equal to 4
≈	approximately equal	approximation	sin(0.01) ≈ 0.01, x ≈ y means x is approximately equal to y
>	strict inequality	greater than	5 > 4 5 is greater than 4
<	strict inequality	less than	4 < 5 4 is less than 5
≥	inequality	greater than or equal to	5 ≥ 4, x ≥ y means x is greater than or equal to y
≤	inequality	less than or equal to	4 ≤ 5, x ≤ y means x is less than or equal to y
( )	parentheses	calculate expression inside first	2 × (3+5) = 16
[ ]	brackets	calculate expression inside first	[(1+2)×(1+5)] = 18
+	plus sign	addition	1 + 1 = 2
−	minus sign	subtraction	2 − 1 = 1
±	plus - minus	both plus and minus operations	3 ± 5 = 8 or -2
±	minus - plus	both minus and plus operations	3 ∓ 5 = -2 or 8
*	asterisk	multiplication	2 * 3 = 6
×	times sign	multiplication	2 × 3 = 6
⋅	multiplication dot	multiplication	2 ⋅ 3 = 6
÷	division sign / obelus	division	6 ÷ 2 = 3
/	division slash	division	6 / 2 = 3
—	horizontal line	division / fraction
mod	modulo	remainder calculation	7 mod 2 = 1
.	period	decimal point, decimal separator	2.56 = 2+56/100
ab	power	exponent	23 = 8
a^b	caret	exponent	2 ^ 3 = 8
√a	square root	√a ⋅ √a  = a	√9 = ±3
3√a	cube root	3√a ⋅ 3√a  ⋅ 3√a  = a	3√8 = 2
4√a	fourth root	4√a ⋅ 4√a  ⋅ 4√a  ⋅ 4√a  = a	4√16 = ±2
n√a	n-th root (radical)		for n=3, n√8 = 2
%	percent	1% = 1/100	10% × 30 = 3
‰	per-mille	1‰ = 1/1000 = 0.1%	10‰ × 30 = 0.3
ppm	per-million	1ppm = 1/1000000	10ppm × 30 = 0.0003
ppb	per-billion	1ppb = 1/1000000000	10ppb × 30 = 3×10-7
ppt	per-trillion	1ppt = 10-12	10ppt × 30 = 3×10-10

Geometry symbols

Symbol	Symbol Name	Meaning / definition	Example
∠	angle	formed by two rays	∠ABC = 30°
	measured angle		ABC = 30°
	spherical angle		AOB = 30°
∟	right angle	= 90°	α = 90°
°	degree	1 turn = 360°	α = 60°
deg	degree	1 turn = 360deg	α = 60deg
′	prime	arcminute, 1° = 60′	α = 60°59′
″	double prime	arcsecond, 1′ = 60″	α = 60°59′59″
	line	infinite line
AB	line segment	line from point A to point B
	ray	line that start from point A
	arc	arc from point A to point B	= 60°
⊥	perpendicular	perpendicular lines (90° angle)	AC ⊥ BC
∥	parallel	parallel lines	AB ∥ CD
≅	congruent to	equivalence of geometric shapes and size	∆ABC≅ ∆XYZ
~	similarity	same shapes, not same size	∆ABC~ ∆XYZ
Δ	triangle	triangle shape	ΔABC≅ ΔBCD
|x-y|	distance	distance between points x and y	| x-y | = 5
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle	c = π⋅d = 2⋅π⋅r
rad	radians	radians angle unit	360° = 2π rad
c	radians	radians angle unit	360° = 2π c
grad	gradians / gons	grads angle unit	360° = 400 grad
g	gradians / gons	grads angle unit	360° = 400 g

