Let me first clarify the distinction I make as between linear and circular (with respect to number systems) on the one hand and quantitative and qualitative on the other.

Now in the conventional Type 1 mathematical approach - though the overall qualitative approach is decidedly linear - both linear and circular notions can be dealt with from a quantitative perspective. The real number system for example is viewed in a linear fashion (as points laid out in a line). Indeed the imaginary number system is viewed in a similar fashion (lying on a line vertical to those of the real). However when it comes to the roots of unity, these lie in the circle of unit radius (in the complex plane). Now the quantitative nature of these roots can be dealt with (without however their true significance being realised).

In the Type 2 mathematical approach both linear and circular notions of logical interpretation can also be brought to bear on these same number systems. Also one clear implication of this approach is that circular notions (in quantitative terms) cannot be properly interpreted in the absence of circular type interpretation (from the qualitative perspective).

Thus the full use of both linear and circular notions requires that they be given both Type 1 (quantitative) and Type 2 (qualitative) interpretations.

The key significance of the Euler Identity is that - though seemingly arising in the context of the Type 1 approach to Mathematics - it actually gives rise to the need for the Type 2 approach.

In a very true sense therefore it arises at the very intersection of Type 1 and Type 2 approaches (where both quantitative and qualitative mathematical notions are fully interdependent).

I have already defined the natural number system with respect to Type 1 and Type 2 interpretation.

The Type 1 approach is qualitatively linear in nature and is literally defined in 1-dimensional terms,

1^1, 2^1, 3^1, 4^1,.......

So here the base number quantity keeps changing while the default dimensional number quality remains fixed as 1.

The Type 2 approach is by contrast quantitatively linear in nature, where the dimensional number quality keeps changing,

1^1, 1^2, 1^3, 1^4,.......

The circular nature of this alternative number system can indirectly be shown

in quantitative terms through obtaining the corresponding root (i.e. the reciprocal of the dimension in question).

Therefore in the circular number system, there is an inverse relationship as between a qualitative dimensional interpretation and corresponding quantitative root.

So once again for example, to properly explain the square root of 1 i.e. 1^(1/2), we need the corresponding qualitative dimensional interpretation of 2 i.e. 1^2!

Now what is fascinating about the fundamental Euler Identity is that it leads directly to this Type 2 Number system

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

Now all other numbers for example in the Type 2 natural number system can be obtained from the expression

e^(2*k*i*pi) where k = 1, 2, 3, 4,....

So in fact e^(2*i*pi) is just the special case where k = 1.

Thus 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.

Now we have already dealt with the ambiguity in terms of conventional interpretation of roots where for example but + 1 and - 1 are both given as the square root of 1, i.e. 1^(1/2).

Indeed I have already argued using Type 2 interpretation that properly - 1 is (unambiguously) the square root of 1 i.e.1^(1/2).

Now this is born out directly through extension of the fundamental Euler Identity.

So when k = 1/2, then

e^(2*k*i*pi) = e^(i*pi) = - 1.

So one valid way of interpreting the Euler Identity (as it is generally presented) is that the square root of 1 i.e. 1^(1/2) = - 1.

However this conflicts directly with conventional reduced (Type 1) interpretation whereby the principle square root of 1 = + 1.

So for example if you take out any calculator, input 1 and then raise this to .5, you will be given the result of 1, which the Euler Identity tells us is erroneous!

One of teh great advantages of the extended Euler Identity is that it provides a ready means for calculating all roots of 1.

So e^(2*k*i*pi) = cos(2*k*pi) + i sin(2*k*pi)

Therefore for example to obtain the cube root of 1 i.e. 1^(1/3) we let k = 1/3

So e^(2*k*i*pi) = e^(2*i*pi)/3 = cos(2*pi)/3 + i sin(2*pi)/3

which represented in degrees (rather than radians) = cos 120 + i sin 120

= {- 1 + 3^(1/2)i}/2 = 0.5 + .8660i (correct to 4 decimal places)

Note once again that just one unambiguous answer corresponds with the cube root of 1 i.e 1^(1/3)!

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