#68 Chapter 1: finite versus infinite, discrete versus continuous, and quantity versus geometrical location; new book 2nd edition: Math a subset of Physics, AP-adics

[plutonium....@gmail.com]

Alright, I had a very good detour there with chapter 2 in ironing out
the correct models of NonEuclidean
Geometry, something I failed to do in the 1st edition but have now
found success in this 2nd edition.
The correct model is so very important because it makes the operations
so very easy to define. To summarize
the correct Model is that there are only two geometries, not three.
There is Euclidean and when you
break the symmetry of Euclidean you have NonEuclidean which is a dual
combination of both
Elliptic and Hyperbolic bound up into one geometry. So we find the
angle sum of a Elliptic triangle
using the concave-outward sides and simply reverse them to be concave-
inwards to
find the associated Hyperbolic triangle.

But now let me return to the characteristics of the Numbers that
compose the New-Reals
and its geometry of Euclidean and the AP-adics which is the geometry
of Elliptic + Hyperbolic.

I interrupted Chapter 1 when I was discussing why the New Reals in a
microscopic interval are
finite and discrete. Let us pick on the New Real of 2.00000....0000
and immediately I can
microscope in on the five numbers preceding 2 and the five numbers
succeeding 2 and they
are:

1.999...9995, 1.9999....9996, 1.999...9997, 1.999...9998, 1.999...999,
2.0000...0000,
2.000...0001, 2.000...0002, 2.000...00003, 2.000...00004,
2.000...00005

So now one can easily see why I call those 11 numbers in the interval
of New-Reals from
1.999...99995 to 2.000...00005 as finite because there are no other
numbers in that interval
and there are only eleven numbers. So they are finite and discrete.

Now I want to use the concepts of discrete compared to continuous as
dual concepts. And use the
concept of finite as the dual of infinite.

But here I want to include another dual concept of Quantity versus the
function of geometric-location.

The numbers 1 and 2 in New Reals fulfill two functions, they speak of
quantity such as 1 apple or
2 oranges and they serve a function of geometric-location of a point
in the Cartesian Coordinate
System where the point 1.000...0000 and 2.000...0000 from 0 point.

So I have some dual concepts at work:
(a) finite versus infinite
(b) discrete versus continuous
(c) quantity function versus geometric-location function

So let me analyze those eleven numbers of New-Reals in the interval
1.999...9995 to
2.000...0005. They are vastly different from the Old-Reals which would
contend that
there are an infinity of Old Reals between 1.999...99995 and
1.999...99996. In the Old
Reals they had the idea that there are an infinity of Old Reals
between any two Old
Reals. Well, those old-timers never had the concept of FrontView and
BackView and
so they were lulled into a huge error of "betweeness".

So the New-Reals and the AP-adics are continuous in intervals if and
only if there are
an infinity of numbers between the endpoints of the interval with the
idea that Continuity
is equivalent to having an infinite supply of numbers to fill the
interval. But given a specific
number in that interval we can also see discreteness because there are
a finite quantity
of numbers in that specific microscopic interval.

So the New Reals and AP-adics are simultaneously continuous and
discrete, simultaneously
infinite and finite depending on the size of the interval in purview.

Now let me talk about the function duality of quantity versus
geometrical location that a Number
plays. And let me pick on probably the very worst mistake in the Old
Reals with their insistence
that 1/3 was a number, an Old-Real Number whose value was 0.333....
This is probably the most
laughable mistake in all of mathematics that has lasted the longest
time in the history of math.
The mistake started by the Ancients calling fractions as rational
numbers. Fractions are really
operations on numbers so when we divide 1 by 3 we go through the
process of division and
3 into ten is 3, remainder 1, carry over the 1 and 3 into another 10
is 3, remainder 1 and finally
make sure the decimal point is in the proper place.

Well the Old Math with the Old Reals could easily fall into that
mistaken trap of thinking that 1/3
was a number when in reality 1/3 was a operation that always had a
remainder of 1 that could not
be hidden or dropped out of the picture.

So we ask where is this Old Real of 1/3 in the New Reals? Is it the
number 0.333...3333
or perhaps the New Real of 0.333...33334 ?

Well the answer is that 1/3 is not a number at all but a division
process. That fractions are not
numbers when they leave a remainder.

More later, have to iron something out.

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

[plutonium....@gmail.com]

It may seem like I strayed too far off course there with discussing
"1/3" and why or why not it is a
number. The problem of 1/3 should be in the chapter of division as
operation. The Old Math would
have 1/3 as 0.333.... and where 1/3 x 0.9999 = 1/3, concluding that
0.999.... is the same as 1.
The New-Reals would have none of that saying that 0.999...9999 is a
different number than 1.000...000
and the predecessor of 1.000...000.

But I should be logical as much as possible and not get sidetracked. I
left the discussion of finite
and infinite and discrete and continuous with trying to fix the
understanding of why the
Real Number 7.0000...00000 is different than the AP-adic Number
0000....0007 and where the
Real is truly finite while our old notions of numbers that are
preceded by zeroes, that we
can ignore the zeroes and pretend as if 000....0007 is a finite 7.

As I mentioned years back, in the 1990s I believe, that what kind of
number is
007658......9832 as a AP-adic Number. The Old Math and the old ways of
thinking
said that if a number ends leftward in a string of zeroes then the
number is finite.
So that number ends in just two zeroes but the number itself is an
infinite integer.
The old way of doing math would say that 000....00007 is a finite
number and is the
number 7.

So I have alot of problems with the old way of math, because the
entire foundation of
the old math was the Peano Axioms in which there was never a mention
of what it
means to be "finite" or "infinite" and what the difference between the
two concepts are.

I defined Numbers as the dual of Geometry and I said the world has
only two types of
Numbers-- Reals and there reverse of AP-adics.
(a) Reals -- infinite strings rightward, finite portion leftwards of
all possible digit arrangements
(b) AP-adics -- infinite strings leftward, finite portion rightwards
of all possible digit
arrangements

So I used the concept of finite and infinite smack in the middle of
the definition of Numbers
themselves.

So being logical, I should define finite and infinite before I defined
Numbers. But maybe I have
to make those definitions axioms themselves.

I said a Number has two functions, two functions that are duals of one
another just as
Numbers are dual to Geometry. Those two functions are to tell a
"quantity" and to
tell a point in a geometry or a geometrical-position.

So the Real Numbers of 5.000...000 and 6.000....0000 tell of two
points at a position from
0 on the x-axis of Euclidean Geometry but also those two Reals tell of
a quantity such as
5 units or 6 units or 5 apples and 6 oranges. Now we ask a question,
is the concept of finite
versus infinite a concept of quantity or a concept of geometrical
position of a point? So it
seems to me that the concepts of finite versus infinite is more to do
with quantity than the
position of a point in geometry.

The point 50000...00000 on the equator of Earth versus the point
999...9999 as one unit
distance away from the South Pole are both infinite integers and yet
they represent quantities
of infinity, neither are finite. Likewise the numbers 000....0005 and
000...0009 represent two
points on the Earth sphere that are very close to the North Pole, and
many people of the
Old Math School would quickly jump in and say those two numbers are
"finite numbers".
And they would argue that they are one and the same as 5 and 9 in
Reals. But the
Reals of 5.0000...0000 and 9.0000...0000 are truly finite numbers
because of the
definition of Reals as a finite portion leftwards.

