N-DIMENSIONAL COMPLEX NUMBERS.
An acquaintance with the established two dimensional complex number system naturally gives rise to the question as to whether this could be extended to a complex number system in a higher number of dimensions, creating complex numbers of the form
a + i b + j c + k d ... + nn
in n + 1 dimensions.
The usual answer to this question indicates that the only mathematically consistent first step in this direction requires the use of quaternions, which are represented as a kind of 4-dimensional complex number system. This answer indicates that a straightforward intermediate step to 3-dimensional complex numbers is a mathematical impossibility.
This notion of mathematical impossibility is supported by the Hurwitz so-called 1,2,4,8 theorem, which proved that numbers associated with a numbers algebra, or 'normed division algebra' cannot occur in any other than in 1, 2, 4, or 8 dimensions.
It follows from this that a straightforward attempt to develop a 3- or higher-dimensional complex number system, of dimensionality other than those mentioned, cannot succeed.
Before considering the method I shall describe here for the creation of 3- and other-dimensional complex numbers in a manner that does not violate the Hurwitz limitation (which therefore requires other than a simple, single algebra), I shall review the difficulties involved in making such an attempt.
INTRODUCTION:
We shall first consider the apparently insoluble problems associated with 3-dimensional complex numbers, as a first step to higher-dimensional forms, being represented, for example, by a number of the form
x + iy + jz
With regard to a 3-dimensional form like x+iy+jz, should we not expect to say that, just as the operator i rotates y from the x to the i direction, the operator j similarly rotates z from the x to the j direction, orthogonal to both the x and i directions? Here, the complex number has three independent, or orthogonal components, represented by three orthogonal directions in a 3-dimensional complex number space. This is the straightforward extrapolation from 2 to 3 dimensions, assuming that the operator j operates orthogonally to i, but is in all other respects identical to i. Then, just as i*i = -1, we have j*j = -1.
A problem immediately arises, however. In the 3-dimensional complex number space, we have three independent planes of rotation of vectors: the x,i plane, the x,j plane, and the i,j plane. What, we may then ask, is to be said of the relationship of the operators i and j in respect of rotations in the i,j plane?
For rotations in this plane to be consistent with those in the other two planes, we should have something like:
jiy = jy, and jjy = jjiy = -y, as describing a rotation of y from the i to the j direction, and then from the j direction to the negative real axis. This, however, is clearly not consistent with the behavioural description of the operators as simply 
. How can we say ji = j? How does the operator i disappear, in
a manner consistent with the operator descriptions?
We might expect ji = -1, which would rotate y in the -x direction, instead of the j direction, which is unsatisfactory. Also, jjy is, according to the operator j, is just -y, whereas jjiy might also be -y, instead of -iy, if ji = j, which does not make sense.
One way out might be to adopt the strange result of forbidding ji to be resolved into anything simpler while, at the same time, requiring the interpretation that ji is in the same direction as j. But then we have the problem about the possible relationship between ji and ij. Also, problems will arise due to the fact that jiji = +1. Another possible resolution is to specify that j operates only on the parameter of i, and not on the operator i itself. Thus, no operations such as ji would be allowed to exist.
There is a further difficulty, however, in that the modulus of numbers must not be altered by operators themselves, which can be a problem when there are two operators i, and j. If an operation on a complex number x + iy by the operator j changes the modulus, it creates an arithmetical effect, which is contrary to the definition of j as a non-arithmetical operator similar to the operator i. This would happen if you had
j(x + iy) = j(x + y)
Then
jj(x + y) = -(x + y)
When i rotates x + iy to -y + ix, the modulus
|x + iy| = |-y + ix|
is unchanged, and the same applies to j(x + jz).
PROBLEMS WITH THE CREATION OF 3-DIMENSIONAL COMPLEX NUMBERS:
We consider the problem in which jiy represents a rotation of y from the i to the j direction, and where the parameter of ji points in the j direction. If we apply this to the two-dimensional complex number x + iy, we have
j(x + iy) = jx + jiy
Both x and y now point in the j direction, so that we have to solve the problem that the modulus of this number can only be something like x+y, which is not satisfactory.
What we require is to preserve the modulus of the original number, x + iy, unchanged. One obvious way to do this would be to simply rotate the number as a whole onto the j axis. In the light of this, it can be seen that many of the former difficulties really rest on, and result from, an unexamined assumption that j must operate on the x and i axes separately. Can we not, therefore, simply discard that assumption?
