# Dual number

In algebra, dual numbers are a sort of hypercomplex numbers, and form an algebraic structure that was introduced in the 19th century. They are expressions of the form a + , where a and b are real numbers, and ε is a symbol whose square is assumed to be zero.

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

${\displaystyle (a+b\varepsilon )(c+d\varepsilon )=ac+(ad+bc)\varepsilon ,}$

which extends by linearity the property ε2 = 0 (this is the only way to define a bilinear operation that satisfies this property).

The dual numbers form a commutative algebra of dimension two over the reals, which is also an Artinian local ring. It is one of the simplest examples of a ring that has a nonzero nilpotent element.

With the rise of abstract algebra at the beginning of 20th century, many rings with nilpotent elements have been studied, and dual numbers are rarely explicitly considered in the main stream of mathematics, except as a starting example.

## 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 + , 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.

## Modern definition

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

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

## Representation in algebras

Let a be an element of a unital algebra A over the reals, such that a2 = 0. The mapping

${\displaystyle \varepsilon \mapsto a}$

induces an algebra isomorphism from the dual numbers to the unital subalgebra of A generated by a.

In other words, as soon one has an element with a zero square in an algebra, one has a "copy" of the dual numbers in the algebra.

In particular, the automorphism of the algebra of dual numbers are all obtained by multiplying ${\displaystyle \varepsilon }$ by a nonzero real number.

In the case of square matrices of dimension two, the matrix ${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}}$ has a zero square. It follows that the matrices of the shape ${\displaystyle {\begin{pmatrix}a&b\\0&a\end{pmatrix}}}$ form a subalgebra isomorphic to the dual numbers through the map

${\displaystyle a+b\varepsilon \mapsto {\begin{pmatrix}a&b\\0&a\end{pmatrix}}.}$

There are many other representations of the dual numbers into the 2×2 matrices, since the matrices of zero square are the matrices

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

with ${\displaystyle a^{2}+bc=0.}$

## Geometry

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

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

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

Let z = a + . 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.

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

that is

${\displaystyle t'=t,\quad x'=vt+x,}$

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 + .

### 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 = αx2} is invariant under the composition of the shear

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

with the translation

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

## Generalization

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 with arbitrary coefficients

There is a more general construction of the dual numbers with more general infinitesimal coefficients. Given a ring ${\displaystyle R}$ and a module ${\displaystyle I}$, there is a ring ${\displaystyle R[I]}$ called the ring of dual numbers which has the following structures:

1. It has the underlying ${\displaystyle R}$-module ${\displaystyle R\oplus I}$
2. The algebra structure is given by ring multiplication ${\displaystyle (r,i)\cdot (r',i')=(rr',ri'+r'i)}$ for ${\displaystyle r,r'\in R}$ and ${\displaystyle i,i'\in I}$

This generalized the previous construction where ${\displaystyle I=R}$ gives the ring ${\displaystyle R[R]}$ which has the same multiplication structure as ${\displaystyle R[\varepsilon ]}$ since any element ${\displaystyle f+\varepsilon g}$ is just a sum of two elements in ${\displaystyle R}$, but the second is indexed in a different position.

## 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 ε2 = 0.

## Differentiation

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:

{\displaystyle {\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′(x)= p1 + 2p2x1 +3p3x2 +... + npnxn-1 is the unique first derivative of P, which is same as we use linear approximation. Similarly, we can compute the second derivate by:

${\displaystyle P(a+b\varepsilon )+P(a-b\varepsilon )-2P(a)=b^{2}P^{\prime \prime }(a)\varepsilon ^{2}}$

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:

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

since all terms of involving 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.

## 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

${\displaystyle {\frac {a+b\varepsilon }{c+d\varepsilon }}}$

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

{\displaystyle {\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

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

The idea of a projective line over dual numbers was advanced by Grünwald[2] and Corrado Segre.[3]

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 + 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 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.[4]

## References

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