Extending the number system from complex numbers, (a+bi), to 4-D(adsbygoogle = window.adsbygoogle || []).push({});

hypercomplex numbers, (a+bi+cj+dk), leads to a multiplication

table such as:

(A) i^2=j^2=-1, ij=ji=k, k^2=+1, ik=ki=-j, jk=kj=-i.

Note that these hypercomplex numbers are commutative and have elementary functions.

We can extend this idea to hypercomplex numbers to any dimension.

Sir W. Hamilton introduced 'quaternions' by presenting the

multiplication table;

(B) i^2=j^2=-1, ij=k, ji=-k, k^2=-1, ik=-j, ki=j, jk=i, kj=-i.

Clearly list (A) is incompatable to list (B).

Is k^2=-1 or is k^2=+1, it cannot be both. k cannot be the

same entity in both cases. I believe Hamilton's algebra

would be consistent with hypercomplex numbers if he had

introduced a Hamilton (H) product such that;

iHi=jHj=-1, iHj=k, jHi=-k, kHk=-1, iHk=-j, kHi=j, jHk=i, kHj=-i

where i,j,k are the same hypercomplex numbers as in (A).

It was misleading and incorrect for Hamilton to consider that

quaternions are entities at all. There are no such things as

quaternions. There is a Hamilton algebra which deals with

the concepts that Hamilton wanted to deal with but they are using

hypercomplex numbers in the context of the Hamilton product (H).

In the 8-D case, (a1+a2i2+a3i3+a4i4+a5i5+a6i6+a7i7+a8i8)

multiplication leads to the entries;

(C) (i2)^2=(i3)^2=(i5)^2=-1, (i2)(i3)=i4, (i2)(i5)=i6, (i3)(i5)=i7,

(i4)(i5)=i8, (i4)^2=+1, (i6)^2=+1, (i7)^2=+1, (i8)^2=-1.

Sir A.Cayley introduced 'octonions' by presenting a multiplication

list containing;

(D) (i2)^2=(i3)^2=(i4)^2=(i5)^2=(i6)^2=(i7)^2=(i8)^2=-1.

Again (C) and (D) are incompatible. (i6)^2=+1 from list (C),

contradicts (i6)^2=-1 from list (D). Cayley makes the same

mistake for 'octonions' that Hamilton made for 'quaternions'

There are no such things as octonions. There is a Cayley algebra,

with a Cayley product (Ca), dealing with 8-D hypercomplex numbers

which expresses what Cayley means.

(i2)Ca(i2)=(i3)Ca(i3)=(i4)Ca(i4)=(i5)Ca(i5)=(i6)Ca(i6)=

(i7)Ca(i7)=(i8)Ca(i8)=-1.

Any opinions?

Owen

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# Quaternions and hypercomplex numbers are incompatible

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