Symbols have been used to represent not only the written word but also logic and concepts. The reason for mathematical and scientific symbols is that they can be combined like language and also convey a lot of meaning in a simple symbol :
The history of mathematical notation^{[1]} includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation’s move to popularity or inconspicuousness. Mathematical notation^{[2]} comprises the symbols used to write mathematical equations andformulas. Notation generally implies a set of welldefined representations of quantities and symbols operators.^{[3]} The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.
The development of mathematical notation can be divided in stages.^{[4]}^{[5]} The “rhetorical” stage is where calculations are performed by words and no symbols are used.^{[6]} The “syncopated” stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. From ancient times through the postclassical age,^{[note 1]} bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light. The “symbolic” stage is where comprehensive systems of notation supersede rhetoric. Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This symbolic system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century,^{[7]} and has continued to develop in the contemporary era.
MATHEMATICAL SYMBOLS:
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples  

Read as  
Category  
{\displaystyle +} 
plus;
add 
4 + 6 means the sum of 4 and 6.  2 + 7 = 9  
the disjoint union of … and …

A_{1} + A_{2} means the disjoint union of sets A_{1} and A_{2}.  A_{1} = {3, 4, 5, 6} ∧ A_{2} = {7, 8, 9, 10} ⇒ A_{1} + A_{2} = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)} 

{\displaystyle }  36 − 11 means the subtraction of 11from 36.  36 − 11 = 25  
negative;
minus; the opposite of 
−3 means the additive inverse of the number 3.  −(−5) = 5  
minus;
without 
A − B means the set that contains all the elements of A that are not in B.
(∖ can also be used for settheoretic complement as described below.) 
{1, 2, 4} − {1, 3, 4} = {2}  
{\displaystyle \pm } \pm 
plus or minus

6 ± 3 means both 6 + 3 and 6 − 3.  The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
Note: 

plus or minus

10 ± 2 or equivalently 10 ± 20%means the range from 10 − 2 to10 + 2.  If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.  
{\displaystyle \mp } \mp 
minus or plus

6 ± (3 ∓ 5) means 6 + (3 − 5) and6 − (3 + 5).  cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).  
{\displaystyle \times } \times{\displaystyle \cdot } \cdot 
times;
multiplied by 
3 × 4 or 3 ⋅ 4 means the multiplication of 3 by 4.  7 ⋅ 8 = 56  
dot

u ⋅ v means the dot product of vectorsu and v  (1, 2, 5) ⋅ (3, 4, −1) = 6  
cross

u × v means the cross product ofvectors u and v  (1, 2, 5) × (3, 4, −1) =
= (−22, 16, −2) 

placeholder
(silent)

A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument.   ·   
{\displaystyle \div } \div{\displaystyle /} 
divided by;
over 
6 ÷ 3 or 6 ⁄ 3 means the division of 6by 3.  2 ÷ 4 = 0.5
12 ⁄ 4 = 3 

mod

G / H means the quotient of group Gmodulo its subgroup H.  {0, a, 2a, b, b + a, b + 2a} / {0, b} = {{0, b}, {a, b + a}, {2a, b + 2a}}  
quotient set
mod

A/~ means the set of all ~equivalence classes in A.  If we define ~ by x ~ y ⇔ x − y ∈ ℤ, thenℝ/~ = {x + n : n ∈ ℤ, x ∈ [0,1)}.  
{\displaystyle \surd }
{\displaystyle {\sqrt {\ }}} 
the (principal) square root of

√x means the nonnegative number whose square is x.  √4 = 2  
the (complex) square root of

If z = r exp(iφ) is represented in polar coordinates with −π < φ ≤ π, then√z = √r exp(iφ/2).  √−1 = i  
{\displaystyle \sum } \sum 
sum over … from … to … of

{\displaystyle \sum _{k=1}^{n}{a_{k}}} means {\displaystyle a_{1}+a_{2}+\cdots +a_{n}}.  {\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30}  
{\displaystyle \int } \int 
indefinite integral of
– OR – the antiderivative of 
∫ f(x) dx means a function whose derivative is f. 
{\displaystyle \int x^{2}dx={\frac {x^{3}}{3}}+C}  
integral from … to … of … with respect to

∫b a f(x) dx means the signed areabetween the xaxis and the graph of the function f between x = a andx = b. 
∫b a x^{2} dx = b^{3} − a^{3}/3 

line/ path/ curve/ integral of … along …

∫ C f ds means the integral of f along the curve C, ∫b a f(r(t))  r‘(t)  dt, where r is a parametrization of C. (If the curve is closed, the symbol ∮ may be used instead, as described below.) 

∮

{\displaystyle \oint } \oint 
Contour integral;
closed line integral contour integral of

Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss’s Law, and while these formulas involve a closedsurface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closedvolume integral, denoted by the symbol ∰ .The contour integral can also frequently be found with a subscript capital letter C, ∮ _{C}, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss’s Law, a subscript capital S, ∮ _{S}, is used to denote that the integration is over a closed surface. 
If C is a Jordan curve about 0, then ∮ _{C} 1/z dz = 2πi. 

{\displaystyle \ldots } \ldots{\displaystyle \cdots } \cdots{\displaystyle \vdots } \vdots {\displaystyle \ddots } 
and so forth
everywhere

Indicates omitted values from a pattern.  1/2 + 1/4 + 1/8 + 1/16 + ⋯ = 1  
{\displaystyle \therefore } \therefore 
therefore;
so; hence everywhere

Sometimes used in proofs beforelogical consequences.  All humans are mortal. Socrates is a human. ∴ Socrates is mortal.  
{\displaystyle \because } \because 
because;
since everywhere

Sometimes used in proofs before reasoning.  11 is prime ∵ it has no positive integer factors other than itself and one.  
{\displaystyle !} 
factorial

n! means the product 1 × 2 × … × n.  {\displaystyle 4!=1\times 2\times 3\times 4=24}  
not

The statement !A is true if and only if Ais false.
A slash placed through another operator is the same as “!” placed in front. (The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation¬A is preferred.) 
!(!A) ⇔ A x ≠ y ⇔ !(x = y) 

˜ 
{\displaystyle \neg } \neg{\displaystyle \sim } 
not

The statement ¬A is true if and only ifA is false.
A slash placed through another operator is the same as “¬” placed in front. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use ! but this is avoided in mathematical texts.) 
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) 

{\displaystyle \propto } \propto 
is proportional to;
varies as everywhere

y ∝ x means that y = kx for some constant k.  if y = 2x, then y ∝ x.  
{\displaystyle \infty } \infty 
infinity

∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.  {\displaystyle \lim _{x\to 0}{\frac {1}{x}}=\infty }  
{\displaystyle \blacksquare } \blacksquare{\displaystyle \Box } \Box{\displaystyle \blacktriangleright } \blacktriangleright 
everywhere

Used to mark the end of a proof.
(May also be written Q.E.D.) 
(1) a + 0 := a (def.) (2) a + succ(b) := succ(a + b) (def.)Proposition. 3 + 2 = 5.Proof. 3 + 2 = 3 + succ(1) (definition of succ) 
Symbols based on equality
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle =} 
is equal to;
equals everywhere

{\displaystyle x=y} means {\displaystyle x} and {\displaystyle y} represent the same math object (Both symbols have the same value).  {\displaystyle 2=2} {\displaystyle 1+1=2} {\displaystyle 365=31} 

{\displaystyle \neq } \ne 
is not equal to;
does not equal everywhere

{\displaystyle x\neq y} means that {\displaystyle x} and {\displaystyle y} do not represent the same math object (Both symbols do not have the same value).
(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 
{\displaystyle 2+2\neq 5} {\displaystyle 365\neq 30} 

{\displaystyle \approx } \approx 
approximately equal
is approximately equal to
everywhere

x ≈ y means x is approximately equal to y.
This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒. 
π ≈ 3.14159  
is isomorphic to

G ≈ H means that group G is isomorphic (structurally identical) to group H.
(≅ can also be used for isomorphic, as described below.) 
Q_{8} / C_{2} ≈ V  
{\displaystyle \sim } \sim 
has distribution

X ~ D, means the random variable X has the probability distribution D.  X ~ N(0,1), the standard normal distribution  
is row equivalent to

A ~ B means that B can be generated by using a series of elementary row operations on A  {\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\sim {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}}  
same order of magnitude

m ~ n means the quantities m and n have the same order of magnitude, or general size.
(Note that ~ is used for an approximation that is poor, otherwise use ≈ .) 
2 ~ 5
8 × 9 ~ 100 but π^{2} ≈ 10 

is similar to^{[1]}

△ABC ~ △DEF means triangle ABC is similar to (has the same shape) triangle DEF.  
is asymptotically equivalent to

f ~ g means {\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}.  x ~ x+1  
are in the same equivalence class
everywhere

a ~ b means {\displaystyle b\in [a]} (and equivalently {\displaystyle a\in [b]}).  1 ~ 5 mod 4  
{\displaystyle =:}
{\displaystyle :=} {\displaystyle \equiv } {\displaystyle :\Leftrightarrow } {\displaystyle \triangleq } {\displaystyle {\overset {\underset {\mathrm {def} }{}}{=}}} {\displaystyle \doteq } 
is defined as;
is equal by definition to everywhere

x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.
(Some writers use ≡ to mean congruence). P ⇔ Q means P is defined to be logically equivalent to Q. P ⇔ Q means if and only if (iff) 
{\displaystyle \cosh x:={\frac {e^{x}+e^{x}}{2}}}
{\displaystyle [a,b]:=a\cdot bb\cdot a} 

≅

{\displaystyle \cong } \cong 
is congruent to

△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.  
is isomorphic to

G ≅ H means that group G is isomorphic (structurally identical) to group H.
(≈ can also be used for isomorphic, as described above.) 
V ≅ C_{2} × C_{2}  
{\displaystyle \equiv } \equiv 
… is congruent to … modulo …

a ≡ b (mod n) means a − b is divisible by n  5 ≡ 2 (mod 3)  
⇔ ↔ 
{\displaystyle \Leftrightarrow } \Leftrightarrow{\displaystyle \leftrightarrow } \leftrightarrow 
if and only if;
iff 
A ⇔ B means A is true if B is true and A is false if B is false.  x + 5 = y + 2 ⇔ x + 3 = y 
:= =: 
{\displaystyle :=} {\displaystyle =:} 
is defined to be
everywhere

A := b means A is defined to have the value b.  Let a := 3, then… f(x) := x + 3 
Symbols that point east or west[edit]
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle <}
{\displaystyle >} 
is less than,
is greater than 
{\displaystyle x<y} means x is less than y.
{\displaystyle x>y} means x is greater than y. 
{\displaystyle 3<4} {\displaystyle 5>4} 

is a proper subgroup of

{\displaystyle H<G} means H is a proper subgroup of G.  {\displaystyle 5\mathrm {Z} <\mathrm {Z} } {\displaystyle \mathrm {A} _{3}<\mathrm {S} _{3}} 

{\displaystyle \ll \!\,}
{\displaystyle \gg \!\,} \ll 
significant (strict) inequality
is much less than,
is much greater than 
x ≪ y means x is much less than y.
x ≫ y means x is much greater than y. 
0.003 ≪ 1000000  
asymptotic comparison
is of smaller order than,
is of greater order than 
f ≪ g means the growth of f is asymptotically bounded by g.
(This is I. M. Vinogradov‘s notation. Another notation is the Big O notation, which looks like f = O(g).) 
x ≪ e^{x}  
absolute continuity
is absolutely continuous with respect to

{\displaystyle \mu \ll \nu } means that {\displaystyle \mu } is absolutely continuous with respect to {\displaystyle \nu },i.e., whenever {\displaystyle \nu (A)=0}, we have {\displaystyle \mu (A)=0}.  If {\displaystyle c} is the counting measure on {\displaystyle [0,1]} and {\displaystyle \mu } is the Lebesgue measure, then {\displaystyle \mu \ll c}.  
{\displaystyle \leq \!\,}
{\displaystyle \geq } \le 
is less than or equal to,
is greater than or equal to 
x ≤ y means x is less than or equal to y.
x ≥ y means x is greater than or equal to y. (The forms <= and >= are generally used in programming languages, where ease of typing and use of ASCII text is preferred.) 
3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 

is a subgroup of

H ≤ G means H is a subgroup of G.  Z ≤ Z A_{3} ≤ S_{3} 

is reducible to

A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction.  If
then


≦ ≧ 
{\displaystyle \leqq \!\,}
{\displaystyle \geqq \!\,} \leqq 
… is less than … is greater than …

10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10  
… is less than or equal… is greater than or equal…

x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y.
x ≧ y means that each component of vector x is greater than or equal to each corresponding component of vector y. It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true. 

