Dual number

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In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form Template:Math, where Template:Mvar and Template:Mvar are real numbers, and Template:Mvar is a symbol taken to satisfy <math>\varepsilon^2 = 0</math> with <math>\varepsilon\neq 0</math>.

Dual numbers can be added component-wise, and multiplied by the formula

<math> (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, </math>

which follows from the property Template:Math and the fact that multiplication is a bilinear operation.

The dual numbers form a commutative algebra of dimension 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.

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 Template:Math, where Template:Mvar is the angle between the directions of two lines in three-dimensional space and Template:Mvar is a distance between them. The Template:Mvar-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.

Modern definitionEdit

In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers <math>(\mathbb{R})</math> by the principal ideal generated by the square of the indeterminate, that is

<math>\mathbb{R}[X]/\left\langle X^2 \right\rangle.</math>

It may also be defined as the exterior algebra of a one-dimensional vector space with <math>\varepsilon</math> as its basis element.

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 evaluate an expression of the form

<math>\frac{a + b\varepsilon}{c + d\varepsilon}</math>

we multiply the numerator and denominator by the conjugate of the denominator:

<math>\begin{align}

 \frac{a + b\varepsilon}{c + d\varepsilon}
   &= \frac{(a + b\varepsilon)(c - d\varepsilon)}{(c + d\varepsilon)(c - d\varepsilon)}\\[5pt]
   &= \frac{ac - ad\varepsilon + bc\varepsilon - bd\varepsilon^2}{c^2 + cd\varepsilon - cd\varepsilon - d^2\varepsilon^2}\\[5pt]
   &= \frac{ac - ad\varepsilon + bc\varepsilon - 0}{c^2 - 0}\\[5pt]
   &= \frac{ac + \varepsilon(bc - ad)}{c^2}\\[5pt]
   &= \frac{a}{c} + \frac{bc - ad}{c^2}\varepsilon

\end{align}</math>

which is defined [[Division by zero|when Template:Mvar is non-zero]].

If, on the other hand, Template:Mvar is zero while Template:Mvar is not, then the equation

<math>{a + b\varepsilon = (x + y\varepsilon) d\varepsilon} = {xd\varepsilon + 0}</math>

has no solution if Template:Mvar is nonzero
is otherwise solved by any dual number of the form Template:Math.

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.

Matrix representationEdit

The dual number <math>a + b \varepsilon</math> can be represented by the square matrix <math>\begin{pmatrix}a & b \\ 0 & a \end{pmatrix}</math>. In this representation the matrix <math>\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}</math> squares to the zero matrix, corresponding to the dual number <math>\varepsilon</math>.

Generally, if <math>\varepsilon</math> is a nilpotent matrix, then B = {x I + y <math>\varepsilon</math>: x, y real} is a subalgebra isomorphic to the algebra of dual numbers. In the case of 2x2 real matrices M(2,R), <math>\varepsilon</math> can be taken as any matrix of the form <math>\begin{pmatrix}a & b \\ c & -a \end{pmatrix}</math> with p = a² + bc = 0.

The dual numbers are one of three isomorphism classes of real 2-algebras in M(2,R). When p > 0 the subalgebra B is isomorphic to split-complex numbers, and when p < 0, B is isomorphic to the complex plane.

Automatic differentiationEdit

Template:Anchor One application of dual numbers is automatic differentiation. Any polynomial

<math>P(x) = p_0 + p_1x + p_2x^2 + \cdots + p_nx^n</math>

with real coefficients can be extended to a function of a dual-number-valued argument,

<math>\begin{align}

P(a + b\varepsilon) &= p_0 + p_1(a + b\varepsilon) + \cdots + p_n(a + b\varepsilon)^n \\[2mu] &= 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^{n-1} b\varepsilon \\[5mu] &= P(a) + bP'(a)\varepsilon, \end{align}</math>

where <math>P'</math> is the derivative of <math>P.</math>

More generally, any (analytic) real function can be extended to the dual numbers via its Taylor series:

<math>f(a + b\varepsilon) = \sum_{n=0}^\infty \frac{f^{(n)} (a)b^n \varepsilon^n}{n!} = f(a) + bf'(a)\varepsilon,</math>

since all terms involving Template:Math or greater powers are trivially Template:Math by the definition of Template:Mvar.