Algebra symbols

Symbol	Symbol Name	Meaning / definition	Example
x	x variable	unknown value to find	when 2x = 4, then x = 2
≡	equivalence	identical to
≜	equal by definition	equal by definition
:=	equal by definition	equal by definition
~	approximately equal	weak approximation	11 ~ 10
≈	approximately equal	approximation	sin(0.01) ≈ 0.01
∝	proportional to	proportional to	y ∝ x when y = kx, k constant
∞	lemniscate	infinity symbol
≪	much less than	much less than	1 ≪ 1000000
≫	much greater than	much greater than	1000000 ≫ 1
( )	parentheses	calculate expression inside first	2 * (3+5) = 16
[ ]	brackets	calculate expression inside first	[(1+2)*(1+5)] = 18
{ }	braces	set
⌊x⌋	floor brackets	rounds number to lower integer	⌊4.3⌋ = 4
⌈x⌉	ceiling brackets	rounds number to upper integer	⌈4.3⌉ = 5
x!	exclamation mark	factorial	4! = 1*2*3*4 = 24
| x |	vertical bars	absolute value	| -5 | = 5
f (x)	function of x	maps values of x to f(x)	f (x) = 3x+5
(f ∘ g)	function composition	(f ∘ g) (x) = f (g(x))	f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)
(a,b)	open interval	(a,b) = {x | a < x < b}	x∈ (2,6)
[a,b]	closed interval	[a,b] = {x | a ≤ x ≤ b}	x ∈ [2,6]
∆	delta	change / difference	∆t = t1 - t0
∆	discriminant	Δ = b2 - 4ac
∑	sigma	summation - sum of all values in range of series	∑ xi= x1+x2+...+xn
∑∑	sigma	double summation
∏	capital pi	product - product of all values in range of series	∏ xi=x1∙x2∙...∙xn
e	e constant / Euler's number	e = 2.718281828...	e = lim (1+1/x)x , x→∞
γ	Euler-Mascheroni constant	γ = 0.5772156649...
φ	golden ratio	golden ratio constant
π	pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle	c = π⋅d = 2⋅π⋅r

Linear Algebra Symbols

Symbol	Symbol Name	Meaning / definition	Example
·	dot	scalar product	a · b
×	cross	vector product	a × b
A⊗B	tensor product	tensor product of A and B	A ⊗ B
	inner product
[ ]	brackets	matrix of numbers
( )	parentheses	matrix of numbers
| A |	determinant	determinant of matrix A
det(A)	determinant	determinant of matrix A
|| x ||	double vertical bars	norm
AT	transpose	matrix transpose	(AT)ij = (A)ji
A†	Hermitian matrix	matrix conjugate transpose	(A†)ij = (A)ji
A*	Hermitian matrix	matrix conjugate transpose	(A*)ij = (A)ji
A -1	inverse matrix	A A-1 = I
rank(A)	matrix rank	rank of matrix A	rank(A) = 3
dim(U)	dimension	dimension of matrix A	dim(U) = 3