≺ ≻ 
{\displaystyle \prec \!\,}
{\displaystyle \succ \!\,} \prec 
is Karp reducible to;
is polynomialtime manyone reducible to 
L_{1} ≺ L_{2} means that the problem L_{1} is Karp reducible to L_{2}.^{[2]}  If L_{1} ≺ L_{2} and L_{2} ∈ P, then L_{1} ∈ P. 
is nondominated by

P ≺ Q means that the element P is nondominated by elementQ.^{[3]}  If P_{1} ≺ Q_{2} then {\displaystyle \forall _{i}P_{i}\leq Q_{i}\land \exists P_{i}<Q_{i}}  
◅ ▻ 
{\displaystyle \triangleleft \!\,}
{\displaystyle \triangleright \!\,} \triangleleft 
is a normal subgroup of

N ◅ G means that N is a normal subgroup of group G.  Z(G) ◅ G 
is an ideal of

I ◅ R means that I is an ideal of ring R.  (2) ◅ Z  
the antijoin of

R ▻ S means the antijoin of the relations R and S, the tuples inR for which there is not a tuple in S that is equal on their common attribute names.  {\displaystyle R\triangleright S=RR\ltimes S}  
⇒ → ⊃ 
{\displaystyle \Rightarrow \!\,}
{\displaystyle \rightarrow \!\,} {\displaystyle \supset \!\,} 
implies;
if … then 
A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.
(→ may mean the same as ⇒, or it may have the meaning forfunctions given below.) (⊃ may mean the same as ⇒,^{[4]} or it may have the meaning forsuperset given below.) 
x = 6 ⇒ x^{2}5 = 365 = 31 is true, butx^{2}5 = 365 = 31 ⇒ x = 6 is in general false (since x could be −6). 
⊆ ⊂ 
{\displaystyle \subseteq \!\,}
{\displaystyle \subset \!\,} 
is a subset of

(subset) A ⊆ B means every element of A is also an element ofB.^{[5]}
(proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.) 
(A ∩ B) ⊆ A
ℕ ⊂ ℚ ℚ ⊂ ℝ 
⊇ ⊃ 
{\displaystyle \supseteq \!\,}
{\displaystyle \supset \!\,} 
is a superset of

A ⊇ B means every element of B is also an element of A.
A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.) 
(A ∪ B) ⊇ B
ℝ ⊃ ℚ 
{\displaystyle \to \!\,} 
function arrow
from … to

f: X → Y means the function f maps the set X into the set Y.  Let f: ℤ → ℕ ∪ {0} be defined by f(x) := x^{2}.  
↦

{\displaystyle \mapsto \!\,} 
function arrow
maps to

f: a ↦ b means the function f maps the element a to the elementb.  Let f: x ↦ x + 1 (the successor function). 
<: <· 
{\displaystyle <:\!\,}
{\displaystyle {<}{\cdot }\!\,} 
is a subtype of

T_{1} <: T_{2} means that T_{1} is a subtype of T_{2}.  If S <: T and T <: U then S <: U (transitivity). 
is covered by

x <• y means that x is covered by y.  {1, 8} <• {1, 3, 8} among the subsets of{1, 2, …, 10} ordered by containment.  
⊧

{\displaystyle \vDash \!\,} 
entails

A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true.  A ⊧ A ∨ ¬A 
⊢

{\displaystyle \vdash \!\,} 
infers;
is derived from 
x ⊢ y means y is derivable from x.  A → B ⊢ ¬B → ¬A 
is a partition of

p ⊢ n means that p is a partition of n.  (4,3,1,1) ⊢ 9, {\displaystyle \sum _{\lambda \vdash n}(f_{\lambda })^{2}=n!}  
⟨

{\displaystyle \langle \ \!\,} 
the bra …;
the dual of … 
⟨φ means the dual of the vector φ⟩, a linear functional which maps a ket ψ⟩ onto the inner product ⟨φψ⟩.  
⟩

{\displaystyle \ \rangle \!\,} 
the ket …;
the vector … 
φ⟩ means the vector with label φ, which is in a Hilbert space.  A qubit‘s state can be represented as α0⟩+β1⟩, where α and β are complex numbers s.t.α^{2} + β^{2} = 1. 
Brackets[edit]
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle {\ \choose \ }} 
n choose k

{\displaystyle {\begin{pmatrix}n\\k\end{pmatrix}}={\frac {n!/(nk)!}{k!}}={\frac {(nk+1)\cdots (n2)\cdot (n1)\cdot n}{k!}}} means (in the case of n = positive integer) the number of combinations of k elements drawn from a set of n elements.(This may also be written as C(n, k), C(n; k), _{n}C_{k}, ^{n}C_{k}, or {\displaystyle \left\langle {\begin{matrix}n\\k\end{matrix}}\right\rangle }.) 
{\displaystyle {\begin{pmatrix}36\\5\end{pmatrix}}={\frac {36!/(365)!}{5!}}={\frac {32\cdot 33\cdot 34\cdot 35\cdot 36}{1\cdot 2\cdot 3\cdot 4\cdot 5}}=376992} {\displaystyle {\begin{pmatrix}.5\\7\end{pmatrix}}={\frac {5.5\cdot 4.5\cdot 3.5\cdot 2.5\cdot 1.5\cdot .5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={\frac {33}{2048}}\,\!} 

{\displaystyle \left(\!\!{\ \choose \ }\!\!\right)} 
u multichoosek

{\displaystyle \left(\!\!{u \choose k}\!\!\right)={u+k1 \choose k}={\frac {(u+k1)!/(u1)!}{k!}}} (when u is positive integer) means reverse or rising binomial coefficient. 
{\displaystyle \left(\!\!{5.5 \choose 7}\!\!\right)={\frac {5.5\cdot 4.5\cdot 3.5\cdot 2.5\cdot 1.5\cdot .5\cdot .5}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7}}={.5 \choose 7}={\frac {33}{2048}}\,\!}  
…

{\displaystyle \ldots \!\,} 
absolute value;
modulus absolute value of; modulus of

x means the distance along the real line (or across the complex plane) between x and zero.  3 = 3
–5 = 5 = 5  i  = 1  3 + 4i  = 5 
Euclidean norm or Euclidean length or magnitude
Euclidean norm of

x means the (Euclidean) length of vector x.  For x = (3,−4) {\displaystyle {\textbf {x}}={\sqrt {3^{2}+(4)^{2}}}=5} 

determinant of

A means the determinant of the matrix A  {\displaystyle {\begin{vmatrix}1&2\\2&9\\\end{vmatrix}}=5}  
cardinality of;
size of; order of 
X means the cardinality of the set X.
(# may be used instead as described below.) 
{3, 5, 7, 9} = 4.  
‖…‖

{\displaystyle \\ldots \\!\,} 
norm of;
length of 
‖ x ‖ means the norm of the element x of a normed vector space.^{[6]}  ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ 
nearest integer to

‖x‖ means the nearest integer to x.
(This may also be written [x], ⌊x⌉, nint(x) or Round(x).) 
‖1‖ = 1, ‖1.6‖ = 2, ‖−2.4‖ = −2, ‖3.49‖ = 3  
{\displaystyle {\{\ ,\!\ \}}\!\,} 
set brackets
the set of …

{a,b,c} means the set consisting of a, b, and c.^{[7]}  ℕ = { 1, 2, 3, … }  
{ : } {  } { ; } 
{\displaystyle \{\ :\ \}\!\,}
{\displaystyle \{\ \ \}\!\,} {\displaystyle \{\ ;\ \}\!\,} 
the set of … such that

{x : P(x)} means the set of all x for which P(x) is true.^{[7]} {x  P(x)} is the same as {x : P(x)}.  {n ∈ ℕ : n^{2} < 20} = { 1, 2, 3, 4 } 
⌊…⌋

{\displaystyle \lfloor \ldots \rfloor \!\,} 
floor;
greatest integer; entier 
⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x.
(This may also be written [x], floor(x) or int(x).) 
⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3 
⌈…⌉

{\displaystyle \lceil \ldots \rceil \!\,} 
ceiling

⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x.
(This may also be written ceil(x) or ceiling(x).) 
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 
⌊…⌉

{\displaystyle \lfloor \ldots \rceil \!\,} 
nearest integer to

⌊x⌉ means the nearest integer to x.
(This may also be written [x], x, nint(x) or Round(x).) 
⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊3.4⌉ = 3, ⌊4.49⌉ = 4 
[ : ]

{\displaystyle [\ :\ ]\!\,} 
the degree of

[K : F] means the degree of the extension K : F.  [ℚ(√2) : ℚ] = 2
[ℂ : ℝ] = 2 [ℝ : ℚ] = ∞ 
{\displaystyle [\ ]\!\,}
{\displaystyle [\ ,\ ]\!\,} {\displaystyle [\ ,\ ,\ ]\!\,} 
the equivalence class of

[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is anequivalence relation.
[a]_{R} means the same, but with R as the equivalence relation. 
Let a ~ b be true iff a ≡ b (mod 5).Then [2] = {…, −8, −3, 2, 7, …}.  
floor;
greatest integer; entier 
[x] means the floor of x, i.e. the largest integer less than or equal to x.
(This may also be written ⌊x⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.) 
[3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4  
nearest integer to

[x] means the nearest integer to x.
(This may also be written ⌊x⌉, x, nint(x) or Round(x). Not to be confused with the floor function, as described above.) 
[2] = 2, [2.6] = 3, [3.4] = 3, [4.49] = 4  
1 if true, 0 otherwise

[S] maps a true statement S to 1 and a false statement S to 0.  [0=5]=0, [7>0]=1, [2 ∈ {2,3,4}]=1, [5 ∈ {2,3,4}]=0  
image of … under …
everywhere

f[X] means { f(x) : x ∈ X }, the image of the function f under the set X⊆ dom(f).
(This may also be written as f(X) if there is no risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) 
{\displaystyle \sin[\mathbb {R} ]=[1,1]}  
closed interval

{\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}.  0 and 1/2 are in the interval [0,1].  
the commutator of

[g, h] = g^{−1}h^{−1}gh (or ghg^{−1}h^{−1}), if g, h ∈ G (a group).
[a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra). 
x^{y} = x[x, y] (group theory).
[AB, C] = A[B, C] + [A, C]B (ring theory). 

