Dual number


In linear algebra, the dual numbers extend the real numbers by adjoining one new element with the property . Thus the multiplication of dual numbers is given by
.
The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form where and 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 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 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.

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, 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, 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, was introduced by Hermann Grassmann in the late 19th century.

Linear representation

Using matrices, dual numbers can be represented as
An alternative representation, noted 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.

Geometry

The "unit circle" of dual numbers consists of those with since these satisfy where. However, note that
so the exponential map applied to the -axis covers only half the "circle".
Let. If and, then is the polar decomposition of the dual number, and the slope is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since.
In absolute space and time the Galilean transformation
that is
relates the resting coordinates system to a moving frame of reference of velocity. With dual numbers representing events 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 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 the real part of, 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, the cycle 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.

Algebraic properties

In abstract algebra terms, the dual numbers can be described as the quotient of the polynomial ring by the ideal generated by the polynomial,
The image of in the quotient is. 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 are not invertible. All elements of this form are zero divisors. The algebra of dual numbers is isomorphic to the exterior algebra of.

Generalization

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

Dual numbers over an arbitrary ring

This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials. Namely, the tangent bundle of a scheme over an affine base can be identified with the points of. For example, consider the affine scheme
Recall that maps are equivalent to maps. Then, every map can be defined as sending the generators
where the relation
holds. This gives us a presentation of as

Explicit tangent vectors

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, the dual number is a unit if and only if is a unit in. In this case, the inverse of is. As a consequence, we see that the dual numbers over any field form a local ring, its maximal ideal being the principal ideal generated by .
A narrower generalization is that of introducing anticommuting generators; these are the Grassmann numbers or supernumbers, discussed below.

Dual numbers with arbitrary coefficients

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 sheaves

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 scheme

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.

Superspace

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 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 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 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 .

Differentiation

One application of dual numbers is automatic differentiation. 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:
where is the derivative of.
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 real function to the dual numbers by looking at its Taylor series:
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 of an -dimensional vector space.

[|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
we multiply the top and bottom by the conjugate of the denominator:
which is defined when is non-zero.
If, on the other hand, is zero while is not, then the equation
  1. has no solution if is nonzero
  2. 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 zero divisors and clearly form an ideal of the associative algebra of the dual numbers.

Projective line

The idea of a projective line over dual numbers was advanced by Grünwald and Corrado Segre.
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.
Suppose is the ring of dual numbers and is the subset with. Then is the group of units of. Let. A relation is defined on B as follows: when there is a in such that and. This relation is in fact an equivalence relation. The points of the projective line over are equivalence classes in under this relation:. They are represented with projective coordinates.
Consider the embedding by. Then points, for, are in but are not the image of any point under the embedding. is mapped onto a cylinder by projection: 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 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.

Applications in mechanics

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. The dualized angles are made of a primitive part, the angles, and a dual part, which has units of length.