Editing Dual number

{{Short description|Real numbers adjoined with a nil-squaring element}}
{{for|dual grammatical number found in some languages|Dual (grammatical number)}}
{{inline|date=April 2023}}

In [[algebra]], the '''dual numbers''' are a [[hypercomplex number|hypercomplex number system]] first introduced in the 19th century. They are [[expression (mathematics)|expressions]] of the form {{math|''a'' + ''bε''}}, where {{mvar|a}} and {{mvar|b}} are [[real number]]s, and {{mvar|ε}} is a symbol taken to satisfy <math>\varepsilon^2 = 0</math> with <math>\varepsilon\neq 0</math>.

Dual numbers can be added component-wise, and multiplied by the formula
: <math> (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, </math>
which follows from the property {{math|1=''ε''{{sup|2}} = 0}} and the fact that multiplication is a [[bilinear operation]].

The dual numbers form a [[commutative algebra (structure)|commutative algebra]] of [[dimension (linear algebra)|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 element|nilpotent elements]].

==History==
Dual numbers were introduced in 1873 by [[William Kingdon Clifford|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 {{math|''θ'' + ''dε''}}, where {{mvar|θ}} is the angle between the directions of two lines in three-dimensional space and {{mvar|d}} is a distance between them. The {{mvar|n}}-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 ring|quotient]] of a [[polynomial ring]] over the real numbers <math>(\mathbb{R})</math> by the [[principal ideal]] generated by the [[square (algebra)|square]] of the [[indeterminate (variable)|indeterminate]], that is
:<math>\mathbb{R}[X]/\left\langle X^2 \right\rangle.</math>

It may also be defined as the [[exterior algebra]] of a one-dimensional [[vector space]] with <math>\varepsilon</math> 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 number|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
:<math>\frac{a + b\varepsilon}{c + d\varepsilon}</math>

we multiply the numerator and denominator by the conjugate of the denominator:
:<math>\begin{align}
  \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{align}</math>

which is defined [[Division by zero|when {{mvar|c}} is non-zero]].

If, on the other hand, {{mvar|c}} is zero while {{mvar|d}} is not, then the equation
:<math>{a + b\varepsilon = (x + y\varepsilon) d\varepsilon} = {xd\varepsilon + 0}</math>
# has no solution if {{mvar|a}} is nonzero
# is otherwise solved by any dual number of the form {{math|{{sfrac|''b''|''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 an [[ideal (ring theory)|ideal]] of the associative [[Algebra over a field|algebra]] (and thus [[Ring (mathematics)|ring]]) of the dual numbers.

==Matrix representation==

The dual number <math>a + b \varepsilon</math> can be represented by the [[square matrix]] <math>\begin{pmatrix}a & b \\ 0 & a \end{pmatrix}</math>. In this representation the matrix <math>\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}</math> squares to the zero matrix, corresponding to the dual number <math>\varepsilon</math>.

Generally, if <math>\varepsilon</math> is a [[nilpotent element|nilpotent]] matrix, then ''B'' = {''x'' I + ''y'' <math>\varepsilon</math>: ''x, y'' real} is a [[subalgebra]] isomorphic to the algebra of dual numbers. In the case of 2x2 real matrices M(2,'''R'''), <math>\varepsilon</math> can be taken as any matrix of the form <math>\begin{pmatrix}a & b \\ c & -a \end{pmatrix}</math> with ''p'' = ''a''<sup>2</sup> + ''bc'' = 0.

The dual numbers are one of three isomorphism classes of real 2-algebras in M(2,'''R'''). When ''p'' > 0 the subalgebra ''B'' is isomorphic to [[split-complex number]]s, and when ''p'' < 0, ''B'' is isomorphic to the [[complex plane]].

==Automatic differentiation==
{{anchor|Differentiation}}
One application of dual numbers is [[automatic differentiation]]. Any polynomial

:<math>P(x) = p_0 + p_1x + p_2x^2 + \cdots + p_nx^n</math>

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

:<math>\begin{align}
P(a + b\varepsilon)
&= p_0 + p_1(a + b\varepsilon) + \cdots + p_n(a + b\varepsilon)^n \\[2mu]
&= p_0 + p_1 a + p_2 a^2 + \cdots + p_n a^n + p_1 b\varepsilon + 2 p_2 a b\varepsilon + \cdots + n p_n a^{n-1} b\varepsilon \\[5mu]
&= P(a) + bP'(a)\varepsilon,
\end{align}</math>

where <math>P'</math> is the derivative of <math>P.</math>

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

:<math>f(a + b\varepsilon) = \sum_{n=0}^\infty \frac{f^{(n)} (a)b^n \varepsilon^n}{n!} = f(a) + bf'(a)\varepsilon,</math>

since all terms involving {{math|''ε''<sup>2</sup>}} or greater powers are trivially {{math|0}} by the definition of {{mvar|ε}}.

