Product (mathematics)

Under a product refers to an arithmetic operation, the two given quantities, a third in the normal case - the product of these two - calculated.

In general, a product is a picture of the shape

Where the product is listed by and mostly as. The term product is selected, if on each of the three quantities involved an addition is already declared and apply the two distributive laws for the picture:

In the event a product is called commutative, if we always have.

Raise Derived from the Latin word producere in meaning (her ) is " product" originally the name of the result of multiplying two numbers (from Latin: multiply multiplicare =). The use of the Malpunktes goes back to Gottfried Wilhelm Leibniz, the alternative symbol on William Oughtred.

• 2.1 residue classes of integers
• 2.2 Function Rings
• 2.3 convolution product
• 2.4 polynomial

• 3.1 Scalar Product
• 3.2 scalar
• 3.3 Cross product in three dimensional space
• 3.4 scalar triple
• 3.5 Composition of linear maps
• 3.6 Product of two matrices
• 3.7 Composition of linear maps as a matrix product
• 3.8 tensor product of vector spaces
• 3.9 imaging matrices as tensors of second stage

• 5.1 Finite products with many factors
• 5.2 The empty product
• 5.3 Infinite products 5.3.1 properties
• 5.3.2 Examples of the lack of convergence

Products of two numbers

Here is always, that is the product of two numbers is again a number. Products are also provided here as associative, meaning that

Product of two natural numbers

If one arranges about game pieces in a rectangular pattern in r rows of s stones, so you need in

Game pieces. The multiplication here is a short notation for the addition of multiple r summands (corresponding to r rows ), which bear all the value s ( in each row are s stones). Can be the total number of well characterized calculate that the number s ( corresponding to the number of the one behind the other in a column of blocks) Total r times (r corresponding to the number of such juxtaposed columns of stones ) is added ( this will require R-1 plus ). Thus the commutativity of multiplication of two natural numbers is already shown.

Counting the number 0 to the natural numbers, so they form a semi-ring. A ring lacking the inverse element with respect to the addition of: there is no integer x to the x = 0 3 Properties

A product in which the number 0 appears as a factor that always has the value zero: An array of zero rows of tiles includes independent of the number of stones per row not a single stone.

Product of two integers

By adding the negative integers to obtain the ring of integers. Two whole numbers are multiplied by multiplying their respective amounts and provides you with the following sign:

In words stating this table:

• Minus times minus gives plus
• Minus times minus gives plus
• Plus minus times minus gives
• Plus times plus gives Plus

For a strictly formal definition on equivalence classes of pairs of natural numbers to compare the article on integers.

Product of two fractions

In the integers, you can add unlimited, subtract and multiply. Division by a non-zero number is only possible, if the dividend is a multiple of the divisor. This restriction can be combined with the transition to the field of rational numbers, ie to the amount of all fractions cancel. The product of two fractions does not require as opposed to their sum the formation of a main denominator:

If necessary, the result can still cut.

Product of two real numbers

As has been able to prove Euclid, there is no rational number whose square gives two. Similarly, the ratio of circumference to diameter, so the π circle number, can not be represented as a quotient of two integers. Both "gaps" are closed by a so-called completion of the transition to the field of real numbers. Since an exact definition of the product in such a short provided herein does not appear possible, the idea was only briefly outlined:

Any real number can be regarded as an infinite decimal. So are about and the rational approximations - about 1.41 and 3.14 - can easily be multiplied. By successively increasing the number of decimal places is obtained - in a non viable in finite time process - a sequence of approximate values ​​for the product

Product of two complex numbers

There are self over the set of real numbers unsolvable equations such as. For both negative and positive values ​​of x the square on the left side is always a positive number. Due to the transition to the field of complex numbers, which is often referred to as adjunction, ie, adding, arises from the real number line, the so-called Gaussian number plane. Two points of this plane, ie two complex numbers are multiplied formally in compliance with:

Geometric interpretation

A complex number can be written in plane polar coordinates:

If, further,

So true because of the addition formulas for sine and cosine

Geometrically this means that multiplication of the lengths while adding the angle.

Product of two quaternions

Even the complex numbers can be expanded even algebraic. The result is a real four-dimensional space, the so-called Hamiltonian quaternions. The corresponding multiplication rules are presented in detail in the article quaternion. In contrast to the above numerical ranges, the quaternion multiplication is not commutative, that is, and are generally different.

