Dual number

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In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε (epsilon) with the property ε2 = 0 (ε is nilpotent). Thus the multiplication of dual numbers is given by

(and addition is done componentwise).

The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers.

The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal. Dual numbers form the coefficients of dual quaternions.

Like the complex numbers and split-complex numbers, the dual numbers form an algebra that is 2-dimensional over the field of real numbers.

HistoryEdit

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, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as ϑ + , where ϑ is the angle between the directions of two lines in three-dimensional space and d is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.

Linear representationEdit

Using matrices, dual numbers can be represented as

 

An alternative representation, noted as  [1] (with the former noted also as  ):

 

The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative within the algebra of dual numbers.

This correspondence is analogous to the usual matrix representation of complex numbers. However, it is not the only representation with 2 × 2 real matrices, as is shown in the profile of 2 × 2 real matrices.

GeometryEdit

The "unit circle" of dual numbers consists of those with a = ±1 since these satisfy zz* = 1 where z* = a. However, note that

 

so the exponential map applied to the ε-axis covers only half the "circle".

Let z = a + . If a ≠ 0 and m = b/a, then z = a(1 + ) is the polar decomposition of the dual number z, and the slope m is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since (1 + )(1 + ) = 1 + (p + q)ε.

In absolute space and time the Galilean transformation

 

that is

 

relates the resting coordinates system to a moving frame of reference of velocity v. With dual numbers t + representing events along one space dimension and time, the same transformation is effected with multiplication by 1 + .

CyclesEdit

Given two dual numbers p and q, they determine the set of z such that the difference in slopes ("Galilean angle") between the lines from z to p and q 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 the real part of z, a cycle is a parabola. The "cyclic rotation" of the dual number plane occurs as a motion of its projective line. According to Isaak Yaglom,[2]:92–93 the cycle Z = {z : y = αx2} is invariant under the composition of the shear

 

with the translation

 

This composition is a cyclic rotation; the concept has been further developed by Kisil.[3]

Algebraic propertiesEdit

In abstract algebra terms, the dual numbers can be described as the quotient of the polynomial ring ℝ[X] by the ideal generated by the polynomial X2,

 

The image of X in the quotient is ε. With this description, it is clear that the dual numbers form a commutative ring with characteristic 0. The inherited multiplication gives the dual numbers the structure of a commutative and associative algebra over the reals of dimension two. The algebra is not a division algebra or field since the elements of the form 0 + are not invertible. All elements of this form are zero divisors (also see the section "Division"). The algebra of dual numbers is isomorphic to the exterior algebra of 1.

GeneralizationEdit

This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient of the polynomial ring R[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above.

Dual numbers over an arbitrary ringEdit

This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms). Namely, the tangent bundle of a scheme over an affine base R can be identified with the points of X(R[ε]). For example, consider the affine scheme

 

Recall that maps Spec(ℂ[ε]) → X are equivalent to maps S → ℂ[ε]. Then, every map φ can be defined as sending the generators

 

where the relation

 

holds. This gives us a presentation of TX as

 

Explicit tangent vectorsEdit

For example, a tangent vector at a point   can be found by restricting

 

and taking a point in the fiber. For example, over the origin,  , this is given by the scheme

 

and a tangent vector   is given by a ring morphism   sending

 

At the point   the tangent space is

 

hence a tangent vector   is given by a ring morphism   sending

 

which is to be expected. Note this only gives one free parameter, compared to the last calculation, showing the tangent space is only of dimension one, as expected since this is a smooth point of dimension one.

Over any ring R, the dual number a + is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + is a−1ba−2ε. As a consequence, we see that the dual numbers over any field (or any commutative local ring) form a local ring, its maximal ideal being the principal ideal generated by ε.

A narrower generalization is that of introducing n anticommuting generators; these are the Grassmann numbers or supernumbers, discussed below.

Dual numbers with arbitrary coefficientsEdit

There is a more general construction of the dual numbers with more general infinitesimal coefficients. Given a ring   and a module  , there is a ring   called the ring of dual numbers which has the following structures:

  1. It has the underlying  -module  
  2. The algebra structure is given by ring multiplication   for   and  

This generalized the previous construction where   gives the ring   which has the same multiplication structure as   since any element   is just a sum of two elements in  , but the second is indexed in a different position.

