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| {{Unreferenced|date=February 2014}}
| | == immediately seems to understand what == |
| {{Calculation results}}
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| In [[mathematics]], a '''product''' is the result of [[Multiplication|multiplying]], or an expression that identifies [[divisor|factor]]s to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and <math>x\cdot (2+x)</math> is the product of <math>x</math> and <math>(2+x)</math> (indicating that the two factors should be multiplied together).
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| | | 相关的主题文章： |
| The order in which [[real number|real]] or [[complex number|complex]] numbers are [[multiplied]] has no bearing on the product; this is known as the [[Commutativity|commutative law]] of multiplication. When [[matrix (mathematics)|matrices]] or members of various other [[associative algebra]]s 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.
| | <ul> |
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| ==Product of two numbers==
| | <li>[http://www.sdlrttl.cn/plus/view.php?aid=118349 http://www.sdlrttl.cn/plus/view.php?aid=118349]</li> |
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| ===Product of two natural numbers===
| | <li>[http://www.gegedan.com/forum.php?mod=viewthread&tid=65396&fromuid=13143 http://www.gegedan.com/forum.php?mod=viewthread&tid=65396&fromuid=13143]</li> |
| [[File:Three by Four.svg|thumb|3 by 4 is 12]]
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| | | <li>[http://www.webngardha.blogdehi.com/shockwave/download/download.cgi http://www.webngardha.blogdehi.com/shockwave/download/download.cgi]</li> |
| Placing several stones into a rectangular pattern with <math>r</math> rows and <math>s</math> columns gives
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| | | </ul> |
| :<math> r \cdot s = \sum_{i=1}^s r = \sum_{j=1}^r s </math>
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| stones.
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| ===Product of two integers===
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| Integers allow positive and negative numbers. The two numbers are multiplied just like natural numbers, except we need an additional rule for the signs:
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| :<math>\begin{array}{|c|c c|}\hline
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| \cdot & - & + \\ \hline
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| - & + & - \\
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| + & - & + \\ \hline
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| \end{array}
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| </math>
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| In words, we have:
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| * Minus times Minus gives Plus
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| * Minus times Plus gives Minus
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| * Plus times Minus gives Minus
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| * Plus times Plus gives Plus
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| ===Product of two fractions===
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| Two fractions can be multiplied by multiplying their numerators and denominators:
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| :<math> \frac{z}{n} \cdot \frac{z'}{n'} = \frac{z\cdot z'}{n\cdot n'}</math> | |
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| ===Product of two real numbers===
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| 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.
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| ===Product of two complex numbers===
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| Two complex numbers can be multiplied by the distributive law and the fact that <math>\mathrm i^2=-1</math>, as follows:
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| :<math>\begin{align}
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| (a + b\,\mathrm i)\cdot (c+d\,\mathrm i)
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| & = a\cdot c + a \cdot d\,\mathrm i + b\cdot c \,\mathrm i + b\cdot d \cdot \mathrm i^2\\
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| & = (a \cdot c - b\cdot d) + (a\cdot d + b\cdot c) \,\mathrm i
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| \end{align}</math>
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| ====Geometric meaning of complex multiplication====
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| [[File:Gaussplane kartesianAndPolar.png|thumb|A complex number in polar coordinates.]] | |
| Complex numbers can be written in [[polar coordinates]]:
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| :<math> a + b\,\mathrm i = r \cdot ( \cos(\varphi) + \mathrm i \sin(\varphi) ) = r \cdot \mathrm e ^{\mathrm i \varphi} </math>
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| Furthermore,
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| :<math> c + d\,\mathrm i = s \cdot ( \cos(\psi) + \mathrm i \sin(\psi) ) = s \cdot \mathrm e ^{\mathrm i \psi} </math>, from which we obtain:
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| :<math> (a \cdot c - b\cdot d) + (a\cdot d + b\cdot c) \,\mathrm i = r\cdot s \cdot ( \cos(\varphi+\psi) + \mathrm i \sin(\varphi+\psi) ) = r\cdot s \cdot \mathrm e ^{\mathrm i (\varphi+\psi)} </math>
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| The geometric meaning is that we multiply the magnitudes and add the angles.
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| ===Product of two quaternions===
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| The product of two quaternions can be found in the article on [[quaternions]]. However, it is interesting to note that in this case, <matH>a\cdot b</math> and <math>b\cdot a</matH> are different.
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| ==Product of sequences==
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| The product operator for the [[Multiplication#Capital Pi notation|product of a sequence]] is denoted by the capital Greek letter Pi <span style="font-family: times, serif; font-size:150%">∏</span> (in analogy to the use of the capital Sigma <span style="font-family: times, serif; font-size:150%">∑</span> 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.
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| ==Further examples for commutative rings==
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| ===Residue classes of integers===
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| Residue classes in the rings <math>\Z/N\Z</math> can be added:
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| :<math> (a+N\Z) + (b+N\Z) = a+b + N\Z</math>
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| and multiplied:
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| :<math> (a+N\Z) \cdot (b+N\Z) = a\cdot b + N\Z</math>
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| ===Rings of functions===
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| Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:
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| :<math>(f+g)(m) : = f(m) + g(m) </math>
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| :<math>(f\cdot g) (m) := f(m) \cdot g(m) </math>
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| ====Convolution====
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| [[Image:Convolucion_Funcion_Pi.gif|thumb|upright=1.5|The convolution of the square wave with itself gives the triangular function]]
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| Two functions from the reals to itself can be multiplied in another way, called the [[convolution]].
