Download On the construction of N-dimensional hypernumbers

Mathematica Aeterna, Vol. 1, 2011, no. 07, 461- 490
On the construction of N-dimensional hypernumbers
John F. Moxnes
Department for Protection
Norwegian Defence Research Establishment
P.O. Box 25
2007 Kjeller, Norway
E-mail: [email protected]
Tel.: +47 63 807514, Fax: +47 63 807509
Kjell Hausken
Faculty of Social Sciences
University of Stavanger
4036 Stavanger, Norway
E-mail: [email protected]
Tel.: +47 51 831632, Tel. Dept: +47 51 831500, Fax: +47 51 831550
Abstract
Complex numbers extend the concept of the 1 dimensional numbers to 2 dimensions.
Quaternions extend numbers to 4 dimensions. Octonions and sedenions are extensions to 8
and 16 dimensions respectively. We study a general form of complex numbers, various
axiomatic structures of 3 dimensional numbers, and finally N dimensional numbers, N  2k ,
k=0,1,2,.... Quaternions, octonions and sedenions are special cases. Two different structures
for addition are studied.
Keywords: complex numbers, hypernumbers, quaternions, octonions, sedenions
1 Introduction
1.1 Background
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J.F. Moxnes and K. Hausken
Man invented natural numbers to count people, sheep, cattle, etc. Fractions (rational numbers)
were probably invented to account for fractions of volumes or areas, e.g. of corn, water or
land. Probably zero was invented together with the construction of a number positioning
system analogous to the abacus. Negative numbers account for dept (something owed to
someone else). The irrational numbers were noticed by the Greeks. The numbers were
necessary to account for the length of all geometric objects defined by Euclidean geometry.
Complex numbers were historically introduced to allow for solutions of certain equations that
have no real number solution, i.e. x^2=-1 (Nahin 1998). Complex numbers extend the concept
of the 1 dimensional numbers to 2 dimensions. When complex numbers are viewed as a
position vector in a 2 dimensional Cartesian system (complex plane), the x-axis is used for the
real part and the vertical axis for the imaginary part. In this way a subspace of the complex
numbers arises, i.e. the numbers on the x-axis are isomorphic to the real numbers. Addition of
complex numbers corresponds to well known vector addition, while complex multiplication
corresponds to multiplying magnitudes of the two vectors and adding their rotational angle
with the x-axis. The commutative and associative laws for multiplication (and addition) are
fulfilled for the complex numbers. Complex numbers are today used in many scientific fields
such as engineering, electromagnetism, quantum physics and applied mathematics. The
addition rule of complex numbers was constructed before vector addition and was most likely
of vital importance for the idea of vector addition. A second type of complex numbers is the
so called split-complex numbers. The square of the imaginary number i is 1 and not -1. These
numbers were introduced by Cockle (1848) and Clifford (1882). For the third well known
type of complex numbers, i.e. dual numbers, the square of the imaginary number is (0,0).
Nonstandard numbers are often referred to as hypernumbers. Along with the imaginary
hypernumber i another hypernumber emerged within physics in the nineteenth century
(Cayley 1845, Clifford 1873, Hamilton 1969). Quaternions are numbers of 4 dimensions
where the base elements are real numbers. Historically quaternions were visualized as the
quotient of two directed lines in a 3 dimensional space. They are associative but not
commutative in multiplication. Quaternions are now used in both theoretical and applied
mathematics and in particular for calculations involving 3 dimensional rotations such as in 3
dimensional computer graphics and computer vision (Evans 1977, Conway et al. 2003). It is
also common for the attitude control system of spacecraft to be commanded in terms of
quaternions. Quaternions can be defined as two pairs of complex numbers with a certain
On the construction of N-dimensional hypernumbers
463
multiplication rule for two such pairs. The 3 dimensional imaginary space corresponds to the
three dimensional vector space of elementary vector calculus.
Octonions are numbers of 8 dimensions. The octonions were developed by Graves (1843),
inspired by the work of his fried Hamilton on quaternions. They were developed
independently by Cayley (1845). Analogously to quaternions, the octonions can be defined as
one pair of quaternions numbers with a certain multiplication rule for the pairs (Baez 2002).
Octonions have some interesting properties related to Lie groups. Octonions have applications
in string theory, special relativity and quantum logic. To day the concept of bi-quaternions are
well known for pairs of quaternions. In general the elements in the bi-quaternions can be
complex, dual or split-complex numbers. All these numbers can be conceived as 8
dimensional numbers. The space of bi-quaternions has a topology in the Euclidean metric on
8 dimensional space. The concepts of special relativity are illustrated through the biquaternions (Girard 1984).
Sedenions are extensions to 16 dimensions (Muses 1976, 1980, 1994, Carmondy 1988,
Carmondy 1997). The geometry of these numbers has been exploited (Carmondy 1997).
Hypernumbers can be used to simplify the algebra of real numbers or extend established
algebraic operations. In the literature the complex algebra extended to 4, 8 or 16 dimensional
algebras are motivated mathematically by various assumptions about the norm and
multiplication rules. No one within the literature has chosen a more generic route which is our
objective in this paper. We do not explore the numbers’ geometric aspects and their
application to solve or simplify physical problems which are topics of future research.
1.2 This paper’s contribution
Earlier research has confined attention to 1,2,4,8,16 dimensional numbers and the Clifford
algebra. We are not aware of research on higher dimensional numbers than 16, and research
not assuming the Clifford algebra. To provide a more generic foundation, we study a general
form of complex numbers, various axiomatic structures of 3 dimensional numbers, and finally
N dimensional numbers, N  2k , k=0,1,2,..., based on the general form of complex numbers.
We proceed outside the Clifford algebra in sections 7 and 8. Our approach is different from
the literature in the sense that a new form of addition different from vector addition,
exemplified with ( x1 , x2 )  ( y1 , y2 ) = ( x1 y2  y1 x2 , x2 y2 ) , is studied and generalized to N  2k ,
k=0,1,2,....
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J.F. Moxnes and K. Hausken
1.3 Organization
In section 2 we give an introduction to well known relations of the complex numbers and also
explore some numerical results in two dimensions. In section 3 we study 3 dimensional
numbers. In section 4 we study 4 dimensional numbers and quaternions. In section 5 we study
8 dimensional numbers and octonions in particular. In section 6 we study N dimensional
numbers that add as vectors. In section 7 we study quite generally numbers that do not add as
vectors. Such numbers have not been studied in the literature. Section 8 provides our most
general
structure
of
N
dimensional
numbers.
Section
9
concludes.
2 Complex numbers and some extensions as an introduction
A complex number is a special type of an ordered pair. We define an ordered pair by
def
( x1 , x2 )  { x1 , { x1 , x2 }}, x1 , x2  R . “def” means definition. The axiomatic structure of the
complex numbers is
mod
( x1 , x2 )  ( y1 , y2 )  ( x1  y1 , x2  y2 ),(a )
(2.1)
mod
( x1 , x2 )  ( y1 , y2 )   x1 y1  x2 y2 , x1 y2  x2 y1  , (b)
where “mod” means model assumption. It follows that
( x1 , 0)  ( y1 , y2 )   x1 y1 , x1 y2   ( y1 , y2 )  ( x1 , 0)
( x, 0)  ( y1 , y2 )   x y1 , x y2   ( y1 , y2 )  ( y1 , y2 )  ( y1 , y2 )  ..., x  N
x times
(0, x2 )  (0, y2 )  ( x2 y2 , 0)   (0,1)  (0,1) 
(2.2)
( x1 , x2 )  ( y1 , y2 )  ( y1 , y2 )  ( x1 , x2 )
(1, 0)  (1, 0)  1, 0 
(0,1)  (0,1)   1, 0 
The distributive law is fulfilled:
 ( x1 , x2 )  ( y1 , y2 )    z1 , z2   ( x1  y1 , x2  y2 )   z1 , z2 
  ( x1  y1 ) z1  ( x2  y2 ) z2 , ( x1  y1 ) z2  ( x2  y2 ) z1 
  x1 z1  x2 z2 , x1 z 2  x2 z1    y1 z1  y2 z2 , y1 z2  y2 z1   ( x1 , x2 )   z1 , z2   ( y1 , y2 )   z1 , z2 
(2.3)
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On the construction of N-dimensional hypernumbers
The Hamilton decomposition can be used, to read
def
def
def
def
e1  (1, 0), e2  (0,1), x   x1 , 0   e1  ( x2 , 0)  e2 , y   y1 , 0   e1  ( y2 , 0)  e2
(2.4)
Assume that we alternatively to (2.1) set the axiomatic structure more generally as
mod
D1:  ( x1 , x2 )  ( y1 , y2 )    z1 , z2   ( x1 , x2 )   z1 , z2   ( y1 , y2 )   z1 , z2 
mod
D 2 : ( x1 , 0)  ( y1 , y2 )   x1 y1 , x1 y2 
(2.5)
mod
D3: (0, x2 )  (0, y2 )  ( x2 y2 , 0)   (0,1)  (0,1) 
It follows that
x  y   x1 , x2    y1 , y2    x1 , 0   e1  ( x2 , 0)  e2   y1 , 0   e1  ( y2 , 0)  e2
   x1 , 0    y1 , 0    e1    x2 , 0    y2 , 0    e2   x1  y1 , 0   e1   x2  y2 , 0   e2   x1  y1 , x2  y2 
x  y    x1 , 0   (0, x2 )     y1 , 0   (0, y2 ) 
  x1 , 0    y1 , 0    x1 , 0   (0, y2 )  (0, x2 )   y1 , 0   (0, x2 )  (0, y2 )
  x1 y1 , 0   (0, x1 y2  x2 y1 )  ( x2 y2 , 0)   (0,1)  (0,1) 
  x1 y1 , 0   e1  ( x1 y2  x2 y1 , 0)  e2  ( x2 y2 , 0)   e2  e2 
(2.6)
Thus the crucial assumption is e2  e2 . For the special case of the complex numbers
e2  e2   e1 . For the so called split-complex numbers e2  e2  e1 , and for the so called dual
numbers e2  e2  (0, 0) .
A question is what is to be called numbers. A strong condition can be applied for what we
indeed will call numbers. We desire that a subspace of the numbers ( x1 , 0) , is isomorphic to
the real numbers. Vectors in 2 dimensions are also ordered pairs. Assume that we define
multiplication as the dot product, to read ( x1 , x2 )  ( y1 , y2 )  ( x1 y1  x2 y2 , 0) . It follows that
( x1 , 0)  ( y1 , 0)  ( x1 y1 , 0) and that ( x1 , 0)  ( y1 , 0)  ( x1  y1 , 0) . Thus these numbers are
isomorphic
to
the
real
numbers.
However,
it
follows
that
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J.F. Moxnes and K. Hausken
( x1 , x2 )  ( x1 , x2 )  (2 x1 , 2 x1 )  (2, 0)  ( x1 , x2 )  (2 x1 , 0) . Thus we could not apply D2 in
equation (2.5). We consider D2 to be important for the definition of what we actually call a
number. However, as we will see in sections 7 and 8, D2 is too restrictive.
Now, consider a quite general multiplication for complex numbers, to read
mod  x y a  x y b  x y c  x y d  e x  f x  g y  h y , 
1 1 1
1 2 1
2 1 1
2 2 1
1 1
1 2
1 1
1 2
( x1 , x2 )  ( y1 , y2 )  

