We have seen that an imaginary number can be interpreted in two different ways (that are ultimately complementary).
(a) As the manner of expressing what is properly an ordinal notion in an indirect cardinal manner (for incorporation in a Type 1 interpretation of number).
Considerable use is now made of complex numbers in Conventional Mathematics; however both parts are are treated strictly as quantities within this approach. Thus the true nature of the imaginary part (as representative of an alternative qualitative relational number system that is ordinal) remains completely unrecognised when treated in an absolute manner.
However when the Type 1 approach is understood in a relative fashion with complex nos. again treated in quantitative terms, implicit in such understanding is the recognition that the imaginary aspect relates properly to the alternative qualitative relational aspect that is now - in a Type 1 context - indirectly given a quantitative expression. And it is this latter type of understanding that is properly consistent with the most comprehensive Type 3 mathematical interpretation.
(b) As the manner of expressing a cardinal notion in an indirect ordinal manner (for incorporation in the Type 2 interpretation of number).
Again in this approach – though still almost entirely unrecognised – complex nos. are interpreted strictly with respect to appreciation of their qualitative (ordinal) nature. However implicit in this is a recognition of the quantitative meaning of the real aspect (within the Type 1 system) that is now indirectly given a qualitative expression within Type 2.
What is again clear from this is that the true significance of complex nos. is entirely missed within the absolute Type 1 framework of Conventional Mathematics. The essential point is that number has both cardinal (independent) and ordinal (relational) meanings which are - relatively - distinct. Therefore to incorporate the ordinal aspect within a real quantitative type approach, it must be treated in an imaginary fashion.
Likewise from the complementary perspective to incorporate cardinal aspect within a - relatively - real ordinal interpretation, the quantitative must be treated as imaginary.
This in Type 3 terms what is imaginary from a quantitative perspective is equally real from the corresponding qualitative perspective; and what is real from a qualitative perspective is imaginary from the corresponding quantitative perspective.
So in this context real and imaginary have ultimately a purely relative meaning.
Higher dimensional interpretations (s > 2), combine both real and imaginary aspects. This entails that 3-dimensional and all higher dimensional interpretations entail number configurations with both cardinal (quantitative) and ordinal (qualitative) features that are properly distinguished.
From a quantitative perspective (again for s > 2) the roots of 1 will entail complex values (with real and imaginary parts) that serve as the quantitative counterpart of real and imaginary interpretation in qualitative terms.
This entails that each dimension is associated with a unique configuration with respect to both analytic (quantitative) and holistic (qualitative) type appreciation. This would mean in turn from a psychological perspective, a unique configuration with respect to both rational and intuitive type processes.
And properly understood both the quantitative and qualitative aspects are complementary.
Thus to properly interpret the quantitative nature of the roots of 1, we need the complementary Type 2 higher dimensional interpretation.
Equally in deriving the structure of these dimensions we require the complementary Type 1 appreciation of corresponding roots.
However though complex values occur at s = 3, the most important occurs for s = 4.
Now looking at the 4 roots of 1 we have 2 real and 2 imaginary.
From a Type 2 perspective this implies a perfect integration of both cardinal and ordinal type meaning. And looking at it from a Type 1 perspective, we have two real and two imaginary roots. However these imaginary roots – though expressed in quantitative terms - are understood as representative of ordinal relationships pertaining to the Type 2 system.
Likewise from the Type 2 perspective, these real and imaginary roots are now given a qualitative Type 2 interpretation with respect to 4–dimensional appreciation, with again perfect matching symmetry.
So then from a Type 3 perspective what is real in one system is imaginary in the other and what is imaginary is real; likewise what is positive in one is negative in the other and vice versa.
We live in a world of 4 dimensions. The deeper understanding of this implies that all reality is subject to opposite polarities in real and imaginary terms, with what is imaginary in terms of one system real in terms of the other and vice versa.
Indeed a clue is given to this in the work of Jung who saw the number 4 as extraordinarily important (from this qualitative perspective).
He also drew attention to the most common forms of mandalas which so often are based on ornate pictorial representations corresponding to the geometrical representation of the four dimensions (four roots of 1) and eight dimensions (eight roots of 1) respectively.
And here we can see the precise mathematical nature for such integration where both the ordinal (relational) and cardinal (independent) nature of number are seen as ultimately identical!
Now one of the important practical implications of this understanding is that it provides an entirely new perspective with which to deal with the Riemann Zeta Function where through using two systems of interpretation, both real and imaginary values become interchangeable in both systems!
For example, the non-trivial zeros (in Type 1 terms) combine a constant real part of 1/2 with varying imaginary values!
This directly implies that we can use these values in Type 2 terms, where now the imaginary aspect is constant at 1/2 and the imaginary parts are now treated as real.
As we have seen we have seen that average mean value of roots of 1 of both cos and sin parts (for any prime number p) approaches 2/pi. However the deviations of actual computed values from this value need to be explained. So just as the non-trivial zeros have a role in Type 1 terms (with reference to their imaginary parts) in precisely predicting the (cardinal) distribution of primes, they likewise have a role in Type 2 terms in precisely predicting this complementary (ordinal) distribution of the primes, with the imaginary aspect now interpreted as real.
So the non-trivial zeros in this context take on an entirely new significance which throws significant light on their true nature!
And the process works both ways as in reverse fashion the Type 2 approach can be used to highlight the wave nature - not of the general distribution of primes - but rather of each individual prime number in Type 1 terms.!