Wednesday, July 13, 2011

Imaginary - Hidden Qualitative Notion

When appropriately understood a number representing a dimension or power is of qualitative nature relative to the number to which it is raised (which is thereby quantitative).

So if we take for example 1^2, in this context 1 is quantitative, whereby 2 is - relatively - of a qualitative nature.

And again in Type 2 Mathematics (i.e. where 1 is raised to a dimensional power) associated with each dimensional number is a unique qualitative result (that is similar in structure to the corresponding quantitative root of the number).

Thus in the important case where the dimension is 2 in Type 2 Mathematics,
1^2 = - 1 and when it is 4, 1^4 = i.

So both - 1 and i thereby have important qualitative interpretations.

Once again, - 1 relates to the negation of rational (unitary) form which is the basis in experience for generating empty (0) intuition. And in a more comprehensive mathematical understanding both reason and intuition must be formally incorporated!

In qualitative terms, i is of a more refined though extremely important nature.

Basically it represents a linear rational attempt to (indirectly) incorporate intuition in interpretation.

So quite literally if we are to attempt to express the 2nd dimension in (reduced) 1-dimensional terms then we obtain the square root of - 1 (which in fact is the definition of an imaginary number).

Therefore i in qualitative terms represents the incorporation of circular 2-dimensional type logic indirectly in a linear rational matter.

So an alternative way to express the fact that comprehensive interpretation requires both rational and intuitive modes of understanding is to say that comprehensive interpretation necessarily requires both real and imaginary rational components!

Now remarkably when 1 is raised to the power of i (i.e. 1^i) a whole range of quantitative type results ensues.

Now again in Type 1 Mathematics - in a complementary manner to the handling of roots - only the 1st result is generally considered and taken as the principle value.

So 1^i = {e^(2*pi*i)}^i = e^(- 2*pi) = .00186744...

However because in Type 1 Mathematics e^(2*pi*i) = 1, therefore {e^(2*pi*i)}^n = 1, where n = 1,2,3,4,....

So for example if n = 2, then 1^i = {e^(2*pi*i)}^i = {e^(4*pi*i)}^i

Therefore in the first case 1^i = .00186744...

However in the second case 1^i = .00000348734....

And a potentially unlimited set of answers can be generated in this manner by using alternative natural number values for n!

However once again from a Type 2 perspective {e^(2*pi*i)} ≠ {e^(4*pi*i)}.

Put another way 1^i ≠ 1^2i.

What is fascinating about this is that failure to recognise the qualitatively distinct nature of a number as dimension, ultimately leads to confusion not only in qualitative but equally in quantitative terms.

Once again both 1^i and 1^2i are distinct expressions with unique results in quantitative terms. However because of the lack of a coherent qualitative basis, it is led to misleadingly concluding that they are equivalent expressions!

Now it is also worth pointing out that i in qualitative terms represents an indirect linear rational means of representing what is inherently of a qualitative nature in quantitative terms.

Therefore in this context when i is used as a dimension (i.e. as power of 1) it takes on a reverse quantitative meaning which leads therefore to the generation of a quantitative result.

So 1 (raised to a real integral power) generates a result that is inherently of a qualitative nature!

However 1 (raised to an imaginary integral power) generates a result that is inherently of a quantitative nature!

And as the number and the dimension to which it is raised are always quantitative and qualitative (and qualitative and quantitative) with respect to each other, this means that when appropriately understood in Type 2 mathematical terms, in the expression,

1^ni (where n = 1, 2, 3,.....), 1 is now inherently of a qualitative nature with its corresponding imaginary power - relatively - quantitative.

So ultimately in a fully comprehensive mathematical understanding (Type 3), both Type 1 and Type 2 interpretation must be integrated to maintain consistency in both quantitative and qualitative terms.

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