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Mathematical Symbols
Page history
last edited
by Dmitry Sokolov 7 years, 10 months ago
http://www.rapidtables.com/math/symbols/Basic_Math_Symbols.htm
Mathematical Symbols
By Frequency of Use
, or just underline ___________ , or ‾‾‾‾‾‾‾‾‾‾‾‾‾‾
or overline: <span style="text-decoration: overline;">1</span>
Δ π Ωω μ
ℓ
⇒ √ ×
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 |
≠ |
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 |
Geometry symbols
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 | |
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 |
|
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
|
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 |
s2 |
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 |
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 |
|
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 |
|
∬ |
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) = {f (t)} |
|
|
Fourier transform |
X(ω) = {f (t)} |
|
δ |
delta function |
|
|
∞ |
lemniscate |
infinity symbol |
|
Numeral symbols
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 alphabet letters
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 |
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 |
See also
Ω
Mathematical Symbols
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