It can still be possible for j to be seen as an extrapolation of the i rotation operator, defined in the same way as i*i = j*j = -1. Only now we will have the case that j is an operator that rotates from the x,i complex number plane to the j axis, which is to say that it treats any resultant vector in the x,i complex number plane as if it were an axis, and is not restricted to acting on the x or i axes only, and certainly does not operate on them independently at the same time. In this way the operator j becomes associated with a phase, defined by the angle a 2-d number vector makes with the x axis.
With the definition of the operator j as operating on the x,i complex number plane as a whole, we now find ourselves in an entirely new environment, which, however, still does not eliminate all difficulties.
The first difficulty is that, if we consider a number x+iy, rotated onto the j axis as j(x + iy), can we not say that the result is simply ja, where the absolute value, or modulus of x + iy is
a = |x + iy|?
This, of course, does represent a true picture of the result, with the exception that ja can also represent a rotation of any other number, a + ib, in the x,i plane, so that it is equally true that
ja = j(x + iy) = j(c + id)
but it can also be true that x 
c, and y d, which means that we cannot unambiguously define what jja might mean
Clearly, for the j operator to be fully defined, the vector ja would have to contain some kind of record of the original vector from which it was rotated. That is, the j operator must be associated with a phase in the x,i complex plane, such that it will specify the plane of rotation of the parameter associated with j. A scalar parameter along the j axis obviously does not contain any indication of such a phase.
The easiest way out of this might be to specify that the form ja is not to be permitted, where a is a scalar, and can be permitted only where a has the form x + iy, i.e., the form of a 2-dimensional complex number, except for the fact that the j axis is one-dimensional, and cannot sensibly have a 2-dimensional parameter.
One way to deal with the fact that the j axis is one-dimensional might be to create a second x,i complex plane infinitesimally above, and separate from, the normal x,i complex plane, which is assigned to the j axis such that it contains the j parameter as a complex number, while the modulus appears on the j axis itself.
While this alternative might at first sight seem promising, it does not support a viable algebra. It cannot provide a complex conjugate which, when multiplied by the original number, gives the modulus squared of the number. Consequently, it also cannot provide an inverse of a 3-d number.
The situation can be improved, however, by restricting such numbers to a subset defined by those in which the plane of rotation of the j operator is restricted to that defined by the 2-dimensional component vector x + iy and the j axis. The resulting numbers can provide a complex conjugate, which can be used to calculate the modulus of the number, and consequently can also be used to create an inverse.
The two alternative possibilities mentioned here are shown in the illustrations below.
But multiplication in such a number system is not associative. That is
(AB)C A(BC)
AN ATTEMPTED SOLUTION:
The reason for describing the failed solutions above is by way of preparation for a description of the following solution, which will make use of, and extend, the idea of associating a phase with the operation of the j operator.
In the following solution, the parameter of the j operator, such as z, in jz, is a scalar, and no longer a complex number. This is because it will not be necessary for the parameter z itself to specify the plane of rotation of j, which was the original purpose of making z complex. The following paragraph describes the rules for the operation of the j operator, which have a degree of complexity due to the fact that the operator can have different phases in the x,i complex plane, and thus different planes of rotation.
The definition of the phase of a 2-d or 3-d number or its operator j:
The phase of a 2-d number is not defined. The phase of its vector, also the phase of a 3-d number, or the operator j, is the angle a 2-d number vector in the i,x complex plane makes with the x axis. If changing a 2-d number, by multiplication etc., rotates its vector in the x,i complex plane, the new phase also becomes the new phase of the 3-d number of which it is a part, and of its j operator.
The phase of a j operator, or 3-d number, or the vector of its 2-d part (which, for convenience, I may call its i-number), can, if necessary, be written explicitly as a subscript.
Rules for the j Operator
The parameter of the j operator is now to be always a scalar, but the j operator still has a phase in the x,i 2-d complex plane, and rotates a complex number as a whole, as its modulus, onto the j axis, so that we may have, for example, (explicitly writing in the phases of the 2-d vector and j operator)
j1(x + iy)X = 0 + jX|x + iy|
where
X = x + iy
and
jjX|x + iy| = -(x + iy)
The operator, here, acquires the phase of the 2-d vector it rotates, rather than the overall 3-d number result. This is because the zero in a resultant number like 0 + jz is regarded as having an ambiguous phase or, equivalently, no phase. Where necessary, the default phase of a j operator is zero (the x axis) and is written as j1, as in the above example.