 x1 y1a2  x1 y2 b2  x2 y1c2  x2 y2 d 2  e2 x1  f 2 x2  g 2 y1  h2 y2 
(2.7)
where a1 , b1 , c1 , d1 , e1 , f1 , g1 , h1 , a2 , b2 , c2 , d 2 , e2 , f 2 , g 2 and h2 are to be determined by
some desirables (DX) of the axiomatic structure. It follows that
D2 :
( x1 , 0)  ( y1 , y2 )
  x1 y1a1  x1 y2 b1  e1 x1  g1 y1  h1 y2 , x1 y1a2  x1 y2b2  e2 x1  g 2 y1  h2 y2 
D2
  x1 y1 , x1 y2   a1  1, b1  0, a2  0, b2  1, e1  e2  g1  g 2  h1  h2  0
(2.8)
D3:
(0, x2 )  (0, y2 )   x2 y2 d1  f1 x2 , x1 y2  x2 y2 d 2  f 2 x2  , (0,1)  (0,1)   d1  f1 , d 2  f 2 
 x2 y2 , 0     0,1   0,1    x2 y2 ,0    d1  f1 , d2  f 2    x2 y2 (d1  f1 ), x2 y2 (d2  f 2 ) 
D3
(0, x2 )  (0, y2 )   x2 y2 , 0     0,1   0,1   f1  f 2  0
We allow for more generality by allowing d1 and d 2 to be arbitrary. For complex numbers
(0,1)  (0,1)   d1 , d 2   (1, 0)  d1  1, d 2  0 . For split-complex numbers d1  1, d 2  0 .
For dual numbers d1  0, d 2  0 .
The addition rule could in general be different from (2.1a). Assume that generally
mod  x y a ' x y b ' x y c ' x y d '  e x ' f ' x  g ' y  h ' y ,

1 1 1
1 2 1
2 1 1
2 2
1
1 1
1
2
1
1
1
2
( x1 , x2 )  ( y1 , y2 )  

 x1 y1a2 ' x1 y2 b2 ' x2 y1c2 ' x2 y2 d 2 ' e2 ' x1  f 2 ' x2  g 2 ' y1  h2 ' y2 
(2.9)
467
On the construction of N-dimensional hypernumbers
where a1 ' , b1 ' , c1 ' , d1 ' , e1 ' , f1 ' , g1 ' , h1 ' , a2 ' , b2 ' , c2 ' , d 2 ' , e2 ' , f 2 ' , g 2 ' , and h2 ' are to be
determined by some desirables (DX) of the axiomatic structure. Assume that we desire that
( x1 , x2 )  ( x1 , x2 )   2, 0   ( x1 , x2 ) . This gives that
 2, 0   ( x1 , x2 )   2 x1 , 2 x2 
 x x a ' x1 x2b1 ' x2 x1c1 ' x2 x2 d1 ' e1 ' x1  f1 ' x2  g1 ' x1  h1 ' x2 , 
 1 1 1

 x1 x1a2 ' x1 x2b2 ' x2 x1c2 ' x2 x2 d 2 ' e2 ' x1  f 2 ' x2  g 2 ' x1  h2 ' x2 
 a1 '  a2 '  b1 '  b2 '  c1 '  c2 '  d1 '  d 2 '  f1 '  f 2 '  h1 '  h2 '  0
(2.10)
e1 '  e2 '  g1 '  g 2 '  1
Thus ( x1 , x2 )  ( y1 , y2 )   x1  y1 , x2  y2  . Indeed, to check whether the general complex
numbers with (0,1)  (0,1)   d1 , d 2  constitute a field we check for the distributive law, to
read
 x , x    y , y    z , z    x  y , x
1
2
1
2
1
2
1
1
2
 y2    z1 , z2 
  ( x1  y1 ) z1  ( x2  y2 ) z2 d1 , ( x1  y1 ) z2  ( x2  y2 ) z2 d 2  ( x2  y2 ) z1 
  ( x1 ) z1  ( x2 ) z 2 d1 , ( x1 ) z2  ( x2 ) z2 d 2  ( x2 ) z1    ( y1 ) z1  ( y2 ) z2 d1 , ( y1 ) z2  ( y2 ) z2  ( y2 ) z1 
  x1 , x2    z1 , z2    y1 , y2    z1 , z2 
(2.11)
 x , x    y , y    z , z    x y
1
2
1
2
1
2
1 1
 x2 y2 d1 , x1 y2  x2 y2 d 2  x2 y1    z1 , z2 
 ( x y  x y d ) z  ( x1 y2  x2 y2 d 2  x2 y1 ) z2 d1 ,

 1 1 2 2 1 1

 ( x1 y1  x2 y2 d1 ) z2  ( x1 y2  x2 y2 d 2  x2 y1 ) z2 d 2  ( x1 y2  x2 y2 d 2  x2 y1 ) z1 
  x1 , x2     y1 , y2    z1 , z2  
Thus we do not need (0,1)  (0,1)  (1, 0) for a field. We define
 z 2  n / 2 , n  2 k
def
z 2 z3




  ... , k  1, 2,3....,  (1, 0), (2, 0), (3, 0),... (2.12)
zn  
Exp
z
z
,
1
 
2 n 1
2!
3!
 z   z , n  2k  1
def
def
def
We write for simplicity that i  (0,1), 1  (1, 0) . We set that i  i  1 . Then we have for z =
i :
468
J.F. Moxnes and K. Hausken
 i 
Exp  i   1  i 
2
3
 i 

 i 

4
 i 

5
 i 

6
 i 

7
..
3!
4!
5!
6!
7!
2
4
6
3
5
7


     
     


 1


 i  


 ..   Cos    i Sin  


2!
4!
6!
3!
5!
7!