the triple scalar product of

[a, b, c] = a × b · c, the scalar product of a × b with c.  [a, b, c] = [b, c, a] = [c, a, b].  
( , ) 
{\displaystyle (\ )\!\,}
{\displaystyle (\ ,\ )\!\,} 
functionapplication
of

f(x) means the value of the function f at the element x.  If f(x) := x^{2}5, then f(6) = 6^{2}5 = 36−5=31. 
image of … under …
everywhere

f(X) means { f(x) : x ∈ X }, the image of the function f under the set X⊆ dom(f).
(This may also be written as f[X] if there is a risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) 
{\displaystyle \sin(\mathbb {R} )=[1,1]}  
precedence grouping
parentheses
everywhere

Perform the operations inside the parentheses first.  (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  
tuple; ntuple;
ordered pair/triple/etc; row vector; sequence everywhere

An ordered list (or sequence, or horizontal vector, or row vector) of values.(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets ⟨ ⟩ instead of parentheses.)  (a, b) is an ordered pair (or 2tuple).(a, b, c) is an ordered triple (or 3tuple).
( ) is the empty tuple (or 0tuple). 

highest common factor;
greatest common divisor; hcf; gcd number theory

(a, b) means the highest common factor of a and b.
(This may also be written hcf(a, b) or gcd(a, b).) 
(3, 7) = 1 (they are coprime); (15, 25) = 5.  
( , ) ] , [ 
{\displaystyle (\ ,\ )\!\,}
{\displaystyle ]\ ,\ [\!\,} 
open interval

{\displaystyle (a,b)=\{x\in \mathbb {R} :a<x<b\}}.(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.)  4 is not in the interval (4, 18).(0, +∞) equals the set of positive real numbers. 
( , ] ] , ] 
{\displaystyle (\ ,\ ]\!\,}
{\displaystyle ]\ ,\ ]\!\,} 
halfopen interval;
leftopen interval 
{\displaystyle (a,b]=\{x\in \mathbb {R} :a<x\leq b\}}.  (−1, 7] and (−∞, −1] 
[ , ) [ , [ 
{\displaystyle [\ ,\ )\!\,}
{\displaystyle [\ ,\ [\!\,} 
halfopen interval;
rightopen interval 
{\displaystyle [a,b)=\{x\in \mathbb {R} :a\leq x<b\}}.  [4, 18) and [1, +∞) 
⟨⟩ ⟨,⟩ 
{\displaystyle \langle \ \rangle \!\,}
{\displaystyle \langle \ ,\ \rangle \!\,} 
inner product of

⟨u,v⟩ means the inner product of u and v, where u and v are members of an inner product space.
Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner product or the linear span. There are many variants of the notation, such as ⟨u  v⟩ and (u  v),which are described below. For spatial vectors, the dot productnotation, x · y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more “keyboard friendly” forms < and > are sometimes seen. These are avoided in mathematical texts. 
The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13 
average
average of

let S be a subset of N for example, {\displaystyle \langle S\rangle } represents the average of all the elements in S.  for a time series :g(t) (t = 1, 2,…)we can define the structure functions S_{q}({\displaystyle \tau }):


(linear) span of;
linear hull of 
⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. ⟨u_{1}, u_{2}, …⟩ is shorthand for ⟨{u_{1}, u_{2}, …}⟩.Note that the notation ⟨u, v⟩ may be ambiguous: it could mean theinner product or the linear span.The span of S may also be written as Sp(S). 
{\displaystyle \left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}}.  
subgroupgenerated by a set
the subgroup generated by

{\displaystyle \langle S\rangle } means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S. {\displaystyle \langle g_{1},g_{2},\ldots ,\rangle } is shorthand for {\displaystyle \langle g_{1},g_{2},\ldots \rangle }. 
In S_{3}, {\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}} and {\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;2\;3))\}}.  
tuple; ntuple;
ordered pair/triple/etc; row vector; sequence everywhere

An ordered list (or sequence, or horizontal vector, or row vector) of values.(The notation (a,b) is often used as well.)  {\displaystyle \langle a,b\rangle } is an ordered pair (or 2tuple).{\displaystyle \langle a,b,c\rangle } is an ordered triple (or 3tuple).
{\displaystyle \langle \rangle } is the empty tuple (or 0tuple). 

⟨⟩ () 
{\displaystyle \langle \ \ \rangle \!\,}
{\displaystyle (\ \ )\!\,} 
inner product of

⟨u  v⟩ means the inner product of u and v, where u and v are members of an inner product space.^{[8]} (u  v) means the same.
Another variant of the notation is ⟨u, v⟩ which is described above. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more “keyboard friendly” forms < and > are sometimes seen. These are avoided in mathematical texts. 
Other nonletter symbols[edit]
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle *\!\,} 
convolution;
convolved with 
f ∗ g means the convolution of f and g.  {\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t\tau )\,d\tau }.  
conjugate

z^{∗} means the complex conjugate of z.
({\displaystyle {\bar {z}}} can also be used for the conjugate of z, as described below.) 
{\displaystyle (3+4i)^{\ast }=34i}.  
the group of units of

R^{∗} consists of the set of units of the ring R, along with the operation of multiplication.
This may also be written R^{×} as described above, orU(R). 
{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\ast }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}  
the (set of) hyperreals

^{∗}R means the set of hyperreal numbers. Other sets can be used in place of R.  ^{∗}N is the hypernatural numbers.  
Hodge dual;
Hodge star 
∗v means the Hodge dual of a vector v. If v is a kvector within an ndimensional oriented inner product space, then ∗v is an (n−k)vector.  If {\displaystyle \{e_{i}\}} are the standard basis vectors of {\displaystyle \mathbb {R} ^{5}}, {\displaystyle *(e_{1}\wedge e_{2}\wedge e_{3})=e_{4}\wedge e_{5}}  
{\displaystyle \propto \!\,} 
is proportional to;
varies as everywhere

y ∝ x means that y = kx for some constant k.  if y = 2x, then y ∝ x.  
Karp reduction^{[9]}
is Karp reducible to;
is polynomialtime manyone reducible to 
A ∝ B means the problem A can be polynomially reduced to the problem B.  If L_{1} ∝ L_{2} and L_{2} ∈ P, then L_{1} ∈ P.  
∖

{\displaystyle \setminus \!\,} 
minus;
without; throw out; not 
A ∖ B means the set that contains all those elements of A that are not in B.^{[5]}
(− can also be used for settheoretic complement as described above.) 
{1,2,3,4} ∖ {3,4,5,6} = {1,2} 
{\displaystyle \!\,} 
given

P(AB) means the probability of the event Aoccurring given that B occurs.  if X is a uniformly random day of the year P(X is May 25  X is in May) = 1/31  
restriction of … to …;
restricted to 
f_{A} means the function f is restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f.  The function f : R → R defined by f(x) = x^{2} is not injective, but f_{R+} is injective.  
such that
such that;
so that everywhere

 means “such that”, see “:” (described below).  S = {(x,y)  0 < y < f(x)} The set of (x,y) such that y is greater than 0 and less than f(x). 

∣ ∤ 
{\displaystyle \mid \!\,}
{\displaystyle \nmid \!\,} 
divides

a ∣ b means a divides b. a ∤ b means a does not divide b.(The symbol ∣ can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar  character is often used instead.) 
Since 15 = 3 × 5, it is true that 3 ∣ 15 and 5 ∣ 15. 
∣∣

{\displaystyle \mid \mid \!\,} 
exactdivisibility
exactly divides

p^{a} ∣∣ n means p^{a} exactly divides n (i.e. p^{a} divides nbut p^{a+1} does not).  2^{3} ∣∣ 360. 
∥ ∦ ⋕ 
{\displaystyle \\!\,} 
is parallel to

x ∥ y means x is parallel to y. x ∦ y means x is not parallel to y. x ⋕ y means x is equal and parallel to y.(The symbol ∥ can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar  characters are often used instead.) 
If l ∥ m and m ⊥ n then l ⊥ n. 
is incomparable to

x ∥ y means x is incomparable to y.  {1,2} ∥ {2,3} under set containment.  
{\displaystyle \#\!\,} 
cardinality of;
size of; order of 
#X means the cardinality of the set X.
(… may be used instead as described above.) 
#{4, 6, 8} = 3  
connected sum of;
knot sum of; knot composition of 
A#B is the connected sum of the manifolds A andB. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition.  A#S^{m} is homeomorphic to A, for any manifold A, and the sphere S^{m}.  
primorial

n# is product of all prime numbers less than or equal to n.  12# = 2 × 3 × 5 × 7 × 11 = 2310  
{\displaystyle :\!\,} 
such that
such that;
so that everywhere

: means “such that”, and is used in proofs and thesetbuilder notation (described below).  ∃ n ∈ ℕ: n is even.  
extends;
over 
K : F means the field K extends the field F.
This may also be written as K ≥ F. 
ℝ : ℚ  
inner productof matrices
inner product of

A : B means the Frobenius inner product of the matrices A and B.
The general inner product is denoted by ⟨u, v⟩, ⟨u v⟩ or (u  v), as described below. For spatial vectors, the dot product notation, x·y is common.See also bra–ket notation. 
{\displaystyle A:B=\sum _{i,j}A_{ij}B_{ij}}  
index of subgroup

The index of a subgroup H in a group G is the “relative size” of H in G: equivalently, the number of “copies” (cosets) of H that fill up G  {\displaystyle G:H={\frac {G}{H}}}  
divided by
over everywhere

A : B means the division of A with B (dividing A byB)  10 : 2 = 5  
⋮

{\displaystyle \vdots \!\,} 
vertical ellipsis
everywhere

Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed.  {\displaystyle P(r,t)=\chi \vdots E(r,t_{1})E(r,t_{2})E(r,t_{3})} 
≀

{\displaystyle \wr \!\,} 
wreath product of … by …

A ≀ H means the wreath product of the group A by the group H.
This may also be written A _{wr} H. 
{\displaystyle \mathrm {S} _{n}\wr \mathrm {Z} _{2}} is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices. 
↯ ※ ⇒⇐ 
contradiction; this contradicts that
everywhere

Denotes that contradictory statements have been inferred. For clarity, the exact point of contradiction can be appended.  x + 4 = x – 3 ※
Statement: Every finite, nonempty, ordered set has a largest element. Otherwise, let’s assume that {\displaystyle X} is a finite, nonempty, ordered set with no largest element. Then, for some {\displaystyle x_{1}\in X}, there exists an {\displaystyle x_{2}\in X}with {\displaystyle x_{1}<x_{2}}, but then there’s also an {\displaystyle x_{3}\in X} with {\displaystyle x_{2}<x_{3}}, and so on. Thus, {\displaystyle x_{1},x_{2},x_{3},…} are distinct elements in {\displaystyle X}. ↯ {\displaystyle X} is finite. 

