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For example:

1. (a+bi+cj+dk) = ((a+bi)+(c+di)j)

where: i<>j, i^2=j^2=-1, ij=ji=k, ik=ki=-j, jk=kj=-i, k^2=+1.

Unlike quaternions, these hypercomplex numbers are:

commutative and associative wrt addition and multiplication, distributive with addition, all multiplicative inverses exist except zero and zero divisors.

And, the elementary functions, e.g. e^(a+bi+cj+dk) are available.

2. (a+b(i2)+c(i3)+d(i4)+e(i5)+f(i6)+g(i7)+h(i8)) =

(a+b(i2)+c(i3)+d(i4)) + (e+f(i2)+g(i3)+h(i4))(i5).

where: i2<>i3, i2<>i5, i3<>i5, (i2)(i3)=(i4), (i2)(i5) =(i6), (i3)(i5)=(i7), (i4)(i5)=(i8).

All other product combinations are easily found granting commutativity and associativity. e.g. (i6)(i4)= (i2)(i5)(i2)(i3)= -(i3)(i5)=-(i7).

Unlike octonions, these hypercomplex numbers are:

commutative and associative wrt addition and multiplication, distributive with addition, all multiplicative inverses exist except zero and zero divisors.

And, the elementary functions,

e.g. e^(a+b(i2)+c(i3)+d(i4)+e(i5)+f(i6)+g(i7)+h(i8)) are available.

Quaternions and Octonions, etc., are produced within these hypercomplex numbers via special product functions.

Complex numbers of any dimention can be constructed in this way.

Whatdoyouthink?