Full Math Symbols List.txt

Full 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
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

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

Name
Western Arabic
Roman
Eastern 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
forty
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

Upper Case Letter
Lower Case Letter
Greek Letter Name
English Equivalent 
Letter Name Pronounce
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