TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a product is the result of
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ... , or an expression that identifies
factor FACTOR (the Foundation to Assist Canadian Talent on Records) is a private non-profit organization "dedicated to providing assistance toward the growth and development of the Music of Canada, Canadian music industry". FACTOR was founded in 1982 by r ... s to be multiplied. For example, 30 is the product of 6 and 5 (the result of multiplication), and $x\cdot \left(2+x\right)$ is the product of $x$ and $\left(2+x\right)$ (indicating that the two factors should be multiplied together). The order in which
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ... numbers are multiplied has no bearing on the product; this is known as the
commutative law In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... of multiplication. When
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
or members of various other
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s are multiplied, the product usually depends on the order of the factors.
Matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... , 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s.

# Product of two numbers

## Product of two natural numbers Placing several stones into a rectangular pattern with $r$ rows and $s$ columns gives :$r \cdot s = \sum_^s r = \underbrace_= \sum_^r s = \underbrace_$ stones.

## Product of two integers

Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule: :$\begin \hline \cdot & - & + \\ \hline - & + & - \\ + & - & + \\ \hline \end$ (This rule is a necessary consequence of demanding
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of multiplication over addition, and is not an ''additional rule''.) In words, we have: * Minus times Minus gives Plus * Minus times Plus gives Minus * Plus times Minus gives Minus * Plus times Plus gives Plus

## Product of two fractions

Two fractions can be multiplied by multiplying their numerators and denominators: :$\frac \cdot \frac = \frac$

## Product of two real numbers

The rigorous definition of the product of two real numbers is a byproduct of the
Construction of the real numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. This construction implies that, for every real number there is a set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
such that is the
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the elements of : :$a=\sup_ x.$ If is another real number that is the least upper bound of , the product $a\cdot b$ is defined as :$a\cdot b=\sup_x\cdot y.$ This definition does not depend of a particular choice of and . That is, if they are changed without changing their least upper bound, then the least upper bound defining $a\cdot b$ is not changed.

## Product of two complex numbers

Two complex numbers can be multiplied by the distributive law and the fact that $i^2=-1$, as follows: :$\begin \left(a + b\, i\right) \cdot \left(c + d\, i\right) &= a \cdot c + a \cdot d\, i + b \, i \cdot c + b \cdot d \cdot i^2\\ &= \left(a \cdot c - b \cdot d\right) + \left(a \cdot d + b \cdot c\right) \, i \end$

### Geometric meaning of complex multiplication Complex numbers can be written in
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... : :$a + b\, i = r \cdot \left( \cos\left(\varphi\right) + i \sin\left(\varphi\right) \right) = r \cdot e ^$ Furthermore, :$c + d\, i = s \cdot \left( \cos\left(\psi\right) + i\sin\left(\psi\right) \right) = s \cdot e^,$ from which one obtains :$\left(a \cdot c - b \cdot d\right) + \left(a \cdot d + b \cdot c\right) i = r \cdot s \cdot e^.$ The geometric meaning is that the magnitudes are multiplied and the arguments are added.

## Product of two quaternions

The product of two
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... s can be found in the article on
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...
. Note, in this case, that $a \cdot b$ and $b \cdot a$ are in general different.

# Product of a sequence

The product operator for the
product of a sequence Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on Computer, computers, by an asterisk ) is one of the four Elementary arithmetic, eleme ...
is denoted by the capital Greek letter pi Π (in analogy to the use of the capital Sigma Σ as
summation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... symbol). For example, the expression $\textstyle \prod_^i^2$is another way of writing $1 \cdot 4 \cdot 9 \cdot 16 \cdot 25 \cdot 36$. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and is equal to 1.

# Commutative rings

Commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
s have a product operation.

## Residue classes of integers

Residue classes in the rings $\Z/N\Z$ can be added: :$\left(a + N\Z\right) + \left(b + N\Z\right) = a + b + N\Z$ and multiplied: :$\left(a + N\Z\right) \cdot \left(b + N\Z\right) = a \cdot b + N\Z$

## Convolution

Two functions from the reals to itself can be multiplied in another way, called the
convolution In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. If :$\int\limits_^\infty , f\left(t\right), \,\mathrmt < \infty\qquad\mbox\qquad \int\limits_^\infty , g\left(t\right), \,\mathrmt < \infty,$ then the integral :$\left(f*g\right) \left(t\right) \;:= \int\limits_^\infty f\left(\tau\right)\cdot g\left(t - \tau\right)\,\mathrm\tau$ is well defined and is called the convolution. Under the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
, convolution becomes point-wise function multiplication.

## Polynomial rings

The product of two polynomials is given by the following: :$\left\left(\sum_^n a_i X^i\right\right) \cdot \left\left(\sum_^m b_j X^j\right\right) = \sum_^ c_k X^k$ with :$c_k = \sum_ a_i \cdot b_j$

# 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
,
exterior product In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
) 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 $\R \times V \rightarrow V$.

