Hypercomplex number in three dimensional spaces. Abdelkarim Assoul

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Abdelkarim Assoul. Hypercomplex number in three dimensional spaces.. 2016. ￿hal-01686021v2￿

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Article: written by Assoul AbdelKarim

Secondry School Maths teacher

Summary

Any point of the real line is the real number image and any point of the R² plane is the complex number image.

Is any point of space R3 a number image, if this number exists is it unique and what is its figure and properties?

In this article we are going to build an algebra in a commutative field R and demonstrate that this one is isomorphic to R so that at the end this lead to the existence of this number its uniqueness also its figure and properties.

The succession of this work wille be the elaboration of this theory so that it will be usefull in applied mathematics,theoretical and physics quantum and espeially using this hypercomplex number for calculating the amplitude and the substitution of the (bit) by the (qubit) in order to have a faster and powerful quantum computer which has also much more capacity.

Review

Quaternions (hypercomplex number in four dimensional spaces) were invented by the Irish William Rovan Hamilton in 1843

Hamilton was resarching for ways to extend the complex numbers (which they could be assimilated to a point of a plane) to a higher dimension of the Euclidean space Rn. He couldn't do it for three dimensions, but the four dimensions produced the quaterions(1).

Kantor and Solodovnikov define the hypercomplex number as an algebra actual unit element (not necessarily associative)

(1)www.fr.wikipedia.org/wiki/Nombre_hypercomplexe

Introduction our work is specially intersted in the hypercomplex numbers in a three dimensional space so we are going to define an algebra (A, +,., ×) on a commutative field K equiped by a binary opération (×) in other words the A Two elements product is an element of A which is bilinear it means for any element x, y, z of A and all elements α, β of K:

 (x+y) × z = (x × z) + (y × z)

 x × (y+z) = (x × y) + (x × z)

 (α.x) × (β.y) = (α. β) (x × y)

Definition: the hypercomplex numbers in three dimensions set marked Ŝ is algebra no associative and not unital on the field of real number R.

Any item of Ŝ is written in unique way under the form: s = x + yi + zj with x, y, z real numbers and i, j purely imaginary numbers such as: i2 = j2 = -1 and i j = j i =0

1. Construction of the set Ŝ.

1.1 we equiped Ŝ of the addition and the external law (.) successively defined by:

(+) : Ŝ × Ŝ Ŝ

((x, y, z), (x`, y`, z`)) (x+x`, y+y`, z+z`)

( . ) : R × Ŝ Ŝ

α. (x, y, z) = (α.x, α.y, α.z)

(Ŝ, +, .) is a vector space on R.

1.2. We equiped Ŝ with binairy opération × defined by;

× : Ŝ × Ŝ Ŝ

((x, y, z), (x`, y`, z`)) (x x`- y y`- z z`, x y` + x`y, x z` + x`z)

× is bilinear and we notes that (Ŝ, +, . , ×) is an R-algebra

2. The imerssion of R in Ŝ

Designate by Ʈ = , the application f: R Ʈ that at any x it associates element is R bijection in Ʈ and also an isomorphisme of the addition and the multiplication

We imerssion R in Ŝ taking R = Ʈ, so we can combine for any x element of R, (x, 0, 0) = x.

But the particular points J (0, 1,0) and K (0, 0,1) are the images of pure imaginary numbers i and j successively, we deduce:

i2 = i × i = (0, 1,0) × (0, 1,0) = (-1, 0,0) = -1

j2 = j × j = (0, 0,1) × (0, 0,1) = (-1, 0,0) = -1

i × j = (0, 1,0) × (0, 0,1) = (0, 0,0) = 0

Note: the algebra base Ŝ on the commutative field R is the elements constitud 1, i, j.

The following multiplication talble is: . 1 i j

1 1 i j i i -1 0 j j 0 -1 3. Properties

If s and s` two elements of Ŝ such as: s = x +y i +z j and s` = x`+y`i +z`j, then

1) s = 0 <=> x = y = z = 0

2) s = s` <=> (x = x`) et (y = y`) et (z = z`)

3) s+ s` = (x+ x`) + (y+ y`) i + (z+ z`) j

4) s × s` = (x x` - y y`- z z`) + (x y` + x`y) i +(x z` + x`z) j

Definition: if s Є Ŝ such as: s = x +y i +z j, then Re (s) = x is the real side of s and Im (s) = y i +z j is the imaginary side of S.

Note: we can consider Im (s) = in the base

4. The hypercomplex number in three dimensional space conjugate

Definition: if s Є Ŝ such as: s = x +y i +z j, then the number s = x - y i - z j is called the conjugate of S.

Note: If s, s` Є Ŝ such as: s = x +y i +z j and s` = x`+y`i +z`j, then:

1) s + s = 2 Re (s),

2) s - s = 2 Im (s),

3) s × s = x2 + y2 + z2

Properties: if s, s` Є Ŝ such as: s = x +y i +z j and s` = x`+y`i +z`j, then:

1) = s

2) (s + s` ) = s + `

3) (s × s ` ) = s × `

4) (s n) = (s ) n , n Є N-

5) (1 / s ) = 1/ ( s) , s ‡0

6) (s / s ` ) = ( s )/( `) , s`‡0

6. The hypercomplex number in a three dimensional space module

Definition: If s Є Ŝ such as: s = x +y i +z j, then: │s│= is called the module of s.

Note: 1) │s│=

2) In the orthonormal reference system (O, I, J, K) of the space, if M(x, y, z) is the image number s = x +y i +z j, then: │s│= OM

Properties: for any numbers s and s` of Ŝ like s = x +y i +z j and s` = x`+y`i +z`j, we have:

1) │s│= 0 s = 0

2) │s│≥ 0

3) │s × s` │ = │s│×│ s` │

4) │sn│ = │s│n, n Є N-

5) │ │= , ‡0

6) │ │= , `‡0

Note: if M(x, y, z) the image number of s = x +y i +z j in the orthonormal reference system (O, I, J, K) of space, then the cylindrical coordinations are in the form of:

Such as │s│= , = , , = and N(x, y, 0)

Thus we conclud: s = x +y i +z j = + + s = ( + e i ) = ( eπ/2 j e i )

7. The hypercomplex number in three dimensional space square roots

Definition: if s Є Ŝ like s = x + y i + z j, we call the square roots of the s number, the number

δ = a + bi + cj such as: = δ2

The Methodology to find the square roots in the hypercomlex number in three dimensional spaces:

If s Є Ŝ such as: s = x + y i + z j is a given number we have two opposite square roots:

2 2 2 δ1 = + i+ j, Δ = x +y +z and δ2 = δ1

Bibliographie :

1. www.fr.wikipedia.org/wiki/Nombre_hypercomplexe 2. (en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Hypercomplex number »(voir la liste des auteurs).

3. A.DONEDDU , Cours De Mathématiques,Tome 1 Structures Fondamentales,LIBRAIRIE VUIBERT PARIS 1984

4. (en) I. L. Kantor et A. S. Solodovnikov, Hypercomplex Numbers : An Elementary Introduction to Algebras, c. 1989, New York: Springer-Verlag, traduit en anglais par A.