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In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and is the product of and (indicating that the two factors should be multiplied together).
The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, and multiplication in other algebras is in general non-commutative.
- 1 Product of two numbers
- 2 Product of sequences
- 3 Further examples for commutative rings
- 4 Products in linear algebra
- 5 Set theoretical product
- 6 Empty product
- 7 Products in category theory
- 8 Other products
- 9 See also
- 10 Notes
- 11 References
- 12 External links
Product of two numbers
Product of two natural numbers
Placing several stones into a rectangular pattern with rows and columns gives
Product of two integers
Integers allow positive and negative numbers. The two numbers are multiplied just like natural numbers, except we need an additional rule for the signs:
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:
Product of two real numbers
The rigorous definition of the product of two real numbers is too complicated for this article. But the idea is that one takes a decimal approximation to each real and multiplies the approximations together, and then take better and better approximations.
Product of two complex numbers
Two complex numbers can be multiplied by the distributive law and the fact that , as follows:
Geometric meaning of complex multiplication
Complex numbers can be written in polar coordinates:
- , from which we obtain:
The geometric meaning is that we multiply the magnitudes and add the angles.
Product of two quaternions
The product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, and are different.
Product of sequences
The product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏ (in analogy to the use of the capital Sigma ∑ as summation symbol). 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, and is equal to 1.
Further examples for commutative rings
Residue classes of integers
Residue classes in the rings can be added:
Rings of functions
Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:
Two functions from the reals to itself can be multiplied in another way, called the convolution.
then the integral
is well defined and is called the convolution.
Under the Fourier transform, convolution becomes multiplication.
The product of two polynomials is given by the following:
Products in linear algebra
By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map .
A scalar product is a bilinear 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, the standard scalar product (called the dot product) is given by:
Cross product in 3-dimensional space
The cross product 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.
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
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 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
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 und W. Let be a basis von U, be a basis of V und 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:
Set theoretical product
In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
The empty product has the value of 1 (the identity element of multiplication) just like the empty sum has the value of 0 (the identity element of addition).
Products in category theory
It is often possible to form the product of two (or more) mathematical objects to form another object of the same kind. Such products are generically called internal products, as they can be described by the generic notion of a monoidal category. Examples include:
- the Cartesian product of sets,
- the product of groups, and also the semidirect product, knit product and wreath product,
- the free product of groups
- the product of rings,
- the product of ideals,
- the product of topological spaces,
- the Wick product of random variables.
- the cap, cup and slant product in algebraic topology.
- the smash product and wedge sum (sometimes called the wedge product) in homotopy.
For the general treatment of the concept of a product, see product (category theory), 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, a category that is the product of categories.
- the ultraproduct, in model theory.
- Hadamard product,
- Kronecker product.
- The product of tensors:
- A function's product integral (as a continuous equivalent to the product of a sequence or 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.
- Here, “formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product.
- Clarke, Francis (2013). Functional analysis, calculus of variations and optimal control. Dordrecht: Springer. pp. 9–10. ISBN 1447148207.
- Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry (2nd ed. ed.). Orlando: Academic Press. p. 200. ISBN 0080874398.
- Moschovakis, Yiannis (2006). Notes on set theory (2nd ed. ed.). New York: Springer. p. 13. ISBN 0387316094.