Product (mathematics)

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 $x\cdot (2+x)$ is the product of $x$ and $(2+x)$ (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.

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 _{i=1}^{s}r=\sum _{j=1}^{r}s

stones.

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:

{\begin{array}{|c|c  c|}\hline \cdot &-&+\\\hline -&+&-\\+&-&+\\\hline \end{array}}

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 {z}{n}}\cdot {\frac {z'}{n'}}={\frac {z\cdot z'}{n\cdot n'}}

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===2+2=5 Two complex numbers can be multiplied by the distributive law and the fact that $\mathrm {i} ^{2}=-1$ , as follows:

{\begin{aligned}(a+b\,\mathrm {i} )\cdot (c+d\,\mathrm {i} )&=a\cdot c+a\cdot d\,\mathrm {i} +b\cdot c\,\mathrm {i} +b\cdot d\cdot \mathrm {i} ^{2}\\&=(a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)\,\mathrm {i} \end{aligned}}

Geometric meaning of complex multiplication

Complex numbers can be written in polar coordinates:

a+b\,\mathrm {i} =r\cdot (\cos(\varphi )+\mathrm {i} \sin(\varphi ))=r\cdot \mathrm {e} ^{\mathrm {i} \varphi }

Furthermore,

c+d\,\mathrm {i} =s\cdot (\cos(\psi )+\mathrm {i} \sin(\psi ))=s\cdot \mathrm {e} ^{\mathrm {i} \psi }

, from which we obtain:

(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 )}

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, $a\cdot b$ and $b\cdot a$ 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 $\mathbb {Z} /N\mathbb {Z}$ can be added:

(a+N\mathbb {Z} )+(b+N\mathbb {Z} )=a+b+N\mathbb {Z}

and multiplied:

(a+N\mathbb {Z} )\cdot (b+N\mathbb {Z} )=a\cdot b+N\mathbb {Z}

Rings of functions

Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:

(f+g)(m):=f(m)+g(m)

(f\cdot g)(m):=f(m)\cdot g(m)

Convolution

Two functions from the reals to itself can be multiplied in another way, called the convolution.

If : $\int \limits _{-\infty }^{\infty }|f(t)|\,\mathrm {d} \,t\;<\;\infty \quad {\mbox{and }}\int \limits _{-\infty }^{\infty }|g(t)|\,\mathrm {d} \,t\;<\;\infty$

then the integral

(f*g)(t)\;:=\int \limits _{-\infty }^{\infty }f(\tau )\cdot g(t-\tau )\,\mathrm {d} \tau

is well defined and is called the convolution.

Under the Fourier transform, convolution becomes multiplication.

Polynomial rings

The product of two polynomials is given by the following:

\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}

with

c_{k}=\sum _{i+j=k}a_{i}\cdot b_{j}

Products in linear algebra

Scalar multiplication

By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map $\mathbb {R} \times V\rightarrow V$ .

Scalar product

A scalar product is a bilinear map:

\cdot :V\times V\rightarrow \mathbb {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 by letting $\|v\|:={\sqrt {v\cdot v}}$ .

The scalar product also allows one to define an angle between two vectors:

\cos \angle (v,w)={\frac {v\cdot w}{\|v\|\cdot \|w\|}}

In $n$ -dimensional Euclidean space, the standard scalar product (called the dot product) is given by:

\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}

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.

The cross product can also be expressed as the formal^[a] determinant:

\mathbf {u\times v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\\end{vmatrix}}

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^[1]

f(t_{1}x_{1}+t_{2}x_{2})=t_{1}f(x_{1})+t_{2}f(x_{2}),\forall x_{1},x_{2}\in V,\forall t_{1},t_{2}\in \mathbb {F} .

If one only considers finite dimensional vector spaces, then

f(\mathbf {v} )=f(v_{i}\mathbf {b_{V}} ^{i})=v_{i}f(\mathbf {b_{V}} ^{i})={f^{i}}_{j}v_{i}\mathbf {b_{W}} ^{j},

in which b_V andb_W denote the bases of V and W, and v_i denotes the component of v on b_Vⁱ, 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

g\circ f(\mathbf {v} )=g({f^{i}}_{j}v_{i}\mathbf {b_{W}} ^{j})={g^{j}}_{k}{f^{i}}_{j}v_{i}\mathbf {b_{U}} ^{k}.

Or in matrix form:

g\circ f(\mathbf {v} )=\mathbf {G} \mathbf {F} \mathbf {v} ,

in which the i-row, j-column element of F, denoted by F_ij, is f^j_i, and G_ij=g^j_i.

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=(a_{i,j})_{i=1\ldots s;j=1\ldots r}\in \mathbb {R} ^{s\times r}

and

B=(b_{j,k})_{j=1\ldots r;k=1\ldots t}\in \mathbb {R} ^{r\times t}

their product is given by

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 \mathbb {R} ^{s\times t}

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 ${\mathcal {U}}=\{u_{1},\ldots u_{r}\}$ be a basis von U, ${\mathcal {V}}=\{v_{1},\ldots v_{s}\}$ be a basis of V und ${\mathcal {W}}=\{w_{1},\ldots w_{t}\}$ be a basis of W. In terms of this basis, let $A=M_{\mathcal {V}}^{\mathcal {U}}(f)\in \mathbb {R} ^{s\times r}$ be the matrix representing f : U → V and $B=M_{\mathcal {W}}^{\mathcal {V}}(g)\in \mathbb {R} ^{r\times t}$ be the matrix representing g : V → W. Then

B\cdot A=M_{\mathcal {W}}^{\mathcal {U}}(g\circ f)\in \mathbb {R} ^{s\times t}

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(v,m)=V(v)W(w),\forall v\in V^{*},\forall w\in W^{*},

where V^* and W^* denote the dual spaces of V and W.^[2]

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.^[3]

Empty product

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.

Other products

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.

Notes

^ 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.

References

^ 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. {{cite book}}: |edition= has extra text (help)
^ Moschovakis, Yiannis (2006). Notes on set theory (2nd ed. ed.). New York: Springer. p. 13. ISBN 0387316094. {{cite book}}: |edition= has extra text (help)

External links

Product on Wolfram Mathworld
"Product". PlanetMath.

[1] 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.

[2] Clarke, Francis (2013). Functional analysis, calculus of variations and optimal control. Dordrecht: Springer. pp. 9–10. ISBN 1447148207.

[3] Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry (2nd ed. ed.). Orlando: Academic Press. p. 200. ISBN 0080874398. {{cite book}}: |edition= has extra text (help)

[4] Moschovakis, Yiannis (2006). Notes on set theory (2nd ed. ed.). New York: Springer. p. 13. ISBN 0387316094. {{cite book}}: |edition= has extra text (help)

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