By computing compositions of these functions over the dual numbers and examining the coefficient of {{mvar|ε}} in the result we find we have automatically computed the derivative of the composition.

A similar method works for polynomials of {{mvar|n}} variables, using the [[exterior algebra]] of an {{mvar|n}}-dimensional vector space.

==Geometry==
The "unit circle" of dual numbers consists of those with {{math|''a'' {{=}} ±1}} since these satisfy {{math|''zz''* {{=}} 1}} where {{math|''z''* {{=}} ''a'' − ''bε''}}. However, note that
: <math> e^{b \varepsilon} = \sum^\infty_{n=0} \frac{\left(b\varepsilon\right)^n}{n!} = 1 + b \varepsilon,</math>
so the [[exponential map (Lie theory)|exponential map]] applied to the {{mvar|ε}}-axis covers only half the "circle".

Let {{math|''z'' {{=}} ''a'' + ''bε''}}. If {{math|''a'' ≠ 0}} and {{math|''m'' {{=}} {{sfrac|''b''|''a''}}}}, then {{math|''z'' {{=}} ''a''(1 + ''mε'')}} is the [[polar decomposition#Alternative planar decompositions|polar decomposition]] of the dual number {{mvar|z}}, and the [[slope]] {{mvar|m}} is its angular part. The concept of a ''rotation'' in the dual number plane is equivalent to a vertical [[shear mapping]] since {{math|(1 + ''pε'')(1 + ''qε'') {{=}} 1 + (''p'' + ''q'')''ε''}}.

In [[absolute space and time]] the [[Galilean transformation]]
:<math>\left(t', x'\right) = (t, x)\begin{pmatrix} 1 & v \\0 & 1 \end{pmatrix}\,,</math>

that is
:<math>t' = t,\quad x' = vt + x,</math>

relates the resting coordinates system to a moving frame of reference of [[velocity]] {{mvar|v}}. With dual numbers {{math|''t'' + ''xε''}} representing [[event (relativity)|event]]s along one space dimension and time, the same transformation is effected with multiplication by {{math|1 + ''vε''}}.

===Cycles===
Given two dual numbers {{mvar|p}} and {{mvar|q}}, they determine the set of {{mvar|z}} such that the difference in slopes ("Galilean angle") between the lines from {{mvar|z}} to {{mvar|p}} and {{mvar|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 {{mvar|z}}, a cycle is a [[parabola]]. The "cyclic rotation" of the dual number plane occurs as a motion of [[#Projective line|its projective line]]. According to [[Isaak Yaglom]],<ref name="yaglom"/>{{rp|92–93}} the cycle {{math|''Z'' {{=}} {''z'' : ''y'' {{=}} ''αx''<sup>2</sup><nowiki>}</nowiki>}} is invariant under the composition of the shear
:<math>x_1 = x ,\quad  y_1 = vx + y </math>

with the [[translation (geometry)|translation]]
:<math>x' = x_1 = \frac{v}{2a}  ,\quad  y' = y_1 + \frac{v^2}{4a}. </math>

==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.<ref>{{Citation|last=Angeles|first=Jorge|title=The Application of Dual Algebra to Kinematic Analysis|date=1998|work=Computational Methods in Mechanical Systems: Mechanism Analysis, Synthesis, and Optimization|volume=161|pages=3–32|editor-last=Angeles|editor-first=Jorge|series=NATO ASI Series|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-662-03729-4_1|isbn=9783662037294|editor2-last=Zakhariev|editor2-first=Evtim}}</ref> See [[screw theory]] for more.

== Algebraic geometry ==
In modern [[algebraic geometry]], the dual numbers over a field <math>k</math> (by which we mean the ring <math>k[\varepsilon]/(\varepsilon^2)</math>) may be used to define the [[tangent vectors]] to the points of a <math>k</math>-[[scheme (mathematics)|scheme]].<ref name=":0" /> Since the field <math>k</math> can be chosen intrinsically, it is possible to speak simply of the tangent vectors to a scheme. This allows notions from [[differential 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 <math> k</math>-[[scheme (mathematics)|scheme]] <math> \operatorname{Spec} (k[\varepsilon]/(\varepsilon^2))</math>.<ref name=":0">{{Citation |last=Shafarevich |first=Igor R. |title=Schemes |date=2013 |url=http://dx.doi.org/10.1007/978-3-642-38010-5_1 |work=Basic Algebraic Geometry 2 |pages=35–38 |access-date=2023-12-27 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-38010-5_1 |isbn=978-3-642-38009-9}}</ref> Then, given a <math> k</math>-scheme <math> X</math>, <math> k</math>-points of the scheme are in 1-1 correspondence with maps <math> \operatorname{Spec} k \to X </math>, while tangent vectors are in 1-1 correspondence with maps <math> \operatorname{Spec} (k[\varepsilon]/(\varepsilon^2)) \to X  </math>.