Now some may say I am just simply mixing, chopping and dicing and
mincing words. But no
I am making mathematics precise, for that is the number one job of
mathematics -- precision.
And one great benefit is already reaped from that precision in that
straightlines of Euclidean
Geometry cannot go to infinity because the leftward portion is finite.
Straightlines in Euclidean
geometry can be parallel, not because they never intersect out to
infinity, but because the
infinite supply of Reals between each finite portion such as between
5.000...0000 to
9.000....00000 maintain a fixed distance is the reason they are
parallel. So already, in perhaps
every textbook in the world that teaches Euclidean Geometry has
mistakes about straightlines
that go to infinity, because no straightlines ever go to infinity in
Geometry because every
Real Number has a finite leftward portion. So being precise, makes a
huge difference in
mathematics.

So I cannot say that the AP-adic of 000...00005 is the same number as
the Real of
5.000....0000 because the AP-adic is an infinite string whereas the 5
digit in that Real
is a "finite portion". Or, someone could say well, why is 1111...11115
a finite 5 or
how about 7777....777779 as a finite 9. Why is 0000...00009, the
zeroes so special
and crafting a "finite specimen". Well the answer is that zeroes are
not special
and that 9.000....0000 has a finite 9 but that 000...00009 is not a
finite 9.

Maybe this has to be axiomatized before I define Numbers as to the
meaning of
finite versus infinite. I think if that is the case, then the
definition of finite versus
infinite would involve the function of Numbers, as this
(a) function as quantity
(b) function as geometrical point position

So that a number is finite only when it is a radix of a AP-adic or
when it is the
leftward portion of a Real. And only finite numbers can give a
"quantity function"
not a geometrical-point-position function.

So the case with 1/3 is a division operation and cannot be the Real
of
0.333...3333 nor 0.333...33334 because the division leaves a remainder
of 1.
But the case of 1/2 as a division does leave us with the Real Number
of
0.5000...000.

So how to resolve that? To say that 1/3 is a quantity seeking and thus
finite
and thus not a Real number but an AP-adic number of radix3 which is
3/10 or radix33 which is 33/100 or radix 333 which is 333/1000. So
Fractions
are begotten from the finite rightwards portion of AP-adics and of
course
1/2 is radix5 which is 5/10.

So in mathematics when dealing with finite, we deal with quantity, the
dual
counterpart of geometric-point-position. And when we deal with
quantity we
can only go to Reals for integer finiteness and have to go to AP-adics
for
Fractional Finiteness.

Now I am not really satisfied with the above, for it does not yet
flow, but it is a
starting point and perhaps with more time I can get it to flow briskly
and logically.

[plutonium....@gmail.com]

Alright for those that tuned in late in this discussion, I am troubled
by 1/3. In Old Reals it was
considered that 1/3 = 0.333.... from thence they erroneously concluded
that 0.999... equals 1
The problem with 1/3 in New Reals is that 0.9999...9999 is a different
number than 1.000...0000
and that the numbers 0.333...3332, 0.3333...3333, 0.333...33334 have
no numbers between
them for a number whose division leaves a endless "remainder of 1"

So that is the problem facing me. Do I thence say Algebra of Old Reals
is wrong? Do I thence
say the New Reals are not a completed-field?

No.

I have a magic trick up my sleeves here. I meantioned this trick in
the first edition of this book
in that perpetual remainders such as 1/3 should be understood not as
0.3333...3333 but as
0.333...333r1 where the "r" stands for radix remainder.

So to solve my problem of what is 1/3 and to preserve all the Algebra
and then some of the
Old Reals, what I do is make a modification of my starting definition
of Numbers. If you remember
my starting definition was this:

New Reals: infinite rightward strings with a finite portion leftwards
of the decimal point denoted
by "d" of All Possible Digit Arrangements

AP-adics: infinite leftward strings with a finite portion rightwards
of the radix denoted by "r"
of All Possible Digit Arrangements

I have to make a small adjustment of those definitions by including a
radix with New Reals
and a decimal point with AP-adics

For example 1/3 as New Reals is 1.0000...0000/ 3.0000....0000 In the
division everything
goes smoothly in the sequence convergence to the number 0.3333....3333
except at the
last decimal-place of 999...99998 since there is still a remainder of
1, so to solve that
the final answer is this

1/3 = 0d333...3333r1

Now that solution also is seen in 1/2 where it is generally written as
0.5000.... in Old Reals
but in New Reals it is written as 0d5000...000 or 0d5000...0000r0

So both the New Reals and AP-adics have a decimal point and a radix
point

Now that helps the AP-adics in addition and subtraction given an
example of
8000...000r8 + 5000...000r7 would have a final answer of
1d3000...0001r5

In other words for the AP-adics the decimal serves as the modulus or
periodicity
in addition or subraction. Like in trigonometry that 2pi is 360
degrees or 0 and that
4pi is 720 degrees or again 0.

So with this small modification I preserve all of Algebra on the New
Reals and AP-adics.

And I correct the lousy error of the past mathematics that believed
that 0.999... was the same
as 1.

I had to solve this 1/3 crisis of a problem because the definition of
both New Reals and
AP-adics was All Possible Digit Arrangements and to say that
0.999...999 is the same
as 1.000...0000 contradicts All Possible Digit Arrangements.

Now I want to spend a brief moment in giving credit to Karl Heuer of
circa 1993-1994 for
this definition that a P-adic of Hensel was an infinite string
leftward with finite portion
rightward. Back in 1993 I called them the Infinite Integers and had
never heard of Hensel
and his P-adics. And so as the months went by, many posters to
sci.math wanted me
to define the P-adics or tell them what they were. And Karl usually
interceded the
questioner and gave that as the definition of a Hensel P-adic as a
infinite string leftward
finite portion rightward with the Reals as the reverse. So I was not
the first to define
Reals and AP-adics as that, but I was the first to include the idea of
All Possible
Digit Arrangements for those strings and now I am the first to include
not only a decimal
point for the Reals but a radix point also for the Reals so they have
two points--
d and r. And likewise for the AP-adics.

By giving them two points of decimal and radix then allows me to keep
All Possible
Digit Arrangements along with all the Complete-Field Algebra of these
two Number
systems.

Today is a good day for mathematics history.

So what is the number 1/3 ? It is a New Real that is 0d333...333r1 and
it fits
in geometry between the two New Reals of 0d333...333 and 0d333...33334

Now the only digits possible for the Real radix are
0,1,2,3,4,5,6,7,8,9, Only ten
possible digits for the radix component of Reals and likewise only ten
possible
digits for the addition and subtraction surplus of its decimal point
to have the
sum on the other side of the globe of Elliptic geometry.

[plutonium....@gmail.com]

>
> So what is the number 1/3 ? It is a New Real that is 0d333...333r1 and
> it fits
> in geometry between the two New Reals of 0d333...333 and 0d333...33334

So what is the number 2/3 in New Reals? In Old Reals 2/3 was
considered
to be 0.666.... in New Reals since it has a perpetual remainder of 2,
the
number 2/3 in New Reals is 0d666...6666r2

What is the number 1/2 in New Reals? It is 0d500...0000r0 or
if one prefers to not use the radix-remainder as simply written
as 0d5000...0000

>
> Now the only digits possible for the Real radix are
> 0,1,2,3,4,5,6,7,8,9, Only ten
> possible digits for the radix component of Reals and likewise only ten
> possible
> digits for the addition and subtraction surplus of its decimal point
> to have the
> sum on the other side of the globe of Elliptic geometry.
>

That is true what I said about the radix-remainder in New Reals is
that
there is a solo digit, only one digit and it can be either
0,1,2,3,4,5,6,7,8,9
The radix remainder has only one place-value.

