Mathematical Symbols

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 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 \frac{6}{2}=3
mod modulo remainder calculation 7 mod 2 = 1
. period decimal point, decimal separator 2.56 = 2+56/100
a3 = 8
a^b caret exponent 2 ^ 3= 8
√a · √9 = ±3
a cube root a · a · a = a 8 = 2
a fourth root 44√4√4√4√16 = ±2
a n-th root (radical) for n=3, 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 Meaning / definition Example
angle ∠ABC = 30º
measured angle ABC = 30º
spherical angle AOB = 30º
right angle º
º degree º
´ arcminute º59′
´´ arcsecond º59’59”
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 º angle) |BC
|| parallel AB || CD
congruent to ≅ ∆XYZ
~ similarity ~ ∆XYZ
Δ triangle ≅ ΔBCD
|xy| distance distance between points x and y | xy | = 5
π pi constant is the ratio between the circumference and diameter of a circle c = π·d = 2·π·r
rad radians º = 2π rad
grad grads º = 400 grad

Algebra symbols


Symbol 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 = 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

a ∙ ba × bA ⊗ B

Symbol Symbol Name Meaning / definition Example
dot scalar product
× cross vector product
AB tensor product tensor product of A and 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 (AT)ij = (A)ji
A Hermitian matrix matrix conjugate transpose (A)ij = (A)ji
A (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(AB) 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 = 4
std(X) standard deviation standard deviation of random variable X std(X) = 2
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
ρ,XY = 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 = (2+5+9) / 3 = 5.333
s2 sample variance population samples variance estimator s2 = 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) = λeλx , x≥0
gamma(c, λ) gamma distribution f (x) = λ c xc-1eλx / Γ(c), x≥0
χ 2(k) chi-square distribution f (x) = xk/2-1ex/2 / ( 2k/2 Γ(k/2) )
F (k2) F distribution
Bin(n,p) binomial distribution f (k) = nCk pk(1-p)n-k
Poisson(λ) Poisson distribution f (k) = λkeλ / 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 Meaning / definition Example
{ } a collection of elements A = {3,7,9,14},
B = {9,14,28}
A ∩ B objects that belong to set A and set B A ∩ B = {9,14}
A ∪ B objects that belong to set A or set B A ∪ B = {3,7,9,14,28}
A ⊆ B subset has fewer elements or equal to the set {9,14,28} ⊆ {9,14,28}
A ⊂ B subset has fewer elements than the set {9,14} ⊂ {9,14,28}
A ⊄ B left set not a subset of right set {9,66} ⊄ {9,14,28}
A ⊇ B set A has more elements or equal to the set B {9,14,28} ⊇ {9,14,28}
A ⊃ B set A has more elements than set B {9,14,28} ⊃ {9,14}
A ⊅ B set A is not a superset of set B {9,14,28} ⊅ {9,66}
2power set all subsets of A
\mathcal{P}(A) all subsets of A
A = B both sets have the same members A={3,9,14},
B={3,9,14},
A=B
Ac all the objects that do not belong to set A
A \ B objects that belong to A and not to B A = {3,9,14},
B = {1,2,3},
A-B = {9,14}
A – B objects that belong to A and not to B A = {3,9,14},
B = {1,2,3},
A-B = {9,14}
A ∆ B 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 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 set membership A={3,9,14}, 3 ∈ A
x∉A no set membership A={3,9,14}, 1 ∉ A
(a,b) collection of 2 elements
A×B set of all ordered pairs from A and B
|A| the number of elements of set A A={3,9,14}, |A|=3
#A the number of elements of set A A={3,9,14}, #A=3
infinite cardinality of natural numbers set
cardinality of countable ordinal numbers set
Ø Ø = { } C = {Ø}
\mathbb{U} set of all possible values
\mathbb{N}0 \mathbb{N}0 = {0,1,2,3,4,…} 0 ∈ \mathbb{N}0
\mathbb{N}1 \mathbb{N}1 = {1,2,3,4,5,…} 6 ∈ \mathbb{N}1
\mathbb{Z} \mathbb{Z} = {…-3,-2,-1,0,1,2,3,…} -6 ∈ \mathbb{Z}
\mathbb{Q} \mathbb{Q} = {x | x=a/b, a,b\mathbb{Z}} 2/6 ∈ \mathbb{Q}
\mathbb{R} \mathbb{R} = {x | -∞ < x <∞} 6.343434 ∈ \mathbb{R}
\mathbb{C} \mathbb{C} = {z | z=a+bi, -∞<a<∞, -∞<b<∞} 6+2i\mathbb{C}

Logic symbols


Symbol Meaning / definition Example
· and and xy
^ caret / circumflex and x ^ y
& ampersand and x & y
+ plus or x + y
reversed caret or xy
| vertical line or x | y
x single quote not – negation x
x
¬ not not – negation ¬ x
! exclamation mark not – negation ! x
circled plus / oplus exclusive or – xor xy
~ tilde negation ~ x
implies
equivalent if and only if
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 – Leibniz’s notation (3x3)’ = 9x2
y second derivative derivative of derivative (3x3)” = 18x
yn times derivation (3x3)(3) = 18
\frac{dy}{dx} derivative derivative – Lagrange’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 notation
time 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 -1 z = 3 + 2i
z* complex conjugate z = a+biz*=abi z* = 3 + 2i
z = a+biz = abi z = 3 + 2i
nabla / del 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

Greek alphabet letters


Greek Symbol Greek Letter Name English Equivalent Pronunciation
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