The field <math>k</math> above can be chosen intrinsically to be a [[residue field]]. To wit: Given a point <math>x</math> on a scheme <math>S</math>, consider the [[stalk (mathematics)|stalk]] <math>S_x</math>. Observe that <math>S_x</math> is a [[local ring]] with a unique [[maximal ideal]], which is denoted <math>\mathfrak m_x</math>. Then simply let <math>k = S_x / \mathfrak m_x</math>.
==Generalizations==
This construction can be carried out more generally: for a [[commutative ring]] {{mvar|R}} one can define the dual numbers over {{mvar|R}} as the [[quotient ring|quotient]] of the [[polynomial ring]] {{math|''R''[''X'']}} by the [[ideal (ring theory)|ideal]] {{math|(''X''<sup>2</sup>)}}: the image of {{mvar|X}} then has square equal to zero and corresponds to the element {{mvar|ε}} from above.

=== Arbitrary module of elements of zero square ===
There is a more general construction of the dual numbers. Given a [[commutative ring]] <math>R</math> and a module <math>M</math>, there is a ring <math>R[M]</math> called the ring of dual numbers which has the following structures:

It is the <math>R</math>-module <math>R \oplus M</math> with the multiplication defined by <math>(r, i) \cdot \left(r', i'\right) = \left(rr', ri' + r'i\right)</math> for <math>r, r' \in R</math> and <math>i, i' \in I.</math>

The algebra of dual numbers is the special case where <math>M = R</math> and <math>\varepsilon = (0, 1).</math>
==Superspace==
Dual numbers find applications in [[physics]], where they constitute one of the simplest non-trivial examples of a [[superspace]]. Equivalently, they are [[Grassmann number|supernumbers]] with just one generator; supernumbers generalize the concept to {{mvar|n}} distinct generators {{mvar|ε}}, each anti-commuting, possibly taking {{mvar|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 {{mvar|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 [[fermion]]s 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&nbsp;{{math|''ε''<sup>2</sup> {{=}} 0}}.

==Projective line==
The idea of a projective line over dual numbers was advanced by Grünwald<ref>{{cite journal|first=Josef|last=Grünwald|date=1906|title=Über duale Zahlen und ihre Anwendung in der Geometrie|journal=Monatshefte für Mathematik|volume=17|pages=81–136|doi=10.1007/BF01697639|s2cid=119840611}}</ref> and [[Corrado Segre]].<ref>{{cite book|first=Corrado|last=Segre|author-link=Corrado Segre|date=1912|chapter=XL. Le geometrie proiettive nei campi di numeri duali|title=Opere}} Also in ''Atti della Reale Accademia della Scienze di Torino'' '''47'''.</ref>

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 (geometry)|cylinder]].<ref name="yaglom">{{cite book|first=I. M.|last=Yaglom|date=1979|title=A Simple Non-Euclidean Geometry and its Physical Basis|publisher=Springer|isbn=0-387-90332-1|mr=520230|url-access=registration|url=https://archive.org/details/simplenoneuclide0000iagl}}</ref>{{rp|pp=149–153}}

Suppose {{mvar|D}} is the ring of dual numbers {{math|''x'' + ''yε''}} and {{mvar|U}} is the subset with {{math|''x'' ≠ 0}}. Then {{mvar|U}} is the [[group of units]] of {{mvar|D}}. Let {{math|''B'' {{=}} {(''a'', ''b'') ∈ ''D'' × ''D'' : ''a'' ∈ U or ''b'' ∈ U}<nowiki/>}}. A [[relation (mathematics)|relation]]  is defined on B as follows: {{math|(''a'', ''b'') ~ (''c'', ''d'')}} when there is a {{mvar|u}} in {{mvar|U}} such that {{math|''ua'' {{=}} ''c''}} and {{math|''ub'' {{=}} ''d''}}. This relation is in fact an [[equivalence relation]]. The points of the projective line over {{mvar|D}} are [[equivalence class]]es in {{mvar|B}} under this relation: {{math|''P''(''D'') {{=}} ''B''/~}}. They are represented with [[projective coordinates]] {{math|[''a'', ''b'']}}.