2!
(2.13)
Thus Exp  i   1 . Assume that instead i  i  1 , which simply is the split-complex numbers.
It follows that
 i 
Exp  i   1  i 
2
 i 

3
 i 

4
 i 

5
 i 

6
 i 

7
..
3!
4!
5!
6!
7!
2
4
6
3
5
7


     
     


 1


 i  


 ..   Cosh    i Sinh  


2!
4!
6!
3!
5!
7!


2!
(2.14)
Thus Exp  i    Cosh    i Sinh   . Assume more generally that i  i  d1 . It follows that
 i 
Exp  i   1  i 
2
 i 

3
 i 

4
 i 

4
 i 

5
 i 

5
 i 

6
 i 

6
 i 

7
 i 

7
 ...
4!
5!
6!
7!
2
4
6
3
5
7


d1   d12   d13  
d1   d12   d13  
(2.15)
 1


 i  


 ... 


2!
4!
6!
3!
5!
7!


2
4
6
3
5
7


d11/ 2   d11/ 2   d11/ 2 
d11/ 3   d12/ 5   d13/ 7 


 1


 i  


 ...  , d1  0


2!
4!
6!
3!
5!
7!


2!
3!
If i  i  id 2 it follows that
 i 
Exp  i   1  i 
2!
 1
d 2  
2!
2

d 2 2  
4!
2
3
 i 

3!
4

d 2 3  
6!
6
..
4!
5!
6!
7!
3
5
7


d 2   d 2 2   d 23  
 i  


 .. 


3!
5!
7!


(2.16)
469
On the construction of N-dimensional hypernumbers
Thus by applying that (0,1)  (0,1)   d1 , d 2  different calculations rules are developed for
Exp  i  .
The division by two complex numbers can be defined by
( x1 , x2 )   a, b    c, d   ( x1 , x2 )   c, d     a, b 
(2.17)
where “   ” defines division. The complex number ( x1 , x2 ) can be found by solving the
equation set
( x1 , x2 )   a, b    x1a  x2b, x1b  x2 a    c, d    x1a  x2b  c, x1b  x2 a  d 
(2.18)
mod
( x1 , x2 )  (a, b)   x1aa1  x1bb1  x2 ac1  x2bd1 , x1aa2  x1bb2  x2 ac2  x2bd 2 
The three most known complex numbers can be given a well known matrix form, to read
def  x
 x2 
1
Complex numbers :  x1 , x2   
,
 x2 x1 
 x  x2   y1  y2   x1  y1  x2  y2 
 x1 , x2    y1 , y2    1


   x1  y1 , x2  y2 
 x2 x1   y2 y1   x2  y2 x1  y1 
 x1  x2   y1  y2   x1 y1  x2 y2  x1 y2  x2 y1 
   x1 y1  x2 y2 , x1 y2  x2 y1 


 x2 x1   y2 y1   x1 y2  x2 y1 x1 y1  x2 y2 
 x1 , x2    y1 , y2   
def  x
x2 
1
Split  Complex numbers :  x1 , x2   
 ,  x1 , x2    y1 , y2    x1  y1 , x2  y2 
 x2 x1 
x x  y y  x y  x y x y  x y 
 x1 , x2    y1 , y2    1 2    1 2    1 1 2 2 1 2 2 1    x1 y1  x2 y2 , x1 y2  x2 y1 
 x2 x1   x2 y1   x1 y2  x2 y1 x1 y1  x2 y2 
def  x
x2 
1
Dual numbers :  x1 , x2   
 ,  x1 , x2    y1 , y2    x1  y1 , x2  y2 
 0 x1 
 x  x2   y1  y2   x1 y1  x1 y2  x2 y1 
 x1 , x2    y1 , y2    1


   x1 y1 , x1 y2  x2 y1 
 0 x1   0 y1   0 x1 y1

(2.19)
470
J.F. Moxnes and K. Hausken
We observe that (0,1)  (0,1)   1, 0  , (0,1)  (0,1)   0, 0  and (0,1)  (0,1)  1, 0  for
complex, dual and split-complex numbers respectively. D1 is trivially satisfied by the matrix
algebra. However, D2 is not obvious.
def
 cx  ex2 ax2  gx1 
Consider a matrix of the form  x1 , x2    1
 . We achieve
 bx2  hx1 dx1  fx2 
 cx  ex2 ax2  gx1 
 cy1  ey2 ay2  gy1 
( x1 , x2 )   1
 , ( y1 , y2 )  

 bx2  hx1 dx1  fx2 
 by2  hy1 dy1  fy2 
D2 :
 cx gx1   cy1  ey2 ay2  gy1 
( x1 , 0)  ( y1 , y2 )   1


 hx1 dx1   by2  hy1 dy1  fy2 
 cx1 (cy1  ey2 )  gx1 (by2  hy1 ) cx1 (ay2  gy1 )  gx1 (dy1  fy2 ) 


 hx1 (cy1  ey2 )  dx1 (by2  hy1 ) hx1 (ay2  gy1 )  dx1 (dy1  fy2 ) 
 cx y  ex1 y2 ax1 y2  gx1 y1 
( x1 y1 , x1 y2 )   1 1

 bx1 y2  hx1 y1 dx1 y1  fx1 y2 
D2
( x1 , 0)  ( y1 , y2 )  ( x1 y1 , x1 y2 )  c  d  1, g  h  0
 ex ax2   ey2 ay2   ex2 ey2  ax2by2 ex2 ay2  ax2 fy2 
D3: (0, x2 )  (0, y2 )   2



 bx2 fx2   by2 fy2   bx2 ey2  fx2 by2 bx2 ay2  fx2 fy2 
2
 x2 y2 0 
 e a   e a   ab  e a (e  f ) 
(0,1)  (0,1)  
,
(
,
0)
x
y






,
2
2



2
 b f   b f   b(e  f ) ab  f 
 0 x2 y2 
2
 x2 y2 0   e2  ab ea  af   (ab  e ) x2 y2 a (e  f ) x2 y2 
( x2 y2 , 0)   (0,1)  (0,1)   
 (2.20)
  
  
2
2
0
x
y
eb

fb
ab

f
b
e

f
x
y
ab

f
x
y
(
)
(
)

2 2 
 
2 2
2 2
 (0, x2 )  (0, y2 )
 ab  e 2 2ae 
We achieve that (0,1)  (0,1)   d1 , d 2   
 2be ab  f 2  . Thus d 2 = 2e which implies


d1  ex2  d1  2e2  ab  e 2 , d1  2 fe  ab  f 2  d1  ab  e 2 , e  f . Thus all matrices of the
form
 x1  ex2 ax2 


 bx2 x1  ex2 
fulfill
D1,
 ab  e 2 2ae 
(0,1)  (0,1)  
 (ab  e 2 , 2e) .
 2be ab  e 2 
d2
d1


