Online Calculators & Tools
List of all mathematical symbols and signs - meaning and examples.
Symbol | Symbol Name | Meaning / definition | Example |
---|---|---|---|
= | equals sign | equality | 5 = 2+3 5 is equal to 2+3 |
≠ | not equal sign | inequality | 5 ≠ 4 5 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 > 4 5 is greater than 4 |
< | strict inequality | less than | 4 < 5 4 is less than 5 |
≥ | inequality | greater than or equal to | 5 ≥ 4, x ≥ ymeans x is greater than or equal to y |
≤ | inequality | less than or equal to | 4 ≤ 5, x ≤ y means xis 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 | |
mod | modulo | remainder calculation | 7 mod 2 = 1 |
. | period | decimal point, decimal separator | 2.56 = 2+56/100 |
a^{b} | power | exponent | 2^{3 }= 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 = 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} |
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, kconstant |
∞ | 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 = t_{1}- t_{0} |
∆ | discriminant | Δ = b^{2} - 4ac | |
∑ | sigma | summation - sum of all values in range of series | ∑ x_{i}= x_{1}+x_{2}+...+x_{n} |
∑∑ | sigma | double summation | |
∏ | capital pi | product - product of all values in range of series | ∏x_{i}=x_{1}∙x_{2}∙...∙x_{n} |
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 |
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 | (A^{T})_{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 |
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 |
σ^{2} | variance | variance of population values | σ^{2 }= 4 |
std(X) | standard deviation | standard deviation of random variable X | std(X) = 2 |
σ_{X} | standard deviation | standard deviation value of random variable X | σ_{X}_{ }= 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_{X,Y} | correlation | correlation of random variables X and Y | ρ_{X,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 = (x_{max}+x_{min})/2 | |
Md | sample median | half the population is below this value | |
Q_{1} | lower / first quartile | 25% of population are below this value | |
Q_{2} | median / second quartile | 50% of population are below this value = median of samples | |
Q_{3} | 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 |
z_{x} | standard score | z_{x} = (x-x) / s_{x} | |
X ~ | distribution of X | distribution of random variable X | X ~ N(0,3) |
N(μ,σ^{2}) | 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-1}e^{-λx}/ Γ(c), x≥0 | |
χ^{ 2}(k) | chi-square distribution | f (x) = x^{k}^{/2-1}e^{-x/2}/ ( 2^{k/2 }Γ(k/2) ) | |
F (k_{1}, k_{2}) | F distribution | ||
Bin(n,p) | binomial distribution | f (k) = _{n}C_{k}p^{k}(1-p)^{n-k} | |
Poisson(λ) | Poisson distribution | f (k) = λ^{k}e^{-λ}/ k! | |
Geom(p) | geometric distribution | f (k) = p(1-p)^{ k} | |
HG(N,K,n) | hyper-geometric distribution | ||
Bern(p) | Bernoulli distribution |
Symbol | Symbol Name | Meaning / definition | Example |
---|---|---|---|
n! | factorial | n! = 1⋅2⋅3⋅...⋅n | 5! = 1⋅2⋅3⋅4⋅5 = 120 |
_{n}P_{k} | permutation | _{5}P_{3} =5! / (5-3)! = 60 | |
_{n}C_{k}
| combination | _{5}C_{3} =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 | 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} |
2^{A} | 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 |
A^{c} | 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, belongs to | 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×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 | ||
_{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 ∈ |
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 | (3x^{3})' = 9x^{2} |
y '' | second derivative | derivative of derivative | (3x^{3})'' = 18x |
y^{(n)} | nth derivative | n times derivation | (3x^{3})^{(3)} = 18 |
derivative | derivative - Leibniz's notation | d(3x^{3})/dx = 9x^{2} | |
second derivative | derivative of derivative | d^{2}(3x^{3})/dx^{2} = 18x | |
nth derivative | n times derivation | ||
time derivative | derivative by time - Newton's notation | ||
time second derivative | derivative of derivative | ||
D_{x }y | derivative | derivative - Euler's notation | |
D_{x}^{2}y | second derivative | derivative of derivative | |
partial derivative | ∂(x^{2}+y^{2})/∂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 |
Re(z) | real part of a complex number | z = a+bi → Re(z)=a | Re(3 - 2i) = 3 |
Im(z) | imaginary part of a complex number | z = a+bi → Im(z)=b | Im(3 - 2i) = -2 |
| z | | absolute value/magnitude of a complex number | |z| = |a+bi| = √(a^{2}+b^{2}) | |3 - 2i| = √13 |
arg(z) | argument of a complex number | The angle of the radius in the complex plane | arg(3 + 2i) = 33.7° |
∇ | 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 |
Name | Western Arabic | Roman | Eastern 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 |
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 |
Upper Case Letter | Lower Case Letter | Greek Letter Name | English Equivalent | Letter Name Pronounce |
---|---|---|---|---|
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 |