## Thursday, July 14, 2011

### The Number System Revisited

Earlier we outlined two number systems that are quantitative (linear) and qualitative (circular) with respect to each other.

The first natural number system which defines Type 1 Mathematics is of a quantitative nature (with a default 1-dimensional interpretation).

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

The second natural number system which defines Type 2 Mathematics is of a qualitative nature (with the default number 1 raised to successive dimensional numbers).

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

This leads to a circular number system that structurally is obtained through taking successive roots of 1 (and then defining results in an appropriate qualitative manner).

Right away this suggests that the two systems are in fact interdependent for the quantitative interpretation of roots is dynamically inseparable from the qualitative interpretation of notion of their corresponding dimensions. So for example the second (square) root of 1 cannot be properly conceived in the absence of corresponding 2-dimensional interpretation!

However we can now widen both Type 1 and Type 2 Mathematics to include imaginary - as well as real - numbers.

So in Type 1 terms the quantitative number system now includes both real and imaginary natural number terms:

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

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

Likewise in Type 2 terms the qualitative number system now includes both real and imaginary natural number terms (as dimensions):

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

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

So both real an imaginary numbers can be given both a quantitative and qualitative interpretation!

What is remarkable however is how - in dynamic interactive terms - interpretations alternate as between both their quantitative and qualitative aspects.

So dynamically speaking, real and imaginary are quantitative and qualitative with respect to each other.

We can see this readily from the fact in Type 2 Mathematics when 1 is raised to a real integer dimensional power it results in a qualitative number interpretation; however when in Type 1 Mathematics, 1 is raised to an imaginary integer dimensional power it results in a corresponding quantitative interpretation.

This once again strongly suggests that Type 1 and Type 2 Mathematics cannot be properly understood in isolation and in fact are interdependent.

So the full integration of both Type 1 and Type 2 Mathematics leads to the most comprehensive approach (which is Type 3 Mathematics).