Dual number

In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form $a + bε$ , where $a$ and $b$ are real numbers, and $ε$ is a symbol taken to satisfy $\varepsilon ^{2}=0$ .

Dual numbers can be added componentwise, and multiplied by the formula

(a+b\varepsilon )(c+d\varepsilon )=ac+(ad+bc)\varepsilon ,

which follows from the property $ε 2 = 0$ 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.

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 $ϑ + dε$ , 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.

Definition in abstract algebra[]

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

\mathbb {R} [X]/\langle X^{2}\rangle .

Representation in algebras[]

The dual number $a+b\epsilon$ can be represented by the matrix ${\begin{pmatrix}a&b\\0&a\end{pmatrix}}$ . This works because the matrix ${\begin{pmatrix}0&1\\0&0\end{pmatrix}}$ squares to the zero matrix, similar to the dual number $\epsilon$ .

There are other ways to represent dual numbers as matrices. Let us consider just the case of $2\times 2$ real matrices. Assuming that the dual number $1$ is represented by the identity matrix, then $\epsilon$ can be represented by any matrix of the form

{\begin{pmatrix}a&b\\c&-a\end{pmatrix}}

where $a^{2}+bc=0,$ except when $a=b=c=0.$

Differentiation[]

One application of dual numbers is automatic differentiation. Consider the real dual numbers above. Given any real polynomial $P (x) = p 0 + p 1 x + p 2 x 2 + ... + p n x n$ , it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result:

{\begin{aligned}P(a+b\varepsilon )={}&p_{0}+p_{1}(a+b\varepsilon )+\cdots +p_{n}(a+b\varepsilon )^{n}\\[5pt]={}&p_{0}+p_{1}a+p_{2}a^{2}+\cdots +p_{n}a^{n}+p_{1}b\varepsilon +2p_{2}ab\varepsilon +\cdots +np_{n}a^{n-1}b\varepsilon \\[5pt]={}&P(a)+bP^{\prime }(a)\varepsilon ,\end{aligned}}

where $P'$ is the derivative of $P$ .

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

f(a+b\varepsilon )=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)b^{n}\varepsilon ^{n}}{n!}}=f(a)+bf'(a)\varepsilon ,

since all terms of involving $ε 2$ 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.

Geometry[]

The "unit circle" of dual numbers consists of those with $a = \pm1$ since these satisfy $zz * = 1$ where $z * = a - bε$ . However, note that

e^{b\varepsilon }=\left(\sum _{n=0}^{\infty }{\frac {\left(b\varepsilon \right)^{n}}{n!}}\right)=1+b\varepsilon ,

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

Let $z = a + bε$ . If $a \neq 0$ and $m = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}b/a$ , then $z = a (1 + mε)$ 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 $(1 + pε)(1 + qε) = 1 + (p + q) ε$ .

In absolute space and time the Galilean transformation

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

that is

t'=t,\quad x'=vt+x,

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

Cycles[]

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,^[1]^: 92–93 the cycle $Z = {z : y = αx 2}$ is invariant under the composition of the shear

x_{1}=x,\quad y_{1}=vx+y

with the translation

x'=x_{1}={\frac {v}{2a}},\quad y'=y_{1}+{\frac {v^{2}}{4a}}.

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 {a+b\varepsilon }{c+d\varepsilon }}

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

{\begin{aligned}{\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{aligned}}

which is defined when $c$ is non-zero.

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

{a+b\varepsilon =(x+y\varepsilon )d\varepsilon }={xd\varepsilon +0}

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

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.

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 (rotoid, rotoid, rotoid, cylindrical). The dualized angles are made of a primitive part, the angles, and a dual part, which has units of length.^[2] See screw theory for more.

Generalizations[]

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 $(X 2)$ : the image of $X$ then has square equal to zero and corresponds to the element $ε$ from above.

Arbitrary module of elements of zero square[]

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

It is the $R$ -module $R\oplus M$ with the multiplication defined by $(r,i)\cdot (r',i')=(rr',ri'+r'i)$ 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, 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 $ε 2 = 0$ .

Projective line[]

The idea of a projective line over dual numbers was advanced by Grünwald^[3] and Corrado Segre.^[4]

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.^[1]^{: 149–153}

Suppose $D$ is the ring of dual numbers $x + yε$ and $U$ is the subset with $x \neq 0$ . Then $U$ is the group of units of $D$ . Let $B = {(a, b) \in D \times D : a \in U or b \in U}$ . 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 $D \to P (D)$ by $z \to [z, 1]$ . Then points $[1, n]$ , for $n 2 = 0$ , 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 ${yε : y \in ℝ}$ , $ε 2 = 0$ . 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 $[1, n]$ , $n 2 = 0$ in the projective line over dual numbers.

References[]

^ ^a ^b Yaglom, I. M. (1979). A Simple Non-Euclidean Geometry and its Physical Basis. Springer. ISBN 0-387-90332-1. MR 0520230.
^ 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, 161, pp. 3–32, doi:10.1007/978-3-662-03729-4_1, ISBN 9783662037294
^ Grünwald, Josef (1906). "Über duale Zahlen und ihre Anwendung in der Geometrie". Monatshefte für Mathematik. 17: 81–136. doi:10.1007/BF01697639. S2CID 119840611.
^ Segre, Corrado (1912). "XL. Le geometrie proiettive nei campi di numeri duali". Opere. Also in Atti della Reale Accademia della Scienze di Torino 47.

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