Mathematical Symbols
Symbol
|
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
|
3 ∓ 5 = -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
|
|
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
|
Symbol
|
Symbol Name
|
Meaning / definition
|
Example
|
∠
|
angle
|
formed by two rays
|
∠ABC = 30º
|
measured angle
|
|||
spherical angle
|
|||
∟
|
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
|
||
|
|
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
|
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 | 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 = 0.527721566...
|
|
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
|
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
|
|
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
|
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
|
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
|
|
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
|
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|
|
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}
|
universal set
|
set of all possible values
|
||
natural numbers / whole numbers set
(with zero)
|
0 ∈
|
||
natural numbers / whole numbers set
(without zero)
|
6 ∈
|
||
integer numbers set
|
-6 ∈
|
||
rational numbers set
|
2/6 ∈
|
||
real numbers set
|
6.343434 ∈
|
||
complex numbers set
|
6+2i ∈
|
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
|
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
|
|
Dx2 y
|
second derivative
|
derivative of derivative
|
|
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+bi → z*=a-bi
|
z* = 3 + 2i
|
z
|
complex conjugate
|
z = a+bi → z = 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) =
|
||
Fourier transform
|
X(ω) =
|
||
δ
|
delta function
|
||
∞
|
lemniscate
|
infinity symbol
|
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 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
|
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
|