Other examples of commutative rings

Residue classes of integers

That the product of two numbers is odd if and only if both factors are odd, is a well known fact. Similar rules apply with respect to the divisibility by an integer N is greater than two. The even numbers are as they come to the multiples of N; an even number is evenly divisible by two. In the odd numbers should distinguish which residue by N remains in the integer division of this number. Modulo 3 - so the speech - there are three residue classes of integers: those that are multiples of three are those that have a remainder of 1 and those with remaining 2 The product of two such numbers has always remaining one modulo three.

The amount of these residue classes, written, has exactly N elements. A typical element has the shape and stands for the set of all integers that when divided by N give the same remainder as the number a on the set of all such cosets is

Addition and by

A multiplication explained. The resulting ring is called the residue class ring modulo N. If and only if N is a prime number, is this even a body. For example, modulo 5, the residual of 2 to 3, as one is the inverse modulo 5 6. The systematic retrieval of multiplicative inverse modulo N by means of the Euclidean algorithm.

Function rings

If the ring R is commutative, the set forms ( the set of all functions of a non-empty set M with values ​​in R ) is also a commutative ring if we define addition and multiplication in component-wise. That is, when

Declared for all.

If you choose the ring R the real numbers with the usual addition and multiplication, and as M as an open subset of, or more generally, the terms continuity and differentiability of functions make sense. The set of continuous and differentiable functions then forms a subring of the function ring, which trivially again must be commutative if and R is commutative.

Convolution product

Be two integrable real functions, the amounts of which have a finite improper integral:

Then the improper integral

For every real number t is also finite. The function f * g defined thereby is called the convolution product or convolution of f and g where f * g can be integrated again with a finite amount uneigentlichem integral. Furthermore, f * g = g * f, that is, the convolution is commutative.

After Fourier transform, the convolution product is up to a constant normalization factor the pointwise defined product (so-called convolution theorem ). The convolution product plays an important role in the mathematical signal processing.

The Gaussian bell curve can be characterized by the fact that its convolution with itself again a slightly drawn in the width of the bell curve results ( see here). It is this characteristic is based on the central limit theorem.

Polynomial

The set of all polynomials in the variable X with real coefficients also forms a so-called polynomial ring. The product is in this case calculated as follows:

With

These rings play an important role in many areas of algebra. Thus, as the field of complex numbers can be formally defined as a factor elegant ring.

In the transition from finite sums to absolutely - convergent series or formal power series is the so-called Cauchy product of the product discussed here.

Products in linear algebra

The linear algebra deals with vector spaces and linear maps between such. In this context occur by different products. The following is the field of real numbers, for simplicity as the main body mostly used.

Scalar product

Already in the definition of a vector space V appeared on the notion of scalar multiplication. This vectors can be generally "stretch" a real factor, and in the case of multiplication by a negative scalar also the direction of the vector is reversed.

The scalar product is a mapping

Scalar product

Of distinguish strictly the term of a scalar product. It is a bilinear map

With the additional requirement that is for all.

Therefore, the expression is always predictable and provides the concept of norm (length ) of a vector.

Similarly, the dot product allows the definition of an angle between two zero vectors v and w:

Polarization formula shows that such a length concept always results in reverse to a scalar, and thus to an angle term.

In any n- dimensional Euclidean space can be explained by orthonormalization an orthonormal system find. If you all vectors as a linear combination with respect to an orthonormal basis is, as can be the scalar product of two such coordinate tuple calculated as the standard scalar product:

Cross product in three dimensions

In, as the standard model of a 3-dimensional Euclidean space, can be another product that define the so-called cross product. It provides excellent services in various problems of analytic geometry in space.

At the cross product is a mapping

Like any Lie - product, it is anticommutative: It is particularly

Triple product

The so-called triple product - also explained only in - not it is a product of two, but three vectors. In modern speech, it coincides with the determinant of three side- written column vectors and is probably easiest according to the rule of Sarrus calculate. Formal is a mapping

Before, which is probably still known today only for historical reasons as a product. Clearly, measures the scalar triple the volume of Spates in the room.

Composition of linear maps

If f: U → V and g: V → W two linear mappings, then their sequential execution

Linear. If we denote the set of all linear maps from U to V, the composition of mappings yields a product

In the special case U = V = W, one obtains the so-called the endomorphism ring of V.

