They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to satisfy
Dual numbers can be added component-wise, and multiplied by the formula which follows from the property ε2 = 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.
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.
by the principal ideal generated by the square of the indeterminate, that is It may also be defined as the exterior algebra of a one-dimensional vector space with
Division of dual numbers is defined when the real part of the denominator is non-zero.
If, on the other hand, c is zero while d is not, then the equation 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.
Any polynomial with real coefficients can be extended to a function of a dual-number-valued argument, where
More generally, any (analytic) real function can be extended to the dual numbers via its Taylor series: since all terms involving ε2 or greater powers are trivially 0 by the definition of ε.
A similar method works for polynomials of n variables, using the exterior algebra of an n-dimensional vector space.
The "unit circle" of dual numbers consists of those with a = ±1 since these satisfy zz* = 1 where z* = a − bε.
However, note that so the exponential map applied to the ε-axis covers only half the "circle".
If a ≠ 0 and m = 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 that is 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.
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 with the translation Dual numbers find applications in mechanics, notably for kinematic synthesis.
In modern algebraic geometry, the dual numbers over a field
can be chosen intrinsically, it is possible to speak simply of the tangent vectors to a scheme.
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 find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace.
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 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.
The idea of a projective line over dual numbers was advanced by Grünwald[4] and Corrado Segre.
[5] 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.
Now take the opposite line on the cylinder for the axis of a pencil of planes.