Dual number explained

In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form, where and are real numbers, and is a symbol taken to satisfy

\varepsilon ^{2} = 0 with \varepsilon ≠ 0 .

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

(a + b\varepsilon)(c + d\varepsilon) = ac + (ad + bc)\varepsilon, which follows from the property 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, 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.

Modern definition

In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers

( R ) by the principal ideal generated by the square of the indeterminate, that isR [X]/\left\langle X ^{2} \right\rangle.

It may also be defined as the exterior algebra of a one-dimensional vector space with

\varepsilon as its basis element.

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 evaluate an expression of the form

(a + b\varepsilon)/(c + d\varepsilon)

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

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

which is defined when is non-zero.

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

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

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

Matrix representation

The dual number

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

There are other ways to represent dual numbers as square matrices. They consist of representing the dual number

1 by the identity matrix, and \epsilon by any matrix whose square is the zero matrix; that is, in the case of matrices, any nonzero matrix of the form\begin{pmatrix}a & b \ c & - a \end{pmatrix} with a ^{2+bc=0.}

Differentiation

One application of dual numbers is automatic differentiation. Any polynomial

P(x) = p _{0} + p _{1x} + p ^{2} _{2 x} + … + p ^{n} _{nx}

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

\begin{align} P(a + b\varepsilon) & = p _{0} + p _{1(a} + b\varepsilon) + … + p _{n(a} + b\varepsilon) ^{n} \\[ 2 mu] & = p _{0} + p _{1} a + p _{2} a ^{2} + … + p _{n} a ^{n} + p _{1} b\varepsilon + 2 p _{2} a b\varepsilon + … + n p _{n} a ^{n-1} b\varepsilon \\[ 5 mu] & = P(a) + bP'(a)\varepsilon, \end{align}

where

P' is the derivative of P.

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

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

since all terms involving or greater powers are trivially 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.

Geometry

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

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

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

that is

t' = t, x' = vt + x,

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 ("Galilean angle") 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