So once again, e

^{2iπ}= 1.

However, it is well known in conventional mathematical terms that when we raise any number to 0 that the resulting answer = 1.

Therefore e

^{0 }= 1.

This then seemed to suggest that as e

^{2iπ }= e

^{0}that therefore 2iπ = 0, which would seem absurd!

Indeed again in conventional terms, e

^{– 2iπ }= 1/e

^{2iπ }= 1/1 = 1.

So this would now seem to suggest that 2iπ = – 2iπ, which again would be absurd from this conventional perspective.

In fact, e

^{2kiπ}= 1, where k = 1, 2, 3, .......n

Therefore, seemingly, e

^{2iπ }= e

^{4iπ }= e

^{6iπ }= .......= 1, implying that 2iπ = 4iπ = 6iπ etc.

It was only that I began to clearly realise something with truly enormous mathematical implications. in that all numbers have two aspects, which depending on context, keep switching in the dynamics of experience.

Now I refer to these two aspects as analytic (quantitative) and holistic (qualitative) with respect to each other.

This is remarkably similar to quantum mechanical reality, where sub-atomic particles can reveal themselves as particles and waves (also depending on context).

In fact the deeper implications here entail the startling realisation that quantum mechanical behaviour is itself inherent in the very nature of the number system (when appropriately understood in a dynamic interactive manner).

So the way we avoid confusion with respect to the issues raised is to now define the natural number system in terms of two complementary aspects (which I refer to as Type 1 and Type 2 respectively).

The Type 1 aspect can be identified with the conventional analytic interpretation of number (i.e. in a merely quantitative manner).

So here each natural number is defined in terms of a default dimensional value of 1

So the natural numbers in Type 1 terms can be represented as

1

^{1}, 2

^{1}, 3

^{1}, 4

^{1},……..

The Type 2 aspect can then be identified with the (unrecognised) holistic interpretation of number (in an intrinsic qualitative manner).

Here the base number remains fixed as 1, whereas the dimensional value now varies over the natural numbers as

1

^{1}, 1

^{2}, 1

^{3}, 1

^{4}, ......

In (Type 1) quantitative terms, 1 for example, represents an independent quantitative unit (that literally lacks any qualitative distinction).

However in (Type 2) qualitative terms, 1 represents a qualitative unit (with the capacity of relating all independent units). In fact this is conventionally referred to as "oneness" (i.e. the quality of 1).

Now quite remarkably - though there is as yet no recognition of this crucial fact in conventional mathematical terms - we naturally keep switching between the Type 1 (analytic) and Type 2 (holistic) aspects of number even with respect to the most commonplace number operations!

For example imagine a class (or group) of 2 objects. Because these are already implicitly recognised as belonging to the same group. we can then give "2" the standard (Type 1) quantitative meaning (as comprised of two independent entities).

However now imagine two separate classes containing in each case 1 object!

Here, the identification of 1 (as now common to both classes) properly requires the unrecognised (Type 2) qualitative meaning of 1.

In other words we are able to identify that 1 is common to both classes through the shared qualitative notion of "oneness" in each case.

So we have now moved from the independent (separate) notion of 1 (in Type 1 terms) to the related or interdependent notion of 1 (in a Type 2 manner).

However in conventional mathematical terms, these two complementary aspects of number, which operate in a dynamic relative manner, are reduced in an absolute - merely quantitative - manner.

When one truly grasps this point, then one profoundly realises that the very foundations of the number system - as currently understood and by extension all Mathematics (and its related sciences) - are hugely inadequate.

For millennia now, we have tried to understand Mathematics in merely quantitative terms.

However despite its great advances this has led to a greatly distorted interpretation (even in quantitative terms).

So the clear message is that in future we will slowly come to realise that both quantitative and qualitative aspects are necessarily involved in even the most trivial mathematical operations and that proper understanding must be based therefore on the dynamic interaction between both poles.

To return to our immediate topic, the amazing discovery that I made is that the Euler Identity in fact directly relates to the (hidden) holistic aspect of the number system!

So to resolve the problems that I raised at the beginning of this entry, we must define number in Type 2 terms.

Therefore the (fundamental) Euler Identity is now expressed as

e

^{2iπ}= 1

^{1}

Then e

^{0 }= 1

^{0}

^{ }e

^{– 2iπ }= 1

^{– 1}

**And e**

^{ }^{2iπ }= 1

^{1}, e

^{4iπ }= 1

^{2}and e

^{6iπ }= 1

^{3}respectively.

So in fact all these expressions which lead to the same value in a Type 1 quantitative manner, are associated with distinctive holistic values from a Type 2 qualitative perspective.

So we will next show more directly how the 3 levels of Band 4 can be associated with the specialised intuitive understanding of these fundamental numerical values.

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