Consider the [[embedding]] {{math|''D'' → ''P''(''D'')}} by {{math|''z'' → [''z'', 1]}}. Then points {{math|[1, ''n'']}}, for {{math|''n''<sup>2</sup> {{=}} 0}}, are in {{math|''P''(''D'')}} but are not the image of any point under the embedding. {{math|''P''(''D'')}} is mapped onto a [[cylinder (geometry)|cylinder]] by  [[projection (mathematics)|projection]]: Take a cylinder tangent to the double number plane on the line {{math|{''yε'' : ''y'' ∈ '''R'''}<nowiki/>}}, {{math|''ε''<sup>2</sup> {{=}} 0}}. Now take the opposite line on the cylinder for the axis of a [[pencil (mathematics)|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 {{math|[1, ''n'']}}, {{math|''n''<sup>2</sup> {{=}} 0}} in the projective line over dual numbers.

==See also==
* [[Smooth infinitesimal analysis]]
* [[Perturbation theory]]
* [[Infinitesimal]]
* [[Screw theory]]
* [[Dual-complex number]]
* [[Laguerre transformations]]
* [[Grassmann number]]
* [[Automatic differentiation#Automatic differentiation using dual numbers|Automatic differentiation]]

==References==
{{Reflist|30em}}

===Further reading===
{{refbegin|30em}}
*{{cite journal |last=Bencivenga |first=Ulderico |date=1946 |title=Sulla rappresentazione geometrica delle algebre doppie dotate di modulo |language=Italian |trans-title=On the geometric representation of double algebras with modulus |journal=Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli |series=3 |mr=0021123 |volume=2 |issue=7}}
*{{cite journal |last=Clifford |first=William Kingdon |author-link=William Kingdon Clifford |date=1873 |title=Preliminary Sketch of Bi-quaternions |journal=[[Proceedings of the London Mathematical Society]] |volume=4 |pages=381–395}}
*{{cite journal |last1=Harkin |first1=Anthony A. |last2=Harkin |first2=Joseph B. |date=April 2004 |title=Geometry of Generalized Complex Numbers |journal=[[Mathematics Magazine]] |doi=10.1080/0025570X.2004.11953236 |s2cid=7837108 |volume=77 |issue=2 |pages=118–129 |url=http://people.rit.edu/harkin/research/articles/generalized_complex_numbers.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://people.rit.edu/harkin/research/articles/generalized_complex_numbers.pdf |archive-date=2022-10-09 |url-status=live}}
*{{cite journal |last1=Miller |first1=William |last2=Boehning |first2=Rochelle |date=1968 |title=Gaussian, Parabolic and Hyperbolic Numbers |journal=[[The Mathematics Teacher]] |volume=61 |issue=4 |pages=377–382|doi=10.5951/MT.61.4.0377 }}
*{{cite book |last=Study |first=Eduard |author-link=Eduard Study |date=1903 |title=Geometrie der Dynamen |page=196 |publisher=B. G. Teubner |url=http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=03150002}} From ''Cornell Historical Mathematical Monographs'' at [[Cornell University]].
*{{cite book |last=Yaglom |first=I. M. |author-link=Isaak Yaglom |others=Translated from Russian by Eric J. F. Primrose |date=1968 |title=Complex Numbers in Geometry |publisher=[[Academic Press]] |location=New York and London |page=[https://archive.org/details/complexnumbersge00yagl_880/page/n20 12]&ndash;18 |url=https://archive.org/details/complexnumbersge00yagl_880 |url-access=limited}}
*{{cite book |last=Brand |first=Louis |date=1947 |title=Vector and tensor analysis |publisher=John Wiley & Sons |location=New York}}
*{{cite book |last=Fischer |first=Ian S. |date=1999 |title=Dual number methods in kinematics, static and dynamics |publisher=CRC Press |location=Boca Raton}}
*{{cite book |last=Bertram |first=W. |date=2008 |title=Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings |series=Memoirs of the AMS |publisher=Amer. Math. Soc. |location=Providence, Rhode Island |volume=192 |number=900}}
*{{cite web |title="Higher" tangent space |website=math.stackexchange.com |url=https://math.stackexchange.com/a/430886}}
{{refend}}

{{Number systems}}
{{Infinitesimals}}

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[[Category:Linear algebra]]
[[Category:Hypercomplex numbers]]
[[Category:Commutative algebra]]
[[Category:Differential algebra]]
[[Category:Nonstandard analysis]]