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 |
a^{3 }= 8 | |||
a^b | caret | exponent | 2 ^ 3= 8 |
√a · √9 = ±3 | |||
^{a} | cube root | ^{a · a · a = a} | ^{8 = 2} |
^{a} | fourth root | ^{4}√4√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 | |
|x–y| | distance | distance between points x and y | | x–y | = 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 | a ≤ x ≤ b} | x ∈ [2,6] |
∆ | delta | change / difference | ∆t = 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 = 0.527721566… | |
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 | |
A⊗B | tensor product | tensor product of A and 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^{ (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(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 }= 4 | |||
std(X) | standard deviation | standard deviation of random variable X | std(X) = 2 |
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_{,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 = (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 = (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_{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 |
Combinatorics Symbols
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 |
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} |
2^{power set} | all subsets of A | |
all subsets of A | ||
A = B | both sets have the same members | A={3,9,14}, B={3,9,14}, A=B |
A^{c} | 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} |
set of all possible values | ||
_{0} | _{0} = {0,1,2,3,4,…} | 0 ∈ _{0} |
_{1} | _{1} = {1,2,3,4,5,…} | 6 ∈ _{1} |
= {…-3,-2,-1,0,1,2,3,…} | -6 ∈ | |
= {x | x=a/b, a,b∈} | 2/6 ∈ | |
= {x | -∞ < x <∞} | 6.343434 ∈ | |
= {z | z=a+bi, -∞<a<∞, -∞<b<∞} | 6+2i ∈ |
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 | x ∨ y |
| | 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 | x ⊕ y |
~ | 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 |
---|---|---|---|
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 | (3x^{3})’ = 9x^{2} |
y ” | second derivative | derivative of derivative | (3x^{3})” = 18x |
y^{n times derivation} | (3x^{3})^{(3)} = 18 | ||
derivative | derivative – Lagrange’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 notation | ||
time second derivative | derivative of derivative | ||
partial derivative | ∂(x^{2}+y^{2})/∂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+bi → z*=a–bi | z* = 3 + 2i |
z = a+bi → z = a–bi | 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 |