True also in AP-adics where the decimal prefix is a solo
digit but in this case of addition it can only be a 0 or 1 digit.
Consider the AP-adics such as
999...9999 + 9999....9999 = 1d99999....99998
The 0 digit indicates
the AP-adic number is in the same hemisphere as the two summed numbers
but if the decimal prefix is a 1 means the sum has trespassed over
into the
other hemisphere

So I added a spicer to the definition of Numbers, that they must
possess two
references of decimal point and radix point. I did it not because I
liked doing it
but because I wanted to preserve the Algebra. The Algebra that every
operation
yields a unique number answer. The only way to preserve the Algebra
and
to preserve All Possible Digit Arrangements is to say that every
number
whether it be a Real Number or an AP-adic Number has both a decimal
point and a radix point.

Now we can see that 0.999...9999 is not equal to 1 but is a different
number
and that 1/3 is not equal to 0.333...3333 but rather equal to
0d333...3333r1
and that 2/3 is not equal to 0.666....6666 but rather is equal to
0d666...6666r2

As I sad so many times, the number one job of mathematics is
precision. To say that
1/3 is 0.333.... ignoring the perpetual remainder of 1 is not
precision and thus leads
to phony mathematics such as the alleged 1 equal to 0.999...999. It is
really funny to
me that phony proofs are pandered off to students when one can easily
run a checkup
test of 1/2 = 0.5000.... and why not 1/2 = 0.4999... in Old Reals. so
we multiply
both sides of 1/2 = 0.4999..... and we have 1 = 0.9999.....8 ?? For
the phony 1/3 proof
it is easy to multiply 0.333.... by 3 and yield 0.9999.... but how
does the phony proof
explain away 0.4999.... the fact that multiplying by 2 cannot get rid
of that "8" digit?

This is what I mean that mathematics has to have precision at all
cost. And if mathematics
cannot provide precision then it has failed.

P.S. I want to also note that in this book I have used Algebra just
once and it starts here
with the inclusion of both a decimal and radix points in the two
number systems. Up until
now, I have not needed Algebra. But I needed it now because I want to
preserve Algebra.
This was the huge fault of mathematics in the 20th century where
Algebra ruled supreme
over all of mathematics. But what this book demonstrates is that the
rulers of mathematics
are not algebra but are Geometry and Numbers. And that the most
important features of
mathematics are Geometry and Numbers and Algebra is of lesser
importance.

[plutonium....@gmail.com]

> As I said (sic) so many times, the number one job of mathematics is

Scratch much of the above for it is wrong. The reason I added a
remainder-radix
to Reals and the reason I added a decimal "d" to AP-adics is because
these
two number systems had no room for division leaving a remainder in
Reals and
addition-subtraction leaving no room for "other side of hemisphere" in
AP-adics.
So I appended a "r" to Reals and a "d" to AP-adics in order to not
contradict
All Possible Digit Arrangements whilst keeping all of a completed-
Field Algebra.

But now I am in violation of All Possible Digit Arrangements if I
allow digits for
remainders in Reals for they do not stay confined to simply ten digits
but sprout
and spring up a whole new entire realm of numbers. So again,
contradicting that
Reals were All Possible Digit Arrangements. The 1/3 has remainder 1
perpetually
and 2/3 has remainder 2 perpetually but what remainders are there for
every
division? So do we end up with a second tier of Reals wedged between
every
two All Possible Digit Arrangements such as the two Reals 0.333...3333
and
0.333...3334?

No. I take a clue from the AP-adics. Its division and multiplication
are completed
algebraic fields because the operations stay on the same hemisphere.
But
addition and subtraction can trespass over into the other hemisphere.
So all I need
is a indicator or prefix that says whether the AP-adic is in the
"usual hemisphere"
or has trespassed over into the unused hemisphere and thus I need only
the digit
"1" for the maximum integer addition is 999...9999+ 9999...9999 =
1.9999...9998

So all I need for AP-adics is a 1d999...99998 to indicate 1.999...9998
and if the
AP-adic is a normal AP-adic on the usual hemisphere I just write it as
9999...9999

Now for Reals, where the operation of division is the cause of
problems since
there is a remainder that is not accounted for I am going to do the
same trick
to preserve both (1) All Possible Digit Arrangements not contradicted
(2)
preserve completed algebraic field.

If a division in Reals such as 1/3 leaves a remainder then it cannot
be
the Real 0.333....3333 nor 0.333...33334 but something in between
those two
Reals and I simply denote it as 0.3333...3333r1 where the r is a
remainder-radix

And now for example the division of 10/30 is again 0.333...3333r1
where 1 is the only
digit that can fit into the r slot. The justification is that 10/30 is
really just 1/(30/10)
so that all fractions can be reduced to a "1" numerator.

So any two Reals divided by each other, if they leave a remainder in
pertuity they have
a "r1". If their division is even such as 1/2 = 0d5000...00000 then
they do not need a
"r1" suffix. The number 2/3 needs the suffix for 2/3 = 0d6666...6666r1

Now I preserved the definition where I said All Possible Digit
Arrangements and preserved
completed algebraic field on these numbers with their operations.

Now an interesting question arises that the prefix of 1d for AP-adics
indicates the other
hemisphere. So does the r1 suffix in Reals have some similar
geometrical meaning? If
the 1d prefix in AP-adics is the second hemisphere in either Elliptic
or Hyperbolic
geometry, then I suspect the r1 suffix to Reals for division must have
some sort of
geometrical meaning in Euclidean geometry? I have to give it some
thought as to
what it may mean geometrically.

[David R Tribble]

Archimedes Plutonium wrote:
> So let me analyze those eleven numbers of New-Reals in the interval
> 1.999...9995 to
> 2.000...0005. They are vastly different from the Old-Reals which would
> contend that
> there are an infinity of Old Reals between 1.999...99995 and
> 1.999...99996. In the Old
> Reals they had the idea that there are an infinity of Old Reals
> between any two Old Reals.

Just to be clear, standard math says that your 1.999...995 and
1.999...996 are not real numbers at all.

[plutonium....@gmail.com]

several weeks ago you said Old Reals were all possible digit
arrangements,
now you say they are not

so how long have you been two-faced

[plutonium....@gmail.com]

plutonium.archime...@gmail.com wrote:

I do not know if the word suffix is the opposite of prefix? The prefix
of
1d on an AP-adic to tell if it is in one hemisphere or the other
hemisphere.
To give the AP-adics a modulus function or a periodicity function as
in trigonometry
where the 2pi is the same as 4pi.

But my post here is about that final question in my last post. By
appending a
decimal point to the AP-adics gives me a marker on the number to tell
whether
the addition or subtraction put the final answer into the same
hemisphere or the
unused hemisphere. That multiplication stays confined to the usual
hemisphere
and the radix point helps in division where the answer is also
confined to the
same hemisphere but in addition and subtraction of AP-adics, the
answer can
trespass over into the unused hemisphere. So I need that prefix
symbolism of a
decimal point and only one digit is needed as "1" since the maximum
sum of integers
is 1.999...99998 and thus the sum of 999...9999 + 999...9999 is
999...99998 in the
other hemisphere and denoted as 1d9999...9998. I need that prefix
notation to
complete the Algebraic Field of AP-adics otherwise I cannot have a
suitable addition
and subtraction.

Simple enough of a correction, but wait until you see the problem of
New Reals
with division. And to remedy that problem I append a suffix of "r1"
where the r
stands for remainder-radix. And here again I need only one digit of 1
since all
divisions can be represented as a "1" for the numerator.

Now why do that? Why not just pretend and behave and act like a Old
Math
timer with the Old Reals, and pretend as if 1/3 is just 0.333.... and
the dots indicating
infinity sweeps all the troubles under the rug. Sweeps away the
annoyance that there
is a remainder of 1 not accounted for. Well, the old math timers were
lazy. Lazy in
logic and lazy in mind and never accepted that mathematics is
precision. And the old
timers Algebra of Galois theory worked just as well with 1/3 sweep
under the rug
because no-one raised questions about the perpetual remainder of 1 or
the perpetual
remainder of 2 in 2/3 as 0.666..... So long as the old timers had
those dots saying
out to infinity, well, precision was chucked out the window as well.