⊕ ⊻ 
{\displaystyle \oplus \!\,}
{\displaystyle \veebar \!\,} 
xor

The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.  (¬A) ⊕ A is always true, A ⊕ A is always false. 
direct sum of

The direct sum is a special way of combining several objects into one general object.
(The bun symbol ⊕, or the coproduct symbol ∐, is used; ⊻ is only for logic.) 
Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) 

{\displaystyle {~\wedge \!\!\!\!\!\!\bigcirc ~}} 
Kulkarni–Nomizu product

Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor. {\displaystyle f=g{\,\wedge \!\!\!\!\!\!\bigcirc \,}h} has components {\displaystyle f_{\alpha \beta \gamma \delta }=g_{\alpha \gamma }h_{\beta \delta }+g_{\beta \delta }h_{\alpha \gamma }g_{\alpha \delta }h_{\beta \gamma }g_{\beta \gamma }h_{\alpha \delta }}.  
□

{\displaystyle \Box \!\,} 
D’Alembertian;
wave operator nonEuclidean Laplacian

It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions.  {\displaystyle \square ={\frac {1}{c^{2}}}{\partial ^{2} \over \partial t^{2}}{\partial ^{2} \over \partial x^{2}}{\partial ^{2} \over \partial y^{2}}{\partial ^{2} \over \partial z^{2}}} 
Letterbased symbols[edit]
Includes upsidedown letters.
Letter modifiers[edit]
Also called diacritics.
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle {\bar {a}}\!\,} 
overbar;
… bar 
{\displaystyle {\bar {x}}} (often read as “x bar”) is the mean (average value of {\displaystyle x_{i}}).  {\displaystyle x=\{1,2,3,4,5\};{\bar {x}}=3}.  
finite sequence, tuple

{\displaystyle {\overline {a}}} means the finite sequence/tuple {\displaystyle (a_{1},a_{2},…,a_{n}).}.  {\displaystyle {\overline {a}}:=(a_{1},a_{2},…,a_{n})}.  
algebraic closure of

{\displaystyle {\overline {F}}} is the algebraic closure of the field F.  The field of algebraic numbers is sometimes denoted as {\displaystyle {\overline {\mathbb {Q} }}} because it is the algebraic closure of the rational numbers {\displaystyle {\mathbb {Q} }}.  
conjugate

{\displaystyle {\overline {z}}} means the complex conjugate of z.
(z^{∗} can also be used for the conjugate of z, as described above.) 
{\displaystyle {\overline {3+4i}}=34i}.  
(topological) closure of

{\displaystyle {\overline {S}}} is the topological closure of the set S.
This may also be denoted as cl(S) or Cl(S). 
In the space of the real numbers, {\displaystyle {\overline {\mathbb {Q} }}=\mathbb {R} } (the rational numbers aredense in the real numbers).  
â

{\displaystyle {\hat {a}}} 
hat

{\displaystyle \mathbf {\hat {a}} } (pronounced “a hat”) is the normalized version of vector {\displaystyle \mathbf {a} }, having length 1.  
estimator for

{\displaystyle {\hat {\theta }}} is the estimator or the estimate for the parameter {\displaystyle \theta }.  The estimator {\displaystyle \mathbf {\hat {\mu }} ={\frac {\sum _{i}x_{i}}{n}}} produces a sample estimate {\displaystyle \mathbf {\hat {\mu }} (\mathbf {x} )} for the mean {\displaystyle \mu }.  
{\displaystyle ‘\!\,} 
… prime;
derivative of 
f ′(x) means the derivative of the function f at the pointx, i.e., the slope of the tangent to f at x.
(The singlequote character ‘ is sometimes used instead, especially in ASCII text.) 
If f(x) := x^{2}, then f ′(x) = 2x.  
{\displaystyle {\dot {\,}}\!\,} 
… dot;
time derivative of 
{\displaystyle {\dot {x}}} means the derivative of x with respect to time. That is {\displaystyle {\dot {x}}(t)={\frac {\partial }{\partial t}}x(t)}.  If x(t) := t^{2}, then {\displaystyle {\dot {x}}(t)=2t}. 
Symbols based on Latin letters[edit]
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle \forall \!\,} 
for all;
for any; for each; for every 
∀ x: P(x) means P(x) is true for all x.  ∀ n ∈ ℕ: n^{2} ≥ n.  
ℂ 
{\displaystyle \mathbb {C} \!\,}
{\displaystyle \mathbf {C} \!\,} 
C;
the (set of) complex numbers 
ℂ means {a + b i : a,b ∈ ℝ}.  i = √−1 ∈ ℂ 
𝔠

{\displaystyle {\mathfrak {c}}\!\,} 
cardinality of the continuum;
c; cardinality of the real numbers 
The cardinality of {\displaystyle \mathbb {R} } is denoted by {\displaystyle \mathbb {R} } or by the symbol {\displaystyle {\mathfrak {c}}} (a lowercase Fraktur letter C).  {\displaystyle {\mathfrak {c}}={\beth }_{1}} 
{\displaystyle \partial \!\,} 
partial;
d 
∂f/∂x_{i} means the partial derivative of f with respect to x_{i}, where f is a function on (x_{1}, …, x_{n}).  If f(x,y) := x^{2}y, then ∂f/∂x = 2xy,  
boundary of

∂M means the boundary of M  ∂{x : x ≤ 2} = {x : x = 2}  
degree of

∂f means the degree of the polynomial f.
(This may also be written deg f.) 
∂(x^{2} − 1) = 2  
E 
{\displaystyle \mathbb {E} }
{\displaystyle \mathrm {E} } 
expected value

the value of a random variable one would “expect” to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained  {\displaystyle \mathbb {E} [X]={\frac {x_{1}p_{1}+x_{2}p_{2}+\dotsb +x_{k}p_{k}}{p_{1}+p_{2}+\dotsb +p_{k}}}} 
∃

{\displaystyle \exists \!\,} 
there exists;
there is; there are 
∃ x: P(x) means there is at least one x such that P(x) is true.  ∃ n ∈ ℕ: n is even. 
∃!

{\displaystyle \exists !\!\,} 
there exists exactly one

∃! x: P(x) means there is exactly one x such that P(x) is true.  ∃! n ∈ ℕ: n + 5 = 2n. 
∈ ∉ 
{\displaystyle \in \!\,}
{\displaystyle \notin \!\,} 
is an element of;
is not an element of everywhere, set theory

a ∈ S means a is an element of the set S;^{[7]} a ∉ S means ais not an element of S.^{[7]}  (1/2)^{−1} ∈ ℕ
2^{−1} ∉ ℕ 
∌

{\displaystyle \not \ni } 
does not contain as an element

S ∌ e means the same thing as e ∉ S, where S is a set ande is not an element of S.  
∋

{\displaystyle \ni } 
such that symbol
such that

often abbreviated as “s.t.”; : and  are also used to abbreviate “such that”. The use of ∋ goes back to early mathematical logic and its usage in this sense is declining.  Choose {\displaystyle x} ∋ 2{\displaystyle x} and 3{\displaystyle x}. (Here  is used in the sense of “divides”.) 
contains as an element

S ∋ e means the same thing as e ∈ S, where S is a set ande is an element of S.  
ℍ 
{\displaystyle \mathbb {H} \!\,}
{\displaystyle \mathbf {H} \!\,} 
quaternions or Hamiltonian quaternions
H;
the (set of) quaternions 
ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}.  
ℕ 
{\displaystyle \mathbb {N} \!\,}
{\displaystyle \mathbf {N} \!\,} 
the (set of) natural numbers

N means either { 0, 1, 2, 3, …} or { 1, 2, 3, …}.
The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author’s definition of N. Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤. 
ℕ = {a : a ∈ ℤ} or ℕ = {a > 0: a ∈ ℤ} 
○

{\displaystyle \circ } 
entrywise product

For two matrices (or vectors) of the same dimensions {\displaystyle A,B\in {\mathbb {R} }^{m\times n}} the Hadamard product is a matrix of the same dimensions {\displaystyle A\circ B\in {\mathbb {R} }^{m\times n}} with elements given by {\displaystyle (A\circ B)_{i,j}=(A)_{i,j}\cdot (B)_{i,j}}.  {\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\circ {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&4\\0&0\\\end{bmatrix}}} 
∘

{\displaystyle \circ \!\,} 
composed with

f ∘ g is the function such that (f ∘ g)(x) = f(g(x)).^{[10]}  if f(x) := 2x, and g(x) := x + 3, then (f ∘ g)(x) = 2(x + 3). 
{\displaystyle O} 
bigoh of

The Big O notation describes the limiting behavior of afunction, when the argument tends towards a particular value or infinity.  If f(x) = 6x^{4} − 2x^{3} + 5 and g(x) = x^{4}, then {\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,}  
{ } 
{\displaystyle \emptyset \!\,}
{\displaystyle \varnothing \!\,} {\displaystyle \{\}\!\,} 
the empty set null set

∅ means the set with no elements.^{[7]} { } means the same.  {n ∈ ℕ : 1 < n^{2} < 4} = ∅ 
ℙ 
{\displaystyle \mathbb {P} \!\,}
{\displaystyle \mathbf {P} \!\,} 
P;
the set of prime numbers 
ℙ is often used to denote the set of prime numbers.  {\displaystyle 2\in \mathbb {P} ,3\in \mathbb {P} ,8\notin \mathbb {P} } 
P;
the projective space; the projective line; the projective plane 
ℙ means a space with a point at infinity.  {\displaystyle \mathbb {P} ^{1}},{\displaystyle \mathbb {P} ^{2}}  
the probability of

ℙ(X) means the probability of the event X occurring.
This may also be written as P(X), Pr(X), P[X] or Pr[X]. 
If a fair coin is flipped, ℙ(Heads) = ℙ(Tails) = 0.5.  
the Power set of

Given a set S, the power set of S is the set of all subsets of the set S. The power set of S0 isdenoted by P(S).  The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2. }}  
ℚ 
{\displaystyle \mathbb {Q} \!\,}
{\displaystyle \mathbf {Q} \!\,} 
Q;
the (set of) rational numbers; the rationals 
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.  3.14000… ∈ ℚ
π ∉ ℚ 
ℝ 
{\displaystyle \mathbb {R} \!\,}
{\displaystyle \mathbf {R} \!\,} 
R;
the (set of) real numbers; the reals 
ℝ means the set of real numbers.  π ∈ ℝ
√(−1) ∉ ℝ 
{\displaystyle {}^{\dagger }\!\,} 
conjugate transpose;
adjoint; Hermitian adjoint/conjugate/transpose/dagger 
A^{†} means the transpose of the complex conjugate of A.^{[11]}
This may also be written A^{∗T}, A^{T∗}, A^{∗}, A^{T} or A^{T}. 
If A = (a_{ij}) then A^{†} = (a_{ji}).  
{\displaystyle {}^{\mathsf {T}}\!\,} 
transpose

A^{T} means A, but with its rows swapped for columns.
This may also be written A′, A^{t} or A^{tr}. 
If A = (a_{ij}) then A^{T} = (a_{ji}).  
{\displaystyle \top \!\,} 
the top element

⊤ means the largest element of a lattice.  ∀x : x ∨ ⊤ = ⊤  
the top type; top

⊤ means the top or universal type; every type in the type system of interest is a subtype of top.  ∀ types T, T <: ⊤  
{\displaystyle \bot \!\,} 
is perpendicular to

x ⊥ y means x is perpendicular to y; or more generally x isorthogonal to y.  If l ⊥ m and m ⊥ n in the plane, then l n.  
orthogonal/ perpendicular complement of;
perp 
W^{⊥} means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W.  Within {\displaystyle \mathbb {R} ^{3}}, {\displaystyle (\mathbb {R} ^{2})^{\perp }\cong \mathbb {R} }.  
is coprime to

x ⊥ y means x has no factor greater than 1 in common withy.  34 ⊥ 55  
is independent of

A ⊥ B means A is an event whose probability is independent of event B.  If A ⊥ B, then P(AB) = P(A).  
the bottom element

