We have seen in recent blog entries, how a number expressed with respect to its Type 1 aspect can then be converted in Type 2 terms.
In reverse we have likewise seen how a number expressed initially with respect to its Type 2 aspect, can be converted in a Type 1 manner.
As the Type 1 and Type 2 aspects related directly to the cardinal (quantitative) and ordinal (qualitative) aspects of number respectively, such conversion as between the two aspects of the number system ultimately relates to the consistency as between both cardinal and ordinal interpretation respectively.
Now as Conventional Mathematics is confined to a merely reduced quantitative interpretation of number, this key issue as to the consistency of both the cardinal and ordinal aspects of number does not even arise.
However, when appropriately understood in a dynamic interactive manner it relates directly to the very nature nature of the Riemann Hypothesis, which in fact is the fundamental condition required to ensure the consistency of both cardinal and ordinal aspects.
What we have shown so far is that the conversion of a Type 1 natural number (on the real line) leads to a corresponding Type 2 number (on the imaginary line).
We have also shown that for the natural numbers, the Type 2 conversions have a negative value.
However the reciprocals of the natural numbers will also lie on the imaginary line (with a positive value).
Now an equally interesting pattern of conversion arises when we attempt to convert a Type 1 circular number i.e. lying on the circle of unit radius in the complex plane) in a corresponding Type 2 fashion.
Now the simplest and perhaps most important example relates to – 1.
So we set (– 1)1 = 1x .
Therefore log (– 1) = x log 1
iπ = x (2 iπ)
Therefore x = 1/2
So (– 1)1 = 11/2
The next most important example relates to i.
So again i is a circular number (with respect to the Type 1 aspect of the number system).
So when we express i in Type 2 terms we get 1/4.
So i1 = 11/4
In general terms therefore when we express a circular number (with respect to Type 1) in Type 2 terms we get a linear fractional number (on the real line)
Alternatively when we express a real linear fraction (with respect to the Type 2), this converts into a circular number (with respect to the Type 1).
And this is all of crucial relevance with respect to the solutions of the Zeta 1 and Zeta 2 zeros respectively.
The Zeta 1 zeros all lie on the imaginary number line; the Zeta 2 zeros (for finite equations) then lie on the unit circle. Then for infinite equations the Zeta 2 solutions = 1/2.
So the requirement that the imaginary line for the Zeta 1 zeros goes through 1/2, is a simple consequence of the Zeta 2 infinite results.