In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a product is the result of
multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
, or an
expression that identifies
objects (numbers or
variables) to be multiplied, called ''factors''. For example, 21 is the product of 3 and 7 (the result of multiplication), and
is the product of
and
(indicating that the two factors should be multiplied together).
When one factor is an
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, the product is called a ''
multiple''.
The order in which
real or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers are multiplied has no bearing on the product; this is known as the ''
commutative law
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...
'' of multiplication. When
matrices or members of various other
associative algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
s are multiplied, the product usually depends on the order of the factors.
Matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
, for example, is non-commutative, and so is multiplication in other algebras in general as well.
There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s.
Product of two numbers
Originally, a product was and is still the result of the multiplication of two or more
number
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s. For example, is the product of and . The
fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is prime or can be represented uniquely as a product of prime numbers, ...
states that every
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...
is a product of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, that is unique
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
the order of the factors.
With the introduction of
mathematical notation
Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...
and
variables at the end of the 15th century, it became common to consider the multiplication of numbers that are either unspecified (
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s and
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s), or to be found (
unknown
Unknown or The Unknown may refer to:
Film and television Film
* The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), Australian silent film
* The Unknown (1915 drama film), ''The Unknown'' (1915 drama film), American silent drama ...
s). These multiplications that cannot be effectively performed are called ''products''. For example, in the
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
the term
denotes the ''product'' of the coefficient
and the unknown
Later and essentially from the 19th century on, new
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
s have been introduced, which do not involve numbers at all, and have been called ''products''; for example, the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
. Most of this article is devoted to such non-numerical products.
Product of a sequence
The product operator for the
product of a sequence
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a '' product''. Multiplication is often d ...
is denoted by the capital Greek letter
pi Π (in analogy to the use of the capital Sigma
Σ as
summation
In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, pol ...
symbol).
For example, the expression
is another way of writing .
The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
, and is equal to 1.
Commutative rings
Commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
s have a product operation.
Residue classes of integers
Residue classes in the rings
can be added:
:
and multiplied:
:
Convolution
Two functions from the reals to itself can be multiplied in another way, called the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
.
If
:
then the integral
:
is well defined and is called the convolution.
Under the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
, convolution becomes point-wise function multiplication.
Polynomial rings
The product of two polynomials is given by the following:
:
with
:
Products in linear algebra
There are many different kinds of products in linear algebra. Some of these have confusingly similar names (
outer product
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...
,
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of ...
) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.
Scalar multiplication
By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map
.
Scalar product
A
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...
is a bi-linear map:
:
with the following conditions, that
for all
.
From the scalar product, one can define a
norm by letting
.
The scalar product also allows one to define an angle between two vectors:
:
In
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, the standard scalar product (called the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
) is given by:
:
Cross product in 3-dimensional space
The
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the
parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
spanned by the two factors.
The cross product can also be expressed as the
formal
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for formal events
* Semi-formal atti ...
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
:
:
Composition of linear mappings
A linear mapping can be defined as a function ''f'' between two vector spaces ''V'' and ''W'' with underlying field F, satisfying
:
If one only considers finite dimensional vector spaces, then
:
in which b
V and b
W denote the
bases of ''V'' and ''W'', and ''v
i'' denotes the
component of v on b
V''i'', and
Einstein summation convention is applied.
Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping ''f'' map ''V'' to ''W'', and let the linear mapping ''g'' map ''W'' to ''U''. Then one can get
:
Or in matrix form:
:
in which the ''i''-row, ''j''-column element of F, denoted by ''F
ij'', is ''f
ji'', and ''G
ij=g
ji''.
The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.
Product of two matrices
Given two matrices
:
and
their product is given by
:
Composition of linear functions as matrix product
There is a relationship between the composition of linear functions and the product of two matrices. To see this, let r = dim(U), s = dim(V) and t = dim(W) be the (finite)
dimensions of vector spaces U, V and W. Let
be a
basis of U,
be a basis of V and
be a basis of W. In terms of this basis, let
be the matrix representing f : U → V and
be the matrix representing g : V → W. Then
:
is the matrix representing
.
In other words: the matrix product is the description in coordinates of the composition of linear functions.