## Scalar product

A
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
is a bi-linear map: :$\cdot : V \times V \rightarrow \R$ with the following conditions, that $v \cdot v > 0$ for all $0 \not= v \in V$. From the scalar product, one can define a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
by letting $\, v\, := \sqrt$. The scalar product also allows one to define an angle between two vectors: :$\cos\angle\left(v, w\right) = \frac$ In $n$-dimensional Euclidean space, the standard scalar product (called the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
) is given by: :$\left\left(\sum_^n \alpha_i e_i\right\right) \cdot \left\left(\sum_^n \beta_i e_i\right\right) = \sum_^n \alpha_i\,\beta_i$

## Cross product in 3-dimensional space

The
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ... of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram 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 theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for forma ...
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... : :$\mathbf = \begin \mathbf & \mathbf & \mathbf \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ \end$

## 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 :$f\left(t_1 x_1 + t_2 x_2\right) = t_1 f\left(x_1\right) + t_2 f\left(x_2\right), \forall x_1, x_2 \in V, \forall t_1, t_2 \in \mathbb.$ If one only considers finite dimensional vector spaces, then :$f\left(\mathbf\right) = f\left\left(v_i \mathbf^i\right\right) = v_i f\left\left(\mathbf^i\right\right) = _j v_i \mathbf^j,$ in which bV and bW denote the bases of ''V'' and ''W'', and ''vi'' denotes the
component Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis *Lumped ele ... of v on bV''i'', and
Einstein summation convention In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 :$g \circ f\left(\mathbf\right) = g\left\left(_j v_i \mathbf^j\right\right) = _k _j v_i \mathbf^k.$ Or in matrix form: :$g \circ f\left(\mathbf\right) = \mathbf \mathbf \mathbf,$ in which the ''i''-row, ''j''-column element of F, denoted by ''Fij'', is ''fji'', and ''Gij=gji''. 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 :$A = \left(a_\right)_ \in \R^$ and $B = \left(b_\right)_\in \R^$ their product is given by :$B \cdot A = \left\left( \sum_^r a_ \cdot b_ \right\right)_ \;\in\R^$

## 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
of vector spaces U, V and W. Let $\mathcal U = \$ be a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of U, $\mathcal V = \$ be a basis of V and $\mathcal W = \$ be a basis of W. In terms of this basis, let $A = M^_\left(f\right) \in \R^$ be the matrix representing f : U → V and $B = M^_\left(g\right) \in \R^$ be the matrix representing g : V → W. Then :$B\cdot A = M^_ \left(g \circ f\right) \in \R^$ is the matrix representing $g \circ f : U \rightarrow W$. 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: :$V \otimes W\left(v, m\right) = V\left(v\right) W\left(w\right), \forall v \in V^*, \forall w \in W^*,$ where ''V*'' and ''W*'' denote the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s of ''V'' and ''W''. For infinite-dimensional vector spaces, one also has the: *
Tensor product of Hilbert spacesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Topological tensor productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. The tensor product,
outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
and
Kronecker product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 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 differently f ...
of all things (of a given
type Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
* The product of
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... s: ** Wedge product or exterior product **
Interior product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
**
Outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
**
Tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... # 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 mathematics, i ...
, a
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is a
mathematical operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
which returns a set (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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ... s —where and . The class of all things (of a given
type Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
on numbers and most
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s has the value of 1 (the identity element of multiplication), just like the
empty sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ... ,
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 mathematics, i ...
,
computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...
and
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
.

# Products over other algebraic structures

Products over other kinds of
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s include: * the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of sets * the
direct product of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and also the
semidirect product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, knit product and
wreath product In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
* the
free product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of groups * the
product of rings In mathematics, it is possible to combine several ring (mathematics), rings into one large product ring. This is done by giving the Cartesian product of a (possibly infinite) family of rings coordinatewise addition and multiplication. The resulting ...
* the
product of ideals In ring theory, a branch of abstract algebra, an ideal of a ring (mathematics), ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction ...
* the product of topological spaces * the
Wick productIn probability theory, the Wick product is a particular way of defining an adjusted product (mathematics), product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a resu ...
of
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s * the
cap A cap is a kind of soft and flat headgear, usually with a visor. Caps have crowns that fit very close to the head. They made their first appearance as early as 3,200BC. Caps typically have a visor, or no brim at all. They are popular in casual and ...
,
cup A cup is an open-top container box. File:Railroad car with container loads.jpg, A Flatcar#Spine car, spine car with a tank container and an open-top intermodal container, intermodal shipping container with canvas cover. A container is any ...
,
Massey Massey may refer to: Places Canada * Massey, Ontario * Massey Island, Nunavut New Zealand * Massey, New Zealand, an Auckland suburb United States * Massey, Alabama * Massey, Iowa * Massey, Maryland People * Massey (surname) Education * ...
and
slant product In algebraic topology the cap product is a method of adjoining a chain (algebraic topology), chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced b ...
in algebraic topology * the
smash product Smash may refer to: People * Smash (wrestler) (born 1959), professional wrestler * Moondog Rex, another professional wrestler who briefly wrestled as the original Smash, before being replaced by the above. * DJ Smash, DJ and music producer Ar ...
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 th ...
(sometimes called the wedge product) in
homotopy In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ... A few of the above products are examples of the general notion of an internal product in a
monoidal category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
; 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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
, 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 In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ... or pullback, * the
product category In the mathematical field of category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objec ... , a category that is the product of categories. * the
ultraproduct The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ( ...
, in
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
. * the internal product of a
monoidal category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, which captures the essence of a tensor product.

# Other products

* A function's
product integralA product integral is any Product_(mathematics), product-based counterpart of the usual Summation, sum-based integral of calculus. The first product integral (''#Type I: Volterra integral, Type I'' below) was developed by the mathematician Vito Volte ...
(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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ... , a theory of elliptic curves.