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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Dual numbers can be added component-wise, and multiplied by the formula : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, which follows from the property and the fact that multiplication is a
bilinear operation In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
. The dual numbers form a
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements.


History

Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician
Eduard Study Eduard Study ( ), more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known f ...
, who used them to represent the dual angle which measures the relative position of two
skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the sa ...
in space. Study defined a dual angle as , where is the angle between the directions of two lines in three-dimensional space and is a distance between them. The -dimensional generalization, the
Grassmann number In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...
, was introduced by
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
in the late 19th century.


Definition in abstract algebra

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the algebra of dual numbers is often defined as the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
over the real numbers (\mathbb) by the
principal ideal In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
generated by the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the indeterminate, that is :\mathbb \left\langle X^2 \right\rangle.


Matrix representation

The dual number a + b \epsilon can be represented by the
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
\begina & b \\ 0 & a \end. In this representation the matrix \begin0 & 1 \\ 0 & 0 \end squares to the zero matrix, corresponding to the dual number \varepsilon. There are other ways to represent dual numbers as square matrices. They consist of representing the dual number 1 by the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, and \epsilon by any matrix whose square is the zero matrix; that is, in the case of matrices, any nonzero matrix of the form :\begina & b \\ c & -a \end with a^2+bc=0.


Differentiation

One application of dual numbers is
automatic differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function s ...
. Consider the real dual numbers above. Given any real polynomial , it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result: : \begin P(a + b\varepsilon) = &p_0 + p_1(a + b\varepsilon) + \cdots + p_n(a + b\varepsilon)^n\\ = &p_0 + p_1 a + p_2 a^2 + \cdots + p_n a^n + p_1 b\varepsilon + 2 p_2 a b\varepsilon + \cdots + n p_n a^ b\varepsilon\\ pt = &P(a) + bP^\prime(a)\varepsilon, \end where is the derivative of . More generally, we can extend any (analytic) real function to the dual numbers by looking at its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
: : f(a + b\varepsilon) = \sum_^\infty \frac = f(a) + bf'(a)\varepsilon, since all terms of involving or greater are trivially 0 by the definition of . By computing compositions of these functions over the dual numbers and examining the coefficient of in the result we find we have automatically computed the derivative of the composition. A similar method works for polynomials of variables, using the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of an -dimensional vector space.


Geometry

The "unit circle" of dual numbers consists of those with since these satisfy where . However, note that : e^ = \sum^\infty_ \frac = 1 + b \varepsilon, so the exponential map applied to the -axis covers only half the "circle". Let . If and , then is the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
of the dual number , and the
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
is its angular part. The concept of a ''rotation'' in the dual number plane is equivalent to a vertical
shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
since . In
absolute space and time Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Before Newton A version of the concept of absolute space (in the sense of a prefe ...
the
Galilean transformation In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
:\left(t', x'\right) = (t, x)\begin 1 & v \\0 & 1 \end\,, that is :t' = t,\quad x' = vt + x, relates the resting coordinates system to a moving frame of reference of
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
. With dual numbers representing
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
s along one space dimension and time, the same transformation is effected with multiplication by .


Cycles

Given two dual numbers and , they determine the set of such that the difference in slopes ("Galilean angle") between the lines from to and is constant. This set is a cycle in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
in the real part of , a cycle is a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
. The "cyclic rotation" of the dual number plane occurs as a motion of its projective line. According to
Isaak Yaglom Isaak Moiseevich Yaglom (russian: Исаа́к Моисе́евич Ягло́м; 6 March 1921 – 17 April 1988) was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom. Yaglom received a Ph.D. from M ...
, the cycle is invariant under the composition of the shear :x_1 = x ,\quad y_1 = vx + y with the
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
:x' = x_1 = \frac ,\quad y' = y_1 + \frac.


Division

Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts. Therefore, to divide an equation of the form :\frac we multiply the top and bottom by the conjugate of the denominator: :\begin \frac &= \frac\\ pt &= \frac\\ pt &= \frac\\ pt &= \frac\\ pt &= \frac + \frac\varepsilon \end which is defined when is non-zero. If, on the other hand, is zero while is not, then the equation : = # has no solution if is nonzero # is otherwise solved by any dual number of the form . This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially)
zero divisors In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
and clearly form an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of the associative
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
(and thus
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
) of the dual numbers.


Applications in mechanics

Dual numbers find applications in
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, notably for kinematic synthesis. For example, the dual numbers make it possible to transform the input/output equations of a four-bar spherical linkage, which includes only rotoid joints, into a four-bar spatial mechanism (rotoid, rotoid, rotoid, cylindrical). The dualized angles are made of a primitive part, the angles, and a dual part, which has units of length. See
screw theory Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...
for more.


Generalizations

This construction can be carried out more generally: for a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
one can define the dual numbers over as the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
by the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
: the image of then has square equal to zero and corresponds to the element from above.


Arbitrary module of elements of zero square

There is a more general construction of the dual numbers. Given a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
R and a module M, there is a ring R /math> called the ring of dual numbers which has the following structures: It is the R-module R \oplus M with the multiplication defined by (r, i) \cdot \left(r', i'\right) = \left(rr', ri' + r'i\right) for r, r' \in R and i, i' \in I. The algebra of dual numbers is the special case where M = R and \varepsilon = (0, 1).


Superspace

Dual numbers find applications in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, where they constitute one of the simplest non-trivial examples of a
superspace Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to distinct generators , each anti-commuting, possibly taking to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions. The motivation for introducing dual numbers into physics follows from the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
for fermions. The direction along is termed the "fermionic" direction, and the real component is termed the "bosonic" direction. The fermionic direction earns this name from the fact that
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s obey the Pauli exclusion principle: under the exchange of coordinates, the quantum mechanical wave function changes sign, and thus vanishes if two coordinates are brought together; this physical idea is captured by the algebraic relation .


Projective line

The idea of a projective line over dual numbers was advanced by Grünwald and
Corrado Segre Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry. Early life Corrado's parents were Abramo Segre and Estella De Ben ...
. Also in ''Atti della Reale Accademia della Scienze di Torino'' 47. Just as the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
needs a north pole
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adj ...
to close up the
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, so a
line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The l ...
succeeds in closing up the plane of dual numbers to a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
. Suppose is the ring of dual numbers and is the subset with . Then is the
group of units In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this ...
of . Let . A
relation Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
is defined on B as follows: when there is a in such that and . This relation is in fact an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
. The points of the projective line over are
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es in under this relation: . They are represented with
projective coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
. Consider the
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
by . Then points , for , are in but are not the image of any point under the embedding. is mapped onto a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
by
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
: Take a cylinder tangent to the double number plane on the line , . Now take the opposite line on the cylinder for the axis of a
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail ...
of planes. The planes intersecting the dual number plane and cylinder provide a correspondence of points between these surfaces. The plane parallel to the dual number plane corresponds to points , in the projective line over dual numbers.


See also

*
Smooth infinitesimal analysis Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of bein ...
*
Perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
*
Infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
*
Screw theory Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...
*
Dual-complex number In this article, we discuss certain applications of the dual quaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we will call the ''planar quaternions''. Th ...
*
Laguerre transformations Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigat ...
*
Grassmann number In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...


References


Further reading

* * * * * From ''Cornell Historical Mathematical Monographs'' at
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...
. * * * * * {{DEFAULTSORT:Dual Number Linear algebra Hypercomplex numbers Commutative algebra Differential algebra