If, however, we have
A = a + ib
and
(A + jA0) + (0 + jX|x + iy|) = A + jA|x + iy|
the phase of j acquires that of the 2-d part of the 3-d number (the i-number vector). This is because the addition rotates the whole 3-d number in the x,i complex plane, and the operator j always has the same phase as that of its i-number vector, and 3-d number as a whole, except in a case where the i-number is zero
The foregoing has an immediate implication for the distributive property of multiplication. That is, the number x + iy is also the addition of 2 numbers
(x + i0) + (0 + iy)
but j(x + iy) does not mean
jx + jiy = j(|x| + |iy|) = j(x + y)
This inevitably means that consistency must require that, where we have two 2-d numbers X + Y, then
j(X + Y) = j|X + Y|
and not
j|X| + j|Y| = j(|X| + |Y|)
But note that here j is a single operator with a single phase.
If, however, we have two numbers j|X| and j|Y|, with the j's having different phases, such as in the multiplication of two 3-d numbers, then we will have, for example
ja|X|X + jb|Y|Y = jX+Y(|X| + |Y|)
but
ja|X|X + ja|Y|Y = jX+Y(|X + Y|)
The reason for this distinction is that, if we have two j-numbers which are multiplied together, the j operator of each cross-multiplies to the 2-d part of the other number. This must not at all include, or imply, any multiplication effect on the 2-d part of its own number. Indeed, it must positively exclude it by the existence of two distinct multiplications.
Note that this principle might, at first sight, appear to be contradicted if a 3-d number is squared. In that case, however, the two different forms are equal: |X|+|X|=|X+X|. So the contradiction is apparent rather than real.
Another difficulty with phases, that arises immediately, is if we multiply a 3-d number by j1, or simply j, with the 1 being understood. If, for example, we have
j(X + jXz) = -zX + jX|X|
the scalar z must be given a phase, the phase of X, and become a complex number after multiplication. The way to convert any 2-d complex number into the equivalent of a phase operator, which rotates a number but does not alter its magnitude, is to divide the number by its modulus, so that it becomes a complex number with a modulus of 1. If we multiply the scalar z by X/|X|, which has a modulus of 1, we will give z the phase of X (its vector), without changing the magnitude of z, to give the result
j(X + jXz) = -z(X/|X|) + jX|X|
in which the phase of the j operator, and its plane of rotation, and the phase of the number as a whole, do not change. The modulus of the number, as a whole, also remains unchanged. This is consistent with the property of the j operator, which simply rotates a 2-d number to its negative by the operation jj.
Referring to the 'phase' of a j operator, or 3-d number, is a way of identifying the plane of rotation of the j operator. A negative i-number can thus be used to identify the same positive phase as its positive counterpart.
Therefore we have the application of an operator phase, to the scalar that operator j has rotated back into the x,i complex plane, defined by the equation.
-zX = -z(X/|X|)
Finally, in certain circumstances only, we will allow the j operator to be given an arbitrary phase.
An illustration of the i- and j-numbers is given below
A REVISED SOLUTION:
In creating a solution, it is necessary to find a way past the restriction imposed by the Hurwitz theorem, which makes a straightforward attempt at a n-dimensional number algebra a futile exercise. We therefore do not attempt to create such a single algebra.
While addition of numbers is straightforward, multiplication, in particular, will not work well. This is because multiplication of complex numbers includes 'multiplication' by operators. So a different concept of multiplication has to be created.
To deal with multiplication, we first split the 3-d number into two 2-d numbers, the result being that we will now have two separate 2-d algebras, each of which is consistent with the Hurwitz limitation. Having thus obtained the two separate 2-d results, we then use an algorithm to unite these two results to create a single result, which will be taken as the overall 3-d number result.
Firstly, we have a number
x + iy + jz
Setting X = x + iy, we now create two 2-d numbers
|X| + jz
and
X
Here is where we may see the usefulness of making the plane of rotation of the j operator arbitrary. The direction of |X|, a scalar, in the x,i complex plane is now made to be arbitrary, or is not specified, so that
|X| plays the role of an x axis, but without being specified to be assigned to the x axis of the x,i complex plane. Therefore |X| + jz is a kind of freely floating 2-d complex number.
