It is perhaps appropriate now to briefly explain the nature of this alternative circular approach to the number system where each number representing a dimension (power or exponent) is given a distinctive qualitative interpretation that is inherently of a dynamic interactive nature.
The basic starting point is that all relationships (and corresponding experience with respect to such relationships) are conditioned by sets of fundamental polar opposites.
The two key sets (with respect to such opposites) relate to the fact that very object (viewed as external) is related to a corresponding subject (that - relatively - is of an internal nature). Likewise all wholes (in collective terms) are necessarily related to corresponding parts (that - are relatively - specific in nature).
Now the relationship between these two sets can best be visualised - like a compass with four co-ordinates - through the unit circle (drawn in the complex plane with horizontal and vertical axes) with the horizontal axis representing the real relationship as between external and internal polarities and the vertical axis the imaginary relationship as between whole and part.
Given the predominance of reductionist linear type logic in scientific understanding, it might initially seem difficult to appreciate why the relationship as between whole and part is of an imaginary nature.
However customary logic leads to the notion of objects as whole-parts (where every object as whole is part of a larger whole) which represents reduced interpretation of their nature.
However more refined appreciation of this relationship (as I have expressed before on these blogs) requires recognition of the role of the unconscious in enabling the dynamic switch as between recognition of whole and part (or alternatively part and whole) in any context to take place. And the qualitative notion of the imaginary is the indirect means of representing in rational terms the nature of the holistic unconscious!
So, as those familiar with Jungian Psychology might appreciate, when this (imaginary) contribution of the unconscious is especially refined in understanding, "objects" can increasingly serve in qualitative terms as infinite archetypal symbols. Here the parts can increasingly be reflected through the whole (in collective terms) or alternatively the whole be reflected through each part (in a unique manner) without undue reductionism.
So the true dynamic relationship as between whole and part is represented through the use of imaginary polarities (that are positive and negative with respect to each other).
However just as a compass can be used to give an ever more detailed notion of co-ordinate directions, in like manner the unit circle (with real and imaginary coordinates) can likewise be used to give ever more detailed directions (representing the relationship as between the fundamental polarities).
So in short each dimension (as number) gives rise to a special direction with respect to the relationship as between polar co-ordinates.
Therefore with higher dimensional numbers (as qualitatively understood) one becomes better enabled to appreciate, in a refined interactive manner, the inherently dynamic relationship as between the fundamental polar coordinates.
And the actual structure of such dimensions is provided through corresponding appreciation of their roots (in quantitative terms).
So for example to find out the nature of 5 as dimensional number (in qualitative terms) we obtain the 5th root of 1 (giving a special configuration with respect to real and imaginary co-ordinates).
In psychological terms this would represent a distinct configuration with of both rational and intuitive type meaning (with respect to interpretation)!
However as we know conventionally with respect to 5 we have 5 roots (and not just one).
So we now express these 5 roots of 1 in a more refined manner as the 5 roots of 1^1, 1^2, 1^3, 1^4 and 1^5 respectively.
Thus with respect to these 5 roots, a unique system of circular interdependence exists.
This means in corresponding qualitative terms that when 1, 2, 3, 4 and 5 are used (in this context of 5) to represent dimensional understanding that a corresponding unique system of circular interdependence exists.
This could be expressed alternatively by saying that where the number 5 (as a cardinal number is concerned) a unique system of circular interdependence exists in ordinal terms as between its 5 members (i.e. 1st, 2nd, 3rd, 4th and 5th).
And right here we are in a position to perhaps appreciate the alternative qualitative significance of prime numbers!
Thus, corresponding to each prime number (as a dimension) is a unique set of ordinal members bound together in a circular interdependent manner that cannot be replicated through any other number.
So in quantitative terms the prime numbers are seen as the basic building blocks of the natural number system! However now in direct complementary fashion from a qualitative perspective, each prime number is seen as composed of a unique set of ordinal members in natural number terms!
Therefore associated with the prime number 5 (in this dimensional qualitative sense) is a unique set of numbers corresponding to its 1st, 2nd 3rd, 4th and 5th roots).
So we started out by seeing prime numbers as the most independent of numbers (with no factors) in quantitative terms.
However now in a complementary manner, the prime numbers are revealed as the most interdependent of numbers, composed of a natural number set of ordinal members (that comprise a unique circle of interdependence with respect to this number).