What are the dual Numbers?

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Double number aˆ is the ordered number of real numbers (a, ao) associated with the actual unit +1 and a dual unit, or operator ε, where ε2 = ε3 = ··· = 0, 0ε = ε0 = 0, 1ε = ε1 = ε. Twice the number is usually defined in the form

                                                           aˆ = a + eao. (1)

The two-digit algebra was invented by Clifford (1873) [1], but The first mechanic applications are based on Kafelnikov (1895) Dual vector algebra provides a simple tool for managing such mathematical organizations such as screws and screws [60]. In fact, helicoidally infinitesimal infinite and finite body hard displacements can be easily made under the dual vector algebra framework. A distinct feature of dual algebra is the shortness of the text. For these reasons, it has been commonly used to search for closed form solutions in the field of migration analysis, kinematic integration and flexible analysis of local methods. Algebraic applications dual research of kinematic and dynamic effects of production and integration of errors in methods are well known.

                                           One of the objectives of this study is the development implementation of algorithms for solution for algebraic problems with a line using double digits. Repeat number solution for non-linear calculations is discussed. Although the algorithms discussed especially related to the solution of kinematic problems, we believe they may be useful and dynamic analysis of processes.

The algorithms presented here can be divided into the following categories:

1. Easy operation involving dual vectors

2. Dual version of basic algorithms for single line algebra

3. Solution for a number of direct and indirect calculations.

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                                                                           Double numbers can be represented as follows:

• Gaussian Representation: a ≡ a + eao.

• Representing a cool place: a ˆ ≡ ρ (1 + εt), where ρ = a and t = ao / a.

• Definition representation: a ˆ ≡ ρeεt, where ρ = a, t = ao / a and eET = 1 + εt.

Acceptance of one representation on behalf of another depends on context.


                                                                     The F function of dual rotation x ˆ = x + εxo can be represented in the correct order

                                          F (x) ˆ = f – x, xo + eg- x, xo

Where f and g are the actual functions of the real variables x and xo. Necessary and sufficient

Conditions for F to be an analysis

                                                             ∂f=xo = 0, ∂f

                                                                 ∂x = ∂g


In the following

                                         f (x) ˆ = f – x + nxa = f (x) + xo ∂f ∂x.


                                                           With reference

                                          Aˆ = a + ε (r 1 × a),

                                          B = b + ε (r 2 × b),

Two vectors represent two different vectors and allow ∗ version The minimum distance between these line vectors is directed from a to b. In such a case, it is necessary to introduce a dual angle concept

                                                    θˆ = θ + es

as a variation required to indicate the relative location and position of vectors of line Aˆ and B. The θ angle θ is measured against the clock mayelana s ∗. The products of the two-and-a-two-vector cross-section are described as follows

                                                Aˆ · Bˆ = a · b + ε

                                         a · (r 2 × b) + b · (r 1 × a)

                                                  (r 1 – r2) · (a × b)

                                                    = ab cos θ – ε – ss

                                                    = ab sin θs

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                                                   = ab [cos θ – εs sin θ] = ab cos θ.ˆ (6)

                                                    Aˆ × Bˆ = a × b + ε

                                                   a × (r 2 × b) + (r 1 × a) × b

                                                      = a × b + ε

                                                     (a · b) (r 2 – r1) + r1 × (a × b) = ab

                                                     s sin θ + ε

                                               s cos θs + sin θ – r 1 × world = abS

                                                (sin θ + εs cos θ) = abSˆ sin θ, ˆ (7)

There is a variety of applications of linear algebra in real life. Basically linear algebra is used in cryptography, road mapping planning, chemical equations balancing, and many more. Especially in electrical engineering we can see how different fields are taking advantages from it.  

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Michelle Gram Smith
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