Dual number

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A variety of dualities in mathematics are listed at duality (mathematics).

For the dual grammatical number found in some languages see dual grammatical number.

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε² = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and b uniquely determined real numbers. The plane of all dual numbers is an "alternative complex plane" that complements the ordinary complex number plane C and the plane of split-complex numbers.

1 Linear representation
2 Geometry
- 2.1 Cycles
3 Generalization
4 Differentiation
5 Superspace
6 Division
7 See also
8 References

[edit] Linear representation

If one is familiar with matrix addition and matrix multiplication, then the phenomena of the dual number can be represented using ε = $\begin{pmatrix}0 & 1 \\0 & 0 \end{pmatrix}$ . The generic dual number then can be taken to be

$a + b\epsilon = \begin{pmatrix}a & b \\ 0 & a \end{pmatrix}$ .

The sum and product of two dual numbers each correspond to the appropriate matrix operation. Both operations are commutative and associative.

[edit] Geometry

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

$\exp(b \epsilon) = 1 + b \epsilon \!$ ,

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

If a ≠ 0 and m = b /a , then z = a(1 + m ε) is the polar coordinate form 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 + p ε)(1 + q ε) = 1 + (p+q) ε.

The dual number plane is used to represent the naive spacetime of Galileo in a study called Galilean invariance since the classical event transformation with velocity v looks like:

$(t',x') = (t,x)\begin{pmatrix}1 & v \\0 & 1 \end{pmatrix}$ , that is $\ \ t'=t,\ \ x' = vt + x \!$ .

[edit] Cycles

Given two dual numbers p, and q, they determine the set of z such that the 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. In the Inversive ring geometry of dual numbers one encounters "cyclic rotation" as a projectivity on the projective line over dual numbers. According to Yaglom (pp. 92,3), the cycle Z = {z : y = α x²} is invariant under the composition of the shear

$x_1 = x ,\ \ y_1 = vx + y \ \$ with the translation

$x' = x_1 = v/2a ,\ \ y' = y_1 + v^2/4a \$ .

This composition is a cyclic rotation; the concept has been further developed by V. V. Kisil.

[edit] 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.

This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms).

Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a⁻¹ − ba⁻²ε. As a consequence, we see that the dual numbers over any field (or any commutative local ring) form a local ring.

[edit] Differentiation

One application of dual numbers is automatic differentiation. Consider the real dual numbers above. Given any real polynomial P(x) = p₀+p₁x+p₂x²+...+p_nxⁿ, it is straightforward to extend the domain of this polynomial from the reals to the dual numbers. Then we have this result: P(a+bε) = P(a)+bP′(a)ε, where P′ is the derivative of P. By computing over the dual numbers, rather than over the reals, we can use this to compute derivatives of polynomials. More generally we may define division of dual numbers and then go on to define transcendental functions of dual numbers by defining f(a+bε) =f(a)+bf′(a)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.

This effect can be explained from the non-standard analysis viewpoint. The imaginary unit ε of dual numbers is a close relative to infinitesimal used in non-standard calculus: indeed the square (or any higher power) of ε is exactly zero and the square of an infinitesimal is almost zero at this infinitesimal's scale (is an infinitesimal of a higher order more precisely).

[edit] Superspace

Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. 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.

[edit] 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:

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

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

$= {(a+b\varepsilon)(c-d\varepsilon) \over (c+d\varepsilon)(c-d\varepsilon)} = {ac-ad\varepsilon+cb\varepsilon-bd\varepsilon^2 \over (c^2+cd\varepsilon-cd\varepsilon-d^2\varepsilon^2)} = {ac-ad\varepsilon+cb\varepsilon-0 \over c^2+0}$