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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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A New Number Theory

Luca Sonaglioni*

Department of Mathematics, University of Bologna, Italy

*Corresponding Author:
Sonaglioni L
Free Professionist, Italy
Tel: 388-0579470
E-mail: [email protected]

Received Date: February 03, 2015; Accepted Date: March 28, 2015; Published Date: April 10, 2015

Citation: Sonaglioni L (2015) A New Number Theory. J Appl Computat Math 4:212. doi: 10.4172/2168-9679.1000212

Copyright: © 2015 Sonaglioni L. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

The article proposes a mathematical method that permits to treat numbers with more than 2 dimensions. The sums and the products must be done by two distinct definitions. Despite this little limitation you have the same algebra of the standard complex numbers. The limitations occur only when you have to really compute the sums and the products between numbers, not in the symbolic algebra. The product and the sum are commutative

Keywords

Quaternions; Number theory; Operator theory; Algebra; Tensor methods

Introduction

Let us consider the 3d space that can be represented by the tern Equation as shown in the Figure 1 below: Equation unit vector in the three-dimensional space

applied-computational-mathematics-3d-space-represented

Figure 1: Title of Figure.

The point P can be written as: P=(x, y, z)= Equation

But also, using the polar notation the point P can be written as Equation

Where Equation

The operator Equation raises the lying vectors in the plane Equation of Equation radians along

The operator Equation rotates the lying vectors in the plane Equation of a radians.

Definition of the sum

Equation

Definition of the product

Equation

The Number

Equation behaves as a complex number with three dimensions provided that: To do the sums it must always be used the definition (1) [cartesian notation] To do the products it must always be used the definition (2) [polar notation] The definition can be extended to 4 dimensions.

The relations between P=(x, y, z) and (r,a,b) are given by the following formulas:

Equation

Equation

Equation

For a=0 the vector is lying in the plane Equation and the polar notation coincides with a vector in the vertical rotation. The y values for a=0 are 0 by default. The transition from one format to another is always possible, because the tern

(x, y, z) always and uniquely identifies the tern (r,a,b) through the formulas (3) The methods of symbolic computation are identical to those of the standard complex numbers.

The operations, of calculation, must always take into account the two rules (1) and (2) above for the sums and products. To assess the calculator expressions you can use the Reverse Polish Notation (RPN) [1].

Calculus

The question is: does the calculus work?

Fixing:

Equation

The definition above, can’t coincides at the infinitesimal to the differential of

Equation

The calculus works using the formulas (3)

Equation

Equation

Equation

Practically the polar notation is useful only to define the concept of the commutative product, necessary for the symbolic operations and for the real calculation of the product between the numbers.

As a last consideration, we can also define Equation and in general:

Equation

Dimensions

Let us consider s as a 4 dimension number as defined above:

Equation

it still true the analysis above? The answer is yes, but we have to introduce the operator Equation which raises the vectors in the cube Equationof λ radians along Equation

The unity vector Equation is orthogonal to Equation and Equation but it can’t be represented graphically [2]; the formulas between the two representation, one cartesian, the other polar, can be detected from an idea given by the Figure 2 below:

applied-computational-mathematics-The-formulas

Figure 2: The formulas between the two representation.

The formulas:

Equation

Equation

Equation

Equation

Equation

Equation

Because Equation can’t be negative, it is clear that Equation must be reduced to Equation When Equation i.e. a pure Equation vector.

Examples: calculating the volume of a sphere

We need a little core of the new algebra, see below.

Here a couple of routines (written in Visual Basic) to estimates the volume of a sphere with the algebra above. The more Kloop is high, the more accurate is the estimation [3].

Sub Sphere A (R0 As Double, Kloop As Integer) Dim S As Complex3d

Dim dAlfa As Double, dBeta As Double, dr As Double Dim I As Long, J As Long, K As Long, KLoop As Long Dim Vol As Double, dv As Double

dAlfa=2 * Pi / KLoop

'Let consider half a sphere dBeta=Pi/2/ Loop

dr=R0/KLoop Vol=0

S=Init_Vector(0, 0, 0) For K=1 To KLoop

For I=1 To KLoop For J=1 To KLoop

‘Vector differential

dv=S.R ^ 2 * dAlfa * Cos(S.Beta) * dBeta * dr

Vol=Vol + dv

S=Rotation_3d(S, dAlfa, 0) Next J

S=Rotation_3d(S, 0, dBeta) Next I

S.Beta=0

S=Rotation_3d(S, 0, 0, dr) Next K

MsgBox "Sphere: estimated Vol: " + Format(2 * Vol) + vbCrLf + "Exact: " + Format(4/3 * Pi * R0^ 3) End Sub