And since Algebra was not affected by the imprecision, noone
questioned the old-timers.

But with FrontView and BackView of Reals, the questions come in
torrent flood, and the
oldtimers of math with their Old Reals can no longer sweep under the
rug and must
account for their lack of precision.

So if 1/3 is not the Real 0.3333.... or 0.3333....3333 then what is
it? Well, it has a remainder
of 1 that is not accounted for in Old Reals and so 1/3 is not
0.3333.... but is the number
0.333...3333r1 and 2/3 is the number 0.6666...666r1 and where 1/2 is
the number
0.5000.... or 0.5000...000

So what is the geometrical meaning of why the Reals need a r1 suffix?
The AP-adics
are the intrinsic numbers of NonEuclidean Geometry and the prefix 1d
is required to
tell if the number is in which hemisphere. The Reals are the intrinsic
numbers of
Euclidean Geometry and the reason a suffix of r1 is needed is to make
every Real
Number evenly divisible at the point of infinity. To make every Real
Number complete
and closed with division requires a suffix that ends all remainders.
The prefix in AP-adics
completes the algebra for addition and subtraction. For the Reals, the
algebra of a
completed field is taken care of for add, subtract and multiply, but
division is left open
with a gaping hole of a remainder at infinity.

The Old Timers of math missed the gaping hole of the Old Reals. They
were not a completed
algebra, because they never addressed the problem of division at the
point of infinity.

The old timers left the Old Reals as not a Completed Field because of
numbers such as
1/3. The 1/3 is not 0.333.... for it is the number 0.333...333r1. The
word "sloppy" is the word
that epitomizes the old timers of mathematics with their completed
Galois Field of the Reals.

So what is the geometric significance of the suffix to Reals? It
allows the algebra to be a
closed completed field on the Reals, otherwise there is internal
contradiction such as
All Possible Digit Arrangements and we have phony proofs such as
0.999... = 1. What
the suffix does is place an additional number between two contiguous
Reals such as
0.333...3333 and 0.333...3334 such that there is a number
0.333...3333r1 between those
two.

And where 0.333...333r1 x 3 = 1 whereas 0.333... x 3 = 0.999...

Now I have a question about Galois Algebra. Why does the AP-adics pick
on the addition
and subtraction whereas the Reals picks on division as troublespots?
Was there a problem
with the Hensel P-adics with addition and subtraction?

So why does Reals and the Euclidean Geometry have troubles with
division? Why division?
And is this trouble somehow connected to Calculus. So the smallest
neighborhood of
zero is this neighborhood:

(-)0.0000...0001, (-)0.0000...000r1, zero, 0.0000...0000r1,
0.000...00001, 0.000...00001r1,
0.000...00002, 0.0000...0002r1, 0.0000...0003,

So the geometrical meaning of the suffix is a end to all division
where every division ends
as a even division with no remainder and thus the number fits between
two numbers
if there had been a remainder. Sort of like a wall for division. The
geometrical meaning of the prefix
for AP-adics was very clear as to the second hemisphere. But the
geometrical meaning of the
suffix was very much subtle, in that division was not closed and
cleaned up, that division was
not fixed at infinity. And it lead to silly sham proofs such as what
is 1/3 and 1 = 0.999...

[plutonium....@gmail.com]

plutonium.archime...@gmail.com wrote:
(all else cut)

>
> So why does Reals and the Euclidean Geometry have troubles with
> division? Why division?
> And is this trouble somehow connected to Calculus. So the smallest
> neighborhood of
> zero is this neighborhood:
>
> (-)0.0000...0001, (-)0.0000...000r1, zero, 0.0000...0000r1,
> 0.000...00001, 0.000...00001r1,
> 0.000...00002, 0.0000...0002r1, 0.0000...0003,
>

I was a bit nervous as to whether every Real between 0 and 1 has
a suffix r1 sandwiched between them, especially 0. A division
where the dividend is 0.0000.... yet with a remainder.
But then, what about 0.000...00001/0.000...00002? It cannot be
a number with a place value of 10^(-)9999...9999 but has to have
a place value of 10^(-)999...9998 so that division is
0.0000...00000r1

Now are these numbers actually Reals? The number 0.000...0001
is a Real and the smallest positive Real because 0.999...9999 is the
largest Real in the open-interval (0,1). But does the suffix r1 change
all that? Is 0.000...0000r1 the smallest Real and 0.999...9999r1 the
largest Real in the open interval (0,1)

I have to think on that perplexion, some would say vexation.

But let me change the subject for the moment. I asked what geometrical
meaning is there to the r1 suffix? I said it closes or completes the
Reals
as algebraic field otherwise no Real represents numbers like 1/3. So
the
geometrical meaning is that there is a Real with a suffix sandwiched
in
between two-contiguous Reals and the geometrical meaning thereof was
a filler or padding in between two contiguous Reals such as
0.3333...333
and 0.3333...334.

But now, let me offer a more lucrative geometrical meaning. In
Calculus,
what makes it work is that you can always get a smaller interval
surrounding
a point and called the limit. So if we have a suffix r1 sandwiched in
between
any two given Reals, then the suffix becomes in essence the limit. So
let
us say we are looking for the derivative at the point 0.333...333 and
the
immediate or contiguous Reals are thus 0.333...3332, 0.3333...3333,
0.3333...3334. So if those were all the Reals in that region, then we
fail
to have a limit concept. But if we pack a suffix r1 between those
contiguous
Reals such as this:

0.333...3332, 0.3333...3332r1,
0.3333...333, 0.333...333r1, 0.3333...3334,
0.3333...3334r1

So what I am thinking is that Calculus exists on the Reals only
because
of a limit concept, but we can only have a limit concept of the Reals
are
not contiguous points but have a buffer zone between each Real.

So this would beg the question as to why division operation is
troublesome
with its remainder at infinity. So a remainder in division at infinity
is related to
the calculus limit function.

More later.

[plutonium....@gmail.com]

Actually what has been done here in this new creative discovery of
mathematics is that a number
has two decimal points, so that decimal points come in pairs, just as
the numbers themselves come
in a leftward string and a rightward string. So why should any of us
feel alarmed that if a number has
two strings that it has also two decimal points.

Now the funny thing is, that it is Algebra leading the charge here,
not Numbers, not Geometry,
not Probability theory, but the old good old Galois Algebra theory of
groups, rings, fields.

So we have the Old Reals with such contraptions as 1/3 = 0.333.....
and where they had phony proofs
that 1=0.9999..... Where they swept infinity under the rug of those
three dots that indicates infinity and
where the old timer mathematician simply stopped thinking. Stopped
thinking because 1/2 is truly
0.5000.... but that 1/3 is not 0.333.....

So Algebra comes to the rescue, and what Algebra does is says we must
preserve All Possible Digit
Arrangements as Reals. We must preserve that, and that means that all
numbers of mathematics
must have two decimal points. For the AP-adics, we have to append a 0d
or 1d to preserve algebraic
addition and subtraction. The 0d indicates one hemisphere whereas the
1d indicates the other hemisphere.
In Reals, the problem is with division. The smallest place-value is 10^
(-)999...9998 so a number such
as 1/3 has a remainder of 1 at the smallest place-value meaning that
1/3 is not the same as
0.3333.... or written with frontview and backview as 0.3333...3333. So
to make the Reals complete
to an Algebraic Field of Galois Algebra the Reals need two decimal
points and I call it a remainder-radix
and its value is either r0 or r1.