⊥ means the smallest element of a lattice.  ∀x : x ∧ ⊥ = ⊥  
the bottom type;
bot 
⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system.  ∀ types T, ⊥ <: T  
is comparable to

x ⊥ y means that x is comparable to y.  {e, π} ⊥ {1, 2, e, 3, π} under set containment.  
𝕌 
{\displaystyle \mathbb {U} \!\,}
{\displaystyle \mathbf {U} \!\,} 
U;
the universal set; the set of all numbers; all numbers considered 
𝕌 means “the set of all elements being considered.” It may represent all numbers both real and complex, or any subset of these–hence the term “universal”. 
𝕌 = {ℝ,ℂ} includes all numbers.
If instead, 𝕌 = {ℤ,ℂ}, then π ∉ 𝕌. 
∪

{\displaystyle \cup \!\,} 
the union of … or …;
union 
A ∪ B means the set of those elements which are either in A, or in B, or in both.^{[5]}  A ⊆ B ⇔ (A ∪ B) = B 
∩

{\displaystyle \cap \!\,} 
intersected with;
intersect 
A ∩ B means the set that contains all those elements that Aand B have in common.^{[5]}  {x ∈ ℝ : x^{2} = 1} ∩ ℕ = {1} 
∨

{\displaystyle \lor \!\,} 
logical disjunction or join in alattice
or;
max; join 
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). 
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. 
∧

{\displaystyle \land \!\,}  The statement A ∧ B is true if A and B are both true; else it is false.
For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)). 
n < 4 ∧ n > 2 ⇔ n = 3 when n is a natural number.  
wedge product;
exterior product 
u ∧ v means the wedge product of any multivectors u and v. In threedimensional Euclidean space the wedge product and the cross product of two vectors are each other’s Hodge dual.  {\displaystyle u\wedge v=*(u\times v)\ {\text{ if }}u,v\in \mathbb {R} ^{3}}  
{\displaystyle \times \!\,} 
times;
multiplied by 
3 × 4 means the multiplication of 3 by 4.
(The symbol * is generally used in programming languages, where ease of typing and use of ASCII text is preferred.) 
7 × 8 = 56  
the Cartesian product of … and …;
the direct product of … and … 
X × Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.  {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}  
cross

u × v means the cross product of vectors u and v  (1,2,5) × (3,4,−1) = (−22, 16, − 2) 

the group of units of

R^{×} consists of the set of units of the ring R, along with the operation of multiplication.
This may also be written R^{∗} as described below, or U(R). 
{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\times }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}  
⊗

{\displaystyle \otimes \!\,} 
tensor product of

{\displaystyle V\otimes U} means the tensor product of V and U.^{[12]} {\displaystyle V\otimes _{R}U}means the tensor product of modules V and U over the ringR.  {1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}} 
⋉ ⋊ 
{\displaystyle \ltimes \!\,}
{\displaystyle \rtimes \!\,} 
the semidirect product of

N ⋊_{φ} H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N ⋊_{φ} H, then G is said to split over N.
(⋊ may also be written the other way round, as ⋉, or as ×.) 
{\displaystyle D_{2n}\cong \mathrm {C} _{n}\rtimes \mathrm {C} _{2}} 
the semijoin of

R ⋉ S is the semijoin of the relations R and S, the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names.  R {\displaystyle \ltimes } S = {\displaystyle \Pi }_{a1,..,an}(R {\displaystyle \bowtie } S)  
⋈

{\displaystyle \bowtie \!\,} 
the natural join of

R ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names.  
ℤ 
{\displaystyle \mathbb {Z} \!\,}
{\displaystyle \mathbf {Z} \!\,} 
the (set of) integers

ℤ means {…, −3, −2, −1, 0, 1, 2, 3, …}.ℤ^{+} or ℤ^{>} means {1, 2, 3, …} . ℤ^{*} or ℤ^{≥} means {0, 1, 2, 3, …} . 
ℤ = {p, −p : p ∈ ℕ ∪ {0}} 
ℤ_{n} ℤ_{p} Z_{n} Z_{p} 
{\displaystyle \mathbb {Z} _{n}\!\,}
{\displaystyle \mathbb {Z} _{p}\!\,} {\displaystyle \mathbf {Z} _{n}\!\,} {\displaystyle \mathbf {Z} _{p}\!\,} 
the (set of) integers modulo n

ℤ_{n} means {[0], [1], [2], …[n−1]} with addition and multiplication modulo n.
Note that any letter may be used instead of n, such as p. To avoid confusion with padic numbers, use ℤ/pℤ or ℤ/(p)instead. 
ℤ_{3} = {[0], [1], [2]} 
the (set of) padic integers

Note that any letter may be used instead of p, such as n or l. 
Symbols based on Hebrew or Greek letters[edit]
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\displaystyle \aleph \!\,} 
aleph

ℵ_{α} represents an infinite cardinality (specifically, the αth one, where α is an ordinal).  ℕ = ℵ_{0}, which is called alephnull.  
{\displaystyle \beth \!\,} 
beth

ℶ_{α} represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ).  {\displaystyle \beth _{1}=P(\mathbb {N} )=2^{\aleph _{0}}.}  
{\displaystyle \delta \!\,} 
Dirac delta of

{\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}  δ(x)  
Kronecker delta of

{\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}  δ_{ij}  
Functional derivative of

{\displaystyle {\begin{aligned}\left\langle {\frac {\delta F[\varphi (x)]}{\delta \varphi (x)}},f(x)\right\rangle &=\int {\frac {\delta F[\varphi (x)]}{\delta \varphi (x’)}}f(x’)dx’\\&=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon f(x)]F[\varphi (x)]}{\varepsilon }}\\&=\left.{\frac {d}{d\epsilon }}F[\varphi +\epsilon f]\right_{\epsilon =0}.\end{aligned}}}  {\displaystyle {\frac {\delta V(r)}{\delta \rho (r’)}}={\frac {1}{4\pi \epsilon _{0}rr’}}}  
∆ ⊖ 
{\displaystyle \vartriangle \!\,}
{\displaystyle \ominus \!\,} 
symmetric difference

A ∆ B (or A ⊖ B) means the set of elements in exactly one of Aor B.
(Not to be confused with delta, Δ, described below.) 
{1,5,6,8} ∆ {2,5,8} = {1,2,6}
{3,4,5,6} ⊖ {1,2,5,6} = {1,2,3,4} 
{\displaystyle \Delta \!\,} 
delta;
change in 
Δx means a (noninfinitesimal) change in x.
(If the change becomes infinitesimal, δ and even d are used instead. Not to be confused with the symmetric difference, written ∆, above.) 
{\displaystyle {\tfrac {\Delta y}{\Delta x}}} is the gradient of a straight line.  
Laplace operator

The Laplace operator is a second order differential operator in ndimensional Euclidean space  If ƒ is a twicedifferentiable realvalued function, then the Laplacian of ƒ is defined by {\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}  
{\displaystyle \nabla \!\,}  ∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (∂f / ∂x_{1}, …, ∂f /∂x_{n}).  If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)  
del dot;
divergence of 
{\displaystyle \nabla \cdot {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}}  If {\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }, then {\displaystyle \nabla \cdot {\vec {v}}=3y+2yz}.  
curl of

{\displaystyle \nabla \times {\vec {v}}=\left({\partial v_{z} \over \partial y}{\partial v_{y} \over \partial z}\right)\mathbf {i} } {\displaystyle +\left({\partial v_{x} \over \partial z}{\partial v_{z} \over \partial x}\right)\mathbf {j} +\left({\partial v_{y} \over \partial x}{\partial v_{x} \over \partial y}\right)\mathbf {k} } 
If {\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }, then {\displaystyle \nabla \times {\vec {v}}=y^{2}\mathbf {i} 3x\mathbf {k} }.  
{\displaystyle \pi \!\,} 
pi;
3.1415926…; ≈355÷113 
Used in various formulas involving circles; π is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14159. It is also the ratio of the circumference to the diameter of a circle.  A = πR^{2} = 314.16 → R = 10  
Projection of

{\displaystyle \pi _{a_{1},\ldots ,a_{n}}(R)} restricts {\displaystyle R} to the {\displaystyle \{a_{1},\ldots ,a_{n}\}} attribute set.  {\displaystyle \pi _{\text{Age,Weight}}({\text{Person}})}  
the nth Homotopy group of

{\displaystyle \pi _{n}(X)} consists of homotopy equivalence classes of base point preserving maps from an ndimensional sphere (with base point) into the pointed space X.  {\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i1}(S^{3})}  
{\displaystyle \prod } 
product over … from … to … of

{\displaystyle \prod _{k=1}^{n}a_{k}} means {\displaystyle a_{1}a_{2}\dots a_{n}}.  {\displaystyle \prod _{k=1}^{4}(k+2)=(1+2)(2+2)(3+2)(4+2)=3\times 4\times 5\times 6=360}  
the Cartesian product of;
the direct product of 
{\displaystyle \prod _{i=0}^{n}{Y_{i}}} means the set of all (n+1)tuples

{\displaystyle \prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}}  
∐

{\displaystyle \coprod \!\,} 
coproduct over … from … to … of

A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the “least specific” object to which each object in the family admits a morphism.  
{\displaystyle \sigma \!\,} 
Selection of

The selection {\displaystyle \sigma _{a\theta b}(R)} selects all those tuples in {\displaystyle R} for which {\displaystyle \theta }holds between the {\displaystyle a} and the {\displaystyle b} attribute. The selection {\displaystyle \sigma _{a\theta v}(R)}selects all those tuples in {\displaystyle R} for which {\displaystyle \theta } holds between the {\displaystyle a}attribute and the value {\displaystyle v}.  {\displaystyle \sigma _{\mathrm {Age} \geq 34}(\mathrm {Person} )} {\displaystyle \sigma _{\mathrm {Age} =\mathrm {Weight} }(\mathrm {Person} )} 

{\displaystyle \sum } 
sum over … from … to … of

{\displaystyle \sum _{k=1}^{n}{a_{k}}} means {\displaystyle a_{1}+a_{2}+\cdots +a_{n}}.  {\displaystyle \sum _{k=1}^{4}{k^{2}}=1^{2}+2^{2}+3^{2}+4^{2}=1+4+9+16=30} 
PHYSICS NOTATION
Symbol  Meaning  SI unit of measure 