Sub SphereB(R0 As Double, Kloop As Integer) Dim S As Complex3d

Dim dAlfa As Double, dBeta As Double, dr As Double Dim I As Long, J As Long, K As Long

Dim Vol As Double, dv As Complex3d dAlfa=2 * Pi/Kloop

dBeta=Pi/2/Kloop dr=R0/Kloop

Vol=0

S=Init_Vector(0, 0, 0) For K=1 To Kloop

‘ Msg "K: " + Format(K) + " di " + Format(Kloop) For I=1 To Kloop

For J=1 To Kloop

dv=Differentiate_Vector_3d(S, dAlfa, dBeta, dr) dv=Project_A_ on_B_3d(dv, S)

Vol=Vol + dv.X * dv.Y * dv.Z

S=Rotation_3d(S, dAlfa, 0) Next J

S=Rotation_3d(S, 0, dBeta) Next I

S.Beta=0

S=Rotation_3d(S, 0, 0, dr) DoEvents

Next K ‘Msg

MsgBox "Sphere: estimated Vol: " + Format(2 * Vol) + vbCrLf + "Exact: " + Format(4 / 3 * Pi * R0 ^ 3) End Sub

Here the code written in Visual Basic that define the sum, the product’s, and s^x…

Option Explicit '-----------------------------------

' CORE 3d ALGEBRA

' V2.4 OPTIMIZED

'-----------------------------------

Public Const Pi=3.14159265358979

-------------------------------------------------------------------------- -----------------------------------

''AVOID THE USE OF SMALL NUMBER IN SIMULATION (OR VERY BIG NUMBERS) 'THE PRECISION IS LIMITED, THE MANTISSA HAVE 15 DIGIT '

Public Const MaxDigit=12, AsZero=10 ^ -12

'

'We can round the results of calculus or not Private Const Round_ Results=True

'-------------------------------------------------------------------------- -----------------------------------

'The definition of the Complex3d type Type Complex3d

X As Double

Y As Double

Z As Double

R As Double

Alfa As Double

BetaAs Double End Type

'The initialization number as in an Algebric notation

Function Init_Algebric(X As Double, Y As Double, Z As Double) As Complex3d Dim R As Complex3d

R.X=X

R.Y=Y

R.Z=Z Calc_Vector_Notation R Init_Algebric=R

End Function

'The initialization number as in a Vector notation

Function Init_Vector(R As Double, Alfa As Double, Beta As Double) As Complex3d Dim S As Complex3d

S.R=R S.Alfa=Alfa S.Beta=Beta

To_Algebric_ Notation S Init_Vector=S

End Function

'The Sum A+B

Function Sum_3d(A As Complex3d, B As Complex3d) As Complex3d

Dim R As Complex3d

R.X=A.X + B.X

R.Y=A.Y + B.Y

R.Z=A.Z + B.Z

Calc_Vector_Notation R

Sum_3d=R

End Function

'The Difference A-B

Function Diff_3d(A As Complex3d, B As Complex3d) As Complex3d

Dim R As Complex3d

R.X=A.X - B.X

R.Y=A.Y - B.Y

R.Z=A.Z-B.Z

Calc_Vector_Notation R

Diff_3d=R

End Function

'The Product A*B

Function Mul_3d(A As Complex3d, B As Complex3d) As Complex3d

Dim R As Complex3d, X As Double

R.R=A.R * B.R

R.Alfa=Modulus(A.Alfa + B.Alfa, 2 * Pi)

R.Beta=Modulus(A.Beta + B.Beta, 2 * Pi)

To_Algebric_Notation R

Mul_3d=R

End Function

'The Division A/B

Function Div_3d(A As Complex3d, B As Complex3d) As Complex3d Dim R As Complex3d, X As Double

'Fai la divisione...

R.R=A.R/B.R

R.Alfa=Modulus(A.Alfa - B.Alfa, 2 * Pi)

R.Beta=Modulus(A.Beta - B.Beta, 2 * Pi) To_Algebric_Notation R

Div_3d=R

End Function

The 1/S

Function Inverse_3d(S As Complex3d) As Complex3d Dim R As Complex3d, X As Double

R.R=1/S.R

R.Alfa=-S.Alfa

R.Beta=-S.Beta To_Algebric_ Notation R Inverse_3d=R

End Function

'S^X; X Real

Function S_elev_X_3d(S As Complex3d, X As Double) As Complex3d Dim R As Complex3d

R.R=S.R ^ X

If Abs(X) <=1 Then R.Alfa=S.Alfa * X R.Beta=S.Beta * X

Else

R.Alfa=Modulus(S.Alfa * X, 2 * Pi)

