There are really two components to this new number system (where numbers are interpreted with respect to their pure dimensional (as opposed to their base quantitative) characteristics.

Once again the linear system - for base quantities - is defined in terms of a fixed dimensional number i.e. 1.

So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as

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

However the corresponding circular system - for dimensional qualitative values - is defined in terms of a fixed base quantity 1.

So the natural numbers 1, 2, 3, 4,... in this system are more fully represented as

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

The circular nature of this latter system comes through raising 1 to the reciprocal of each dimension thus obtaining a quantitative value that lies on the circle of unit radius.

So when we raise 1 for example to the reciprocal of 4 i.e. 1/4 we obtain in quantitative terms i, which lies on the circle of unit radius!

Because there is a direct relationship as between each dimension (as quality) and its reciprocal (in quantitative terms), this means that the 4 as dimension is associated with the qualitative (i.e. holistic) interpretation of i.

Thus rather that just one valid interpretation of mathematical symbols, which in conventional terms is associated with the default value of 1, potentially an infinite set of possible interpretations exists for all all mathematical symbols, relationships etc.

So whereas Type 1 Mathematics is associated merely with the (reduced) quantitative aspects of mathematical symbols, Type 2 is associated with appropriate qualitative interpretation of these same symbols.

Thus i for example has not merely a quantitative, but also an important qualitative meaning. However this qualitative dimension is completely ignored in Type 1 conventional terms.

This makes no sense for ultimately the quantitative results that are derived for example in complex analysis are somewhat meaningless in the absence of appropriate qualitative interpretation!

Type 3 Mathematics - which is easily the most refined and demanding in nature, then involves consistently relating both quantitative (Type 1) and qualitative (Type 2) interpretation.

However just as the base quantitative system has an imaginary counterpart, likewise the dimensional qualitative counterpart has an imaginary counterpart.

So again the natural numbers in the first system would be

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

whereas in the second system the corresponding imaginary version is

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

Now because in Type 1 Mathematics the second system is not formally recognised this entails with respect to the real part that

1^1 = 1^2 = 1^3 = 1^4 =.....= 1^n.

As we have seen this leads to the misleading conclusion that for example + 1 and - 1 are both the square root of 1.

(Through use of the Type 2 system we can see that - 1 is the square root of 1^1 and + 1 the square root of 1^2 (which are distinct in Type 2 terms).

In other words we cannot properly divorce here proper quantitative from proper qualitative interpretation!)

Also because in Type 1 Mathematics the second system is not recognised this entails with respect to the imaginary part that

1^i = 1^2i= 1^3i = 1^4i =.....= 1^n.

This comes from the corresponding assumption that

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

This then leads to the misleading conclusion that 1^i for example can have an unlimited number of possible quantitative solutions.

However because properly speaking in a Type 2 approach

1^i, 1^2i, 1^3i, 1^4i, =.....= 1^n are all distinct,

this means that 1^i has indeed just one unique quantitative value!

So once again - this type directly with respect to a quantitative result - we cannot properly divorce here quantitative from qualitative interpretation!

As we have seen 1^i = e^(- 2pi) = .00186744....

However as i = 1^(1/4), this means that i^i = 1^(i/4) = e^{(- pi)/2} = .2078795763...

It must be stressed that in accordance with Type 1 Mathematics than - as with 1^i - an infinite set of possible values exists for i^i.

So in Type 1 terms, 1^i = e^(2*i*pi)^i = e^(4*i*pi)^i = e^(6*i*pi)^i = e^(8*i*pi)^i =....

By this logic, for example 1^i = e^(2*i*pi)^i = e^(8*i*pi)^i

= e^(- 2*pi) = e^(- 8*pi)

So i^i = e^(- pi/2) = e^(- 2pi)

However this would therefore suggest that i^i = 1^i (which makes little sense).

Therefore Type 2 interpretation needs to be included to avoid such confusion!

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