So all numbers have both a d and a r decimal points whether they are
AP-adics or Reals.

In the case of 1/2 it is 0d50000.....00r0 in the case of 1/3 it is
0d333...3333r1

In the case of 0d9999....9999r + 0d000...0002r = 1d0000....0001r
which means that the sum is one metric unit distance across the South
Pole

Now I said that the Northpole was the appended number "e" and the
South Pole was
the appended number "pi". I reserve the right to change my mind
frequently, and in this
case, I think that the North Pole on a Elliptic geometry sphere is 0d
and the South Pole
is 1d, and where all the AP-adic numbers on the one side hemisphere
have a 0d prefix
and all the other AP-adics on the other hemisphere are 1d prefix.

Now I believe the Old Reals worked with Calculus limit function
because they erroneously
believed an infinitude of Reals between any two Reals. But in All
Possible Digit Arrangements
as the Reals does not allow that concept of betweeness that the old
timer mathematicians
used and thus the Calculus of limit breaks down and where they
conveniently had fake proofs
that 1 = 0.999...

So to restore the Reals, to restore the Calculus, and to restore the
Galois Field for Reals, is
all taken care of by the simple insertion of not only one decimal
point but that every Real has
two decimal points. The first one is well known and has been used
throughout history, but the second
one called r1 or r0 was discovered by me in the last few days. It is
to allow All Possible Digit Arrangements
for the Reals and to simultaneously save the Reals as a Algebraic
Field. Without the appended
second decimal point of either r1 or r0, then division breaks down
because 1/3 is not
0.3333..... and 2/3 is not 0.6666..... and 1 is not 0.9999......

But that 1/3 is 0d3333....3333r1 and 1/2 is 0d5000...000r0 and 2/3 is
0d666...666r1

Why only a digit of either "0" or "1". Because in Reals all numerators
are converted to a 1
so we only need a 0 and 1 to work for Reals. And in AP-adics we need
only a 1 for the
largest integer sum is 1.999...9998.

So the prefix is only either 0d or 1d and the suffix is either r0 or
r1.

That saves All Possible Digit Arrangements and makes the largest place-
value as
10^9999...9998. And that saves Algebra of its Field for both Reals and
AP-adics
by saving division in Reals and saving addition and subtraction on AP-
adics.

Now I wonder if 0d and 1d are the poles in both Positive and Negative
AP-adics?? I wonder
if the poles are better served as 0d and 1d rather than as "e" or "pi"
or "2pi"??

Maybe with time and reflection I can discern which is the better.

[spudnik]

so, you really killfiled me, without saying, so?

anyway, all of this alleged math & the ambiguity
of your New Real format is alleviated,
simply by using base-one. perhaps, though, then,
it appears as a rather simple abSURDity.

> > Just to be clear, standard math says that your 1.999...995 and
> > 1.999...996 are not real numbers at all.
>
> several weeks ago you said Old Reals were all possible digit
> arrangements,
> now you say they are not

thus:
yeah, or a "hyperpoint," or a hypoplane?

> is a line segment the closed convex hull of a finite set in R?

thus:
you should study the brachistochrone of Liebniz, Bornoulli,
Huygens, Fermat et al, because it is all downhill from
"Forsooth, I have got a formula." just because I reply, does not mean
that I believe that you are in any way serious or sincere. if you
insist,
the Copenhagenschool parody of Schroedinger's Headgasket is not the
sine
qua non plus ultra of "existential predeterminacy," mod Heisenberg's,
may be I'll just laugh & quit this thread.

I am not the least practiced in Liebniz' calculus, but
I can still not pretend that you are, either,
based upon your limpwristed premonition
of fuzzy logic. now, if the "space deforms according
the existential function, f, or the integral, F,"
you would still have to do some math and
establish this mysterious operator.

all mathematical objects are ideals;
one reifies the math at one's own peril -- or
the general laughter of the Peanut Gallery.

> It would be very easy to calculate relativistic corrections using
> existential indeterminacy. It is an almost elementary exercise, as is
> the precession of planetary perihelions. These things become very easy
> to calculate.

--only 24 hours to impeach Trickier Dick from the N.Admin,
metaphorically typing, or Cheeny & Zbiggy, fo'mo' years;
Good Morning, Afghanistan!
... Good Afternoon, Sudan!
http://tarpley.net/bush12.htm
http://wlym.com/campaigner/8011.pdf -- Brits hate Shakes, Why?
http://www.wlym.com/~seattle/dynamis/
http://www.21stcenturysciencetech.com/current.html
http://www.rwgrayprojects.com/synergetics/plates/plates.html
http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=3163
http://wlym.com/campaigner/8011.pdf

[plutonium....@gmail.com]

plutonium.archime...@gmail.com wrote:

>
> So all numbers have both a d and a r decimal points whether they are
> AP-adics or Reals.
>
> In the case of 1/2 it is 0d50000.....00r0 in the case of 1/3 it is
> 0d333...3333r1
>
> In the case of 0d9999....9999r + 0d000...0002r = 1d0000....0001r
> which means that the sum is one metric unit distance across the South
> Pole
>

What a quirk of fate the above examples, where I should written the
last
example as -- In the case of AP-adics of 0d999...999r + 0d0000...0002r
=
1d0000...0001r which means that the sum is one metric unit distance
across the South Pole.

I should have included "In the case of AP-adics" because a reader
could
easily mistake the sum as a sum on Reals thinking I was writing
about 0.999...999 + 0.000...0002 = 1.000...00001.

So one can see that we can easily be confused whether we are writing
about
a Real in the interval 0 to 1 or writing about an AP-adic.

Of course if I wrote 14d999...9999r1 there is no mistake that it is a
Real since
it is between 14 and 15.

And no mistake that 1d9999...9999r2 is a AP-adic since the Reals r
suffix
can only be r0 or r1.

But let me designate the two decimal points so that never an ambiguity
arises.
Let me call the Real decimal point, the one we are familar with when
we
write say 1.5, and let me call the decimal point a "d"

Let me call the AP-adics radix point an "r" in the tradition of the
Hensel P-adics
which called it a radix

Let me call the AP-adics prefix decimal point a "p" for pole.

And that finally leaves the second decimal point in Reals begot from
remainder
carryover in division and let me call it a "c" for carryover.

So the Real 1/3 looks like this 0d333...333c1, and the Real 1/4 looks
like this
0d25000...000c0

And the AP-adic of the largest AP-adic integer looks like this in the
first hemisphere
0p9999...9999r0 and in the second hemisphere looks like this
1p9999...9999r0
and the smallest integer in the first hemisphere looks like this
0p000...0001r0 and the smallest integer in the second hemisphere is
this 1p0000..00001r0 Now the AP-adic that simulates the Real 1.5 in
the
first hemisphere looks like this 0p0000...0001r5

Now the interesting question is whether a Real such as say 2/3 which
would be
0.666.... in Old Reals but would be 0.666...666c1 in New Reals, the
question is
whether that is a genuine valid number? Because we defined them as All
Possible
Digit Arrangements with one decimal point so can a second decimal
point render
a valid number? And this is what I mean by Algebra Field theory taking
the lead.
If that number for 2/3 as 0.666...666c1 is not a valid legitimate
number as is the
number 1 or the number 1.5 or the number 100, well, then we do not
have the
Reals as an Algebraic Field because division falls apart. The Old
Reals never
questioned division and said that 2/3 = 0.666.... covers up all the
ills and dirty
trash of the Reals. That we pretend that the dots of infinity removes
all questions
about the carryover of "2" in the infinity division. So the old math
left the problem
of division unanswered. They left mathematics as imprecise. So to
solve that
imprecision we need to have two decimal points to every number that
exists.
By the suffix c decimal point which can have only a value of either c0
for an
even division with no carryover or with a c1 suffix where there is a
carryover.