The advantage of this is that, if we have two 3-d numbers,
x + iy + jz, and a + ib + jc
which can be written
X + jz and A + jc
the two numbers |X| + jz and |A| + jc, which both have arbitrary phase in the x,i complex plane, can be rotated into the same plane, so that |X| and |A| can be taken to be in the same direction. This allows any normal complex 2-d algebra to be carried out between the two numbers |X| + jz and |A| + jc, thus creating a result assigned to a single complex plane.
After carrying out a separate 2-d multiplication on the two i-numbers, X and A (by contrast, a number of the form |X| + jz can be conveniently referred to as a j-number). An algorithm is then used to combine the two 2-d results, to create an overall, unambiguous 3-d number result.
Commutative Property
It follows immediately that multiplication will be commutative. The two 2-d component algebras are both commutative, and the algorithmic step, uniting their results, takes place as a final step, and does not alter either result.
MULTIPLICATION:
In this example we will multiply two 3-d numbers
(x + iy + jz)(a + ib + jc)
also written as
(X + jXz)(A + jAc)
where X and A have the same meaning as before. We begin with the two 2-d component j-number multiplications, with arbitrary phase
(|X| + j?z)(|A| + j?c) = (|X||A| - zc) + j?(|A|z + |X|c)
and the two i-number multiplication
XA = (a + ib)(x + iy) = (ax - by) + i(ay + bx)
Note that the parameter of j is |A|z + |X|c, and not |Az + Xc|, because, according to the rule, A and C are multiplied by two j operators with different original planes of rotation, although these have been given arbitrary phases when converted to j numbers. The rule for the use of j? is not identical to the normal, phased j operator, and depends on the context in which it was originally made to be arbitrary.
Now, in the first result, |X||A| = |XA|, for 2-d numbers, so we rewrite the first result as
|XA|(1 - zc/|XA|) + j(|A|z + |X|c)
We now create an algorithmic unification of the two results by replacing |XA|, outside the bracket only, by the complex result, XA, of the i-number multiplication, to give the final result
XA(1 - zc/|XA|) + jxa(|A|z + |X|c)
which, written out fully, is
(1 - zc/|XA|){(ax - by) + i(ay + bx)} + j(|A|z + |X|c)
Note here that the zc term, which was contributed by a jj operation, is multiplied by XA/|XA|, which is a phase setting complex number divided by its modulus, as mentioned previously. This imposes a phase on the component zc and makes it to have the same phase as the 2-d complex component XA vector. zc is a scalar, and any scalar (which would otherwise be associated with the x axis) that results from the operator multiplication jj must be given the phase of the overall 3-d number result, which is the phase of the 2-d component vector, and also the phase acquired by the 3-d number j operator.
In some cases, the phase to be assigned to the j operator is not obviously indicated by the result in this way. For example, if we multiply the above result by the operator j alone, which is the number (0 + i0 + j1), the i-number part of the result will be
[XA(1 - zc/|XA|)](0 + i0) = 0
and the j-number part of the result will be
- (|A|z + |X|c) + j(|XA| - zc)
Here the i-number part acquires a result of zero, which cannot be combined with the first part to produce an overall phase for the 3-d number. To overcome this, there must be a phase explicitly assigned to the j operator, and the number as a whole.