Dual numbers of sheavesEdit

If we have a topological space   with a sheaf of rings   and a sheaf of  -modules  , there is a sheaf of rings   whose sections over an open set   are  . This generalizes in an obvious way to ringed topoi in Topos theory.

Dual numbers on a schemeEdit

A scheme is a special example of a ringed space  . The same construction can be used to construct a scheme   whose underlying topological space is given by   but whose sheaf of rings is  .

SuperspaceEdit

Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to n distinct generators ε, each anti-commuting, possibly taking n 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 for fermions. The direction along ε is termed the "fermionic" direction, and the real component is termed the "bosonic" direction. The fermionic direction earns this name from the fact that fermions 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 ε2 = 0.

DifferentiationEdit

One application of dual numbers is automatic differentiation. Consider the real dual numbers above. Given any real polynomial P(x) = p0 + p1x + p2x2 + ... + pnxn, it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result:

 

where P is the derivative of P.[4]

By computing over the dual numbers, rather than over the reals, we can use this to compute derivatives of polynomials.

More generally, we can extend any (analytic) real function to the dual numbers by looking at its Taylor series:

 

since all terms of involving ε2 or greater are trivially 0 by the definition of ε.

By computing compositions of these functions over the dual numbers and examining the coefficient of ε in the result we find we have automatically computed the derivative of the composition.

A similar method works for polynomials of n variables, using the exterior algebra of an n-dimensional vector space.

DivisionEdit

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

 

we multiply the top and bottom by the conjugate of the denominator:

 

which is defined when c is non-zero.

If, on the other hand, c is zero while d is not, then the equation

 
  1. has no solution if a is nonzero
  2. is otherwise solved by any dual number of the form b/d + .

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 and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers.

Projective lineEdit

The idea of a projective line over dual numbers was advanced by Grünwald[5] and Corrado Segre.[6]

Just as the Riemann sphere needs a north pole point at infinity to close up the complex projective line, so a line at infinity succeeds in closing up the plane of dual numbers to a cylinder.[2]:149–153

Suppose D is the ring of dual numbers x + and U is the subset with x ≠ 0. Then U is the group of units of D. Let B = {(a,b) ∈ D × D : a ∈ U or b ∈ U}. A relation is defined on B as follows: (a,b) ~ (c,d) when there is a u in U such that ua = c and ub = d. This relation is in fact an equivalence relation. The points of the projective line over D are equivalence classes in B under this relation: P(D) = B/~. They are represented with projective coordinates [a, b].

Consider the embedding DP(D) by z → [z, 1]. Then points [1, n], for n2 = 0, are in P(D) but are not the image of any point under the embedding. P(D) is mapped onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line { : y ∈ ℝ}, ε2 = 0. Now take the opposite line on the cylinder for the axis of a pencil 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 [1, n], n2 = 0 in the projective line over dual numbers.

Applications in mechanicsEdit

Dual numbers find applications in mechanics, 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.[7]

See alsoEdit

ReferencesEdit

  1. ^ Dattoli, G.; Licciardi, S.; Pidatella, R. M.; Sabia, E. (July 2018). "Hybrid Complex Numbers: The Matrix Version". Advances in Applied Clifford Algebras. 28 (3): 58. doi:10.1007/s00006-018-0870-y. ISSN 0188-7009.
  2. ^ a b Yaglom, I. M. (1979). A Simple Non-Euclidean Geometry and its Physical Basis. Springer. ISBN 0-387-90332-1. MR 0520230.
  3. ^ Kisil, V. V. (2007). "Inventing a Wheel, the Parabolic One". arXiv:0707.4024 [math].
  4. ^ Berland, Håvard. "Automatic differentiation" (PDF). Retrieved 13 May 2013.
  5. ^ Grünwald, Josef (1906). "Über duale Zahlen und ihre Anwendung in der Geometrie". Monatshefte für Mathematik. 17: 81–136.
  6. ^ Segre, Corrado (1912). "XL. Le geometrie proiettive nei campi di numeri duali". Opere. Also in Atti della Reale Accademia della Scienze di Torino 47.
  7. ^ Angeles, Jorge (1998), Angeles, Jorge; Zakhariev, Evtim (eds.), "The Application of Dual Algebra to Kinematic Analysis", Computational Methods in Mechanical Systems: Mechanism Analysis, Synthesis, and Optimization, NATO ASI Series, Springer Berlin Heidelberg, pp. 3–32, doi:10.1007/978-3-662-03729-4_1, ISBN 9783662037294

Further readingEdit