final solutions on second decimal (a*L + b) where sqrt2 = 1d414....(2L + 0)


Last night I realized what is at stake here. If I fail then Godel
Incompleteness
has some truth value. And that Incompleteness extends not only to
"statements
of mathematics" but extends all the way down into operations versus
digit
arrangements. That there are some operations for which it is
impossible to
have a closure to number digit arrangements.

We see this already in Old Reals in that 1/3 has no digit arrangement
representative in Old Reals. They glided around in textbooks in
classrooms
in public that 1/3 = 0.3333..... and used those dots to sweep under
the rug
the fact that there is a missing carryover of 1.

The AP-Reals understands this sweeping of mistakes where 1/3
= 0d3333....33333(L + 1/3)

1/2 is cool for it is 0d5000...0000L where nothing more is needed to
fix it up proper.

But 2/3 again, requires fixing up at the repair shop of L
2/3 = 0d666...6666(L + 2/3)

This repair shop has been known since ancient times as the
infinitesimal.
It is tiny but important to fix numbers so that the operation is
complete.
So important is this tiny number that the Calculus pivots on its well
being.

If I fail, then there is some truth to Godel Incompleteness but that
we
lose Calculus. If I succeed, then Calculus is saved and where the
limit
concept in mathematics is redundant. The second decimal point L
is the Limit.

I may have hit upon the solution for me, at the moment anyway. This
stuff is extremely difficult for it requires me to blindside my
misconceptions
and to work with infinity and to create and assemble something new. So
there are enormous stumbling blocks in this path.

The solution I hit upon is to change perception that there is no
digital solution
pre-existing but that I have to force upon the digital strings a
solution. So it was
a false perception on my part.

Here are the first hundred smallest positive AP-Reals

0d000...0001
0d000...0002
0d000...0003
0d000...0004
0d000...0005
0d000...0006
0d000...0007
0d000...0008
,
,
,
0d000...0097
0d000...0098
0d000...0099
0d000...0100

I need a square root operation to be complete on them. So I need
a square-root of 0d000...0002

Obviously there is no AP-Real with a digit arrangement that could
satisfy
that request.

I have as ally the format of the Complex-Number as (aY + bZ) and I
use a version of it as (a*L + b) I think that is sufficient.

So I say that the square-root of 0d000...0002 is equal to

0d000...0001(2L + 0) where the "a" = 2 and the "b" = 0

If I square that number I get 0d000...00001(4L +0)

If I take the square root of that number I get 0d000...0001(2L + 0)

Now let me see if it works on our familar sqrt2 of the Old Real 2.

sqrt2 = 1.414213.........

In AP-Reals it would be sqrt2 = 1d414213.....(2L + 0)

Squaring that number we have 2d0000....(4L + 0)
Taking the square root of that number we have
1d414213.....(2L + 0)

Does it work? Appears to work.

Let me try some familar number

sqrt49 = 7d0000....0000L
Squaring it we have 49d0000....0000L

So, yes, it works and the square-root operation is Complete in AP-
Reals

0d000...0001
0d000...0002
0d000...0003
0d000...0004
0d000...0005
0d000...0006
0d000...0007
0d000...0008
,
,
,
0d000...0097
0d000...0098
0d000...0099
0d000...0100

So for example sqrt0d000...0005

Would be 0d000...0002(5L + 0)

Squaring that number we have 0d000...0005(25L + 0)

But I am far from being out of the woods in problems.
What about the most ugliest of operations I could dream up
that would not allow for the (a*L + b) to fix a digit string?

How about this for an Operation--

9234100582942
- 859500003841
______________

Now the operation is that you subtract normally except where you
have to carryover and in those spots you divide.

So the first subtraction is normal and we have 2 - 1 = 1
But the second term of 4-4 requires a carryover so we divide
and that is 4/4 so our new number so far is going to look like this

......11

And the next digit is easy in 9-8 = 1, so our answer so far
is ....111

But now we run into difficulty because we have 2-3 and are
forced to divide and we have 2/3 and so what digit symbol
do we put for an answer in terms of .....111

So, here is the underlying question, is this L second decimal
point capable of handling every Operation thrown at it such as
the above ugly operation?

The stakes are high here. The stakes are that if (a*L + b) can
handle any and all such operations means the AP-Reals are
complete to all operations. If not, then Godel had some truth
value.

I would bet on the Complex Number format of (a*L + b)

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

.

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