D2
and
D3.
We
achieve
that
471
On the construction of N-dimensional hypernumbers
3 Ordered triplets and 3 dimensional numbers
We define 3 dimensional numbers through the ordered triplet ( x1 , x2 , x3 ) ={ x1 ,{ x1 , x2 },{ x1 ,
x2 , x3 }}. We forecast some desirables that these triplets should fulfill to be called a number.
Analogously to equation (2.5) we set that
mod
D1:  ( x1 , x2 , x3 )  ( y1 , y2 , y3 )   ( z1 , z2 , z3 )  ( x1 , x2 , x3 )  ( z1 , z2 , z3 )  ( y1 , y2 , y3 )  ( z1 , z2 , z3 )
mod
D 2 : ( x1 ,0)  ( y1 , y2 , y3 )   x1 y1 , x1 y2 , x1 y3 
(3.1)
mod
D3 : (0, x2 , 0)  (0, y2 , 0)  ( x2 y2 , 0, 0)   (0,1, 0)  (0,1, 0) 
mod
D 4 : (0, 0, x3 )  (0, 0, y3 )  ( x3 y3 , 0, 0)   (0, 0,1)  (0, 0,1) 
We
use
the
Hamilton
decomposition
and
set
that
x   x1 , 0, 0   e1  ( x2 , 0, 0)  e2  ( x3 , 0, 0)  e3 . If we seek correspondence with the complex
numbers e2  e2 = e1 . Assume that we choose that e3  e3 =  e1 . Assume that we apply that
factors commute and set that e3  e2 = e2  e3 =  e2 (Alt 1) or e3  e2 = e2  e3 =  e3 (Alt 2).
Thus we can construct Table 1.
Table 1: Table for ei  e j , i, j  1, 2,3
.
e1
e2
e3
e1
e1
e2
e3
e2
e2
- e1
 e2 (-+ e3 )
e3
e3
 e2 (-+ e3 )
 e1
We observe that the numbers of the type ( x1 , y1 , 0) are isomorphic to the complex numbers. It
follows that
472
J.F. Moxnes and K. Hausken
Alt1: e2  e3   e2
 e2  e3   e2   e2   e2  me1 , e2   e3  e2   e2   e2   me1
 e2  e3   e3   e2   e3  e2 , e2   e3  e3   e2   e1   e2
(3.2)
Alt 2 : e2  e3   e3
 e2  e3   e2   e3   e2  e3 , e2   e3  e2   e2    e3   e3
 e2  e3   e3   e3   e3  me1 , e2   e3  e3   e2   e1   e2
Although the commutative law for multiplication is satisfied, the associative law is only
satisfied if e2  e3   e2 and e3  e3  e1 .
To
determine
the
general
algorithm,
we
write
for
simplicity
that
x  x1e1  x2 e2  x3e3 , y  y1e1  y2 e2  y3 e3 , to read
Alt1:  e2  e3    e2 , e3  e3   e1
x  y  x1e1  x2 e2  x3 e3  y1e1  y2 e2  y3e3   x1  y1 , x2  y2 , x3  y3 
x  y   x1e1  x2 e2  x3e3    y1e1  y2 e2  y3 e3 
(3.3)
  x1 y1  x2 y2  x3 y3  e1   x1 y2  y1 x2  ( x2 y3  x3 y2 )  e2   x1 y3  x3 y1  e3
Alt 2 :  e2  e3    e3 , e3  e3   e1
x  y   x1 y1  x2 y2  x3 y3  e1   x1 y2  x2 y1  e2   x1 y3  x3 y1  ( x2 y3  x3 y2 )  e3
def
def
We write that i  (0,1, 0), 1  (1, 0, 0), i  i  1 . It follows easily that Exp  i   1 . If
def
j  (0, 0,1), j  j  1 it follows easily that Exp  j   1 . We can also calculate
Exp  i  j  . Assume that we set i  j  i . It follows that
 i  j 
Exp  i  j   1  i  j 
2!
2
3
4
 i  j 

3!
5
3
 i  j 

4!
6
4
5
 i  j 

5!
 i  j 

6!

6
 i  j 

7!
7
..
7
i     i     i  




..
2!
3!
4!
5!
6!
7!
2
4
6
3
5
7


     
     


 1


 i  


 ..   Cos    i Sin  


2!
4!
6!
3!
5!
7!


 1  i 
 
2
(3.4)
473
On the construction of N-dimensional hypernumbers
Thus Exp  i  j   1 .
It would be nice if we could define a 3 dimensional number as a pair of a 1 dimensional and 2
dimensional complex number. We consider the ordered pair
 x , 0 ,  x , x    x , x , x 
1
2
3
1
2
3
which we conceive as 3 dimensional number. We now use sum and multiplication rules as
defined for the complex numbers, to read
mod
  x , 0 ,  x , x      y , 0 ,  y , y      x
  x , 0 ,  x , x      y , 0  ,  y , y  
mod

1
2
3
1
2
3
1
2
3
1
2
3
1
 y1 , 0  ,  x2 , x3    y2 , y3      x1  y1 , 0  ,  x2  y2 , x3  y3  
  x , 0    y , 0    x , x    y , y  ,  x , 0   y , y    x , x    y , 0  
1
1
2
3
2
3
1
2
3
2
3
1
(3.5)
Thus it follows that
  x , 0 ,  x , x      y , 0  ,  y , y  
   x , 0   y , 0    x , x    y , y  ,  x , 0    y , y    x , x    y , 0 
  x y , 0   x y  x y , x y  x y  ,  x y  x y , x y  x y 
1
2
1
3
1
1
2
1 1
2
2
2
3
3 3
3
2
3
2 3
3
1
2
2
1 2
3
2 1
2
3 1
3
1
(3.6)
1 3



   x1 y1  x2 y2  x3 y3 ,  x2 y3  x3 y2  ,  x1 y2  x2 y1 , x3 y1  x1 y3  




0


But this is not a 3 dimensional number. However, if we use the multiplication rule for the dual
numbers we achieve that
mod
  x , 0 ,  x , x      y , 0 ,  y , y      x
1
2
3
1
2
3
1
 y1 , 0  ,  x2 , x3    y2 , y3      x1  y1 , 0  ,  x2  y2 , x3  y3  
mod
  x , 0 ,  x , x      y , 0  ,  y , y      x ,0    y , 0  ,  x , 0   y , y    x , x    y , 0  
   x y , 0 ,  x y  x y , x y  x y     x y , x y  x y , x y  x y 
1
2
1 1
3
1 2
1
2 1
2
3 1
3
1
1 3
1 1
1
1 2
1
2 1
2
3 1
3
2
3
1
1 3
(3.7)
Thus the dual numbers allows for a viable construction of 3 dimensional numbers from a pair
of two numbers. However, note that  x2 , x3  and  y2 , y3  do not need to be dual. We could
use complex or split-complex numbers also, but with the multiplication rule for pairs as for
dual numbers. Equation (3.7) applies for all three kinds of complex numbers. We achieve
Table 2.
474
J.F. Moxnes and K. Hausken
Table 2: Table for ei  e j , i, j  1, 2,3
.
e1
e2
e3
e1
e1 , 0(c)
e2 , e3 (c)
e3 , - e2 (c)
e2
e2 ,- e3 (c)
0, 0(c)
0, e1 (c)
e3
e3 , e2 (c)
0,- e1 (c)
0, 0(c)
In 3 dimensions the well known cross product between vectors is defined. Indeed this is a
special type of multiplication for ordered triplets. In Table 2 the cross product relations are
denoted by (c). This gives that
  x , 0 ,  x , x      y , 0  ,  y , y     x y
1
2
3
1
2
3
2 3
 x3 y2 , x3 y1  x1 y3 , x1 y2  x2 y1  , (a )
(3.8)
 x1 , 0, 0    y1 , y2 , y3    0,  x1 y3 , x1 y2  , (b)
The
cross
product
in
(3.8b),
and
the
triplet
with
the
dot
product
 x1 , x2 , x3    y1 , y2 , y3    x1 y1  x2 y2  x3 y3 , 0, 0  , violate D2. The implications are that the dot
product and the cross product do not apply for numbers.
4 4 dimensional numbers and the quaternions
The quaternions are most simply defined by the Hamilton decomposition, see Table 3.
Table 3: Table for quaternions (q) ei  e j , i, j  1, 2,3, 4
x
e1
e2
e3
e4
e1
e1
e2
e3
e4
e2
e2
- e1
e4
- e3
e3
e3
- e4
- e1
e2
e4
e4
e3
- e2
- e1
The numbers of the type  x1 , x2 , 0, 0  correspond to the 2 dimensional complex numbers. Due
to the relation e3  e2   e4 the numbers of the type
 x1 , x2 , x3 , 0  do
not correspond to 3
475
On the construction of N-dimensional hypernumbers
dimensional numbers. Thus the product of two 3 dimensional numbers (e.g. e3  e2 ) gives a
number with a fourth component ( e4 ).
We note that the commutative law is not fulfilled. To check the associative law we calculate
 e2  e3   e4  e4  e4  e1 , e2   e3  e4   e2  e2  e1
 e4  e2   e3  e3  e3  e1 , e4   e2  e3   e4  e4  e1
 e4  e2   e4  e3  e4  e2 , e4   e2  e4   e4   e3   e2
 e2  e3   e2  e4  e2  e3 , e2   e3  e2   e2   e4   e3
 e2  e3   e3  e4  e3  e2 , e2   e3  e3   e2
(4.1)
The quaternions are associative but not commutative in multiplication.
If would be nice if we could construct 4 dimensional numbers from pairs of complex
numbers, to read
  x , x  ,  x , x     x , x , x , x  . To show that this is possible, we apply
1
2
3
4
1
2
3
4
for pairs analogously to the multiplication of complex numbers
mod
 x , x  ,  x , x    y , y  ,  y , y    x  y , x
 x , x  ,  x , x    y , y  ,  y , y 
mod

