Mathematical Symbols

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
not equal sign inequality 5 ≠ 4
> strict inequality greater than 5 > 4
< strict inequality less than 4 < 5
inequality greater than or equal to 5 ≥ 4
inequality less than or equal to 4 ≤ 5
( ) 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 and -2
minus - plus both minus and plus operations 35 = -2 and 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 \frac{6}{2}=3
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
3a cube root 3a · 3a  · 3a  = a 38 = 2
4a fourth root 4a · 4a  · 4a  · 4a  = a 416 = ±2
na n-th root (radical)   for n=3, n8 = 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 = 30º
measured angle   ABC = 30º
spherical angle   AOB = 30º
right angle = 90º α = 90º
º degree 1 turn = 360º α = 60º
´ arcminute 1º = 60´ α = 60º59'
´´ arcsecond 1´ = 60´´ α = 60º59'59''
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 = 60º
| perpendicular perpendicular lines (90º 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 360º = 2π rad
grad grads grads angle unit 360º = 400 grad

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

f(x) g(x)

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 | single vertical bar 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 | axb} 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 = 0.527721566...  
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
AB tensor product tensor product of A and B A B
\langle x,y \rangle 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  
A T 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

rank(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(AB) probability of events intersection probability that of events A and B P(AB) = 0.5
P(A B) probability of events union probability that of events A or B P(AB) = 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 _{n}P_{k}=\frac{n!}{(n-k)!} 5P3 = 5! / (5-3)! = 60
nCk

 

combination _{n}C_{k}=\binom{n}{k}=\frac{n!}{k!(n-k)!} 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 subset has fewer elements or equal to the set {9,14,28} {9,14,28}
A B proper subset / strict subset subset has fewer elements than the set {9,14} {9,14,28}
A B not subset left set not a subset of right set {9,66} {9,14,28}
A B superset set A has more elements or equal to the set B {9,14,28}{9,14,28}
A B proper superset / strict superset set A has more elements than set B {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  
\mathcal{P}(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}
aA element of set membership  A={3,9,14}, 3 A
xA 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| 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
aleph-null infinite cardinality of natural numbers set  
aleph-one cardinality of countable ordinal numbers set  
Ø empty set Ø = { } C = {Ø}
\mathbb{U} universal set set of all possible values  
\mathbb{N}0 natural numbers / whole numbers  set (with zero) \mathbb{N}0 = {0,1,2,3,4,...} 0 \mathbb{N}0
\mathbb{N}1 natural numbers / whole numbers  set (without zero) \mathbb{N}1 = {1,2,3,4,5,...} 6 \mathbb{N}1
\mathbb{Z} integer numbers set \mathbb{Z} = {...-3,-2,-1,0,1,2,3,...} -6 \mathbb{Z}
\mathbb{Q} rational numbers set \mathbb{Q} = {x | x=a/b, a,b\mathbb{Z}} 2/6 \mathbb{Q}
\mathbb{R} real numbers set \mathbb{R} = {x | -∞ < x <∞} 6.343434 \mathbb{R}
\mathbb{C}  complex numbers set \mathbb{C} = {z | z=a+bi, -∞<a<∞,      -∞<b<∞} 6+2i \mathbb{C}

Logic symbols

Symbol Symbol Name Meaning / definition Example
· and and x · y
^ caret / circumflex and x ^ y
& ampersand and x & y
+ plus or x + y
reversed caret or x y
| vertical line or x | y
x' single quote not - negation x'
x bar not - negation x
¬ not not - negation ¬ x
! exclamation mark not - negation ! x
circled plus / oplus exclusive or - xor x y
~ tilde negation ~ x
implies    
equivalent if and only if (iff)  
equivalent if and only if (iff)  
for all    
there exists    
there does not exists    
therefore    
because / since    

Calculus & analysis symbols

Symbol Symbol Name Meaning / definition Example
\lim_{x\to x0}f(x) 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
\frac{dy}{dx} derivative derivative - Leibniz's notation d(3x3)/dx = 9x2
\frac{d^2y}{dx^2} second derivative derivative of derivative d2(3x3)/dx2 = 18x
\frac{d^ny}{dx^n} nth derivative n times derivation  
\dot{y} time derivative derivative by time - Newton's notation  
time second derivative derivative of derivative  
Dx y derivative derivative - Euler's notation  
Dx2 y second derivative derivative of derivative  
\frac{\partial f(x,y)}{\partial x} partial derivative   ∂(x2+y2)/∂x = 2x
integral opposite to derivation  
double integral integration of function of 2 variables  
triple integral integration of function of 3 variables  
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+biz*=a-bi z* = 3 + 2i
z complex conjugate z = a+biz = a-bi z = 3 + 2i
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

Name European Roman Hindu Arabic Hebrew
zero 0   ٠  
one 1 I ١ א
two 2 II ٢ ב
three 3 III ٣ ג
four 4 IV ٤ ד
five 5 V ٥ ה
six 6 VI ٦ ו
seven 7 VII ٧ ז
eight 8 VIII ٨ ח
nine 9 IX ٩ ט
ten 10 X ١٠ י
eleven 11 XI ١١ יא
twelve 12 XII ١٢ יב
thirteen 13 XIII ١٣ יג
fourteen 14 XIV ١٤ יד
fifteen 15 XV ١٥ טו
sixteen 16 XVI ١٦ טז
seventeen 17 XVII ١٧ יז
eighteen 18 XVIII ١٨ יח
nineteen 19 XIX ١٩ יט
twenty 20 XX ٢٠ כ
thirty 30 XXX ٣٠ ל
fourty 40 XL ٤٠ מ
fifty 50 L ٥٠ נ
sixty 60 LX ٦٠ ס
seventy 70 LXX ٧٠ ע
eighty 80 LXXX ٨٠ פ
ninety 90 XC ٩٠ צ
one hundred 100 C ١٠٠ ק

 

Greek alphabet letters

Greek Symbol Greek Letter Name English Equivalent Pronunciation
Upper Case Lower Case
Α α Alpha a al-fa
Β β Beta b be-ta
Γ γ Gamma g ga-ma
Δ δ Delta d del-ta
Ε ε Epsilon e ep-si-lon
Ζ ζ Zeta z ze-ta
Η η Eta h eh-ta
Θ θ Theta th te-ta
Ι ι Iota i io-ta
Κ κ Kappa k ka-pa
Λ λ Lambda l lam-da
Μ μ Mu m m-yoo
Ν ν Nu n noo
Ξ ξ Xi x x-ee
Ο ο Omicron o o-mee-c-ron
Π π Pi p pa-yee
Ρ ρ Rho r row
Σ σ Sigma s sig-ma
Τ τ Tau t ta-oo
Υ υ Upsilon u oo-psi-lon
Φ φ Phi ph f-ee
Χ χ Chi ch kh-ee
Ψ ψ Psi ps p-see
Ω ω Omega o o-me-ga

Roman numerals

Number Roman numeral
0 not defined
1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10 X
11 XI
12 XII
13 XIII
14 XIV
15 XV
16 XVI
17 XVII
18 XVIII
19 XIX
20 XX
30 XXX
40 XL
50 L
60 LX
70 LXX
80 LXXX
90 XC
100 C
200 CC
300 CCC
400 CD
500 D
600 DC
700 DCC
800 DCCC
900 CM
1000 M
5000 V
10000 X
50000 L
100000 C
500000 D
1000000 M

 


See also

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