Dual number - Citizendia

A variety of dualities in mathematics are listed at duality (mathematics). In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution.

For the dual grammatical number found in some languages see dual grammatical number. Dual is a Grammatical number that some languages use in addition to singular and Plural.

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε² = 0 (ε is nilpotent). Linear algebra is the branch of Mathematics concerned with In Mathematics, the real numbers may be described informally in several different ways In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real

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

Linear representation

If one is familiar with matrix addition and matrix multiplication, then the phenomena of the dual number can be represented using ε = $\begin{pmatrix}0 & 1 \\0 & 0 \end{pmatrix}$ . In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix 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.

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. Slope is used to describe the steepness incline gradient or grade of a straight line. 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. In Mathematics, a shear or transvection is a particular kind of Linear mapping.

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 $\ \ t'=t,\ \ x' = vt + x \!$ . Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial

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. Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial 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 Mathematics, a quadratic equation is a Polynomial Equation of the second degree. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In the Inversive ring geometry of dual numbers one encounters "cyclic rotation" as a projectivity on the projective line over dual numbers. In Mathematics, inversive ring geometry is the extension to the context of Associative rings of the concepts of Projective line, Homogeneous According to Yaglom (pp. Isaak Moiseevich Yaglom (Иссак Моисеевич Яглом ( 6 March 1921, Kharkov — 17 April 1988, Moscow) was 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 \$ . In Euclidean geometry, a translation is moving every point a constant distance in a specified direction

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

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. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms). In Mathematics, Kähler differentials provide a generalization of Differential forms to arbitrary Commutative rings (or schemes. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is

Over any ring R, the dual number a + bε is a unit (i. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, 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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called

Differentiation

One application of dual numbers is automatic differentiation. In Mathematics and Computer algebra, automatic differentiation, or AD sometimes alternatively called algorithmic differentiation, is a method to numerically 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. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number 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). Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have In Mathematics, non-standard calculus is the application of Non-standard analysis techniques to differential and integral calculus

Superspace

Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. " Superspace " has had two meanings in physics The word was first used by John Wheeler to describe the Configuration space of General relativity; for example 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. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

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:

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

$= {ac + \varepsilon(cb - ad) \over c^2}$

$= {a \over c} + {(cb - ad) \over c^2}\varepsilon$

Which is defined when c is non-zero.

References

Isaak Yaglom (1968) Complex Numbers in Geometry
V. In Mathematics, Clifford algebras are a type of Associative algebra. This article describes perturbation theory as a general mathematical method Isaak Moiseevich Yaglom (Иссак Моисеевич Яглом ( 6 March 1921, Kharkov — 17 April 1988, Moscow) was V. Kisil (2007) "Inventing the Wheel, the Parabolic One" arXiv:0707.4024

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