1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
1
2
 y2  ,  x3  y3 , x4  y4  
(4.2)
 x , x    y , y    x , x    y , y  ,  x , x    y , y    x , x    y , y 
1
2
1
2
3
4
3
4
1
2
3
4
3
4
1
2
Thus it follows if the pairs are complex numbers that
 x1 , x2 , x3 , x4    y1 , y2 , y3 , y4 
  x1 , x2    y1 , y2    x3 , x4    y3 , y4  ,  x1 , x2    y3 , y4    x3 , x4    y1 , y2  
   x1 y1  x2 y2 , x1 y2  x2 y1   ( x3 y3  x4 y4 , x3 y4  x4 y3 ),  x1 y3  x2 y4 , x1 y4  x2 y3    x3 y1  x4 y2 , x4 y1  x3 y2  
   x1 y1  x2 y2  x3 y3  x4 y4 , x1 y2  x2 y1  x3 y4  x4 y3  ,  x1 y3  x3 y1  x2 y4  x4 y2 , x1 y4  x4 y1  x2 y3  x3 y2  
(4.3)
This means that
476
J.F. Moxnes and K. Hausken
e1  e1  e1 , e2  e2  e1 , e3  e3   e1 , e4  e4  e1
e3  e4  e4  e3  e2 ,
(4.4)
e2  e4  e4  e2  e3
e2  e3  e3  e2  e4
The relationship above does not apply for the quaternions. The quaternions do not commute
for e4  e3 = e2 and e2  e3 = e4 . Another difference is that e4  e4   e1 for quaternions. To
compare we construct the Table 4 where the m1 numbers are denoted by m1 where they are
different from the quaternions.
Table 4: Quaternions and m1 4 dimensional numbers, ei  e j , i, j  1, 2,3, 4
.
e1
e2
e3
e4
e1
e1
e2
e3
e4
e2
e2
- e1
e4
- e3
e3
e3
- e4
- e1
e2
e4 (m1)
e4
e4
e3
- e3 (m1)
- e2 (m1)
- e2
- e1
e1 (m1)
The m1 numbers are associative, to read
 e2  e3   e4  e4  e4  e1 , e2   e3  e4   e2  (e2 )  e1
 e4  e2   e3  (e3 )  e3  e1 , e4   e2  e3   e4  e4  e1
 e4  e2   e4  (e3 )  e4  e2 , e4   e2  e4   e4   e3   e2
 e2  e3   e2  e4  e2  e3 , e2   e3  e2   e2   e4   e3
(4.5)
m1 numbers are both commutative and associative. If we apply that the pairs are dual
numbers, and apply the multiplication rule corresponding to dual numbers, we achieve that
477
On the construction of N-dimensional hypernumbers
 x1 , x2 , x3 , x4    y1 , y2 , y3 , y4     x1 , x2    y1 , y2  ,  x1 , x2    y3 , y4    x3 , x4    y1 , y2  
   x1 y1 , x1 y2  x2 y1  ,  x1 y3 , x1 y4  x2 y3    x3 y1 , x4 y1  x3 y2  
   x1 y1 , x1 y2  x2 y1  x3 y4  x4 y3  ,  x1 y3  x3 y1 , x1 y4  x4 y1  x2 y3  x3 y2  
  x1 y1 , x1 y2  x2 y1  x3 y4  x4 y3 , x1 y3  x3 y1 , x1 y4  x4 y1  x2 y3  x3 y2 
(4.6)
To construct the quaternions form pairs of complex numbers we apply the intuitive but ad hoc
multiplication rule
 x , x  ,  x , x    y , y  ,  y , y 
1
2
3
4
1
2
3
4
mod
  x , x    y , y    x , x    y , y  *,  x , x    y , y    x , x    y , y  *
  x , x    y , y    x , x    y ,  y  ,  x , x    y , y    x , x    y ,  y 

1
1
2
2
1
1
2
2
3
3
4
4
3
3
4
4
1
1
2
2
3
3
4
4
3
3
4
4
1
1
(4.7)
2
2
This gives the quaternions, to read
 x1 , x2 , x3 , x4    y1 , y2 , y3 , y4 
  x1 , x2    y1 , y2    x3 , x4    y3 ,  y4  ,  x1 , x2    y3 , y4    x3 , x4    y1 ,  y2  
   x1 y1  x2 y2 , x1 y2  x2 y1   ( x3 y3  x4 y4 ,  x3 y4  x4 y3 ),  x1 y3  x2 y4 , x1 y4  x2 y3    x3 y1  x4 y2 , x4 y1  x3 y2  
   x1 y1  x2 y2  x3 y3  x4 y4 , x1 y2  x2 y1  x3 y4  x4 y3  ,  x1 y3  x3 y1  x2 y4  x4 y2 , x1 y4  x4 y1  x2 y3  x3 y2  
  x1 y1  x2 y2  x3 y3  x4 y4 , x1 y2  x2 y1  x3 y4  x4 y3 , x1 y3  x3 y1  x2 y4  x4 y2 , x1 y4  x4 y1  x2 y3  x3 y2 
(4.8)
Note that a different set of quaternions can be constructed by allowing
 x1 , x2 
to be split
complex numbers or dual numbers. For split-complex and dual numbers we achieve that
478
J.F. Moxnes and K. Hausken
Split  Complex :
 x1 , x2 , x3 , x4    y1 , y2 , y3 , y4 
   x1 , x2    y1 , y2    x3 , x4    y3 ,  y4  ,  x1 , x2    y3 , y4    x3 , x4    y1 ,  y2  
   x1 y1  x2 y2 , x1 y2  x2 y1   ( x3 y3  x4 y4 ,  x3 y4  x4 y3 ),  x1 y3  x2 y4 , x1 y4  x2 y3    x3 y1  x4 y2 , x4 y1  x3 y2  
   x1 y1  x2 y2  x3 y3  x4 y4 , x1 y2  x2 y1  x3 y4  x4 y3  ,  x1 y3  x3 y1  x2 y4  x4 y2 , x1 y4  x4 y1  x2 y3  x3 y2  
 x1 y1  x2 y2  x3 y3  x4 y4 , x1 y2  x2 y1  x3 y4  x4 y3 , x1 y3  x3 y1  x2 y4  x4 y2 , x1 y4  x4 y1  x2 y3  x3 y2 
Dual :
 x1 , x2 , x3 , x4    y1 , y2 , y3 , y4 
   x1 , x2    y1 , y2  ,  x1 , x2    y3 ,  y4    x3 , x4    y1 ,  y2  
   x1 y1 , x1 y2  x2 y1  ,  x1 y3 ,  x1 y4  x2 y3    x3 y1 , x4 y1  x3 y2  
  x1 y1 , x1 y2  x2 y1 , x1 y3  x3 y1 ,  x1 y4  x4 y1  x2 y3  x3 y2 
(4.9)
The quaternions can be defined by a matrix
def
 x1  e2 x2 x3  e2 x4 

  x3  e2 x4 x1  e2 x2 
 x1 , x2 , x3 , x4   
x e x x e x   y e y y e y 
 x1 , x2 , x3 , x4    y1 , y2 , y3 , y4    1 2 2 3 2 4    1 2 2 3 2 4 
  x3  e2 x4 x1  e2 x2    y3  e2 y4 y1  e2 y2 
The multiplication rule in equation (4.3) gives a new type of 4 dimensional numbers.
Before concluding this section we state another type of useful quaternions, to read
Table 5: 4 dimensional numbers
.
e1
e2
e3
e4
e1
e1
e2
e3
e4
e2
e2
- e1
e4
- e3
- e2 (- e3 ) (m2)
e3
e3
- e4
e4 (m1)
- e1
e2
- e2 (m1)
(4.10)
479
On the construction of N-dimensional hypernumbers
- e2 (- e3 ) (m2)
e4
e4
e3
- e2
- e1
- e3 (m1)
e1 (m1)
5 The 8 dimensional numbers and octonions
The octonions are defined by Table 6.
Table 6: Octonions ei  e j , i, j  1, 2,3, 4,5, 6, 7,8
.
e1
e2
e3
e4
e5
e6
e7
e8
e1
e1
e2
e3
e4
e5
e6
e7
e8
e2
e2
- e1
- e4
- e3
- e6
- e5
- e8
e7
e3
e3
- e4
- e1
e2
e7
e8
- e6
- e6
e4
e4
e3
- e2
- e1
e8
- e7
e6
- e5
e5
e5
- e6
- e7
- e8
- e1
e2
e3
e4
e6
e6
e5
- e8
e7
- e2
- e1
- e4
e3
e7
e7
e8
e5
- e6
- e3
e4
- e1
- e2
e8
e8
- e7
e6
e5
- e4
- e3
e2
- e1
The octonions can be constructed by applying the multiplication rule in equation (4.6) with
quaternions as basic elements, to read
 x , x , x , x  ,  x , x , x , x    y , y , y , y  ,  y , y , y , y 
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
  x1 , x2 , x3 , x4    y1 , y2 , y3 , y4    x5 , x6 , x7 , x8    y5 , y6 , y7 , y8  *, 

  x , x , x , x    y , y , y , y    x , x , x , x    y , y , y , y  * 
5
6
7
8
5
6
7
8
1
2
3
4
 1 2 3 4

