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Encyclopedia > Table of mathematical symbols

The following table lists many specialized symbols commonly used in mathematics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

Symbol Name Explanation Examples
Category
=
equality x = y means x and y represent the same thing or value. 1 + 1 = 2
is equal to; equals
everywhere

<>

!=
inequation x ≠ y means that x and y do not represent the same thing or value.

(The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)
1 ≠ 2
is not equal to; does not equal
everywhere
<

>

strict inequality x < y means x is less than y.

x > y means x is greater than y.

x ≪ y means x is much less than y.

x ≫ y means x is much greater than y.
3 < 4
5 > 4.

0.003 ≪ 1000000 In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ... For the use of the < and > signs in punctuation, see Bracket. ...

is less than, is greater than, is much less than, is much greater than
order theory

<=

>=
inequality x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.

(The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to; varies as
everywhere
+
addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
disjoint union A1 + A2 means the disjoint union of sets A1 and A2. A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
the disjoint union of ... and ...
set theory
subtraction 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
minus
arithmetic
negative sign −3 means the negative of the number 3. −(−5) = 5
negative; minus
arithmetic
set-theoretic complement A − B means the set that contains all the elements of A that are not in B.

∖ can also be used for set-theoretic complement as described below.
{1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
multiplication 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
times
arithmetic
Cartesian product 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)}
the Cartesian product of ... and ...; the direct product of ... and ...
set theory
cross product u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
·
multiplication 3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56
times
arithmetic
dot product u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6
dot
vector algebra
÷

division 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
divided by
arithmetic
±
plus-minus 6 ± 3 means both 6 + 3 and 6 - 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
plus or minus
arithmetic
plus-minus 10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.
plus or minus
measurement
minus-plus 6 ± (3 ∓ 5) means both 6 + (3 - 5) and 6 - (3 + 5). cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
minus or plus
arithmetic
square root x means the positive number whose square is x. √4 = 2
the principal square root of; square root
real numbers
complex square root if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2). √(-1) = i
the complex square root of …

square root
complex numbers
|…|
absolute value or modulus |x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5|

i | = 1

| 3 + 4i | = 5
absolute value (modulus) of
numbers
Euclidean distance |x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5
Euclidean distance between; Euclidean norm of
Geometry
Determinant |A| means the determinant of the matrix A $begin{vmatrix} 1&2 2&4 end{vmatrix} = 0$
determinant of
Matrix theory
|
divides A single vertical bar is used to denote divisibility.
a|b means a divides b.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
Number Theory
!
factorial n ! is the product 1 × 2× ... × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
T
transpose Swap rows for columns Aij = (AT)ji
transpose
matrix operations
~
probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution
statistics
Row equivalence A~B means that B can be generated by using a series of elementary row operations on A $begin{bmatrix} 1&2 2&4 end{bmatrix} sim begin{bmatrix} 1&2 0&0 end{bmatrix}$
is row equivalent to
Matrix theory

material implication AB 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 for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below.
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if … then
propositional logic, Heyting algebra

material equivalence 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
if and only if; iff
propositional logic
¬

˜
logical negation The statement ¬A is true if and only if A 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.)
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
logical conjunction or meet in a lattice The statement AB 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.
and; min
propositional logic, lattice theory
logical disjunction or join in a lattice The statement AB 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.
or; max
propositional logic, lattice theory

exclusive or The statement AB is true when either A or B, but not both, are true. AB means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).

Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = )
direct sum of
Abstract algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n.
for all; for any; for each
predicate logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even.
there exists
predicate logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n.
there exists exactly one
predicate logic
:=

:⇔
definition x := y or x ≡ y means x is defined to be another name for y

(Some writers useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
cosh x := (1/2)(exp x + exp (−x))

A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
is defined as
everywhere
congruence △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is congruent to
geometry
congruence relation a ≡ b (mod n) means a − b is divisible by n 5 ≡ 11 (mod 3)
... is congruent to ... modulo ...
modular arithmetic
{ , }
set brackets {a,b,c} means the set consisting of a, b, and c. ℕ = { 1, 2, 3, …}
the set of …
set theory
{ : }

{ | }
set builder notation {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4}
the set of … such that
set theory

{ }
empty set means the set with no elements. { } means the same. {n ∈ ℕ : 1 < n2 < 4} =
the empty set
set theory

set membership a ∈ S means a is an element of the set S; a  S means a is not an element of S. (1/2)−1 ∈ ℕ

2−1
is an element of; is not an element of
everywhere, set theory

subset (subset) A ⊆ B means every element of A is also element of B.

