This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(April 2023) (Learn how and when to remove this message)

Inalgebra, thedual numbers are aquadratic algebra first introduced in the 19th century. They areexpressions of the forma +bε, wherea andb arereal numbers, andε is a symbol taken to satisfy${\displaystyle {\epsilon }^2=0}$ with${\displaystyle \epsilon \ne 0}$.

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

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

which follows from the propertyε² = 0 and the fact that multiplication is abilinear operation.

The dual numbers form acommutative algebra ofdimension two over the reals, and also anArtinian local ring. They are one of the simplest examples of a ring that has nonzeronilpotent elements.

Dual numbers were introduced in 1873 byWilliam Clifford, and were used at the beginning of the twentieth century by the German mathematicianEduard Study, who used them to represent the dual angle which measures the relative position of twoskew lines in space. Study defined a dual angle asθ +dε, whereθ is the angle between the directions of two lines in three-dimensional space andd is a distance between them. Then-dimensional generalization, theGrassmann number, was introduced byHermann Grassmann in the late 19th century.

In modernalgebra, the algebra of dual numbers is often defined as thequotient of apolynomial ring over the real numbers${\displaystyle (\mathbb{R})}$ by theprincipal ideal generated by thesquare of theindeterminate,^[1] that is

${\displaystyle \mathbb{R}[X]/⟨X^2⟩.}$

It may also be defined as theexterior algebra of a one-dimensionalvector space with${\displaystyle \epsilon }$ as its basis element.

Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous tocomplex 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

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

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

which is definedwhenc is non-zero.

If, on the other hand,c is zero whiled is not, then the equation

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

1. has no solution ifa is nonzero
2. is otherwise solved by any dual number of the form⁠b/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 anideal of the associativealgebra (and thusring) of the dual numbers.

The dual number${\displaystyle a+b\epsilon }$ can be represented by thesquare matrix. In this representation the matrix squares to the zero matrix, corresponding to the dual number${\displaystyle \epsilon }$.

Generally, if${\displaystyle \epsilon }$ is anilpotent matrix, thenB = {x I +y${\displaystyle \epsilon }$:x, y real} is asubalgebra isomorphic to the algebra of dual numbers. In the case of 2x2 real matrices M(2,R),${\displaystyle \epsilon }$ can be taken as any matrix of the form withp =a² +bc = 0.

The dual numbers are one of three isomorphism classes of real 2-algebras in M(2,R). Whenp > 0 the subalgebraB is isomorphic tosplit-complex numbers, and whenp < 0,B is isomorphic to thecomplex plane.

One application of dual numbers isautomatic differentiation. Any polynomial

${\displaystyle P(x)=p_0+p_{1}x+p_2{x}^2+\cdots +p_n{x}^n}$

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

where${\displaystyle P^{\prime }}$ is the derivative of${\displaystyle P.}$

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

${\displaystyle f(a+b\epsilon )=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)b^n{\epsilon }^n}{n!}=f(a)+b{f}^{\prime }(a)\epsilon ,}$

since all terms involvingε² or greater powers are trivially0 by the definition ofε.

By computing compositions of these functions over the dual numbers and examining the coefficient ofε in the result we find we have automatically computed the derivative of the composition.

A similar method works for polynomials ofn variables, using theexterior algebra of ann-dimensional vector space.

The "unit circle" of dual numbers consists of those witha = ±1 since these satisfyzz* = 1 wherez* =a −bε. However, note that

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

so theexponential map applied to theε-axis covers only half the "circle".

Letz =a +bε. Ifa ≠ 0 andm =⁠b/a⁠, thenz =a(1 +mε) is thepolar decomposition of the dual numberz, and theslopem is its angular part. The concept of arotation in the dual number plane is equivalent to a verticalshear mapping since(1 +pε)(1 +qε) = 1 + (p +q)ε.

Inabsolute space and time theGalilean transformation

that is

${\displaystyle t^{\prime }=t,x^{\prime }=vt+x,}$

relates the resting coordinates system to a moving frame of reference ofvelocityv. With dual numberst +xε representingevents along one space dimension and time, the same transformation is effected with multiplication by1 +vε.

Given two dual numbersp andq, they determine the set ofz such that the difference in slopes ("Galilean angle") between the lines fromz top andq is constant. This set is acycle in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is aquadratic equation in the real part ofz, a cycle is aparabola. The "cyclic rotation" of the dual number plane occurs as a motion ofits projective line. According toIsaak Yaglom,^{[2]: 92–93} the cycleZ = {z :y =αx²} is invariant under the composition of the shear

${\displaystyle x_1=x,y_1=vx+y}$

with thetranslation

${\displaystyle x^{\prime }=x_1=\frac{v}{2a},y^{\prime }=y_1+\frac{v^2}{4a}.}$

Dual numbers find applications inmechanics, 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.^[3] Seescrew theory for more.