So these numbers are just as valid as the number 1.5 or 12 or 8.200...

Now I need to research the Calculus for these two decimal points in
Reals
makes a different picture of the Limit concept in Calculus. And one
would
almost say that the carryover remainder allows the Limit concept to
exist.
In the old-math, the limit concept was shrouded in the prophecy that
between
any two Reals exists an infinity of more Reals. Well that contradicts
All
Possible Digit Arrangements. Because there are no more Reals between
0d333...3333c0 and 0d333...333c1 and 0d333...3334c0 Between those
three Reals are two holes or two gaps which no other Real can plug
into
those two gaps. So that when we do a derivative or integral at the
point
in Reals of 0.333...333 the neighboring Real on the right side is
0d333...333c1 and on the leftside is the Real 0d333...3332c1

Now one can visualize that a number such as 0d333...3333c1 was begot
by
an infinite divisions of different numbers which all ended in the
string
333...3333 and yet had a carryover where 1/3 is such a division and
3/9 another such division. Whether that has any meaning as to the
Limit
concept of Calculus is up for question.

[plutonium....@gmail.com]

I am going to write a separate chapter in this book on how the
Calculus Limit concept is contained within
the second decimal-point of the suffix c0 or c1 of infinite division
with remainder. In other words Calculus
comes out of Algebra as a completed Field and where division requires
a second decimal point to get
rid of all remainders.

And that makes commonsense when you think about it, that the Calculus
was all along a process
of infinite division, both derivative and integral. And one of the
reasons that there is no Calculus in
NonEuclidean geometry, only Euclidean geometry, is because of the
second decimal point suffix
c exists only on Reals.

But let me spend a good amount of time on the foundation concept of
Numbers as All Possible Digit
Arrangements. I have not spent enough time telling of how important
that single concept is. Perhaps
the most important concept in this entire book.

The Old Math timers with their Old Reals were wrong on these features
of Mathematics:
(a) they were schizophrenic on the concept of All Possible Digit
Arrangements because in
one instance they would say that their Old Reals possess All Possible
Digit Arrangements
and yet in the next minute they would blurt out that between every
Real y and z exists
an infinity supply of more Reals. Well, if you understand and believe
in All Possible Digit
Arrangements, the concept of Betweenness no longer is valid, and the
idea that between
any two Reals supplies and infinity of more Reals is absurd and
contradictory. So the moment
anyone says Reals have to be All Possible Digit Arrangements,
logically forces them to
understand that between the Real Numbers 0.333....333 and
0.333...33334 exists no other
Real Number in All Possible Digit Arrangements, unless you include the
second decimal point
suffix, and even then, there are gaps and holes.

So, the Old timers of math and the Old Reals were schizophrenic. They
wanted their Reals to
be All Possible Digit Arrangements but they failed to understand what
that meant. Because if
they sat down for a minute and reflected on the concept of All
Possible Digit Arrangements they
would understand that it means there are tiny holes between every two
Reals and that the
concept of Betweenness of an infinitude of Reals between every two
Reals is contradictory.
So, you either have that Reals are All Possible Digit Arrangements or
you have that Reals
are infinite supply between any two Reals. You cannot have both. One
must go.

(b) The old timers of math also failed to realize that All Possible
Digit Arrangements also forces
the Reals to be Countable and thus the world has only one type of
infinity. In the first edition of this
book someone (Dik Winter, I think) posted what Cantor wrote for his
denumerable Reals, claiming
that Cantor knew of the concept of All Possible Digit Arrangements. I
find that somewhat difficult to
believe that Cantor simultaneously understood the concept of All
Possible Digit Arrangements as
well as one-to-one correspondence. I can understand that Cantor was
confused and knew
of a concept of All Possible Digit Arrangements but did not understand
what All Possible Digit
Arrangements means. Many of us, probably all of us have this
experience where we talk and use
some concept but do not fully grasp what the concept actually means.
This is a difference between
being "aware or knowing" and the difference in actually
"understanding". Being aware and knowing
can sometimes be a foggy state of awareness or knowing. To understand
something means there
is some form of clarity in the knowing. So that Cantor was probably
aware of a concept of All Possible
Digit Arrangements but Cantor had no handle on what it really meant.

Here is a sample of the 1st edition of this book where Cantor talks
about All Possible Digit Arrangements
as translated by Dik Winter:

> "Sind nämlich m und n irgend zwei einander ausschliessende Charaktere, so
> betrachten wir einen Inbegriff M von elementen
> E = (x1, x2, ..., xv, ...)
> welche von unendliche vielen Koordinaten x1, x2, ..., xv, ... abhängen,
> wo jede dieser Koordinaten entweder m oder w ist. M sei die Gesamtheit
> aller Elemente E."
> And if you wish, in translation:
> "Namely, let m and n be two different characters, and consider a set
> M of elements
> E = (x1, x2, ..., xv, ...)
> which depend on infinitely many coordinates x1, x2, ..., xv, ..., and
> where each of the coordinates is either m or w. Let M be the totality
> of all elements E."
> In what way do you think that here M is *not* the set of all possible
> arrangements?

I do not know how reliable that translation is.

But I do know this with certainty, that you cannot have a
understanding of
All Possible Digit Arrangements and then end up with the idea that the
Reals are
uncountable.

If anyone understands All Possible Digit Arrangement will end up with
the conclusion that
the Reals are Countable. All Possible Digit Arrangements and
Uncountable are contradictory.

So if Cantor truly understood All Possible Digit Arrangements then he
would never have
accepted transfinites, never accepted a hierarchy of infinities.

(c) Thirdly the concept of All Possible Digit Arrangements demands a
smallest number
and a largest number and these would be 0.000....00001 and
0.9999...9999 in Reals between
0 and 1. And by doing so, forces there to be a largest and smallest
decimal place-value
as 10^9999...99998 and 10^(-)9999....99998

(d) And finally, perhaps the most egregious error of the old timers of
math with their Old Reals was
that none of them seemed to ever raise voice about 1/3 or 2/3 or
0.9999..... And this should be
a harbinger to all scientists in all subjects of science. If your
science subject harbors a "defect of
logic" or a "imprecision" or a conundrum or a point of contention,
then your subject is flawed.

It is obvious to anyone, whether a old-timer stuck in his ways or a
student in High School. That
if you convert 1/2 to a decimal number it is a breeze of 0.5000...0000
but when you convert
1/3 there is that remainder carryover of 1 so that it cannot be
0.333...., and that it must be
0.33333....3333c1.

All Possible Digit Arrangements forces these things:
(A) Reals are continuous in an interval where there are an infinite
supply of Reals such as
0.1000....0000 to 0.2000...0000 interval. But the Reals are discrete
as we microscope into
a tiny interval such as 0.099999...999, 0.1000...0000, 0.1000...0001
where we have three
sequential Reals and two gaps between them.
(B) Reals are Countable because All Possible Digit Arrangements forces
them to be
1 to 1 correspondence with the Counting Numbers
(C) Reals have a largest Real and smallest Real in the open interval 0
to 1
(D) Reals have a upper bound and lower bound on place-value imposed by
All Possible
Digit Arrangements
(E) Reals in order to be a completed Algebraic Field must have a
second decimal point as
suffix c, whose value can be either c0 or c1 and which is the
remainder carryover in division
(F) This second decimal point of a remainder carryover is the Calculus
Limit concept.