To do this we may note that we must conclude that the overall number phase has not been altered by the multiplication by j, which only rotates the number within the existing plane of rotation. Since, therefore, the original phase of the number was XA, this must also be the phase of the final result. This will give the result
- (|A|z + |X|c)(XA/|XA|) + jXA(|XA| - zc)
Associative Property
That multiplication is associative follows from the fact that 2-d number multiplication is associative. Both the j-number and i-number parts of the multiplication are associative. The algorithmic creation of the overall result does not alter this property, since it is used only to create the final step in the calculation. If, for example, we have the j-number multiplication
(|D| + jf)(|X| + jz)(|A| + jc)
written as
D'X'A'
to represent the 3-d numbers. We now multiply
D'(X'A')
and begin with (X'A'), to give the j-number result
|XA| - zc + j(|A|z + |X|c)
in which the |XA| is not taken outside a bracket. Now, multiplying by D', we have
|D||XA| - |D|zc - f|A|z - f|X|c+j(|D||A|z+|D||X|c+f|XA| - fzc)
= |DXA| - |D|zc - f|A|z - f|X|c+j(|DA|z+|DX|c+f|XA| - fzc)
We now multiply
(D'X')A'
and begin with (D'X'), to give the j-number result
|DX| - fz + j(|X|f + |D|z)
Now, multiplying by A', we have
|A||DX| - |A|fz - c|X|f - c|D|z+j(|A||X|f+|A||D|z+c|DX| - cfz)
= |ADX| - |A|fz - c|X|f - c|D|z+j(|AX|f+|AD|z+c|DX| - cfz)
The resultant j-numbers for the two multiplications are the same, which demonstrates that, for the j-number part
D'(X'A') = (D'X')A'
It is also true, for the corresponding 2-d complex i-numbers, that
D(XA) = (DX)A
We now convert the j-number result
|ADX|-|A|fz-c|X|f-c|D|z+ j(|AX|f+|AD|z+c|DX|-cfz)
= |ADX|(1-(|A|fz-c|X|f-c|D|z)/|ADX|)+j(|AX|f+|AD|z+ c|DX|-cfz)
with the algorithm that replaces |ADX| with the complex i-number result ADX to give, finally
ADX(1-(|A|fz-c|X|f-c|D|z)/|ADX|)+j(|AX|f+|AD|z+ c|DX|-cfz)
Note that the jj rotated parameters all are given the same final phase of the overall number, by being multiplied by the phase-setting 2-d number ADX/|ADX|
Distributive Property
We have the following multiplication, which is in the form D'X' + D'A'
(d + ie + jf){(x + iy + jz) + (a + ib + jc)}
= (D + jf){(X + jz) + (A + jc)}
Converting to j numbers, with 2-d moduli
(|D| + jf){(|X| + jz) + (|A| + jc)}
= (|DX| - zf) + j(f|X| + |D|z) + (|DA| - fc) + j(f|A| + |D|c)
= |D|(|X| + |A|) - f(z + c) + j{f|X + A| + |D|(z + c)}
Note that jf|X| + jf|A| has been converted to the result jf|X + A|, in accordance with the rule that a single instance of the operator j, operating on two 2-d complex numbers added together, or their j-number moduli, acts on the sum, and not on each separately. It doesn't matter whether or not they were initially converted into j numbers as |X| + |A| or as |X + A|. This rule does not apply to existing j parameters, or those resulting from a double application of the j operator (jj)
Now, calculating according to the alternative form, D'(X' + A'), we have
(|D| + jf){(|X + A|) + j(z + c)}
= |D|(|X + A|) - f(z + c) + j{f|X + A| + |D|(z + c)}
MULTIPLICATION EXAMPLE:
We will add the numbers
(x+iy+jz) + (a+ib+jc) = (X+jz) + (A+jc)
We shall have the two 2-d component additions
(|X|+jz) + (|A|+jc)
= (|X|+|A|) + j(z+c)
and
x + iy + a + ib = (x + a) + i(y + b) = X + A
The algorithm used is to simply replace, from the first result, |X| + |A|, with complex value, X + A, from the second result, which will provide the resultant 3-d number
(x + a) + i(y + b) + j(z + c)
Here, the phase of rotation of the operator j, which is arbitrary, is now taken to be that defined by the 2-d number
(x + a) + i(y + b)
Now, clearly, this result could have been obtained immediately, without splitting the 3-d number at all. However, it is relevant to do it this way, here, by way of illustration and comparison with the method used here for other complex operations.
ADDITION EXAMPLE:
The complex conjugate of a 3-D complex number
is
x - iy - jz = X* - jz
With X = x + iy, the modulus of the 2-d component is |X|, with X* and |X*| representing the complex conjugate. Multiplying the 3-d number by its complex conjugate will give
(|X| + jz)(|X*| - jz) = |XX*|2 + z2
Now, according to the multiplication algorithm, |X||X*| = |XX*|, is replaced by the 2-d complex multiplication XX*, to give
XX* + z2 = x2 + y2 + z2
which is the modulus squared of the 3-d number
THE 3-D COMPLEX CONJUGATE:
The identity element for 2-d complex numbers is simply 1, as for ordinary integers or, if preferred, 1 + i0. This is because, in the case of multiplication of a complex number by the identity element, both parameters of the complex number are multiplied, by definition, by the parameter 1 in 1 + i0, which leaves the number unchanged. The same applies to 3-d complex numbers, so that the identity element is 1 + i0 + j0.
From the identity element we can specify what the inverse must be, in that we must have
A/A = I
where 1/A is the inverse of A, and I is the identity element.