Product of two matrices

Given two matrices and. Since the number of the columns of A to the number of lines matches B, can be the matrix product

Form. In the special case r = s = t square matrices, this creates the matrix ring.

Composition of linear maps as a matrix product

Is a close relationship between the composition of linear maps and the product of two matrices. Be to r = dim ( U), s = dim (V ) and t = dim ( W) the (finite) dimensions of the involved vector spaces U, V and W. Let further a basis of U, a basis of V and a base of W. With respect to these bases are the matrix representing f: U → V and the performing matrix of g: V → W. Then

Performing the matrix of.

In other words, the matrix product provides the coordinate -dependent description of the composition of two linear maps.

Tensor product of vector spaces

The tensor product of two real vector spaces V and W is a kind of product of two vector spaces. It is therefore similar to that discussed below quantitative theoretical product. In contrast to this but it is not is the categorical product in the category of real vector spaces. It can still take on a universal property with respect to bilinear pictures categorical. Then the canonical embedding

So to speak, the "mother of all on V and W definable products". Any other real - bilinear product

With values ​​in some vector space Y is namely downstream connection through a clearly defined linear map

About.

Imaging matrices as tensors of second stage

The vector space Hom (V, W ) of all linear maps between two vector spaces V and W can be attributed to ( bifunktoriell ) naturally as a tensor product of the dual space V * of V conceive with W:

Here is a decomposable tensor, ie, a functional f: V → R and a vector w in W, the linear map g: V → W with

Assigned. Can so any linear map from V to W is obtained? No, but just is not any tensor decomposable. Like any tensor can be written as the sum of decomposable tensors, as can also be any linear map from V to W is obtained as the sum of figures such as the g defined above.

The fact that Hom (V, W) is isomorphic in a natural way to the tensor of the dual space of V with W, also means that it is the matrix representing a linear map g: V → W is a simple contravariant and simply covariant tensor. This is also expressed in the transformation behavior of representing arrays in a change of basis from.

Theoretical amount of product

The Cartesian product M × N of two sets M and N fits at first glance does not casual in the presented product term one. Nevertheless, a connection exists not only in the word " product": The product of two natural numbers m and n was explained above as the cardinality of the Cartesian product of an m -element with an n- element set. Furthermore, certain forms of distributive apply.

The Cartesian product is also the categorical product in the category of sets.

Finite and infinite products

Finite products with many factors

The factorial of a natural number n (! Written as n ) describes the number of possible arrangements of n distinct objects in a row:

The product character is modeled based on the first letter of the word product of the Greek majuscule Pi; as is ajar used to the Sigma as the sum of characters.

As the product of natural numbers is commutative, one can also use an index set (and thus the order of the factors can be indefinite)

Here is an animation to product notation:

The empty product

The empty product has the value one ( the neutral element of multiplication) - always zero results as well as the empty sum ( the neutral element of addition).

Infinite products

John Wallis in 1655 discovered the startling fact that

Applies (see Wallis cal product). What exactly is but meant by the infinite product on the right side? One regards to the sequence of finite partial products

If this sequence converges to a real number P, then we define

More was a sequence of numbers. The infinite product

Is said to be convergent if the following conditions are met:

( The validity of the last two conditions is regardless of what you have chosen in the first ). In this case, is

This limit exists because either is at least one factor and from then all partial zero or one can choose wlog in the second condition.

Core number criterion ( convergence criterion for infinite products ): The following statements are equivalent:

• An infinite product converges absolutely positive nuclei.
• The core series converges absolutely.

Properties

• A convergent infinite product is then exactly zero if one of the factors is zero. Without the third condition, this statement would be false.
• The factors of a convergent product converge to 1 ( necessary condition ).

Examples of the lack of convergence

Although the sequence of partial products converges ( to zero ), infinite products are not labeled as the following convergent:

• Infinitely many factors are zero, the first condition is violated.
• : You have to choose. But if the first factor is omitted, converges the partial product sequence does not ( diverges determined against ). The second condition is violated.
• The sequence of the partial products is converged to zero, however, so that the third condition is violated.

These three examples also do not meet the above necessary criterion. Although the product fulfills the necessary criteria, the sequence of the partial products but not converged: The product of the first factors.