12/19/2007, 08:42 AM

May be this makes it easier to move forward with the problem at hand:

I have come to one conclusion based on my tetration thinking about what is infinitesimals and i.

Let me put it like this:

1) Imagine infinitesimal of some grade, structure. Then (1/this infinitesimal) is infinity of the same grade, so that

Infinitesimal* ( 1/infinitesimal) is = 1. This is simple stuff known from Euler's times, and accepted in non-standard analysis.

Now ask a Question: what is

Infinitesimal^(1/infinitesimal) = infinitesimal^ (infinity of the same grade)? It is not 0 as infinitesimal is NOT 0, as it is bigger than infinitesimal squared, which is also not zero.

Or, alternatively, what is (1/infinitesimal) ^ (infinitesimal) = (infinity of the same grade)^ infinitesimal? It is not 1 since infinitesimal is NOT 0.

2) By some logic, we would expect that ( I can not prove it yet):

infinitesimal^(1/infinitesimal) = 1/infinitesimal^infinitesimal ( all infinitesimals infinities are of the same grade, in the SAME SCALE of infinitesimals).

Now look at this- these are simple identities once You use i=e^(ipi/2), -i = e^-i(pi/2):

i^-i = i^(1/i) = -i^i = (1/i)^i = e^( pi/2)

We also know that (1+ 1/infinitesimal)^infinitesimal = e, so (1-i)^i =e as well. Is it?

What is the conclusion/conjecture?

At any given scale (grade) of infinitesimals , what ever it is, i is a notation for infinitesimal, and it has a property in this scale that

i^1/i = infinitesimal^ 1/infinitesimal= e^pi/2.

So complex numbers, or imaginary unit appear as a result of MANDATORY need to involve infinitesimals in any GIVEN scale we operate our math. Since we use to jump over scales of infinitesimals in mathematics ( everything was just small, in case of limit approach, we do not even notice them at all), we could not decipher it earlier.

3) At any given infinitesimal scaleSCALE is THE word, and SCALES are relative):

i^2=-1

i^3=-i

i^4=1 etc telling that by definition, infinitesimals of the same grade has very limiting constrictions of how they can be organized. And that scales 1 and -1 are created with some meaning at ANY scale - may be in the one we happen to be in, may be lower, may be higher.

Even if infinitesimal scale is something like infinity^sgrt(7) smaller than some other, once we enter that scale, and define the size of infinitesimal by dividing previous scale with infinity^sgrt(7) it becomes i on that scale, and the cyclical properties of i holds.

4) The difference between scales of infinitesimals is the level of complexity. In our case it is Monster. In lower scales- less. In higher? Are there scales more complex than Human? I do not know.

5) Huh, finally I see the light once imaginary unit is somehow placed where it should be. I ( imaginary unit) is a universal notation of infinitesimal in ONE infinitesimal scale which is more or less INFINITY away from next scale.

From here we can notice that i^3= -i = infinity in that scale. 1 is clear, more or less, - 1 still a little bit of a mystery.

6) As we go further with powers of i knowing that i is infinitesimal, we may ask whether or not as we rotate something via e^i(phi) = e^ h( e^pi/2) *phi what we are really doing? Are we not adding infinitesimal 3RD dimension on top of complex plane by each rotation, so that it is UNNOTICEABLE in the scale we operate, but in infinitesimal space, there is a spiral structure growing which we do not notice directly, but it INFLUENCES what happens in our scale.

I have come to one conclusion based on my tetration thinking about what is infinitesimals and i.

Let me put it like this:

1) Imagine infinitesimal of some grade, structure. Then (1/this infinitesimal) is infinity of the same grade, so that

Infinitesimal* ( 1/infinitesimal) is = 1. This is simple stuff known from Euler's times, and accepted in non-standard analysis.

Now ask a Question: what is

Infinitesimal^(1/infinitesimal) = infinitesimal^ (infinity of the same grade)? It is not 0 as infinitesimal is NOT 0, as it is bigger than infinitesimal squared, which is also not zero.

Or, alternatively, what is (1/infinitesimal) ^ (infinitesimal) = (infinity of the same grade)^ infinitesimal? It is not 1 since infinitesimal is NOT 0.

2) By some logic, we would expect that ( I can not prove it yet):

infinitesimal^(1/infinitesimal) = 1/infinitesimal^infinitesimal ( all infinitesimals infinities are of the same grade, in the SAME SCALE of infinitesimals).

Now look at this- these are simple identities once You use i=e^(ipi/2), -i = e^-i(pi/2):

i^-i = i^(1/i) = -i^i = (1/i)^i = e^( pi/2)

We also know that (1+ 1/infinitesimal)^infinitesimal = e, so (1-i)^i =e as well. Is it?

What is the conclusion/conjecture?

At any given scale (grade) of infinitesimals , what ever it is, i is a notation for infinitesimal, and it has a property in this scale that

i^1/i = infinitesimal^ 1/infinitesimal= e^pi/2.

So complex numbers, or imaginary unit appear as a result of MANDATORY need to involve infinitesimals in any GIVEN scale we operate our math. Since we use to jump over scales of infinitesimals in mathematics ( everything was just small, in case of limit approach, we do not even notice them at all), we could not decipher it earlier.

3) At any given infinitesimal scaleSCALE is THE word, and SCALES are relative):

i^2=-1

i^3=-i

i^4=1 etc telling that by definition, infinitesimals of the same grade has very limiting constrictions of how they can be organized. And that scales 1 and -1 are created with some meaning at ANY scale - may be in the one we happen to be in, may be lower, may be higher.

Even if infinitesimal scale is something like infinity^sgrt(7) smaller than some other, once we enter that scale, and define the size of infinitesimal by dividing previous scale with infinity^sgrt(7) it becomes i on that scale, and the cyclical properties of i holds.

4) The difference between scales of infinitesimals is the level of complexity. In our case it is Monster. In lower scales- less. In higher? Are there scales more complex than Human? I do not know.

5) Huh, finally I see the light once imaginary unit is somehow placed where it should be. I ( imaginary unit) is a universal notation of infinitesimal in ONE infinitesimal scale which is more or less INFINITY away from next scale.

From here we can notice that i^3= -i = infinity in that scale. 1 is clear, more or less, - 1 still a little bit of a mystery.

6) As we go further with powers of i knowing that i is infinitesimal, we may ask whether or not as we rotate something via e^i(phi) = e^ h( e^pi/2) *phi what we are really doing? Are we not adding infinitesimal 3RD dimension on top of complex plane by each rotation, so that it is UNNOTICEABLE in the scale we operate, but in infinitesimal space, there is a spiral structure growing which we do not notice directly, but it INFLUENCES what happens in our scale.