In the case of 2-d complex numbers, we may note that, where A* is the complex conjugate of A:
AA* = |A|2 = |A*|2
so that
A(A*/|A*|2) = I
and therefore
A*/|A*|2
can be taken as the inverse of A
For 3-d complex numbers, the same argument
applies, without alteration, because, as has been shown above, for 3-d complex numbers, it is also true that
A'A'* = |A'|2 = |A'*|2
and therefore the inverse of a 3-d complex number, A', is also
A'*/|A'*|2
Thus, since the complex conjugate of the 3-d complex number
A' = A + jc
is
A'* = A* - jc
the inverse is
1/(A + jc) = (A* - jc)/(|A* - jc|2) = A'*/|A'*|2
In general, then, we can define, for complex numbers of 2 or 3 dimensions, the same form
1/A = A*/|A*|2
and
1/A* = A/|A|2
THE INVERSE AND THE IDENTITY:
Division, which is multiplication by the inverse of a number, is just another form of multiplication, and therefore has the same properties as multiplication.
Thus, where A' and B' are 3-d numbers, the division
of A' by B' is
A'/B' = A'(B'*/|B'*|2)
DIVISION:
If X' and A' are 3-d numbers
1) |X'A'| = |X'||A'|
2) (X' + A')* = (X'* + A'*)
3) (X'A')* = X'*A'*
4) X'*A' = (X'A'*)*
1) Firstly, it is well known that, where X and A are 2-d complex numbers
|XA| = |X||A|
If, now
X' = X + jz
A' = A + jc
are 3-d numbers, with X and A the same 2-d numbers as before, we have, using j-numbers (we do not need to use i-numbers for this calculation because they will not change anything in the result)
(|X'||A'|)2 = (|X|2 + z2)(|A|2 + c2)
= (|X||A|)2 + (zc)2 + (|X|c)2 + (|A|z)2
and
X'A' = (|X||A|) - zc + j(|X|c + |A|z)
|X'A'|2 = {(|X||A|) - zc}2 +{(|X|c + |A|z)}2
= (|X||A|)2 + (zc)2 + (|A|z)2 + (|X|c)2
+ 2(|X||A|)zc - 2(|X||A|)zc
= (|X||A|)2 + (zc)2 + (|X|c)2 + (|A|z)2
= (|X'||A'|)2
so
|X'A'| = |X'||A'|
In what follows, keep in mind that the moduli of 2-d numbers are equal to the moduli of their complex conjugates.
2) Next, to show that, where X and A are 2-d complex numbers
(X + A)* = (X* + A*)
where
X = x + iy and A = a + ib
we have
(X + A) = (x + a) + i(y + b)
(X + A)* = (x + a) - i(y + b)
= (x - iy) + (a - ib)
= (X* + A*)
If X' and A' are 3-d complex numbers, and
X' = x + iy + jz, A' = a + ib + jc
X = x + jy and A = a + jb
we have
(X' + A') = (X + A) + j(z + c)
(X' + A')* = (X + A)* - j(z + c)
= (X* + A*) - j(z + c)
from the 2-d example above. Now
(X'* + A'*) = (X* - jz) + (A* - jc)
= (X* + A*) - j(z + c)
=(X' + A')*
3) Next, to show that (XA)* = X*A*
for 2-d complex numbers we have
XA = (x + iy)(a + ib)
= (xa - yb) + i(ya + bx)
(XA)* = (xa - yb) - i(ya + bx)
and
X*A* = (x - iy)(a - ib)
= (xa - yb) - i(ya + bx) = (XA)*
For 3-d complex numbers we have, using j-numbers
X'A' = (|X| + jz)(|A| + jc)
= (|X||A| - zc) + j(|X|c + |A|z)
= XA(1 - zc/|XA|) + j(|X|c + |A|z)
after including the i-number, and
(X'A')* = (XA)*(1 - zc/|XA|) - j(|X|c + |A|z)
Now
X'*A'* = (|X*| - jz)(|A*| - jc)
= (|X*||A*| - zc) - j(|X*|c + |A*|z)
= (XA)*(1 - zc/|XA|) - j(|X|c + |A|z) = (X'A')*
4) Finally, to show that
A'*X' = (A'X'*)*
for 2-d complex numbers
A*X = (a - ib)(x + iy) = (ax + by) - i(bx - ay)
AX* = (a + ib)(x - iy) = (ax + by) - i(ay - bx)
(AX*)* = (ax + by) + i(ay - bx)
= (ax + by) - i(bx - ay) = A*X
For 3-d complex numbers
A'*X' = (|A*| - jc)(|X| + jz)
= (|A*||X| + zc) - j(|X|c - |A*|z)
= A*X(1 + zc/|AX|) - j(|X|c - |A*|z)
Then we have
(A'X'*)= AX*(1 + zc/|AX|) - j(|A|z - |X*|c)
(A'X'*)*= (AX*)*(1 + zc/|AX|) - j(|X*|c - |A|z)
= A'*X'
SOME EXTRA FORMULAE:
The 3-dimensional number used above can serve as a model for higher-dimensional numbers. If we consider 4 dimensional numbers for example (NOT quaternions), having the form a + ib + jc + kd, we can write it in the following forms:
A'' = A' + kd
A' = A + jc
A = a + ib