Probability and statistics symbols

Symbol	Symbol Name	Meaning / definition	Example
P(A)	probability function	probability of event A	P(A) = 0.5
P(A ⋂ B)	probability of events intersection	probability that of events A and B	P(A⋂B) = 0.5
P(A ⋃ B)	probability of events union	probability that of events A or B	P(A⋃B) = 0.5
P(A | B)	conditional probability function	probability of event A given event B occured	P(A | B) = 0.3
f (x)	probability density function (pdf)	P(a ≤ x ≤ b) = ∫ f (x) dx
F(x)	cumulative distribution function (cdf)	F(x) = P(X≤ x)
μ	population mean	mean of population values	μ = 10
E(X)	expectation value	expected value of random variable X	E(X) = 10
E(X | Y)	conditional expectation	expected value of random variable X given Y	E(X | Y=2) = 5
var(X)	variance	variance of random variable X	var(X) = 4
σ2	variance	variance of population values	σ2 = 4
std(X)	standard deviation	standard deviation of random variable X	std(X) = 2
σX	standard deviation	standard deviation value of random variable X	σX  = 2
	median	middle value of random variable x
cov(X,Y)	covariance	covariance of random variables X and Y	cov(X,Y) = 4
corr(X,Y)	correlation	correlation of random variables X and Y	corr(X,Y) = 0.6
ρX,Y	correlation	correlation of random variables X and Y	ρX,Y = 0.6
∑	summation	summation - sum of all values in range of series
∑∑	double summation	double summation
Mo	mode	value that occurs most frequently in population
MR	mid-range	MR = (xmax+xmin)/2
Md	sample median	half the population is below this value
Q1	lower / first quartile	25% of population are below this value
Q2	median / second quartile	50% of population are below this value = median of samples
Q3	upper / third quartile	75% of population are below this value
x	sample mean	average / arithmetic mean	x = (2+5+9) / 3 = 5.333
s 2	sample variance	population samples variance estimator	s 2 = 4
s	sample standard deviation	population samples standard deviation estimator	s = 2
zx	standard score	zx = (x-x) / sx
X ~	distribution of X	distribution of random variable X	X ~ N(0,3)
N(μ,σ2)	normal distribution	gaussian distribution	X ~ N(0,3)
U(a,b)	uniform distribution	equal probability in range a,b	X ~ U(0,3)
exp(λ)	exponential distribution	f (x) = λe-λx , x≥0
gamma(c, λ)	gamma distribution	f (x) = λ c xc-1e-λx / Γ(c), x≥0
χ 2(k)	chi-square distribution	f (x) = xk/2-1e-x/2 / ( 2k/2 Γ(k/2) )
F (k1, k2)	F distribution
Bin(n,p)	binomial distribution	f (k) = nCk pk(1-p)n-k
Poisson(λ)	Poisson distribution	f (k) = λke-λ / k!
Geom(p)	geometric distribution	f (k) =  p(1-p) k
HG(N,K,n)	hyper-geometric distribution
Bern(p)	Bernoulli distribution

Combinatorics Symbols

Symbol	Symbol Name	Meaning / definition	Example
n!	factorial	n! = 1⋅2⋅3⋅...⋅n	5! = 1⋅2⋅3⋅4⋅5 = 120
nPk	permutation		5P3 = 5! / (5-3)! = 60
nCk	combination		5C3 = 5!/[3!(5-3)!]=10

Set theory symbols

Symbol	Symbol Name	Meaning / definition	Example
{ }	set	a collection of elements	A = {3,7,9,14}, B = {9,14,28}
A ∩ B	intersection	objects that belong to set A and set B	A ∩ B = {9,14}
A ∪ B	union	objects that belong to set A or set B	A ∪ B = {3,7,9,14,28}
A ⊆ B	subset	A is a subset of B. set A is included in set B.	{9,14,28} ⊆ {9,14,28}
A ⊂ B	proper subset / strict subset	A is a subset of B, but A is not equal to B.	{9,14} ⊂ {9,14,28}
A ⊄ B	not subset	set A is not a subset of set B	{9,66} ⊄ {9,14,28}
A ⊇ B	superset	A is a superset of B. set A includes set B	{9,14,28} ⊇ {9,14,28}
A ⊃ B	proper superset / strict superset	A is a superset of B, but B is not equal to A.	{9,14,28} ⊃ {9,14}
A ⊅ B	not superset	set A is not a superset of set B	{9,14,28} ⊅ {9,66}
2A	power set	all subsets of A
	power set	all subsets of A
A = B	equality	both sets have the same members	A={3,9,14}, B={3,9,14}, A=B
Ac	complement	all the objects that do not belong to set A
A \ B	relative complement	objects that belong to A and not to B	A = {3,9,14}, B = {1,2,3}, A-B = {9,14}
A - B	relative complement	objects that belong to A and not to B	A = {3,9,14}, B = {1,2,3}, A-B = {9,14}
A ∆ B	symmetric difference	objects that belong to A or B but not to their intersection	A = {3,9,14}, B = {1,2,3}, A ∆ B = {1,2,9,14}
A ⊖ B	symmetric difference	objects that belong to A or B but not to their intersection	A = {3,9,14}, B = {1,2,3}, A ⊖ B = {1,2,9,14}
a∈A	element of, belongs to	set membership	A={3,9,14}, 3 ∈ A
x∉A	not element of	no set membership	A={3,9,14}, 1 ∉ A
(a,b)	ordered pair	collection of 2 elements
A×B	cartesian product	set of all ordered pairs from A and B	A×B = {(a,b)|a∈A , b∈B}
|A|	cardinality	the number of elements of set A	A={3,9,14}, |A|=3
#A	cardinality	the number of elements of set A	A={3,9,14}, #A=3
|	vertical bar	such that	A={x|3<x<14}
	aleph-null	infinite cardinality of natural numbers set
	aleph-one	cardinality of countable ordinal numbers set
Ø	empty set	Ø = { }	C = {Ø}
	universal set	set of all possible values
0	natural numbers / whole numbers  set (with zero)	0 = {0,1,2,3,4,...}	0 ∈ 0
1	natural numbers / whole numbers  set (without zero)	1 = {1,2,3,4,5,...}	6 ∈ 1
	integer numbers set	= {...-3,-2,-1,0,1,2,3,...}	-6 ∈
	rational numbers set	= {x | x=a/b, a,b∈}	2/6 ∈
	real numbers set	= {x | -∞ < x <∞}	6.343434∈
	complex numbers set	= {z | z=a+bi, -∞<a<∞,      -∞<b<∞}	6+2i ∈