Summary: If a person really and truly understands the concept All
Possible Digit Arrangements
then it is in contradiction to both "uncountable" and to "infinite
betweeness microscopically"

P.S. I should write a whole chapter on that the Calculus comes into
existence due to
All Possible Digit Arrangements and the need to have Reals a completed
Field as per
division such as 1/3 = 0.333...3333c1. In that chapter I will build
the Calculus from scratch
and never use the Limit concept but use only graphs of functions. In
other words, the Limit
was never an essential part of mathematics.

[plutonium....@gmail.com]

Well, I never thought I would one day be revising Calculus and
offering the best way to teach the
subject. And I would not embark on such a journey unless I had made
some large change
to a subject. The New Reals does make a huge change on Calculus,
impacting the concept of
Limit.

Did you ever get the feeling in reading any College Calculus book that
they never seemed to
explain it with clarity or with understanding? Did you ever get the
feeling that every Calculus
book says the derivative is the inverse of integral, just as add is
the inverse of subtract and
multiply the inverse is divide? Oh, sure, we all can understand the
inverse of add is subtract
and the inverse is multiply is divide, but who can say, well, that
calculus book was so
good that it finally made a visual image in one's mind how the
derivative is the inverse
of integral? None. There is not a calculus book in the world at
present that can teach
why the derivative is the inverse of integral. Not a one because all
of them rely on the
Limit concept which is a rather phony concept because the Old Reals
were phony
with contraptions like 1/3 = 0.3333.....

I am going to make a separate chapter on the Calculus but I am going
to guide the way
for that chapter as a preview.

Because the New Reals are All Possible Digit Arrangments and because
they have
two decimal points for taking care of division such that 1/3 =
0.333...3333c1. Then
that gives a new meaning to the Limit Concept. It is no longer needed.
We know
exactly what numbers lie next to another Real. The Limit Concept was
needed in the
old phony math of Old-Reals because they drenched themselves with a
silly idea that
between the Real 0.9999.... and 1.000....0001 was an infinite supply
of more Reals
when in fact there are exactly three Reals between them:

0.9999..... , then 0.9999....c1, then 1.000..... , then 1.0000....c1,
and then 1.0000...00001

So let me preliminarily start the chapter on Calculus, the way it
should be taught
in all schools, and which cuts out the phoniness.

Since I am limited with graphics on the computer posts, I will refer
to a graph on a page of "Calculus" Gilbert
Strang, 1991.

The basic idea of Calculus is not the Limit concept, but the concept
that rate-of-change is the
inverse of area, or spoken as, differentiation is the inverse of
integration.

The reason the old timers brought in the Limit was because they never
had a "good understanding
of the Reals". With the New Reals, we do not need a Limit, because we
can immediately tell
what the neighbors are for any Real Number. The Reals have no
transcendental numbers for that
was phony baloney also.

Now let me show you how Calculus education should be conducted by all
schools:

We start Calculus with the aim and goal of showing that the Derivative
or Differentiation is the
inverse of Integration. That is our end goal. If a student goes
through this course and understands
how those two are inverses, then they have learned the Calculus. So
how do we start to teach
Calculus to arrive at that endgoal?

We start with the area of a Right triangle is 1/2 base X altitude. And
we start with a specific
right-triangle that is 1/2 the area of a square.

And we start with the world's most simple function of f(x) = x, or we
use the terminology
that y substituted for f(x). So the world's most simple function of y
= x
and is a diagonal line that bisects the 1st quadrant of the Cartesian
Coordinate System.
I will talk about functions only in the first quadrant.

So already one can perceive a connection. A connection with a Right-
Triangle and its
area as 1/2ba or in the right triangle of a square is 1/2x^2, and the
connection with the
function y = x. Look at that function and look at the right-triangle.
Can you see where
the diagonal that bisects the 1st quadrant is a right-triangle all
along the x-axis.

Now I bring in the picture of a picket fence for integrals. Where we
turn every point of the
function into a picket-fence and add up all the picket-fences to find
the area. Only to get
the proper sharp pointed end of the picket fence depends on the slope
or tangent. So this
dependency of the slope or tangent to form the picket fence is the
derivative and how it is
the inverse of the area.

Now the derivative of the function y = x is simply just 1. This is
what all the calculus books
teach us these easy rules to find the derivative as x^n becomes n(x^
(n-1)) and the integral
of x^n becomes (x^(n +1))/(n+1)

So the derivative of the world's most simple function y = x is 1, and
the integral is
(1/2)x^2

This could be teaching Calculus all in one page.

Notice that the integral was nothing but the right-triangle in the 1st
quadrant.

Notice that the slope of the line graphed by y = x is a line whose
slope is 1. Its tangent
to the line is the line itself with slope 1. Just as the right-
triangle that is 1/2 of a square has
slope of 1.

Now make any function that lies in the 1st quadrant, make some that
curve upwards fast,
or make some that lie low to the x-axis or make some that bobb up and
down. The point is
that the Calculus merely imposes that simple function of y = x over
all functions and that
the accompanying right-triangle is used on all the functions, so that
the slope is adjusted for
the new function and thus the picket-fence that is formed is adjusted.
Now Gilbert Strang is
trying to explain this on page 94 with his graphic. But I can explain
it so much easier. Given
any function other than the y = x function, that we impose the y = x
function on the new
function and simply adjust the slopes at each point, or inversely,
adjust the picket fence area.

So, never a word or mention needed of a neighborhood of any points. We
already know what the
neighbors of a point are because the New Reals tells us the neighbors
of any Real Number.

And now we can begin to see how derivative is the inverse of
intregral, in that the hypotenuse of
the right-triangle that fits the graph of the function at a point
determines the picket-fence of the
area of that picket-fence.

And we see also why division is not a completed-field algebra in Old
Reals because of numbers
such as 1/3 cannot be 0.3333.... but must be 0.333...c1

Area is multiplication and tangent slope is division. Multiplication
is the inverse of division,
and so Integral is the inverse of derivative.

In the Old Reals, such as what Strang is trying to teach from his book
shrouds the idea
of the slope as inverse to the area, and shrouded because of the
arcane idea of a limit.

A picket fence requires a slope to have a base times altitude. The
slope is provided by that
second-decimal point, for instance the Real 0.333....3333 has its
immediate neighbor of
0.3333...3333c1 to make two points to provide a slope line on the
picket fence. And so,
now the area under the point is provided.

The above is a preliminary and the chapter should be vastly more
clear.

[plutonium....@gmail.com]

This book has alot of new concepts, some of them earthshattering for
mathematics, for it
uncovers those areas of mathematics that are phony and it provides new
techniques in
mathematics.

I feel it important to stop and give the history of these new ideas,
not for any personal
reason but for to see how creative discovery process works. Some
people do not have
the time to read all of my posts which number in the thousands and
they would like to
research how it was that I came to those beautiful new concepts. So
here is an attempt
to outline the discoveries.

In this book I would rank the discoveries as following:

(1) Euclidean geometry = Elliptic geometry union Hyperbolic geometry,
circa 1991
(2) Euclidean geometry = Reals circa 1991
(3) Elliptic and Hyperbolic geometries are formed from AP-adics,
started 1993,
only with P-adics and eventually in 2007 became AP-adics
(4) Reals are All Possible Digit Arrangements, circa 1991
(5) Frontview and Backview on all numbers, 2007
(6) Every number has two decimal points, a prefix and suffix decimal,
circa 2007
and fully bloomed 2008

So I would like to outline a history of those new discoveries and give
a representative
post to the Internet on the discovery.