(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
is a subset of
set theory

superset A ⊇ B means every element of B is also 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
is a superset of
set theory
set-theoretic union (exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both.
"A or B, but not both."

(inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B.
"A or B or both".
A ⊆ B  ⇔  (A ∪ B) = B (inclusive)
the union of … and …

union
set theory
set-theoretic intersection A ∩ B means the set that contains all those elements that A and B have in common. {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
intersected with; intersect
set theory
Δ
symmetric difference AΔB means the set of elements in exactly one of A or B. {1,5,6,8} Δ {2,5,8} = {1,2,6}
symmetric difference
set theory
set-theoretic complement A ∖ B means the set that contains all those elements of A that are not in B.

− can also be used for set-theoretic complement as described above.
{1,2,3,4} ∖ {3,4,5,6} = {1,2}
minus; without
set theory
( )
function application f(x) means the value of the function f at the element x. If f(x) := x2, then f(3) = 32 = 9.
of
set theory
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
parentheses
everywhere
f:XY
function arrow fX → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ be defined by f(x) := x2.
from … to
set theory,type theory
o
function composition fog is the function, such that (fog)(x) = f(g(x)). if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
composed with
set theory

N
natural numbers N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. ℕ = {|a| : a ∈ ℤ, a ≠ 0}
N
numbers

Z
integers ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. ℤ = {p, -p : p ∈ ℕ} ∪ {0}
Z
numbers

Q
rational numbers ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ

π ∉ ℚ
Q
numbers

R
real numbers ℝ means the set of real numbers. π ∈ ℝ

√(−1) ∉ ℝ
R
numbers

C
complex numbers ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ
C
numbers
arbitrary constant C can be any number, most likely unknown; usually occurs when calculating antiderivatives. if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)
C
integral calculus
𝕂

K
real or complex numbers K means the statement holds substituting K for R and also for C.
$x^2inmathbb{C},forall xin mathbb{K}$

because

$x^2inmathbb{C},forall xin mathbb{R}$

and

$x^2inmathbb{C},forall xin mathbb{C}$.
K
linear algebra
infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. limx→0 1/|x| = ∞
infinity
numbers
||…||
norm || x || is the norm of the element x of a normed vector space. || x  + y || ≤  || x ||  +  || y ||
norm of

length of
linear algebra
summation

$sum_{k=1}^{n}{a_k}$ means a1 + a2 + … + an. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... The infinity symbol âˆž in several typefaces. ... The extended real number line is obtained from the real number line R by adding two elements: +&#8734; and &#8722;&#8734; (which are not considered to be real numbers). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... A number is an abstract idea used in counting and measuring. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... For evaluation of sums in closed form see evaluating sums. ...

$sum_{k=1}^{4}{k^2}$ = 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic
product

$prod_{k=1}^na_k$ means a1a2···an. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...

$prod_{k=1}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic
Cartesian product

$prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...

(y0, …, yn).

$prod_{n=1}^{3}{mathbb{R}} = mathbb{R}timesmathbb{R}timesmathbb{R} = mathbb{R}^3$

the Cartesian product of; the direct product of
set theory
coproduct
coproduct over … from … to … of
category theory

derivative f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.