The conjugate of  x1 , x2 , x3 , x4  * is defined as  x1 ,  x2 ,  x3 ,  x4  .
We consider the pair of two m1- quaternions
(5.1)
480
J.F. Moxnes and K. Hausken
 x , x , x , x  ,  x , x , x , x    x , x , x , x , x , x , x , x 
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
(5.2)
8
It would be nice if we can consider one pair of 4 dimensional numbers as one 8 dimensional
number. We apply the complex multiplication rule
 x , x , x , x  ,  x , x , x , x     y , y , y , y  ,  y , y , y , y 
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
  x1 , x2 , x3 , x4    y1 , y2 , y3 , y4    x5 , x6 , x7 , x8    y5 , y6 , y7 , y8  , 


  x1 , x2 , x3 , x4    y5 , y6 , y7 , y8    x5 , x6 , x7 , x8    y1 , y2 , y3 , y4  
(5.3)
This gives that
 x , x , x , x  ,  x , x , x , x     y , y , y , y  ,  y , y , y , y 
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
   x1 y1  x2 y2  x3 y3  x4 y4 , x1 y2  x2 y1  x3 y4  x4 y3  ,  

 
   x1 y3  x3 y1  x2 y4  x4 y2 , x1 y4  x4 y1  x2 y3  x3 y2   


   x5 y5  x6 y6  x7 y7  x8 y8 , x5 y6  x6 y5  x7 y8  x8 y7  ,  
 
  
x
y

x
y

x
y

x
y
,
x
y

x
y

x
y

x
y


  5 7 7 5 6 8 8 6 5 8 8 5 6 7 7 6 
  x y  x y  x y  x y , x y  x y  x y  x y , 
   1 5 2 6 3 7 4 8 1 8 2 5 3 8 4 7   
   x1 y7  x3 y5  x2 y8  x4 y6 , x1 y8  x4 y5  x2 y7  x3 y6   

 
   x5 y1  x6 y2  x7 y3  x8 y4 , x5 y2  x6 y1  x7 y4  x8 y3  ,  


   x5 y3  x7 y1  x6 y4  x8 y2 , x5 y4  x8 y1  x6 y3  x7 y2   

 
and
8
(5.4)
481
On the construction of N-dimensional hypernumbers
 x , x , x , x  ,  x , x , x , x     y , y , y , y  ,  y , y , y , y 
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
   x1 y1  x2 y2  x3 y3  x4 y4  x5 y5  x6 y6  x7 y7  x8 y   

, 
   , x1 y2  x2 y1  x3 y4  x4 y3  x5 y6  x6 y5  x7 y8  x8 y7   
 x y  x y  x y  x y  x y  x y  x y  x y ,  
   1 3 3 1 2 4 4 2 5 7 7 5 6 8 8 6   
   x1 y4  x4 y1  x2 y3  x3 y2  x5 y8  x8 y5  x6 y7  x7 y6   


   x1 y5  x2 y6  x3 y7  x4 y8  x5 y1  x6 y2  x7 y3  x8 y4 ,   
 
,
    x1 y8  x2 y5  x3 y8  x4 y7  x5 y2  x8 y1  x7 y4  x8 y3   
  x y  x y  x y  x y  x y  x y  x y  x y , 
  1 7 3 5 2 8 4 6 5 3 7 1 6 4 8 2  
   x1 y8  x4 y5  x2 y7  x3 y6  x5 y4  x8 y1  x6 y3  x7 y2   