{\displaystyle A}  area  meter squared (m^{2}) 
magnetic vector potential  
amplitude  meter  
{\displaystyle \mathbf {a} }  acceleration  meters per second squared (m/s^{2}) 
{\displaystyle \mathbf {B} }  magnetic flux density also called the magnetic field density or magnetic induction 
tesla (T), or equivalently, weber per square meter (Wb/m^{2}) 
{\displaystyle C}  capacitance  farad (F) 
heat capacity  joule per kelvin (J K^{−1}), or equivalently, joule per degree Celsius (J °C^{−1})  
constant of integration  varied depending on context  
{\displaystyle c}  speed of light (in vacuum)  299,792,458 meter per second (m/s) 
speed of sound  340.29 meter per second (m/s)  
specific heat capacity  joule per kilogram per kelvin (J kg^{−1} K^{−1}), or equivalently, joule perkilogram per degree Celsius (J kg^{−1} °C^{−1})  
viscous damping coefficient  kilogram per second (kg/s)  
{\displaystyle \mathbf {D} }  electric displacement field also called the electric flux density 
coulomb per square meter (C/m^{2}) 
{\displaystyle D}  density  kilograms per cubic meter (kg/m^{3}) 
{\displaystyle d}  distance  meter (m) 
direction  unitless  
impact parameter  meter (m)  
diameter  meter (m)  
differential(e.g. {\displaystyle dx})  
{\displaystyle d\mathbf {A} }  differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S 
square meter (m^{2}) 
{\displaystyle dV}  differential element of volume V enclosed by surface S  cubic meter (m^{3}) 
{\displaystyle \mathbf {E} }  electric field  newton per coulomb (N C^{−1}), or equivalently, volt per meter (V m^{−1}) 
{\displaystyle E}  energy  joule (J) 
Young’s Modulus  Pascal (Pa) or Newton per square meter (N/m^{2)} or kilogram per meter persecond squared (kg·m^{−1}·s^{−2)}  
{\displaystyle e}  eccentricity  unitless 
2.71828… (base of the natural logarithm), electron, elementary charge  
{\displaystyle \mathbf {F} }  force  newton (N) 
{\displaystyle f}  frequency  hertz (Hz) 
function  
friction  newton (N)  
{\displaystyle G}  the gravitational constant  newton meter squared per kilogram squared (N m^{2}/kg^{2}) 
{\displaystyle g}  acceleration due to gravity  meter per second squared (m/s^{2}), or equivalently, newton per kilogramme(N/kg) 
{\displaystyle \mathbf {H} }  magnetic field strength also called just magnetic field 
ampere per meter (A/m) 
{\displaystyle H}  Hamiltonian  joule (J) 
{\displaystyle h}  height  meter (m) 
Planck’s constant  joule second (J s)  
{\displaystyle \hbar }  reduced Planck’s constant {\displaystyle \textstyle \left({\frac {h}{2\pi }}\right)}  joule second (J s) 
{\displaystyle I}  action  joule second (J s) 
intensity  watt per square meter (W/m^{2})  
sound intensity  watt per square meter (W/m^{2})  
electric current  ampere (A)  
moment of inertia  kilogram meter squared (kg m^{2})  
{\displaystyle i}  intensity  watt per square meter (W/m^{2}) 
imaginary unit  
{\displaystyle \mathbf {\hat {i}} }  Cartesian xaxis basis unit vector  unitless 
{\displaystyle \mathbf {J} }  free current density, not including polarization or magnetization currents bound in a material 
ampere per square meter (A/m^{2}) 
impulse  kilogram meter per second (kg m/s)  
{\displaystyle \mathbf {\hat {j}} }  Cartesian yaxis basis unit vector  unitless 
{\displaystyle K}  kinetic energy2π/λ  joule (J) 
{\displaystyle k}  Boltzmann constant  joule per kelvin (J/K) 
wavenumber  radians per meter (m^{−1})  
{\displaystyle \mathbf {\hat {k}} }  Cartesian zaxis basis unit vector  unitless 
{\displaystyle L}  inductance  henry (H) 
luminosity  watt (W)  
angular momentum  newton meter second (N m s or kg m^{2} s^{−1})  
{\displaystyle l}  length  meter (m) 
{\displaystyle M}  magnetization  ampere per meter (A/m) 
moment of force often simply called moment or torque 
newton meter (N m)  
{\displaystyle m}  mass  kilogram (kg) 
{\displaystyle N}  normal vector  unit varies depending on context 
atomic number  unitless  
{\displaystyle n}  refractive index  unitless 
principal quantum number  unitless  
{\displaystyle P}  power  watt (W) 
{\displaystyle \mathbf {p} }  momentum  kilogram meter per second (kg m/s) 
pressure  pascal (Pa)  
{\displaystyle Q}  electric charge  coulomb (C) 
Heat  joule (J)  
{\displaystyle q}  electric charge  coulomb (C) 
{\displaystyle R}  electrical resistance  ohm (Ω) 
Ricci tensor  unitless  
radiancy  
gas constant  joule per kilogramme kelvin (J/kgK)  
{\displaystyle \mathbf {r} }  radius vector (position)  meter (m) 
{\displaystyle r}  radius of rotation or distance between two things such as the masses inNewton’s law of universal gravitation  meter (m) 
{\displaystyle S}  surface area  m^{2} 
entropy  joule per kelvin (J/K)  
action  
{\displaystyle s}  arc length  meter (m) 
displacement  
{\displaystyle T}  period  second (s) 
thermodynamic temperature also called absolute temperature 
kelvin (K)  
{\displaystyle t}  time  second (s) 
{\displaystyle \mathbf {U} }  fourvelocity  meter per second (m/s) 
{\displaystyle U}  potential energy  joule (J) 
internal energy  joule (J)  
{\displaystyle u}  relativistic mass  kilogram (kg) 
energy density  joule per cubic meter (J/m^{3}) or joule per kilogram (J/kg) depending on the context  
{\displaystyle V}  voltage also called electric potential difference 
volt (V) 
volume  cubic meter (m^{3})  
shear force  
{\displaystyle \mathbf {v} }  velocity  meter per second (m/s) 
{\displaystyle W}  mechanical work  joule (J) 
{\displaystyle w}  width  meter (m) 
{\displaystyle x}  a generic unknown  varied depending on context 
displacement  meter (m)  
{\displaystyle Z}  electrical impedance  ohm (Ω) 
Greek characters[edit]
Symbol  Name  Meaning  SI unit of measure 

{\displaystyle \alpha }  alpha  angular acceleration  radian per second squared (rad/s^{2}) 
{\displaystyle \beta }  beta  velocity in terms of the speed of light c  unitless 
{\displaystyle \gamma }  gamma  Lorentz factor  unitless 
photon  
gamma ray  
shear strain  
Heat capacity ratio  unitless  
{\displaystyle \Delta }  delta  a change in a variable (e.g. {\displaystyle \Delta x})  unitless 
Laplace operator  
{\displaystyle \delta }  delta  displacement (usually small)  
{\displaystyle \epsilon }  epsilon  permittivity  farad per meter (F/m) 
strain  unitless  
ε_{0}  epsilonnought  The vacuum permittivity constant  
{\displaystyle \zeta }  zeta  damping ratio  unitless 
{\displaystyle \eta }  eta  energy efficiency  unitless 
coefficient of viscosity also called simply viscosity 
pascal second (Pa s)  
{\displaystyle \theta }  theta  angular displacement  radian (rad) 
{\displaystyle \mathrm {K} }  kappa  torsion coefficient also called torsion constant 
newton meter per radian (N m/rad) 
{\displaystyle \Lambda }  lambda  cosmological constant  per second squared (s^{−2}) 
{\displaystyle \lambda }  wavelength  meter (m)  
{\displaystyle \mathbf {\mu } }  mu  magnetic moment  ampere square meter (A m^{2}) 
coefficient of friction  unitless  
dynamic viscosity  Pascal second (Pa s)  
permeability (electromagnetism)  Henry per meter (H/m)  
reduced mass  kilogram (kg)  
{\displaystyle \nu }  nu  frequency  hertz (Hz) 
kinematic viscosity  meters squared per second (m^{2}/s)  
{\displaystyle \pi }  pi  3.14159… (irrational number)  
{\displaystyle \rho }  rho  mass density usually simply called density 
kilogram per cubic meter (kg/m^{3}) 
free electric charge density, not including dipole charges bound in a material 
coulomb per cubic meter (C/m^{3})  
resistivity  Ohm meter ({\displaystyle \Omega } m)  
{\displaystyle \Sigma }  sigma  summation operator  
{\displaystyle \sigma }  sigma  electrical conductivity  Siemens per meter (S/m) 
normal stress  
{\displaystyle \tau }  tau  torque  newton meter (N m) 
shear stress  
time constant  second (s)  
6.28318… (2π)  
{\displaystyle \Phi }  phi  field strength  unit varies depending on context 
magnetic flux  Weber (Wb)  
{\displaystyle \phi }  phi  electric potentialHiggs Field  
{\displaystyle \Psi }  psi  wave function  unitless 
{\displaystyle \omega }  omega  angular frequency2πf  radian per second (rad/s) 
{\displaystyle \Omega }  omega  electric resistance  ohm 
Other characters[edit]
Symbol  Name  Meaning  SI unit of measure 

{\displaystyle \nabla \cdot }  nabla dot  the divergence operator often pronounced “del dot” 
per meter (m^{−1}) 
{\displaystyle \nabla \times }  nabla cross  the curl operator often pronounced “del cross” 
per meter (m^{−1}) 
{\displaystyle \nabla }  nabla  del (differential operator)  
{\displaystyle \Delta }  Delta  the Laplace operator  per square meter (m^{−2}) 
{\displaystyle \partial }  “der”, “dow”, “die”, “partial” or simply “d”  partial derivative (e.g. {\displaystyle \partial y/\partial x})  
{\displaystyle \Box }  D’Alembert operator  {\displaystyle \nabla ^{2}\partial _{t}^{2}} 
Greek letters used in mathematics, science, and engineering
Greek letters  