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

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

GeometryEdit

The "unit circle" of dual numbers consists of those with Template:Math since these satisfy Template:Math where Template:Math. However, note that

<math> e^{b \varepsilon} = \sum^\infty_{n=0} \frac{\left(b\varepsilon\right)^n}{n!} = 1 + b \varepsilon,</math>

so the exponential map applied to the Template:Mvar-axis covers only half the "circle".

Let Template:Math. If Template:Math and Template:Math, then Template:Math is the polar decomposition of the dual number Template:Mvar, and the slope Template:Mvar is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since Template:Math.

In absolute space and time the Galilean transformation

<math>\left(t', x'\right) = (t, x)\begin{pmatrix} 1 & v \\0 & 1 \end{pmatrix}\,,</math>

that is

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

CyclesEdit

Given two dual numbers Template:Mvar and Template:Mvar, they determine the set of Template:Mvar such that the difference in slopes ("Galilean angle") between the lines from Template:Mvar to Template:Mvar and Template:Mvar 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 Template:Mvar, 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,<ref name="yaglom"/>Template:Rp the cycle Template:Math is invariant under the composition of the shear

with the translation

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.<ref>Template:Citation</ref> See screw theory for more.

Algebraic geometryEdit

In modern algebraic geometry, the dual numbers over a field <math>k</math> (by which we mean the ring <math>k[\varepsilon]/(\varepsilon^2)</math>) may be used to define the tangent vectors to the points of a <math>k</math>-scheme.<ref name=":0" /> Since the field <math>k</math> can be chosen intrinsically, it is possible to speak simply of the tangent vectors to a scheme. This allows notions from differential geometry to be imported into algebraic geometry.

In detail: The ring of dual numbers may be thought of as the ring of functions on the "first-order neighborhood of a point" – namely, the <math> k</math>-scheme <math> \operatorname{Spec} (k[\varepsilon]/(\varepsilon^2))</math>.<ref name=":0">Template:Citation</ref> Then, given a <math> k</math>-scheme <math> X</math>, <math> k</math>-points of the scheme are in 1-1 correspondence with maps <math> \operatorname{Spec} k \to X </math>, while tangent vectors are in 1-1 correspondence with maps <math> \operatorname{Spec} (k[\varepsilon]/(\varepsilon^2)) \to X </math>.

The field <math>k</math> above can be chosen intrinsically to be a residue field. To wit: Given a point <math>x</math> on a scheme <math>S</math>, consider the stalk <math>S_x</math>. Observe that <math>S_x</math> is a local ring with a unique maximal ideal, which is denoted <math>\mathfrak m_x</math>. Then simply let <math>k = S_x / \mathfrak m_x</math>.

GeneralizationsEdit

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

Arbitrary module of elements of zero squareEdit

There is a more general construction of the dual numbers. Given a commutative ring <math>R</math> and a module <math>M</math>, there is a ring <math>R[M]</math> called the ring of dual numbers which has the following structures:

It is the <math>R</math>-module <math>R \oplus M</math> with the multiplication defined by <math>(r, i) \cdot \left(r', i'\right) = \left(rr', ri' + r'i\right)</math> for <math>r, r' \in R</math> and <math>i, i' \in I.</math>

The algebra of dual numbers is the special case where <math>M = R</math> and <math>\varepsilon = (0, 1).</math>

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 Template:Mvar distinct generators Template:Mvar, each anti-commuting, possibly taking Template:Mvar 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 Template:Mvar 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 Template:Math.

Projective lineEdit

The idea of a projective line over dual numbers was advanced by Grünwald<ref>Template:Cite journal</ref> and Corrado Segre.<ref>Template:Cite book Also in Atti della Reale Accademia della Scienze di Torino 47.</ref>

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.<ref name="yaglom">Template:Cite book</ref>Template:Rp

Suppose Template:Mvar is the ring of dual numbers Template:Math and Template:Mvar is the subset with Template:Math. Then Template:Mvar is the group of units of Template:Mvar. Let Template:Math. A relation is defined on B as follows: Template:Math when there is a Template:Mvar in Template:Mvar such that Template:Math and Template:Math. This relation is in fact an equivalence relation. The points of the projective line over Template:Mvar are equivalence classes in Template:Mvar under this relation: Template:Math. They are represented with projective coordinates Template:Math.

Consider the embedding Template:Math by Template:Math. Then points Template:Math, for Template:Math, are in Template:Math but are not the image of any point under the embedding. Template:Math is mapped onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line Template:Math, Template:Math. 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 Template:Math, Template:Math in the projective line over dual numbers.

ReferencesEdit

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