 
(5.5)
Thus we achieve that
e2  e2  e1 , e3  e3  e1 , e4  e4  e1 , e5  e5  e1 , e6  e6  e1 , e7  e7  e1 , e8  e8  e1
e3  e4   e2 , e5  e6   e2 , e7  e8  e2
e2  e4  e3 , e5  e7  e3 , e6  e8  e3
e2  e3  e4 , e5  e8  e4 , e6  e7   e4 ,
(5.6)
e2  e6  e5 , e3  e7   e5 , e4  e8  e5 ,
e2  e5  e6 , e3  e8  e6 , e4  e7  e6 ,
e3  e5  e7 , e2  e8  e7 , e4  e6   e7 ,
e4  e5  e8 , e2  e7  e8 , e3  e6  e8 ,
This gives Table 7.
Table 7: 8 dimensional numbers. ei  e j , i, j  1, 2,3, 4,5, 6, 7,8
.
e1
e2
e3
e4
e5
e6
e7
e8
e1
e1
e2
e3
e4
e5
e6
e7
e8
e2
e2
- e1
- e4
- e3
- e6 (o)
- e5
- e8
e7
e4 (m1)
- e3 (m1)
e6 (m1) -
e8 (m1) -
e5 (m1)
e3
e3
- e4
e4 (m1)
- e1
e7 (m1)
- e6 (o)
e2
e7
e8
- e2 (m1)
e7 (m1)
e8 (m1) -
- e6
-
482
e4
e5
e6
J.F. Moxnes and K. Hausken
e4
e5
e6
e3 (q)
- e2
- e1
e8
- e3 (m1)
- e2 (m1)
e1 (m1)
e8 (m1) -
- e6
- e7
- e8
e6 (m1)
e7 (m1)
e8 (m1)
- e1
- e7
e8
e7
e8
e6 (m1)
e6
- e5
-
e5 (m1)
e7 (m1)
e6 (m1)
e2
e3
e4
- e2
-
-
e3 (m1)
e4 (m1)
- e4
e3
e5
- e8
e7
- e2
- e1
- e5 (m1)
e8 (m1)
- e7 (m1)
-
e1 (m1) -
e2 (m1)
e7
e5 (m1)
e3 (m1)
e4 (m1)
e8
e5
- e6
- e3
e4
- e1
- e2
e8 (m1)
- e5 (m1)
e6 (m1)
-
-
e1 (m1)
e2 (m1)
e3 (m1)
e4 (m1)
- e1
- e7
e6
e5
- e4
- e3
e2
- e7 (m1)
- e6 (m1)
e5 (m1)
-
e3 (m1)
e2 (m1)
e4 (m1)
The sedenions can be constructed by applying two octonions as a pair.
6 N-dimensional numbers which add as vectors
Assuming that numbers add as vectors, in this section we construct N dimensional numbers
quite generally, where N  2k , k=0,1,2,....
For 2 dimensions we set that
mod
( x1 , x2 )  ( y1 , y2 )   x1  y1 , x2  y2  , xi  R
mod
( x1 , x2 )  ( y1 , y2 )   x1 y1  x2 y1c1  x2 y2 d1 , x1 y2  x2 y1c2  x2 y2 d 2 
(6.1)
On the construction of N-dimensional hypernumbers
483
If we apply that multiplication commutes, it follows that c1  0, c2  1 . We observe that
( x1 , 0)  ( y1 , y2 )   x1 y1 , x1 y2  , (0,1)  (0,1)   d1 , d 2  . To construct higher dimensions we set
that
mod
N  2 :  x1 , x2    y1 , y2    x1  y1 , x2  y2 
mod
 x1 , x2    y1 , y2    x1 y1  x2 y1c1  x2 y2 d1 , x1 y2  x2 y1c2  x2 y2 d 2 
N  4 :  x1 , x2 , x3 , x4     x1 , x2  ,  x3 , x4     X 21 , X 2 2 
mod
 X 21 , X 2 2   Y 21 , Y 2 2    X 21  Y 21 , X 2 2  Y 22 
mod
 X 21 , X 2 2   Y 21 , Y 22    X 21  Y 21  X 22  Y 21c1  X 2 2  Y 22 d1 , X 21  Y 22  X 2 2  Y 21c2  X 2 2  Y 2 2 d2 
N  8 :  x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8     x1 , x2 , x3 , x4  ,  x5 , x6 , x7 , x8     X 41 , X 4 2 
X
X
mod
4
1
, X 4 2   Y 41 , Y 4 2    X 41  Y 41 , X 4 2  Y 4 2 
4
1
, X 4 2    Y 41 , Y 4 2    X 41  Y 41  X 4 2  Y 41c1  X 4 2  Y 4 2 d1 , X 41  Y 4 22  X 4 2  Y 41c2  X 4 2  Y 4 2 d 2 
mod
N  2k  n :
X
X
mod
n
1
, X n 2   Y n1 , Y n 2    X n1  Y n1 , X n 2  Y n 2 
n
1
, X n 2    Y n1 , Y n 2    X n1  Y n1  X n 2  Y n1c1  X n 2  Y n 2 d1 , X n1  Y n 22  X n 2  Y n1c2  X n 2  Y n 2 d 2 
mod
(6.2)
In this way we construct all high dimensional numbers of the type N  2k . For odd
dimensions where N  2k  1 we set that c1  d1  0 . Thus
mod
( x1 , x2 )  ( y1 , y2 )   x1 y1 , x1 y2  x2 y1c2  x2 y2 d 2 
 x1 , x2 , x3 ,....x2 k , x2k 1     x1 , 0  ,  x2 , x3 , x4 ,...., x2 k     X 1 , X 2 k 2 
 X , X   Y , Y    X  Y , X
 X , X   Y , Y    X Y , X  Y
2k
1
2k
2
1
2k
1
2k
2
1
2
1 1
1
2k
2
1
1
2k
2 Y 2 
2k
2
(6.3)
 X 2 k 2  Y1  d 2 X 2 k 2  Y 2 k 2 
7 Other types of numbers; the rational numbers of higher dimension
Contrary to the earlier sections, in this section we do not assume that numbers add as vectors.
We denote the whole numbers together with zero as Z 0 . We define multiplication and
addition of a 2 dimensional number by
484
J.F. Moxnes and K. Hausken
def
( x1 , x2 )  ( y1 , y2 )  ( x1 y2  y1 x2 , x2 y2 ), (a ), x1 , x2 , y1 , y2  Z 0
(7.1)
def
( x1 , x2 )  ( y1 , y2 )  ( x1 y1 , x2 y2 ), (b),
It is easily observed that the set of numbers of the type ( x1 ,1) is isomorphic to the numbers in
Z 0 since
( x1 ,1)  ( y1 ,1)  ( x1  y1 ,1), ( x1 ,1)  ( y1 ,1)  ( x1 y1 ,1)
(7.2)
(0,1) is the zero element. Notice that 1 instead of 0 acts as the crucial second element to
construct a subgroup that gives numbers isomorphic to
Z 0 . Also notice that
( x1 ,1)  ( y1 , y2 )  ( x1 y1 , y2 ) . Thus we do not multiply x1 with y2 .
However we can also study the numbers of the type (0, x2 ) , to read
(0, x2 )  (0, y2 )  (0, x2 y2 ), (a ),
(0, x2 )  (0, y2 )  (0, x2 y2 ), (b),
(7.3)
These numbers have the peculiar property that the sum of two numbers equals the product of
two numbers. The zero element is (0, 0) . However, to use these numbers we must know how
to calculate x2 y2 . We do not need to know how to sum two numbers x2  y2 .
Returning to equation (7.1), the distributive law is not fulfilled since
 ( x1 , x2 )  ( y1 , y2 )   ( z1 , z2 )  ( x1 y2  y1 x2 , x2 y2 )  ( z1 , z2 )
  ( x1 y2  y1 x2 ) z1 , x2 y2 z2    x1 y2 z1  y1 x2 z1 , x2 y2 z2 
(7.4)
( x1 , x2 )  ( z1 , z2 )  ( y1 , y2 )  ( z1 , z2 )  ( x1 z1 , x2 z2 )  ( y1 z1 , y2 z2 )
  x1 y2 z1 z 2  y1 x2 z1 z2 , x2 y2 z2 z2 
485
On the construction of N-dimensional hypernumbers
In particular we have that (2,1)  ( x1 , x2 )  (2 x1 , x2 ) , and that ( x1 , x2 )  ( x1 , x2 )  (2 x1 x2 , x2 2 ) . It
would be nice if these two numbers could be conceived as equal and also that the distributive
law is fulfilled. To show that these two requirements are satisfied, we define
def
x1 / x2  {( x1 x, x2 x)}, x  Z 0 . We then have
x1 / x2  y1 / y2  ( x1 x, x2 x)  ( y1 y, y2 y )  ( x1 y2 xy  y1 x2 xy , x2 y2 xy )  ( x1 y2  y1 x2 ) /( x2 y2 )
x1 / x2  y1 / y2  ( x1 x, x2 x )  ( y1 y, y2 y )  ( x1 y1 xy, x2 y2 dxy )  x1 y1 /( x2 y2 )
(7.5)
It is easily verified that the distributive law is fulfilled. For simplicity we write
 x1 , x2    x1 x, x2 x  = x1 / x2 ) which is called a rational number. We have that
1, 2   1, 2    2  2, 2 2    4, 4   1,1
(7.6)
So one half + one half is indeed one. However, we have that
1,1  1, 2    2  1, 2    3, 2 
(7.7)
This is a number different from  x,1 . We like to put the rational number on the so called real
x-axis. Consider a minimum stick to measure lengths. Consider that straight lines are
constructed between two arbitrary points in space (Moxnes and Hausken 2011). We define the
straight line to be the x-axis which is constructed by discrete points (0,1), (1,1), (2,1), .... We
would like to put (1,2) between (0,1) and (1,1), but there are actually no points between (0,1)
and (1,1).
However, assume that we transport (1,1) to (2,1)  (1,1) = (2,1) and (2,1) to (2,1)  (2,1) =
(4,1). Then we have that (1,2) to (2,1)  (1,2)= (1,1). Thus we can put (1,2) between (0,1) and
(1,1).
More
generally
we
use
the
general
( x1 ,1)  ( y1 , y2 )  ( x1 y1 , y2 ) it follows that
formalism
(2.7).
If
we
desire
that
486
J.F. Moxnes and K. Hausken
mod  x y a  x y b  x y c  x y d  e x  f x  g y  h y , 
1 1
1 2
( x1 , x2 )  ( y1 , y2 )   1 1 1 1 2 1 2 1 1 2 2 1 1 1 1 2

