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22.5 Summary & Conclusions

In summary this Notebook deals with the foundation of numbers themselves. It examines a family of series whose limits includes all rational roots of the rational numbers. This number system of iterative series includes both rationals and irrationals within its family.

System based upon iterative functions

There are a number of interesting features to this number system. While our most familiar number systems are either integer or fraction based, this number system is based upon iterative functions. Because of its iterative nature, zero is the only number that is exact, while all of the rest are approximations.

Generation Patterns independent of 'seed' or 'starting point'

Another intriguing feature is that the generation patterns of our iterative functions are independent of 'seed' or 'starting point'. This means that no matter where we start, we always end up the same place. A corollary to this feature is that the generation patterns are self-correcting. (This aspect is explored more thoroughly in the next Notebook.) We will find that even if corrupted data is introduced, for instance due to mechanical limitations, the pattern corrects itself, always heading for the proper destination. This feature seems quite Life-like. It is like the Seed, which generates the Plant almost independent of polluting environmental influences.

Implications of binomialization and inverted dimensions

In the last section of this paper, we binomialized our Series to achieve a more precise form. OchhamÕs Razor: Simplicity wins. This binomialization produced some interesting characters. One was the infinitesimal, while the other was the dimensional generator. The first was a necessary placeholder, while the other acknowledged the dimensional nature of Roots. It seemed that each successive Root generated a new inverted Dimension. While normal dimensions are based upon content-based spatial dimensions, these inverted dimensions are based upon contextually based feedback dimensions. With each new inverted dimension, another layer of feedback is added. Square Roots are based upon one layer of feedback; cube roots on two layers and fourth roots on three layers. While these results are intriguing, they are little understood.

The existence of bogus equations

Another feature of our iterative equations is the existence of bogus equations. These are equations that yield the proper result under 'normal' circumstances, but fall apart under 'exceptional' circumstances. Why do they fall apart and why do they yield the right result? We do not know. You will have to read the next Notebook, 'Root Beings', to find out answers to these questions. While this Notebook is primarily equation based, the next Notebook is primarily graph based. It is very visual. We get to see these unusual pattern-based equations in action. Lots of pictures and few equations.


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