Tensor product of vector spaces
Given two finite dimensional vector spaces ''V'' and ''W'', the tensor product of them can be defined as a (2,0)-tensor satisfying:
:
where ''V
*'' and ''W
*'' denote the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
s of ''V'' and ''W''.
For infinite-dimensional vector spaces, one also has the:
*
Tensor product of Hilbert spaces
*
Topological tensor product.
The tensor product,
outer product
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...
and
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vector ...
all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its
intrinsic definition. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).
The class of all objects with a tensor product
In general, whenever one has two mathematical
objects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the
internal product of a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
. That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do. More precisely, a monoidal category is the
class
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
of all things (of a given
type) that have a tensor product.
Other products in linear algebra
Other kinds of products in linear algebra include:
*
Hadamard product
*
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vector ...
* The product of
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s:
**
Wedge product or exterior product
**
Interior product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterio ...
**
Outer product
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...
**
Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
Cartesian product
In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
is a
mathematical operation
In mathematics, an operation is a function from a set to itself. For example, an operation on real numbers will take in real numbers and return a real number. An operation can take zero or more input values (also called "'' operands''" or "argu ...
which returns a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
(or product set) from multiple sets. That is, for sets ''A'' and ''B'', the Cartesian product is the set of all
ordered pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
s —where and .
The class of all things (of a given
type) that have Cartesian products is called a
Cartesian category. Many of these are
Cartesian closed categories. Sets are an example of such objects.
Empty product
The
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
on numbers and most
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s has the value of 1 (the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
of multiplication), just like the
empty sum has the value of 0 (the identity element of addition). However, the concept of the empty product is more general, and requires special treatment in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
,
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
,
computer programming
Computer programming or coding is the composition of sequences of instructions, called computer program, programs, that computers can follow to perform tasks. It involves designing and implementing algorithms, step-by-step specifications of proc ...
and
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
.
Products over other algebraic structures
Products over other kinds of
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s include:
* the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of sets
* the
direct product of groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is o ...
, and also the
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product:
* an ''inner'' sem ...
,
knit product and
wreath product
* the
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...
of groups
* the
product of rings
* the
product of ideals
* the
product of topological spaces
* the
Wick product of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s
* the
cap
A cap is a flat headgear, usually with a visor. Caps have crowns that fit very close to the head. They made their first appearance as early as 3200 BC. The origin of the word "cap" comes from the Old French word "chapeau" which means "head co ...
,
cup
A cup is an open-top vessel (container) used to hold liquids for drinking, typically with a flattened hemispherical shape, and often with a capacity of about . Cups may be made of pottery (including porcelain), glass, metal, wood, stone, pol ...
,
Massey and
slant product
In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q\leq p, to form a composite chain of degree p-q. It was introduced by Eduard Čech in 1936, and independently by Hass ...
in algebraic topology
* the
smash product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) and is the quotient of the product space under the identifications for all in and in . The smash prod ...
and
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the ...
(sometimes called the wedge product) in
homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
A few of the above products are examples of the general notion of an
internal product in a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
; the rest are describable by the general notion of a
product in category theory.
Products in category theory
All of the previous examples are special cases or examples of the general notion of a product. For the general treatment of the concept of a product, see
product (category theory)
In category theory, the product of two (or more) object (category theory), objects in a category (mathematics), category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product ...
, which describes how to combine two
objects of some kind to create an object, possibly of a different kind. But also, in category theory, one has:
* the
fiber product or pullback,
* the
product category
In the mathematical field of category theory, the product of two categories ''C'' and ''D'', denoted and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bif ...
, a category that is the product of categories.
* the
ultraproduct, in
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
.
* the
internal product of a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
, which captures the essence of a tensor product.
Other products
* A function's
product integral (as a continuous equivalent to the product of a sequence or as the multiplicative version of the normal/standard/additive integral. The product integral is also known as "continuous product" or "multiplical".
*
Complex multiplication, a theory of elliptic curves.
See also
*
*
Indefinite product
*
Infinite product
*
*
Notes
References
Bibliography
*
{{DEFAULTSORT:Product (Mathematics)
Multiplication