R.Beta=Modulus(S.Beta * X, 2 * Pi) End If

To_Algebric_Notation R S_elev_X_3d=R

End Function

'Square Root of S

Function Sqr_3d(S As Complex3d) As Complex3d

Dim R As Complex3d

R.R=Sqr(S.R)

R.Alfa=S.Alfa/2

R.Beta=S.Beta/2

To_Algebric_Notation R

Sqr_3d=R

End Function

'Rotation and Elongation

Function Rotation_3d(S As Complex3d, dAlfa As Double, dBeta As Double, Optional dr As Double=0) As Complex3d Dim R As Complex3d

R=S

If Near0(R.R)=0 And Near0(dr)=0 Then Rotation_3d=R

Exit Function End If

R.R=R.R + dr

R.Alfa=Modulus(S.Alfa + dAlfa, 2 * Pi)

R.Beta=Modulus(S.Beta + dBeta, 2 * Pi) To_Algebric_Notation R

Rotation_3d=R End Function

'Creates ds from a vector S and dAlfa,dBeta and dr

Function Differentiate_Vector_3d(S As Complex3d, dAlfa As Double, dBeta As Double, dr As Double) As Complex3d Dim dx As Double, dy As Double, dz As Double, ds As Complex3d

dz=dr * Sin(S.Beta) + S.R * Cos(S.Beta) * dBeta

dy=dr * Cos(S.Beta) * Sin(S.Alfa) - S.R * Sin(S.Beta) * Sin(S.Alfa) * dBeta + S.R * Cos(S.Beta) * Cos(S.Alfa) * dAlfa dx=dr * Cos(S.Beta) * Cos(S.Alfa) - S.R * Sin(S.Beta) * Cos(S.Alfa) * dBeta - S.R * Cos(S.Beta) * Sin(S.Alfa) * dAlfa ds=Init_Algebric(dx, dy, dz)

Differentiate_ Vector_3d=ds End Function

'Internal product

Function A_V_B_3d(A As Complex3d, B As Complex3d) As Double A_V_B_3d=A.X * B.X + A.Y * B.Y + A.Z * B.Z

End Function

'External product

Function A_X_B_3d(A As Complex3d, B As Complex3d) As Complex3d Dim X As Double, Y As Double, Z As Double, R As Complex3d

X=A.Y * B.Z - A.Z * B.Y

Y=A.Z * B.X - A.X * B.Z

Z=A.X * B.Y - A.Y * B.X

R=Init_Algebric(X, Y, Z) A_X_B_3d=R

End Function

'Versor of S

Function Versor_3d(S As Complex3d) As Complex3d Dim R As Complex3d, R0 As Double

If Near0(S.R)=0 Then GoTo Set_To_Zero R=S

R0=R.R

R.R=1 '----------------------------------

'Optimization 'To_ Algebric_Notation R R.X=R.X/R0

R.Y=R.Y/R0

R.Z=R.Z/R0 Check_Algebric_Zero R '-----------------------------------

Versor_3d=R

Exit Function

Set_To_Zero:

R.X=0

R.Y=0

R.Z=0

R.R=0

R.Alfa=0

R.Beta=0 Versor_3d=R

End Function

Return vector A along components on B axes; B new Real axes

Function Project_A_on_B_3d(A As Complex3d, B As Complex3d) As Complex3d

Dim Wx As Complex3d, Wy As Complex3d, Wz As Complex3d, R As Complex3d, R0 As Double Dim X As Double, Y As Double, Z As Double

Dim BVx As Double, BVy As Double, BVz As Double

If Near0(B.R)=0 Then GoTo Set_To_Zero

If Near0(A.R)=0 Then GoTo Set_To_Zero

'Versors Wx, Wy and Wz the new base Wx=Versor_3d(B) '-------- --------------------------------------------

' Optimization

'Wy=Init_Algebric_3d(-Wy.Y, Wy.X, 0)

Wy.X=-Wx.Y

Wy.Y=Wx.X

Wy.Z=0

'Wy=Versor_3d(Wy)

R0=Sqr(Wy.X ^ 2 + Wy.Y ^ 2)

If Near0(R0)=0 Then R0=1

Wy.X=Wy.X/R0

Wy.Y=Wy.Y/R0 '---------------------------------------------------- '

'----------------------------------------------------

'consider Wz as

'Wz=A_X_B_3d(Wx, Wy)

Wz.X=Wx.Y * Wy.Z - Wx.Z * Wy.Y Wz.Y=Wx.Z * Wy.X - Wx.X * Wy.Z Wz.Z=Wx.X * Wy.Y - Wx.Y * Wy.X 'Wz=Versor_3d(Wz)