The dot notation indicates a time derivative. That is $dot{x}(t)=frac{partial}{partial t}x(t)$. In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... For a non-technical overview of the subject, see Calculus. ... Look up Slope in Wiktionary, the free dictionary. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

If f(x) := x2, then f ′(x) = 2x
… prime

derivative of
calculus
indefinite integral or antiderivative ∫ f(x) dx means a function whose derivative is f. x2 dx = x3/3 + C
indefinite integral of

the antiderivative of
calculus
definite integral ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. 0b x2  dx = b3/3;
integral from … to … of … with respect to
calculus
contour integral or closed line integral 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 this formula involves a closed surface integral, the representation describes only the first or initial 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 closed volume integral, denoted by the symbol ∰.

contour integral of
calculus
gradient f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
vector calculus
divergence $nabla cdot vec v = {partial v_x over partial x} + {partial v_y over partial y} + {partial v_z over partial z}$ If $vec v := 3xymathbf{i}+y^2 zmathbf{j}+5mathbf{k}$, then $nabla cdot vec v = 3y + 2yz$.
del dot, divergence of
vector calculus
curl $nabla times vec v = left( {partial v_z over partial y} - {partial v_y over partial z} right) mathbf{i} + left( {partial v_x over partial z} - {partial v_z over partial x} right) mathbf{j} + left( {partial v_y over partial x} - {partial v_x over partial y} right) mathbf{k}$ If $vec v := 3xymathbf{i}+y^2 zmathbf{j}+5mathbf{k}$, then $nablatimesvec v = -y^2mathbf{i} - 3xmathbf{k}$.
curl of
vector calculus
partial differential With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. If f(x,y) := x2y, then ∂f/∂x = 2xy
partial, d
calculus
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
perpendicular xy means x is perpendicular to y; or more generally x is orthogonal to y. If lm and mn then l || n.
is perpendicular to
geometry
bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
||
parallel x || y means x is parallel to y. If l || m and mn then ln.
is parallel to
geometry
entailment AB means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. AA ∨ ¬A
entails
model theory
inference xy means y is derived from x. AB ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
normal subgroup NG means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
/
quotient group G/H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
mod
group theory
quotient set A/~ means the set of all ~ equivalence classes in A. If we define ~ by x~y ⇔ x-y∈Z, then
R/~ = {{x+n : nZ} : x ∈ (0,1]}
mod
set theory
isomorphism GH means that group G is isomorphic to group H Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
approximately equal xy means x is approximately equal to y π ≈ 3.14159
is approximately equal to
everywhere
~
same order of magnitude m ~ n, means the quantities m and n have the general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar

poorly approximates
Approximation theory
〈,〉

( | )

< , >

·

:
inner product x,y〉 means the inner product of x and y as defined in an inner product space.

For spatial vectors, the dot product notation, x·y is common.

The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
〈x, y〉 = 2×−1 + 3×5 = 13
 A:B = ∑ AijBij i,j
inner product of
linear algebra
tensor product VU means the tensor product of V and U. {1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor product of
linear algebra
*
convolution f * g means the convolution of f and g. $(f * g )(t) = int f(tau) g(t - tau), dtau$
convolution, convoluted with
functional analysis
$bar{x}$
mean $bar{x}$ (often read as "x bar") is the mean (average value of xi). $x = {1,2,3,4,5}; bar{x} = 3$.
overbar, … bar
statistics
$overline{z}$
complex conjugate $overline{z}$ is the complex conjugate of z. $overline{3+4i} = 3-4i$
conjugate
complex numbers
$triangleq$
delta equal to $triangleq$ means equal by definition. When $triangleq$ is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. $p(x_1,x_2,...,x_n) triangleq prod_{i=1}^n p(x_i | x_{pi_i})$.
equal by definition
everywhere

In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Mathematical alphanumeric symbols are modifications of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles (one example is blackboard bold, or double-struck (in Unicode terminology)). Unicode now includes many such symbols (in the range U+1D400 . ... In logic, a set of symbols is frequently used to express logical constructs. ... Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ... ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. ... Many Roman letters, both capital and small, are used in mathematics, science and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, physical entities. ... Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. ... In probability theory and statistics, some special forms of mathematical notation are of interest : Random variables (for example, the height of students) are written in upper case. ... In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ... ISO 31 Category: ...

Results from FactBites:

 Article about "Mathematics" in the English Wikipedia on 24-Apr-2004 (1412 words) Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences.
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