Creating exact replicas of the complex number system in n dimensions

[radn]

Exact replicas of complex number system in n dimensions using basic algebra

Not knowing hypercomplex numbers, and using only basic algebra, I found a method of creating n-dimensional spaces exactly analogous to the complex numbers’ 2D space. These number systems exhibit the feature of multiple imaginary units, multiplication by which causes a visit to each of the axes in turn, once on the plus side and once on the minus. This is exactly analogous to the behavior of i in the complex number system.

The method is described here:

http://rebirthofreason.com/Forum/GeneralForum/0935.shtml

R. Rawlings

Edited August 13, 2006 by radn

[Eric]

Unfortunately, this doesn't really give an "exact replica" of the complex number system in higher dimensions.

The problem is division: many of your "numbers" do not have multiplicative inverses. (For example, in 3D, 1 + j*k^2 has no inverse.)

Also, Hamilton originally set out to find a 3-dimensional space that contained the complex numbers and that had many of the same features as the complexes. Your "numbers" do not contain the complexes, so you did not solve Hamilton's problem. In fact, it is easy to show that there can be no solution.

- Eric

[Starblade Enkai]

Exact replicas of complex number system in n dimensions using basic algebra

Not knowing hypercomplex numbers, and using only basic algebra, I found a method of creating n-dimensional spaces exactly analogous to the complex numbers’ 2D space. These number systems exhibit the feature of multiple imaginary units, multiplication by which causes a visit to each of the axes in turn, once on the plus side and once on the minus. This is exactly analogous to the behavior of i in the complex number system.

The method is described here:

http://rebirthofreason.com/Forum/GeneralForum/0935.shtml

R. Rawlings

After seeing you mention this, I decided to try something akin to what you are doing. I called the third dimensional numbers "Legendary numbers". Unfortunately, I ran into some problems with the concept of L.

Suppose that we adopt the normal rules, such that L * 0 = 0, L * 1 = L, and L * -1 = -L. Suppose also that we retain the typical algebraic rules. What is L * i? Let's assume that L multiplied by i produces three unique numbers in the three dimensional coordinate system, or rather, Li = a+bi+cL. Now L * i * i = - L = ai + bi^2 + cLi. Since Li = a+bi+cL and i^2 = -1, we get ai - b + c(a+bi+cL), or ac-b + (a+bc)i + c^2 * L. This is already problematic and c^2 = - 1 already contradicts the rules I put forth, since it was assumed that we got three unique numbers and the only solutions to c^2 = -1 are c = i or c = -i. Furthermore, we have ac-b =0 and a+bc = 0. This can easily be turned into abc-b^2 = 0 and abc+a^2 = 0, or a^2+b^2 = 0. Either a and b must both be zero, or one of them must be real and the other imaginary, or they both must be complex. Taking them to be zero, which is the least problematic, and accepting the nonreal solution to c leaves us with Li = cLi, which is redundant. We have accomplished nothing.

If anybody wants to help me develop this idea with different ideas as for how real and imaginary numbers combine with legendary numbers, let me know!

[Robert J. Kolker]

If anybody wants to help me develop this idea with different ideas as for how real and imaginary numbers combine with legendary numbers, let me know!

First read

http://en.wikipedia.org/wiki/Hypercomplex_number

See also:

http://en.wikipedia.org/wiki/Multicomplex_numbers

Bob Kolker