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| If :<math>\int\limits_{-\infty}^\infty |f(t)|\,\mathrm{d}\,t \;<\;\infty\quad\mbox{und }
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| \int\limits_{-\infty}^\infty |g(t)|\,\mathrm{d}\,t \;<\; \infty</math>
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| then the integral
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| :<math> (f*g) (t) \;:= \int\limits_{-\infty}^\infty f(\tau)\cdot g(t-\tau)\,\mathrm{d}\tau </math>
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| is well defined and is called the convolution.
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| Under the [[Fourier transform]], convolution becomes multiplication.
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| ===Polynomial rings===
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| The product of two polynomials is given by the following:
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| :<math> \left(\sum_{i=0}^n a_i X^i\right) \cdot \left(\sum_{j=0}^m b_j X^j\right) = \sum_{k=0}^{n+m} c_k X^k </math>
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| with
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| :<math> c_k = \sum_{i+j=k} a_i \cdot b_j </math>
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| ==Products in linear algebra==
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| ===Scalar multiplication===
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| By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map <math> \R \times V \rightarrow V </math>.
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| ===Scalar product===
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| A [[scalar product]] is a bilinear map:
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| :<math> \cdot : V \times V \rightarrow \R </math>
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| with the following conditions, that <math> v\cdot v > 0</math> for all <math> 0 \not= v \in V </math>.
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| From the scalar product, one can define a [[Norm (mathematics)|norm]] by letting <math>\|v\| := \sqrt{v\cdot v} </math>.
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| The scalar product also allows one to define an angle between two vectors:
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| :<math> \cos \angle (v,w) = \frac{v\cdot w}{\|v\| \cdot \|w\|} </math>
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| In <math>n</math>-dimensional Euclidean space, the standard scalar product (called the [[dot product]]) is given by:
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| :<math> \left(\sum_{i=1}^n \alpha_i e_i \right) \cdot \left(\sum_{i=1}^n \beta_i e_i \right) = \sum_{i=1}^n \alpha_i\,\beta_i </math>
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| ===Cross product in 3-dimensional space===
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| 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.
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| The cross product can also be expressed as the [[formal calculation|formal]]{{Efn|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.}} [[determinant]]:
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| :<math>\mathbf{u\times v}=\begin{vmatrix}
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| \mathbf{i}&\mathbf{j}&\mathbf{k}\\
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| u_1&u_2&u_3\\
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| v_1&v_2&v_3\\
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| \end{vmatrix}</math>
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| ===Composition of linear maps===
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| ===Product of two matrices===
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| Given two matrices
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| :<math> A = (a_{i,j})_{i=1\ldots s;j=1\ldots r} \in \R^{s\times r}</math> and <math> B = (b_{j,k})_{j=1\ldots r;k=1\ldots t}\in \R^{r\times t}</math>
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| their product is given by
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| :<math> B \cdot A = \left( \sum_{j=1}^r a_{i,j} \cdot b_{j,k} \right)_{i=1\ldots s;k=1\ldots t} \;\in\R^{s\times t} </math>
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| ===Composition of linear functions as matrix product===
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| ===Tensor product of vector spaces===
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| ==Set theoretical product==
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| In set theory, a '''Cartesian product''' is a [[mathematical operation]] which returns a [[set (mathematics)|set]] (or '''product set''') from multiple sets. That is, for sets ''A'' and ''B'', the Cartesian product {{nowrap|''A'' × ''B''}} is the set of all [[ordered pair]]s {{nowrap|(a, b)}} where {{nowrap|a ∈ ''A''}} and {{nowrap|b ∈ ''B''}}.
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| ==Empty product==
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| 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).
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| ==Other products==
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| Many different kinds of products are studied in mathematics:
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| * Products of the various classes of [[number]]s
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| * The product of [[Matrix (mathematics)|matrices]] and [[Euclidean vector|vectors]]:
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| ** [[scalar multiplication]],
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| ** [[matrix multiplication]],
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| ** [[dot product]],
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| ** [[cross product]],
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| ** [[Hadamard product (matrices)|Hadamard product]],
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| ** [[Kronecker product]].
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| * The product of [[tensor]]s:
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| **[[Exterior algebra|Wedge product or exterior product]]
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| **[[Interior product]]
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| **[[Outer product]]
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| **[[Tensor product]]
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| * The [[pointwise product]] of two [[function (mathematics)|functions]].
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| * 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".
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| * It is often possible to form the product of two (or more) mathematical [[object (category theory)|objects]] to form another object of the same kind. Such products are generically called [[internal product]]s, as they can be described by the generic notion of a [[monoidal category]]. Examples include:
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| ** the [[Cartesian product]] of sets,
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| ** the [[product of groups]], and also the [[semidirect product]], [[knit product]] and [[wreath product]],
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| ** the [[free product]] of groups
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| ** the [[product of rings]],
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| ** the [[product of ideals]],
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| ** the [[product topology|product of topological spaces]],
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| ** the [[Wick product]] of [[random variable]]s.
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| ** the [[cap product|cap]], [[cup product|cup]] and [[slant product]] in algebraic topology.
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| ** the [[smash product]] and [[wedge sum]] (sometimes called the wedge product) in homotopy.
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| * 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:
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| ** the [[fiber product]] or pullback,
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| ** the [[product category]], a category that is the product of categories.
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| ** the [[ultraproduct]], in [[model theory]].
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| * [[Complex multiplication]], a theory of elliptic curves.
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| ==See also== | |
| * [[Pi (letter)]]
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| * [[Iterated binary operation]]
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| == Notes ==
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| {{Notelist}}
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| ==External links==
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| *[http://mathworld.wolfram.com/Product.html Product] on [[Mathworld|Wolfram Mathworld]]
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| * {{planetmath reference|id=7710|title=Product}}
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| {{DEFAULTSORT:Product (Mathematics)}}
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| [[Category:Multiplication]]
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