'----------------------------------------------------'

'Project A on Wx, Wy, Wz, Wt BVx=A_V_B_3d(A, Wx) BVy=A_V_B_3d(A, Wy) BVz=A_V_B_3d(A, Wz)

R=Init_Algebric(BVx, BVy, BVz)

Project_A_on_B_3d=R

Exit Function

Set_To_Zero:

R.X=0

R.Y=0

R.Z=0

R.R=0

R.Alfa=0

R.Beta=0

Project_A_on_B_3d=R

End Function

'THE TRASFORMATION FROM ALGEBRIC TO POLAR NOTATION Private Sub Calc_Vector_Notation (S As Complex3d) [4]

Dim SinBeta As Double, CosBeta As Double, SinAlfa As Double, CosAlfa As Double

Check_Algebric_Zero S

'Calc R

S.R=Sqr(S.X ^ 2 + S.Y ^ 2 + S.Z ^ 2)

If Near0(S.R)=0 Then GoTo Set_To_Zero

'Solve Beta....

SinBeta=S.Z / S.R

CosBeta=Sqr(S.X ^ 2 + S.Y ^ 2)/S.R 'CosBeta is always >=0

'SinBeta can be <=0

If Round(CosBeta, MaxDigit)=0 Then

If Round(SinBeta, MaxDigit)=0 Then GoTo Set_To_Zero

S.Beta=Pi/2 * Sgn(S.Z)

Else

S.Beta=ArcSin(SinBeta)

End If

'Solve Alfa....

If Round(CosBeta, MaxDigit) <> 0 Then SinAlfa=S.Y/S.R/CosBeta CosAlfa=S.X/S.R/CosBeta 'CosAlfa can be <=0 ...

If Round(CosAlfa, MaxDigit)=0 Then If Round(SinAlfa, MaxDigit)=0 Then

S.Alfa=0 Else

S.Alfa=Pi/2* Sgn(S.Y) End If

Else

S.Alfa=ArcSin(SinAlfa) If CosAlfa<0 Then

S.Alfa=(Pi-Abs(S.Alfa)) * Sgn(S.Y)

Else

S.Alfa=Pi

End If

End If

End If

End If

Exit Sub

Set_To_Zero:

S.X=0

S.Y=0

S.Z=0

S.R=0

S.Alfa=0

S.Beta=0

End Sub

'THE TRASFORMATION FROM VECTOR TO ALGEBRIC Private Sub To_Algebric_Notation(S As Complex3d)

Dim CosBeta As Double

If Near0(S.R)=0 Then GoTo Set_To_Zero

'Solve X,Y,Z

S.Z=S.R * Sin(S.Beta)

CosBeta=Cos(S.Beta)

If Near0(CosBeta)=0 Then ‘a w vector! S.Y=0

S.X=0

'If CosBeta=0 Alfa is irrelevant (?)

S.Alfa=0 Else

S.Y=S.R * CosBeta * Sin(S.Alfa) S.X=S.R* CosBeta * Cos(S.Alfa)

End If

Check_Algebric_Zero S

Exit Sub

Set_To_Zero:

S.X=0

S.Y=0

S.Z=0

S.R=0

S.Alfa=0

S.Beta=0

End Sub

Private Sub Check_Algebric_Zero(S As Complex3d)

S.Z=Near0(S.Z)

S.Y=Near0(S.Y)

S.X=Near0(S.X)

End Sub

Function Modulus(X As Double, Y As Double) As Double

Dim Resto As Double

Resto=X/Y-Fix(X/Y)

If Round(Resto * Y, MaxDigit)=0 Then

Modulus=0

Else

Modulus=Resto * Y

End If

End Function

Function ArcSin(X As Double) As Double

If Round(Abs(X), MaxDigit)=1 Then

ArcSin=Pi/2 * Sgn(X)

Exit Function

End If

ArcSin=Atn(X/Sqr(1-X^2))

End Function

Function ArcCos(X As Double) As Double

If Round(Abs(X), MaxDigit)=1 Then

If X>0 Then

ArcCos=0

Exit Function

Else

ArcCos=Pi

Exit Function

End If

End If

ArcCos=Atn(-X/Sqr(1-X^2)) + 2 * Atn(1)

End Function

Function Near0(X As Double) As Double

Dim R As Double

R=X

If Round_Results Then R=Round(R, MaxDigit)

Near0=R

If Abs(R) <=AsZero Then

Near0=0

End If

End Function

'-----------------------------------

'END CORE 3d ALGEBRA '-----------------------------------

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