Logic symbols

Calculus & analysis symbols

Symbol	Symbol Name	Meaning / definition	Example
	limit	limit value of a function
ε	epsilon	represents a very small number, near zero	ε → 0
e	e constant / Euler's number	e = 2.718281828...	e = lim (1+1/x)x , x→∞
y '	derivative	derivative - Lagrange's notation	(3x3)' = 9x2
y ''	second derivative	derivative of derivative	(3x3)'' = 18x
y(n)	nth derivative	n times derivation	(3x3)(3) = 18
	derivative	derivative - Leibniz's notation	d(3x3)/dx = 9x2
	second derivative	derivative of derivative	d2(3x3)/dx2 = 18x
	nth derivative	n times derivation
	time derivative	derivative by time - Newton's notation
	time second derivative	derivative of derivative
Dx y	derivative	derivative - Euler's notation
Dx2y	second derivative	derivative of derivative
	partial derivative		∂(x2+y2)/∂x = 2x
∫	integral	opposite to derivation	∫ f(x)dx
∫∫	double integral	integration of function of 2 variables	∫∫ f(x,y)dxdy
∫∫∫	triple integral	integration of function of 3 variables	∫∫∫ f(x,y,z)dxdydz
∮	closed contour / line integral
∯	closed surface integral
∰	closed volume integral
[a,b]	closed interval	[a,b] = {x | a ≤ x ≤ b}
(a,b)	open interval	(a,b) = {x | a < x < b}
i	imaginary unit	i ≡ √-1	z = 3 + 2i
z*	complex conjugate	z = a+bi → z*=a-bi	z* = 3 - 2i
z	complex conjugate	z = a+bi → z = a-bi	z = 3 - 2i
Re(z)	real part of a complex number	z = a+bi → Re(z)=a	Re(3 - 2i) = 3
Im(z)	imaginary part of a complex number	z = a+bi → Im(z)=b	Im(3 - 2i) = -2
| z |	absolute value/magnitude of a complex number	|z| = |a+bi| = √(a2+b2)	|3 - 2i| = √13
arg(z)	argument of a complex number	The angle of the radius in the complex plane	arg(3 + 2i) = 33.7°
∇	nabla / del	gradient / divergence operator	∇f (x,y,z)
	vector
	unit vector
x * y	convolution	y(t) = x(t) * h(t)
	Laplace transform	F(s) = {f (t)}
	Fourier transform	X(ω) = {f (t)}
δ	delta function
∞	lemniscate	infinity symbol

Numeral symbols

Greek alphabet letters

Roman numerals