In the last contribution, I showed the relationship as between the extended Euler Identity and the Type 2 Number System.

Thus once again,

where k = 1, e^(2*k*i*pi) = e^(2*i*pi) = 1^1

where k = 2, e^(2*k*i*pi) = e^(4*i*pi) = 1^2

where k = 3, e^(2*k*i*pi) = e^(6*i*pi) = 1^3

where k = 4, e^(2*k*i*pi) = e^(8*i*pi) = 1^4,

and so on.

As there is no recognition in Type 1 Conventional Mathematics of the qualitative dimensional aspect of interpretation, a reduced and - ultimately - faulty understanding is given of the Euler relationships.

So just as in Type 1 terms, 1^1 = 1^2 = 1^3 = 1^4 =......,

likewise in reduced quantitative terms,

e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =.......

However this misleading interpretation can be shown to lead to a problem which is very revealing in its consequences.

Because e^(2*i*pi) = 1^1, then when we raise both sides to the power of i, we get

e^(2*i*pi)^i = 1^i

Therefore e^(- 2*pi) = 1^i

So 1/{e^(2*pi)} = 1^i

Therefore 1^i = .0018674427....

However according to Type 1 interpretation,

e^(2*i*pi) for example = e^(4*i*pi)

So therefore in Type 1 terms,

e^(4*i*pi)^i = 1^i

Thus e^(- 4*pi) = 1^i

And 1/{{e^(4*pi)} = 1^i

Thus 1^i = .00000348734...

And because in Type 1 terms,

e^(2*i*pi) = e^(4*i*pi) = e^(6*i*pi) = e^(8*i*pi) =....... ad infinitum,

this implies that we can have an infinite number of valid quantitative results for 1^i!

Now in Type 1 terms this myriad of embarrassing riches is handled in a merely pragmatic unconvincing fashion. Just as with the many possible (circular type) roots of a number the positive real numbered root is considered as the principle root (though strictly it does not represent the correct root), likewise in this situation the quantitative result pertaining to e^(2*i*pi) is taken as the principle value with the other possible values (of an infinite set) effectively ignored.

However by employing Type 2 interpretation we can easily resolve this problem

So from a Type 2 interpretation

e^(2*i*pi) ≠ e^(4*i*pi)≠ e^(6*i*pi) ≠ e^(8*i*pi)... and so on,

Rather e^(2*i*pi) = 1^1

e^(4*i*pi) = 1^2

e^(6*i*pi) = 1^3

e^(8*i*pi) = 1^4, and so on

Therefore for example whereas

e^(2*i*pi) = 1^i,

e^(4*i*pi) = 1^2i, and so on

Therefore when seen from this perspective 1^i does indeed have one unique answer.

The second answer that we calculated above i.e. .00000348734... does not correspond in fact with the value of 1^i but rather 1^2i!

What is remarkable here is that we have now used Type 2 interpretation - not alone to show a qualitative distinction - when 1 is raised to a real dimensional number, but now in reverse fashion to show that a quantitative distinction is likewise involved when 1 is raised to an imaginary dimensional number.

This also strongly hints at the true nature of the imaginary number i.e. as of a qualitative holistic nature (expressed indirectly in a real quantitative manner)!

So when we raise 1 to a real rational number (as dimension), the result will fall on the circle (of unit radius).

However when we raise 1 to an imaginary rational number (as dimension), the result will fall on the straight line!

Though I had for many years recognised that there was a qualitatively distinct approach to Mathematics (which I refer to as Type 2), For some time I considered that these two separate aspects could be conducted in relative independence of each other.

In other words I did not directly consider that Type 2 interpretation would have a direct relevance with respect to derivation of quantitative results!

However appreciation of the extended use of the Euler Identity has changed all this for its real message is that both quantitative and qualitative type interpretation are inextricably linked!

So ultimately we cannot have consistent type interpretation of quantitative results without corresponding consistency in qualitative terms.

Therefore in my own evolution appreciation of the true nature of the Euler Identity (from both a quantitative and qualitative perspective) was to prove a key landmark in eventually unravelling the true nature of the Riemann Hypothesis which is essentially the same message i.e. that both quantitative and qualitative type interpretation are inseparable!

However with respect to the Riemann Hypothesis, this poses insuperable problems as within Conventional Mathematics there is - as yet - no formal recognition of its equally important qualitative aspect!

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