Now, considering just the case of multiplication, which is the most problematic, we again divide the number into two numbers, a 2-d number, which I will call a k-number, and a 3-d number. The k operator rotates the modulus of a whole 3-d number from the 3-d complex space onto the fourth dimension k axis. As with other operators, kk = -1. The 2-d and 3-d numbers are
|A'| + kd
which is the k-number, and
A'
which is the 3-d number. Two k-numbers can be multiplied, as before, with the k operators being given arbitrary 3-d phases in the 3-d complex space. The multiplication
(|A'| + kd)(|X'| + kw)
gives
|A'||X'| - dw + k(|A'|w + |X'|d)
= |A'X'|(1 - dw/|A'X'|) + k(|A'|w + |X'|d)
since we have found that, for 3-d numbers, |A'||X'| = |A'X'|. Once again, the |A'X'| term, outside the bracket, will be replaced by the 3-d complex number result A'X', to give
A'X'(1 - dw/|A'X'|) + k(|A'|w + |X'|d)
Note that the parameter dw is given the phase of the vector representation of the 3-d vector in 3-d complex space, by being multiplied by A'X'/|A'X'|, which is a 3-d complex number with modulus 1. This is also the overall phase of the 4-d number.
A phase in 3-d complex space, of course, requires two angles to specify it: the angle the i-number makes with the x axis, and the angle the j-number makes with the x,i complex plane. However, these do not need to be specified independently, since they are both automatically specified by a 3-d complex number with modulus 1.
Now, the 3-d result A'X' will have been calculated in the manner already shown for 3-d numbers, which involves the use of two 2-d algebras. Thus, a 4-d number uses altogether three 2-d algebras for a
calculation.
Note also that, while the dw parameter has the phase of the 4-d number as a whole, the zc parameter, which will be calculated as part of the 3-d number, will still have the phase of the 3-d number as a whole, and not that of the 4-d number.
It can be shown, in the same way as for 3-d numbers, that, for 4-d numbers, it is also true that
|A''||X''| = |A''X''|
which is an important equation to enable moving to higher dimensions.
The same method used to move from 3-d numbers to 4-d numbers can now be used to move from 4-d numbers to 5-d numbers, and so on, to an n-dimensional number in n-dimensional complex space. An n-dimensional number can be written in the form
An-1 + inc
where An-1 is an n-1-dimensional number, and the subscripts refer to numbers of dimensions, and not phases. Multiplying together two n-dimensional numbers
(An-1 + inc)(Xn-1 + inz)
would give the result
An-1Xn-1(1 - zc/|An-1Xn-1|) + in(|An-1|z + |Xn-1|c)
The zc parameter now has the phase of the vector of the n-1-dimensional number in n-1-dimensional complex space, by being multiplied by an n-1-dimensional complex number of modulus 1, as before.
The n-1-dimensional complex number An-1Xn-1 is calculated by dividing it into a 2-dimensional number and an n-2-dimensional number. This process will be continued until we reach the division of the 3-dimensional number component into a j-number and an i-number, to complete the calculation.
The reason why it is possible to achieve the same results for any number of dimensions is that all numbers are divided up into 2-d algebras. A 3-dimensional number has two 2-d algebras; a 4-dimensional number has three 2-d algebras; and so on up to n dimensions, in which an n-dimensional number will have n-1 2-d algebras.
I suppose, with n-dimensional numbers, there may be a conflict between subscripts indicating phases and numbers of dimensions, which might need some kind of modification to the notation.
HIGHER DIMENSIONS:
© Alen, March 2007; update Dec 2010.
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alen.1@bigpond.com
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