Name  TeX  HTML  Name  TeX  HTML  Name  TeX  HTML  Name  TeX  HTML  Name  TeX  HTML  
Alpha  {\displaystyle \mathrm {A} \,\alpha \,}  Α α  Digamma  {\displaystyle \digamma \,}  Ϝ ϝ  Kappa  {\displaystyle \mathrm {K} \,\kappa \,\varkappa \,}  Κ κ ϰ  Omicron  {\displaystyle \mathrm {O} \,\mathrm {o} \,}  Ο ο  Upsilon  {\displaystyle \Upsilon \,\upsilon \,}  Υ υ  
Beta  {\displaystyle \mathrm {B} \,\beta \,}  Β β  Zeta  {\displaystyle \mathrm {Z} \,\zeta \,}  Ζ ζ  Lambda  {\displaystyle \Lambda \,\lambda \,}  Λ λ  Pi  {\displaystyle \Pi \,\pi \,\varpi \,}  Π π ϖ  Phi  {\displaystyle \Phi \,\phi \,\varphi \,}  Φ ϕ φ  
Gamma  {\displaystyle \Gamma \,\gamma \,}  Γ γ  Eta  {\displaystyle \mathrm {H} \,\eta \,}  Η η  Mu  {\displaystyle \mathrm {M} \,\mu \,}  Μ μ  Rho  {\displaystyle \mathrm {P} \,\rho \,\varrho \,}  Ρ ρ ϱ  Chi  {\displaystyle \mathrm {X} \,\chi \,}  Χ χ  
Delta  {\displaystyle \Delta \,\delta \,}  Δ δ  Theta  {\displaystyle \Theta \,\theta \,\vartheta \,}  Θ θ ϑ  Nu  {\displaystyle \mathrm {N} \,\nu \,}  Ν ν  Sigma  {\displaystyle \Sigma \,\sigma \,\varsigma \,}  Σ σ ς  Psi  {\displaystyle \Psi \,\psi \,}  Ψ ψ  
Epsilon  {\displaystyle \mathrm {E} \,\epsilon \,\varepsilon \,}  Ε ϵ ε  Iota  {\displaystyle \mathrm {I} \,\iota \,}  Ι ι  Xi  {\displaystyle \Xi \,\xi \,}  Ξ ξ  Tau  {\displaystyle \mathrm {T} \,\tau \,}  Τ τ  Omega  {\displaystyle \Omega \,\omega \,}  Ω ω 
Concepts represented by a Greek letter[edit]
Αα (alpha)[edit]
 {\displaystyle \alpha } represents:
 the first angle in a triangle, opposite the side A
 one root of a quadratic equation, where β represents the other
 the ratio of collector current to emitter current in a bipolar junction transistor (BJT) in electronics
 the statistical significance of a result
 the false positive rate in statistics (“Type I” error)
 the reciprocal of the sacrifice ratio
 the fine structure constant in physics
 the angle of attack of an aircraft
 an alpha particle (He^{2+})
 angular acceleration in physics
 the linear thermal expansion coefficient
 the thermal diffusivity
 In organic chemistry the αcarbon is the backbone carbon next to the carbonyl carbon, most often for amino acids
 right ascension in astronomy
 the brightest star in a constellation
 Iron ferrite and numerous phases within materials science
 the return in excess of the compensation for the risk borne in investment
 the αconversion in lambda calculus
 the independence number of a graph
Ββ (beta)[edit]
 Β represents the beta function
 β represents:
 the thermodynamic beta, equal to (k_{B}T)^{−1}, where k_{B} is Boltzmann’s constant and T is the absolute temperature.
 the second angle in a triangle, opposite the side B
 one root of a quadratic equation, where α represents the other
 the standardized regression coefficient for predictor or independent variables in linear regression (unstandardized regression coefficients are represented with the lowercase Latin b, but are often called “betas” as well)
 the ratio of collector current to base current in a bipolar junction transistor (BJT) in electronics (current gain)
 the false negative rate in statistics (“Type II” error)
 the beta coefficient, the nondiversifiable risk, of an asset in mathematical finance
 the sideslip angle of an airplane
 the firstorder effects of variations in Coriolis force with latitude in planetary dynamics
 a beta particle (e^{−})
 sound intensity
 velocity divided by the speed of light in special relativity
 the beta brain wave in brain or cognitive sciences
 ecliptic latitude in astronomy
 The ratio of plasma pressure to magnetic pressure in plasma physics
 βreduction in lambda calculus
 The ratio of the velocity of an object to the speed of light as used in the Lorentz factor
 In organic chemistry, β represents the second carbon from a functional group
Γγ (gamma)[edit]
 Γ represents:
 the circulation in fluid dynamics
 the reflection coefficient of a transmission or telecommunication line.
 the confinement factor of an optical mode in a waveguide
 the gamma function, a generalization of the factorial
 the upper incomplete gamma function
 the modular group, the group of fractional linear transformations
 the gamma distribution, a continuous probability distribution defined using the gamma function
 secondorder sensitivity to price in mathematical finance
 the Christoffel symbols of the second kind
 the neighbourhood of a vertex in a graph
 the stack alphabet in the formal definition of a pushdown automaton
 γ represents:
 the circulation strength in fluid dynamics
 the partial safety factors applied to loads and materials in structural engineering
 the specific weight of substances
 the lower incomplete gamma function
 the third angle in a triangle, opposite the side C
 the Euler–Mascheroni constant in mathematics
 gamma rays and the photon
 the heat capacity ratio in thermodynamics
 the Lorentz factor in special relativity
 the damping constant (kg/s)
Δδ (delta)[edit]
 Δ represents:
 a finite difference
 a difference operator
 a symmetric difference
 the Laplace operator
 the angle that subtends the arc of a circular curve in surveying
 the determinant of an inverse matrix^{[1]}
 the maximum degree of any vertex in a given graph
 the difference or change in a given variable, e.g. ∆v means a difference or change in velocity
 sensitivity to price in mathematical finance
 distance to Earth, measured in astronomical units
 heat in a chemical formula
 the discriminant in the quadratic formula which determines the nature of the roots
 the degrees of freedom in a nonpooled statistical hypothesis test of two population means
 δ represents:
 percent error
 a variation in the calculus of variations
 the Kronecker delta function
 the Feigenbaum constant
 the force of interest in mathematical finance
 the Dirac delta function
 the receptor which enkephalins have the highest affinity for in pharmacology^{[2]}
 the Skorokhod integral in Malliavin calculus, a subfield of stochastic analysis
 the minimum degree of any vertex in a given graph
 a partial charge. δ− represents a negative partial charge, and δ+ represents a positive partial charge chemistry (See also: Solvation)
 the Chemical shift of an atomic nucleus in NMR spectroscopy. For protons, this is relative to tetramethylsilane = 0.
 stable isotope compositions
 declination in astronomy
 the Turner function in computational material science
 depreciation in macroeconomics
 noncentrality measure in statistics^{[3]}
Εε (epsilon)[edit]
 ε represents:
 a small positive quantity; see limit
 a random error in regression analysis
 the absolute value of an error ^{[4]}
 in set theory, the limit ordinal of the sequence {\displaystyle \omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\dots }
 in computer science, the empty string
 the LeviCivita symbol
 in electromagnetics, dielectricpermittivity
 emissivity
 strain in continuum mechanics
 permittivity
 the Earth’s axial tilt in astronomy
 elasticity in economics
 expected value in probability theory and statistics
 electromotive force
 in chemistry, the molar extinction coefficient of a chromophore.
 set membership symbol ∈ is based on ε
Ϝϝ (digamma)[edit]
 Ϝ is sometimes used to represent the digamma function, though the Latin letter F (which is nearly identical) is usually substituted.
 A hypothetical particle Ϝ speculated to be implicated in the 750 GeV diphoton excess, now known to be simply a statistical anomaly
Ζζ (zeta)[edit]
 ζ represents:
 the Riemann zeta function and other zeta functions in mathematics
 the coefficient of viscous friction in polymer dynamics
 the damping ratio
 relative vertical vorticity in fluid dynamics
Ηη (eta)[edit]
 Η represents:
 the Eta function of Ludwig Boltzmann‘s Htheorem (“Eta” theorem), in statistical mechanics
 Information theoretic (Shannon) entropy
 η represents:
 the intrinsic wave impedance of a medium (e.g. the impedance of free space)
 the partial regressioncoefficient in statistics
 elasticities in economics
 the absolute vertical vorticity (relative vertical vorticity + Coriolis effect) in fluid dynamics
 an index of refraction
 the eta meson
 viscosity
 energy conversion efficiency
 efficiency (physics)
 the Minkowski metric tensor in relativity
 noise in communication system models
 ηconversion in lambda calculus
 Costpush supply side shocks in the Phillips curve equation (economics)^{[citation needed]}
 A right angle, i.e., π/2, as a followup to the tau/pi argument ^{[5]}
Θθ (theta)[edit]
 Θ (uppercase) represents:
 an asymptotically tight bound related to big O notation.
 Debye temperature in solid state physics
 sensitivity to the passage of time in mathematical finance
 in set theory, a certain ordinal number
 in econometrics and statistics, a space of parameters from which estimates are drawn
 θ (lowercase) represents:
 a plane angle in geometry
 the angle to the x axis in the xyplane in spherical or cylindrical coordinates (mathematics)
 the angle to the z axis in spherical coordinates (physics)
 Bragg’s angle of diffraction
 the potential temperature in thermodynamics
 the mean time between failure in reliability engineering
 soil water contents in soil science
 in mathematical statistics, an unknown parameter
 theta functions
 the angle of a scattered photon during a Compton scattering interaction
 ϑ (“script theta”), the cursive form of theta, often used in handwriting, represents
 the first Chebyshev function in number theory
Ιι (iota)[edit]
 ι represents:
 an inclusion map in set theory
 the index generator function in APL (in the form ⍳)
 the orbital inclination in celestial mechanics.
Κκ (kappa)[edit]
 Κ represents:
 the Kappa number, indicating lignin content in pulp
 κ represents:
 the Von Kármán constant, describing the velocity profile of turbulent flow
 the kappa curve, a twodimensional algebraic curve
 the condition number of a matrix in numerical analysis
 the connectivity of a graph in graph theory
 curvature
 dielectric constant{\displaystyle (\varepsilon /\varepsilon _{0})}
 thermal conductivity (usually a lowercase Latin k)
 thermal diffusivity
 a spring constant (usually a lowercase Latin k)
 the heat capacity ratio in thermodynamics (usually γ)
 the receptor which dynorphins have the highest affinity for in pharmacology^{[2]}
Λλ (lambda)[edit]
 Λ represents:
 the von Mangoldt function in number theory
 the set of logical axioms in the axiomatic method of logical deduction in firstorder logic
 the cosmological constant
 the lambda baryon
 a diagonal matrix of eigenvalues in linear algebra
 the permeance of a material in electromagnetism
 a lattice
 λ represents:
 one wavelength of electromagnetic radiation
 the decay constant in radioactivity
 function expressions in the lambda calculus
 a general eigenvalue in linear algebra
 the expected number of occurrences in a Poisson distribution in probability
 the arrival rate in queueing theory
 the average lifetime or rate parameter in an exponential distribution (commonly used across statistics, physics, and engineering)
 the failure rate in reliability engineering
 the fundamental length of a fabrication process in VLSI design
 the mean or average value (probability and statistics)
 the latent heat of fusion
 the lagrange multiplier in the mathematical optimization method, known as the shadow price in economics
 the Lebesgue measure denotes the volume or measure of a Lebesgue measurable set
 longitude in geodesy
 linear density
 ecliptic longitude in astronomy
 the Liouville function in number theory
 the Carmichael function in number theory
 a unit of measure of volume equal to one microlitre (1 μL) or one cubic millimetre (1 mm³)
 the empty string in formal grammar
 binding of a variable in a function in lambda calculus in computer science.
Μμ (mu)[edit]
 μ represents:
 the Möbius function in number theory
 the ring representation of a representation module
 the population mean or expected value in probability and statistics
 a measure in measure theory
 micro, an SI prefix denoting 10^{−6} (one millionth)
 the coefficient of friction in physics
 the service rate in queueing theory
 the dynamic viscosity in physics
 magnetic permeability in electromagnetics
 a muon
 reduced mass
 chemical potential in condensed matter physics
 the ion mobility in plasma physics
 the Standard gravitational parameter in celestial mechanics
Νν (nu)[edit]
 ν represents:
 frequency in physics in hertz (Hz)
 degrees of freedom in statistics
 Poisson’s ratio in material science
 a neutrino
 kinematic viscosity of liquids
 stoichiometric coefficient in chemistry
 dimension of nullspace in mathematics
 true anomaly in celestial mechanics
 the matching number of a graph
Ξξ (xi)[edit]
 Ξ represents:
 the original Riemann Xi function, i.e. Riemann’s lower case ξ, as denoted by Edmund Landau and currently
 the grand canonical ensemble found in statistical mechanics
 the xi baryon
 ξ represents:
 the original Riemann Xi function
 the modified definition of Riemann xi function, as denoted by Edmund Landau and currently
 a random variable
 the extent of a chemical reaction
 coherence length
 the damping ratio
 universal set
Οο (omicron)[edit]
 Ο represents:
 big O notation (may be represented by an uppercase Latin O)
 o represents:
 small o notation (may be represented by a lowercase Latin o)
Ππ (pi)[edit]
 Π represents:
 the product operator in mathematics
 a plane
 the unary projection operation in relational algebra
 osmotic pressure
 π represents:
 Archimedes’ constant, the ratio of a circle‘s circumference to its diameter
 the primecounting function
 profit in microeconomics and game theory
 inflation in macroeconomics, expressed as a constant with respect to time
 the state distribution of a Markov chain
 in reinforcement learning, a policy function defining how a software agent behaves for each possible state of its environment
 a type of covalent bond in chemistry (pi bond)
 a pion (pi meson) in particle physics
 in statistics, the population proportion
 nucleotide diversity in molecular genetics
 in electronics, a special type of small signal model is referred to as a hybridpi model
 in relational algebra for databases, represents projection
 ϖ (a graphic variant, see pomega) represents:
 angular frequency of a wave, in fluid dynamics (angular frequency is usually represented by {\displaystyle \omega } but this may be confused with vorticity in a fluid dynamics context)
 longitude of pericenter, in astronomy^{[6]}
 comoving distance, in cosmology^{[7]}
Ρρ (rho)[edit]
 Ρ represents:
 one of the Gegenbauer functions in analytic number theory (may be replaced by the capital form of the Latin letter P).
 ρ represents:
 one of the Gegenbauer functions in analytic number theory.
 the Dickmande Bruijn function
 the radius in a polar, cylindrical, or spherical coordinate system
 the correlation coefficient in statistics
 the sensitivity to interest rate in mathematical finance
 density (mass or charge per unit volume; may be replaced by the capital form of the Latin letter D)
 resistivity
 the shape and reshape operators in APL (in the form ⍴)
 the utilization in queueing theory
 the rank of a matrix
 the rename operator in relational algebra
Σσς (sigma)[edit]
 Σ represents:
 the summation operator
 the covariance matrix
 the set of terminal symbols in a formal grammar
 σ represents:
 Stefan–Boltzmann constant in blackbody radiation
 the divisor function in number theory
 the real part of the complex variable s = σ + it in analytic number theory
 the sign of a permutation in the theory of finite groups
 the population standard deviation, a measure of spread in probability and statistics
 a type of covalent bond in chemistry (sigma bond)
 the selection operator in relational algebra
 stress in mechanics
 electrical conductivity
 area density
 nuclear cross section
 uncertainty
 utilization in operations management
 surface charge density for microparticles
Ττ (tau)[edit]
 τ (lowercase) represents:
 torque, the net rotational force in mechanics
 the elementary tau lepton in particle physics
 a mean lifetime, of an exponential decay or spontaneous emission process
 the time constant of any device, such as an RC circuit
 proper time in relativity
 one turn: the constant ratio of a circle‘s circumference to its radius, with value 2π (6.283…).^{[8]}
 Kendall tau rank correlation coefficient, a measure of rank correlation in statistics
 Ramanujan’s tau function in number theory
 a measure of opacity, or how much sunlight cannot penetrate the atmosphere
 the intertwining operator in representation theory
 shear stress in continuum mechanics
 an internal system step in transition systems
 a type variable in type theories, such as the simply typed lambda calculus
 path tortuosity in reservoir engineering
 in topology, a given topology
 the tau in biochemistry, a protein associated to microtubules
 the golden ratio 1.618… (although φ (phi) is more common)
 the number of divisors of highly composite numbers (sequence A000005 in the OEIS)
 in proton NMR spectroscopy, τ was formerly used for physical shift
Υυ (upsilon)[edit]
 Υ represents:
 the upsilon meson
 υ represents:
 frequency in physics textbooks
Φφ (phi)[edit]
 Φ represents:
 the work function in physics; the energy required by a photon to remove an electron from the surface of a metal
 magnetic flux
 the cumulative distribution function of the normal distribution in statistics
 phenyl functional group
 the reciprocal of the golden ratio (represented by φ, below), also represented as 1/φ
 the value of the integration of information in a system (based on integrated information theory)
 note: a symbol for the empty set, {\displaystyle \varnothing }, resembles Φ but is not Φ
 φ represents:
 the golden ratio 1.618… in mathematics, art, and architecture
 Euler’s totient function in number theory
 a holomorphic map on an analytic space
 the argument of a complex number in mathematics
 the value of a plane angle in physics and mathematics
 the angle to the z axis in spherical coordinates (mathematics)
 the angle to the x axis in the xyplane in spherical or cylindrical coordinates (physics)
 latitude in geodesy
 a scalar field
 radiant flux
 electric potential
 the probability density function of the normal distribution in statistics
 a feature of a syntactic node giving that node characteristics such as gender, number and person in syntax
 the diameter of a vessel (engineering)
 capacity reduction factor of materials in structural engineering
Χχ (chi)[edit]
 χ represents:
 the chi distribution in statistics ({\displaystyle \chi ^{2}} is the more frequently encountered chisquared distribution)
 the chromatic number of a graph in graph theory
 the Euler characteristic in algebraic topology
 electronegativity in the periodic table
 the Rabi frequency
 the spinor of a fundamental particle
 the Fourier transform of a linear response function
 a character in mathematics; especially a Dirichlet character in number theory
 the Sigma vectors in the unscented transform used in the unscented Kalman filter
 sometimes the mole fraction
 a characteristic or indicator function in mathematics
 the Magnetic susceptibility of a material in physics
Ψψ (psi)[edit]
 Ψ represents:
 water potential
 a quaternary combinator in combinatory logic
 ψ represents:
 the wave function in the Schrödinger equation of quantum mechanics
 the stream function in fluid dynamics
 yaw angle in vehicle dynamics
 the angle between the xaxis and the tangent to the curve in the intrinsic coordinates system
 the reciprocal Fibonacci constant
 the second Chebyshev function in number theory
 the polygamma function in mathematics
 load combination factor in structural engineering
 the Supergolden Ratio ^{[9]}
Ωω (omega)[edit]
 Ω represents:
 the SI unit measure of electrical resistance, the ohm
 angular velocity / radian frequency (rev/min)
 the right ascension of the ascending node (RAAN) or Longitude of the ascending node in astronomy and orbital mechanics
 the rotation rate of an object, particularly a planet, in dynamics
 the omega constant 0.5671432904097838729999686622…
 an asymptotic lower bound related to big O notation
 in probability theory and statistical mechanics, the set of possible distinct system states
 a solid angle
 the omega baryon
 the arithmetic function counting a number’s prime factors
 the density parameter in cosmology
 ω represents:
 angular velocity / radian frequency (rad/sec)
 the argument of periapsis in astronomy and orbital mechanics
 a complexcuberoot of unity — the other is ω² — (used to describe various ways of calculating the discrete Fourier transform)
 the differentiability class (i.e.{\displaystyle C^{\omega }}) for functions that are infinitely differentiable because they are complex analytic
 the first infiniteordinal
 the omega meson
 the set of natural numbers in set theory (although {\displaystyle \mathbb {N} } or N is more common in other areas of mathematics)
 an asymptotically dominant quantity related to big O notation
 in probability theory, a possible outcome of an experiment
 in economics, the total wealth of an agent in general equilibrium theory
 vertical velocity in pressurebased coordinate systems (commonly used in atmospheric dynamics)
 the arithmetic function counting a number’s distinct prime factors
 a differential form (esp. on an analytic space)
 the symbol ϖ, a graphic variant of π, is sometimes construed as omega with a bar over it; see π
 the last carbon atom of a chain of carbon atoms is sometimes called the ω (omega) position, reflecting that ω is the last letter of the Greek alphabet. This nomenclature can be useful in describing unsaturated fatty acids.
Notation in probability and statistics
Probability theory
 Random variables are usually written in upper case roman letters: X, Y, etc.
 Particular realizations of a random variable are written in corresponding lower case letters. For example x_{1}, x_{2}, …, x_{n}could be a sample corresponding to the random variable X and a cumulative probability is formally written {\displaystyle P(X>x)}to differentiate random variable from realization.
 The probability is sometimes written {\displaystyle \mathbb {P} } to distinguish it from other functions and measure P so as to avoid having to define ” P is a probability” and {\displaystyle \mathbb {P} (A)} is short for {\displaystyle P(\{\omega :X(\omega )\in A\})}, where {\displaystyle \omega } is an event and {\displaystyle X(\omega )} a corresponding random variable.
 {\displaystyle \mathbb {P} (A\cap B)} or {\displaystyle \mathbb {P} [A\cap B]} indicates the probability that events A and B both occur.
 {\displaystyle \mathbb {P} (A\cup B)} or {\displaystyle \mathbb {P} [A\cup B]} indicates the probability of either event A or event B occurring (“or” in this case means one or the other or both).
 σalgebras are usually written with upper case calligraphic (e.g. {\displaystyle {\mathcal {F}}} for the set of sets on which we define the probability P)
 Probability density functions (pdfs) and probability mass functions are denoted by lower case letters, e.g. f(x).
 Cumulative distribution functions (cdfs) are denoted by upper case letters, e.g. F(x).
 Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:{\displaystyle {\overline {F}}(x)=1F(x)}
 In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(z).
 Some common operators:

 E[X] : expected value of X
 var[X] : variance of X
 cov[X, Y] : covariance of X and Y
 X is independent of Y is often written {\displaystyle X\perp Y} or {\displaystyle X\perp \!\!\!\perp Y}, and X is independent of Y given W is often written
 {\displaystyle X\perp \!\!\!\perp Y\,\,W} or
 {\displaystyle X\perp Y\,\,W}
 {\displaystyle \textstyle P(A\mid B)}, the posterior probability, is the probability of {\displaystyle \textstyle A} given {\displaystyle \textstyle B}, i.e., {\displaystyle \textstyle A} after {\displaystyle \textstyle B} is observed.^{[citation needed]}
Statistics[edit]
 Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
 A tilde (~) denotes “has the probability distribution of”.
 Placing a hat, or caret, over a true parameter denotes an estimator of it, e.g., {\displaystyle {\widehat {\theta }}} is an estimator for {\displaystyle \theta }.
 The arithmetic mean of a series of values x_{1}, x_{2}, …, x_{n} is often denoted by placing an “overbar” over the symbol, e.g. {\displaystyle {\bar {x}}}, pronounced “x bar”.
 Some commonly used symbols for sample statistics are given below:
 the sample mean {\displaystyle {\bar {x}}},
 the sample variance s^{2},
 the sample standard deviation s,
 the sample correlation coefficient r,
 the sample cumulants k_{r}.
 Some commonly used symbols for population parameters are given below:
 the population mean μ,
 the population variance σ^{2},
 the population standard deviation σ,
 the population correlation ρ,
 the population cumulants κ_{r}.
Critical values[edit]
The αlevel upper critical value of a probability distribution is the value exceeded with probability α, that is, the value x_{α} such that F(x_{α}) = 1 − α where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:
 z_{α} or z(α) for the Standard normal distribution
 t_{α,ν} or t(α,ν) for the tdistribution with ν degrees of freedom
 {\displaystyle {\chi _{\alpha ,\nu }}^{2}} or {\displaystyle {\chi }^{2}(\alpha ,\nu )} for the chisquared distribution with ν degrees of freedom
 {\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}} or F(α,ν_{1},ν_{2}) for the Fdistribution with ν_{1} and ν_{2} degrees of freedom
Linear algebra[edit]
 Matrices are usually denoted by boldface capital letters, e.g. A.
 Column vectors are usually denoted by boldface lower case letters, e.g. x.
 The transpose operator is denoted by either a superscript T (e.g. A^{T}) or a prime symbol (e.g. A′).
 A row vector is written as the transpose of a column vector, e.g. x^{T} or x′.
Abbreviations[edit]
Common abbreviations include:
 a.e. almost everywhere
 a.s. almost surely
 cdf cumulative distribution function
 cmf cumulative mass function
 df degrees of freedom (also {\displaystyle \nu })
 i.i.d. independent and identically distributed
 pdf probability density function
 pmf probability mass function
 r.v. random variable
 w.p. with probability; wp1 with probability 1
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