 x1 y1a2  x1 y2 b2  x2 y1c2  x2 y2 d 2  e2 x1  f 2 x2  g 2 y1  h2 y2 
x y a  x y b  yc  y d e x  f  g y h y , 
( x1 ,1)  ( y1 , y2 )   1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 2   ( x1 y1 , y2 )
 x1 y1a2  x1 y2 b2  y1c2  y2 d 2  e2 x1  f 2  g 2 y1  h2 y2 
 a1  d 2 , a2  0, b1  b2  0, c1  c2  0, d1  0  e1  e2  f1  f 2  g1  g 2  h1  h2
(7.8)
( x1 , x2 )  ( y1 , y2 )   x1 y1a1 , x2 y2 d 2 
For the addition rule we would like that (2,1)  ( x1 , x2 )  (2 x1a1 , x2 d 2 ) = ( x1 , x2 )  ( x1 , x2 ) . Using
the general equation (2.9) we achieve that
mod  x x a ' x x b ' x x c ' x x d '  e ' x  f ' x  g ' x  h ' x , 
1 1 1
1 2 1
2 1 1
2 2
1
1
1
1
2
1
1
1
2
( x1 , x2 )  ( x1 , x2 )  
  (2 x1a1 , x2 d 2 )
 x1 x1a2 ' x1 x2 b2 ' x2 x1c2 ' x2 x2 d 2 ' e2 ' x1  f 2 ' x2  g 2 ' x1  h2 ' x2 
 b1 '  c1 '  a1 , d 2 '  d 2 , a1 '  a2 '  c2 '  d '1  e1 '  e2 '  f1 '  f 2 '  g1 '  g 2 '  h1 '  h2 '  0
(7.9)
Only this solution is feasible if we assume symmetry during summation. Thus we have the
general construction
( x1 , x2 )  ( y1 , y2 )   ( x1 y2  x2 y1 )a1 , x2 y2 d 2  , ( x1 , x2 )  ( y1 , y2 )   x1 y1a1 , x2 y2 d 2 
(7.10)
However, although these numbers have some interesting properties that can be explored, we
will further set that a1 = d 2 = 1. This gives that ( x1 ,1)  ( y1 ,1)   ( x1  y1 ),1 and
( x1 ,1)  ( y1 ,1)   x1 y1 ,1 .
To construct a 4 dimensional number we use that ( x1 , x2 , x3 , x4 )   ( x1 , x2 ), ( x3 , x4 )  . Thus we
have when we use the rules in (7.1) that
On the construction of N-dimensional hypernumbers
487
( x1 , x2 , x3 , x4 )   ( x1 , x2 ), ( x3 , x4 ) 
( x1 , x2 , x3 , x4 )  ( y1 , y2 , y3 , y4 )   ( x1 , x2 ), ( x3 , x4 )    ( y1 , y2 ), ( y3 , y4 ) 
  ( x1 , x2 )  ( y1 , y2 ), ( x3 , x4 )  ( y3 , y4 ) 
  ( x1 y1 , x2 y2 ), ( x3 x4 , x4 y4 )    x1 y1 , x2 y2 , x3 y3 , x4 y4 
(7.11)
( x1 , x2 , x3 , x4 )  ( y1 , y2 , y3 , y4 )   ( x1 , x2 ), ( x3 , x4 )    ( y1 , y2 ), ( y3 , y4 ) 
  ( x1 , x2 )  ( y3 , y4 )  ( y1 , y2 )  ( x3 , x4 ), ( x3 , x4 )  ( y3 , y4 ) 
  ( x1 y3 , x2 y4 )  ( y1 x3 , y2 x4 ), ( x3 y3 , x4 y4 ) 
  ( x1 y3 y2 x4  y1 x3 x2 y4 , x2 y4 y2 x4 ), ( x3 y3 , x4 y4 ) 
  x1 y3 y2 x4  y1 x3 x2 y4 , x2 y4 y2 x4 , x3 y3 , x4 y4 
In this way we can construct all numbers of dimension N  2k , to read
N 2
 x1 , x2    y1 , y2    x1 y2  y1 x2 , x2 y2 
 x1 , x2    y1 , y2    x1 y1 , x2 y2 
N  4, ( x1 , x2 , x3 , x4 )   ( x1 , x2 ), ( x3 , x4 )    X 21 , X 2 2 
X
X
2
1
, X 2 2   Y 21 , Y 2 2    X 21  Y 2 2  Y 21  X 2 2 , X 2 2  Y 2 2 
2
1
, X 2 2    Y 21 , Y 2 2    X 21  Y 21 , X 2 2  Y 2 2 
(7.12)
N  2k  n
X
X
n
1
, X n 2   Y n1 , Y n 2    X n1  Y n 2  Y n1  X n 2 , X n 2  Y n 2 
n
1
, X n 2    Y n1 , Y n 2    X n1  Y n1 , X n 2  Y n 2 
We can check whether all lower dimensional numbers are special cases. We in particular
check whether the 3 dimensional numbers are of the type ( x1 , x2 , x3 ,1) and whether the 2
dimensional numbers are of the type ( x1 , x2 ,1,1) , to read
( x1 , x2 , x3 ,1)  ( y1 , y2 , y3 ,1)   x1 y1 , x2 y2 , x3 y3 ,1
( x1 , x2 , x3 ,1)  ( y1 , y2 , y3 ,1)   x1 y3 y2  y1 x3 x2 , x2 y2 , x3 y3 ,1
(7.13)
( x1 , x2 ,1,1)  ( y1 , y2 ,1,1)   x1 y1 , x2 y2 ,1,1
( x1 , x2 ,1,1)  ( y1 , y2 ,1,1)   x1 y2  y1 x2 , x2 y2 ,1,1
488
J.F. Moxnes and K. Hausken
Thus ( x1 , x2 , x3 ,1) and ( x1 , x2 ,1,1) are viable 3 and 2 dimensional numbers. We like to position
(1,2) on the real x axis of the Euclidean geometry. We position (1,1,2)= (1,1,2,1) as follows
(1,1, 2,1)  (1,1, 2,1)  (2  2,1, 4,1)  (4,1, 4,1)  (1,1,1,1)
(1, 2,1,1)  (1, 2,1,1)  (2  2, 4,1,1)  (4, 4,1,1)  (1,1,1,1)
(7.14)
As a mixed structure assume that the pairs in equation (7.11) are complex numbers. We then
achieve that
( x1 , x2 , x3 , x4 )   ( x1 , x2 ), ( x3 , x4 ) 
 ( x1 , x2 ), ( x3 , x4 )    ( y1 , y2 ), ( y3 , y4 )    ( x1 , x2 )  ( y1 , y2 ), ( x3 , x4 )  ( y3 , y4 ) 
  ( x1 y1  x2 y2 , x1 y2  x2 y1 ),( x3 y3  x4 y4 , x3 y4  x4 y3 )    x1 y1  x2 y2 , x1 y2  x2 y1 , x3 y3  x4 y4 , x3 y4  x4 y3 
 ( x1 , x2 ), ( x3 , x4 )    ( y1 , y2 ), ( y3 , y4 ) 
  ( x1 , x2 )  ( y3 , y4 )  ( y1 , y2 )  ( x3 , x4 ), ( x3 , x4 )  ( y3 , y4 ) 
  ( x1 y3  x2 y4 , x1 y4  x2 y3 )  ( y1 x3  y2 x4 , y1 x4  y2 x3 ), ( x3 y3  x4 y4 , x3 y4  x4 y3 ) 
  x1 y3  x2 y4  y1 x3  y2 x4 , x1 y4  x2 y3  y1 x4  y2 x3 , x3 y3  x4 y4 , x3 y4  x4 y3 
(7.15)
8 The most general construction
In this section we construct an even more general structure for N dimensional numbers, where
N  2k , k=0,1,2,.... Assume that in 2 dimensions where the elements x1 , x2 , y1 , y2  R
mod
( x1 , x2 )  ( y1 , y2 )  ( f1  x1 , x2 , y1 , y2  , f 2  x1 , x2 , y1 , y2 ), (a ),
mod
(8.1)
( x1 , x2 )  ( y1 , y2 )  ( g1  x1 , x2 , y1 , y2  , g 2  x1 , x2 , y1 , y2 ), (b),
where f1 , f 2 , g1 g 2 are arbitrary functions.
Assume that we desire that ( x1 , x2 )  ( y1 , y2 ) = ( y1 , y2 )  ( x1 , x2 ) and that ( x1 , x2 )  ( y1 , y2 ) =
( y1 , y2 )  ( x1 , x2 ) . This gives that
489
On the construction of N-dimensional hypernumbers
( x1 , x2 )  ( y1 , y2 )  ( f1  x1 , x2 , y1 , y2  , f 2  x1 , x2 , y1 , y2 )  ( f1  y1 , y2 , x1 , x2  , f 2  y1 , y2 , x1 , x2 ),
 f1  x1 , x2 , y1 , y2   f1  y1 , y2 , x1 , x2  , f 2  x1 , x2 , y1 , y2   f 2  y1 , y2 , x1 , x2 
(8.2)
( x1 , x2 )  ( y1 , y2 )  ( g1  x1 , x2 , y1 , y2  , g 2  x1 , x2 , y1 , y2 )  ( g1  y1 , y2 , x1 , x2  , g 2  y1 , y2 , x1 , x2 )
 g1  x1 , x2 , y1 , y2   g1  y1 , y2 , x1 , x2  , g 2  x1 , x2 , y1 , y2   g 2  y1 , y2 , x1 , x2  ,
However, a very strong condition can be applied for what we indeed will call numbers. This
means that for some number b
D1: ( x1 , b)  ( y1 , b)  ( x1  y1 , b),
D 2 : ( x1 , b)  ( y1 , b)  ( x1 y1 , b)
(8.3)
D3: ( x1 , x2 )  ( x1 , x2 )  (2, b)  (ax1 , ax2 ), a  R
This gives that
D1: ( x1 , b)  ( y1 , b)  ( f1  x1 , b, y1 , b  , f 2  x1 , b, y1 , b )   x1  y , b 
D 2 : ( x1 , b)  ( y1 , b)  ( g1  x1 , b, y1 , b  , g 2  x1 , b, y1 , b )   x1 y1 , b 
D3 :  f1  x1 , x2 , x1 , x2  , f 2  x1 , x2 , x1 , x2    (2, b)  (ax1 , ax2 )  ( g1  2, b, ax1 , ax2  , g 2  2, b, ax1 , ax2 )
(8.4)
For N= 2^k and N=2k+1 dimensions we can construct these by using pairs analogous to the
construction in the earlier sections.
9 Conclusion
Complex numbers extend the concept of the 1 dimensional numbers to 2 dimensions.
Quaternions extend numbers to 4 dimensions. Octonions and sedenions are extensions to 8
and 16 dimensions respectively. We study a general form of complex numbers, various
axiomatic structures of 3 dimensional numbers, and finally N dimensional numbers, N  2k ,
k=0,1,2,.... Quaternions, octonions and sedenions are special cases. Two different structures
for addition are studied.
490
J.F. Moxnes and K. Hausken
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