In modernalgebraic geometry, the dual numbers over a field${\displaystyle k}$ (by which we mean the ring${\displaystyle k[\epsilon ]/({\epsilon }^2)}$) may be used to define thetangent vectors to the points of a${\displaystyle k}$-scheme.^[4] Since the field${\displaystyle k}$ can be chosen intrinsically, it is possible to speak simply of the tangent vectors to a scheme. This allows notions fromdifferential geometry to be imported into algebraic geometry.

In detail: The ring of dual numbers may be thought of as the ring of functions on the "first-order neighborhood of a point" – namely, the${\displaystyle k}$-scheme${\displaystyle \mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))}$.^[4] Then, given a${\displaystyle k}$-scheme${\displaystyle X}$,${\displaystyle k}$-points of the scheme are in 1-1 correspondence with maps${\displaystyle \mathrm{Spec}k\to X}$, while tangent vectors are in 1-1 correspondence with maps${\displaystyle \mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))\to X}$.

The field${\displaystyle k}$ above can be chosen intrinsically to be aresidue field. To wit: Given a point${\displaystyle x}$ on a scheme${\displaystyle S}$, consider thestalk${\displaystyle S_x}$. Observe that${\displaystyle S_x}$ is alocal ring with a uniquemaximal ideal, which is denoted${\displaystyle {\mathfrak{m}}_x}$. Then simply let${\displaystyle k=S_x/{\mathfrak{m}}_x}$.

This construction can be carried out more generally: for acommutative ringR one can define the dual numbers overR as thequotient of thepolynomial ringR[X] by theideal(X²): the image ofX then has square equal to zero and corresponds to the elementε from above.

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

It is the${\displaystyle R}$-module${\displaystyle R\oplus M}$ with the multiplication defined by${\displaystyle (r,i)\cdot \left(r^{\prime },i^{\prime }\right)=\left(r{r}^{\prime },r{i}^{\prime }+r^{\prime }i\right)}$ for${\displaystyle r,r^{\prime }\in R}$ and${\displaystyle i,i^{\prime }\in I.}$

The algebra of dual numbers is the special case where${\displaystyle M=R}$ and${\displaystyle \epsilon =(0,1).}$

Dual numbers find applications inphysics, where they constitute one of the simplest non-trivial examples of asuperspace. Equivalently, they aresupernumbers with just one generator; supernumbers generalize the concept ton distinct generatorsε, each anti-commuting, possibly takingn to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions.

The motivation for introducing dual numbers into physics follows from thePauli exclusion principle for fermions. The direction alongε is termed the "fermionic" direction, and the real component is termed the "bosonic" direction. The fermionic direction earns this name from the fact thatfermions 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^[5] andCorrado Segre.^[6]

Just as theRiemann sphere needs a north polepoint at infinity to close up thecomplex projective line, so aline at infinity succeeds in closing up the plane of dual numbers to acylinder.^{[2]: 149–153}

SupposeD is the ring of dual numbersx +yε andU is the subset withx ≠ 0. ThenU is thegroup of units ofD. LetB = {(a,b) ∈D ×D :a ∈ U orb ∈ U}. Arelation is defined on B as follows:(a,b) ~ (c,d) when there is au inU such thatua =c andub =d. This relation is in fact anequivalence relation. The points of the projective line overD areequivalence classes inB under this relation:P(D) =B/~. They are represented withprojective coordinates[a,b].

Consider theembeddingD →P(D) byz → [z, 1]. Then points[1,n], forn² = 0, are inP(D) but are not the image of any point under the embedding.P(D) is mapped onto acylinder byprojection: Take a cylinder tangent to the double number plane on the line{yε :y ∈R},ε² = 0. Now take the opposite line on the cylinder for the axis of apencil 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.

• Smooth infinitesimal analysis
• Perturbation theory
• Infinitesimal
• Screw theory
• Dual-complex number
• Laguerre transformations
• Grassmann number
• Automatic differentiation

• Linear algebra
• Hypercomplex numbers
• Commutative algebra
• Differential algebra
• Nonstandard analysis

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from April 2023
• All articles lacking in-text citations
• CS1 Italian-language sources (it)