Math Symbols List
List of all mathematical symbols and signs - meaning and examples.
Basic math symbols
Symbol Symbol Name Meaning / definition Example
5 = 2+3
= equals sign equality
5 is equal to 2+3
5≠4
≠ not equal sign inequality
5 is not equal to 4
sin(0.01) ≈ 0.01,
≈ approximately equal approximation
x ≈ y means x is approximately equal to y
5>4
> strict inequality greater than
5 is greater than 4
4<5
< strict inequality less than
4 is less than 5
5 ≥ 4,
≥ inequality greater than or equal to
x ≥ y means x is greater than or equal to y
4 ≤ 5,
≤ inequality less than or equal to
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 6 / 2 = 3 (/= horizontal line)
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 = 3.141592654...
π pi constant is the ratio between the circumference and diameter c = π⋅d = 2⋅π⋅r
of a circle
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 = 3.141592654...
π pi constant is the ratio between the circumference and diameter of a c = π⋅d = 2⋅π⋅r
circle
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
cumulative distribution function
F(x) F(x) = P(X≤ x)
(cdf)
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
standard deviation value of random variable
σX standard deviation σX = 2
X
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 - sum of all values in range of
∑ summation
series
∑∑ double summation double summation
value that occurs most frequently in
Mo mode
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
50% of population are below this value =
Q2 median / second quartile
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
s2 sample variance population samples variance estimator s2 = 4
population samples standard deviation
s sample standard deviation s=2
estimator
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
n Pk permutation 5P3 = 5! / (5-3)! = 60
nCk combination 5C3 = 5!/[3!(5-3)!]=10
Set theory symbols
Symbol Symbol Name Meaning / definition Example
A = {3,7,9,14},
{} set a collection of elements
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
P(A) power set all subsets of A
A={3,9,14},
A=B equality both sets have the same members B={3,9,14},
A=B
Ac complement all the objects that do not belong to set A
A = {3,9,14},
A\B relative complement objects that belong to A and not to B B = {1,2,3},
A-B = {9,14}
A = {3,9,14},
A-B relative complement objects that belong to A and not to B B = {1,2,3},
A-B = {9,14}
A = {3,9,14},
objects that belong to A or B but not to their
A∆B symmetric difference B = {1,2,3},
intersection
A ∆ B = {1,2,9,14}
A = {3,9,14},
objects that belong to A or B but not to their
A⊖B symmetric difference B = {1,2,3},
intersection
A ⊖ B = {1,2,9,14}
element of,
a∈A set membership A={3,9,14}, 3 ∈ A
belongs to
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
natural numbers / whole numbers set
(with zero)
0 = {0,1,2,3,4,...} 0∈ 0
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 constant / Euler's
e e = 2.718281828... e = lim (1+1/x)x , x→∞
number
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
y‘ time derivative derivative by time - Newton's notation
y“ 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
real part of a complex
Re(z) z = a+bi → Re(z)=a Re(3 - 2i) = 3
number
imaginary part of a
Im(z) z = a+bi → Im(z)=b Im(3 - 2i) = -2
complex number
absolute value/magnitude
|z| |z| = |a+bi| = √(a2+b2) |3 - 2i| = √13
of a complex number
argument of a complex
arg(z) The angle of the radius in the complex plane arg(3 + 2i) = 33.7°
number
∇ nabla / del gradient / divergence operator ∇f (x,y,z)
velocity, mom velocity, momentum, force, electromagnetic
vector fields and weight. entum, force, electromagnetic fields and
weight
a vector that has a magnitude of one and it is also known as
unit vector
a direction vector
x*y convolution y(t) = x(t) * h(t)
L Laplace transform F(s) = L {f (t)}
F Fourier transform X(ω) = F {f (t)}
he difference between two values of a variable is often called
δ or Δ delta function its "delta": that is, given two points on the plane, the "delta y"
is the difference between their y-values. For example
∞ lemniscate infinity symbol
