Mathematical symbols
                     




= equals sign equality 5 = 2+35 is equal to 2+3

≠ not equal sign inequality 5 ≠ 45 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 > 45 is greater than 4

< strict inequality less than 4 < 54 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 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

∠ 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

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 | 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 = 0.5772156649...  

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

· 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 rank(U) = 3

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 probabilityfunction
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 σ
2 = 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 varianceestimator 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 x

c-1e-λx / Γ(c), x≥0  

χ 2(k) chi-square distribution f (x) = x
k/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   
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 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 Bbut 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  

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 ∈ 

⋅ 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   

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
∇ nabla / del gradient / divergence ∇f (x,y,z)


operator
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  
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 אי
twelve 12 XII ١٢ בי
thirteen 13 XIII ١٣ גי
fourteen 14 XIV ١٤ די
fifteen 15 XV ١٥ וט
sixteen 16 XVI ١٦ זט
seventeen 17 XVII ١٧ זי
eighteen 18 XVIII ١٨ חי
nineteen 19 XIX ١٩ טי
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

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
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