#######

(1) Euclidean geometry = Elliptic geometry union Hyperbolic geometry,
circa 1991
(2) Euclidean geometry = Reals circa 1991
(3) Elliptic and Hyperbolic geometries are formed from AP-adics,
started 1993,
only with P-adics and eventually in 2007 became AP-adics

This concept came to me while trying to proof outstanding conjectures
in math
in 1991 and the idea came to me that fitting a ball on top of a saddle
shape
should yield a cube. And how unsatisfactory is mathematics that they
have
to borrow the Reals to talk about arithmetic on NonEuclidean geometry.
That
they should have their own numbers, native to them. And when I began
posting
to sci.math in August of 1993, I soon learned there was P-adics of
Hensel
and that the numbers have a "natural arching around back to where they
started from." This natural arching around, spurred me further to the
idea
that infinite strings leftward (opposite of Reals) were those numbers
for
NonEuclidean geometry.

The below is an early representative post of mine to sci.math
where I outline my concept that Euclidean Geometry is the union
of Elliptic and Hyperbolic geometry with their associated intrinsic
and native numbers. I remember specifically a post asking Karl
Heuer about the idea of Eucl = Ellipt + Hyperbolic
Only in those years of 1993 and beyond I called the formula
Euclidean = Riemannian + Lobachevskian

Newsgroups: sci.math
From: Ludwig.P...@dartmouth.edu (Ludwig Plutonium)
Subject: PROOF OF THE POINCARE CONJECTURE
Message-ID: (CBv18...@dartvax.dartmouth.edu>
Date: Mon, 16 Aug 1993 22:24:08 GMT
Lines: 159

The stereographic projection of the circle unto a line,
and the sphere unto a plane is seen in graph 3. This is
the first mistake topologists have made in the statement of
Poincare's conjecture. They have wished and assumed this
projection is a homeomorphism, but it is not, for as bizarre
and counterintuitive as it is, the sphere has one more point
which is never mapped unto the plane (likewise circle with
line). For positive and negative infinity are not points,
but concepts, yet the polar point is a point which exists
materialized. To wish that infinity (and different types of
infinity-- positive and negative at that) is a number, and
is completed at a point is only wishful thinking and never
mathematics.
Graph 3: Stereographic projections of a circle onto the
Real number line, and a sphere onto the Real number plane.
I quote from MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN
TIMES 2: "The question of dimension, as already noted, was
raised by Cantor's demonstration of a one-to-one
correspondence of line and plane and by Peano's curve, which
fills out a square. Frechet (already working with abstract
spaces) and Poincare saw the need for a definition of
dimension that would apply to abstract spaces and yet grant
to line and plane the dimensions usually assumed for them.
The definition that had been tacitly accepted was the number
of coordinates needed to fix the points of a space. This
definition was not applicable to general spaces.
In 1912 Poincare gave a recursive definition. A
continuum (a closed connected set) has dimension n if it can
be separated into two parts whose common boundary consists
of continua of dimension n-1. Brouwer pointed out that the
definition does not apply to the cone with two nappes,
because the nappes are separated by a point. Poincare's
definition was improved by Brouwer,Urysohn,and Menger.
(Continued). Another widely accepted definition is due to
Lebesgue. (Continued)."
The definition of dimension used by topologists is a
second mistake. What is wrong with the definition of
dimension is analogous to the definition of an electron as
an electron planet going around a nuclear sun. Later
quantum theory corrected this Bohr atom model. Likewise,
the definition of dimension used in Poincare's conjecture
and what we usually think of dimension before this book,
is wrong, because it is not based on quantum physics, but
rather based on a simplistic notion of Euclidean geometry
leaving out Riem,Loba,complex, hypercomplex geometries and
so on. The proper definition of dimension comes from atom
geometries using the quantum superposition principle and
is as follows: dimension 1 is Riemannian geometry;
dimension 2 is Lobachevskian geometry; dimension 3 is
Euclidean geometry which contains the superposition of
Riem. and Loba. simultaneously; dimension 4 involves the
complex numbers and so contains the superposition of Riem,
Loba, Eucl,simultaneously; dimension 5 is hypercomplex
geometry and involves those types of numbers, and so on.
To each of these geometries or dimensions corresponds a set
of numbers, for Riemannian it is the set of all positive
real numbers, Loba., the set of all the negative real
numbers, Eucl., the set of all the reals which includes
the zero concept, and so on. A proper analysis for the
meaning of different dimensions is parallel to the
construction of the different types of numbers. See my
proof of The Infinitude of Geometries. The construction
starts with the positive integers as given, and they
produce the positive rational numbers which then produces
the positive real numbers. The positive real numbers
forms positive curvature geometry which is Riem geometry.
Then the mathematical operation of subtraction produces
negative numbers, negative numbers are different types of
numbers than positive numbers. Then the mathematical
operation of roots produces complex numbers and so on.
The coordinate system for Euclidean geometry was
formulated by Descartes and is well known as the cartesian
coordinate system. But the cartesian coordinate system is
not intrinsic to Riem geometry. Riem. geometry has no
concept of betweenness and so Dedekind cuts are not
intrinsically generated from Riem. geometry. The intrinsic
coordinate system for Riem. is a logarithmic spiral
starting at zero, but excluding 0, and spiraling out to
include all the positive Reals numbers (positive
Rationals).

#######

ALL POSSIBLE DIGIT ARRANGEMENTS history
(4) Reals are All Possible Digit Arrangements, circa 1991

This concept I had whilst trying to prove those outstanding
conjectures
in 1991 and I cannot remember which conjecture I thought of this
concept or had need for this concept. It may have been the Infinitude
of
Twin Primes conjecture or it may have been Fermat's Last Theorem. But
I was certain that I had the concept in 1991 and then when I entered
sci.math
in 1993, I would on occasion blurt it out as in this below post of
1994 shows.

Newsgroups: sci.math, alt.sci.physics.plutonium
From: Ludwig.Pluton...@dartmouth.edu (Ludwig Plutonium)
Date: 14 Feb 1994 12:35:47 GMT
Local: Mon, Feb 14 1994 6:35 am
Subject: 3 PROOFS THAT PEANO AXIOMS ARE FLAWED
Reply to author | Forward | Print | Individual message | Show original
| Report this message | Find messages by this author
Below, I am offering these 3 proofs that the Peano Axioms are flawed
and because of these flaws, proofs such as FLT, Goldbach, infinitude
of
perfect numbers, infinitude of twin primes, infinitude of
constructible
regular n-sided polygons, and other proofs of infinitude were
impossible to prove--not because the proofs were difficult but because
the mathematical foundation underpinning these conjectures were fuzzy
and flawed with error. GC and FLT especially are perhaps the first
math
theorems which told mankind to look not for a clever proof method for
no proof exists so long as we have a flawed foundation. Proving FLT
from Peano Axioms is analogous to trying to build the Eiffel Tower in
mid-air, impossibility because in violation of physical law.

Plutonium Axioms:
P1) There exists two numbers 2 and 1.
P2) 2 =1+1
P3) From P2, having been given equality and addition, new numbers are
manufactured by adding 1.
P4) If a set of numbers contains 2=1+1, and all the numbers
manufactured by adding 1, then it is the set of all Whole Numbers.

Definition of Whole numbers: all possible infinite digit strings
leftward of the radix point and where all strings rightward are 0's.
Notice that all of those infinite leftward strings from the radix
point
of P-adics and N-adics are subsets of the Whole numbers. (P-adics
narrowly defined with all finite rightward strings as 0's.) Notice
that
all the finite numbers which we call positive integers are infinite
numbers with endless leftward 0's.

Three proof arguments follow that the "old archaic finite integers,
what we call the Counting or Natural numbers" are infinite integers.
That is, the Peano Axioms yield P-adics (narrowly defined).
(1) Consider the Naturals with their endless strings of 0's to the
left of the last digit. Observe that the Naturals are a proper subset
of the P-adics. For example the Naturals 1, 94, and 231 are P-adics
...001,and ...0094. and ...00231. respectively.
The Naturals never have a largest member by the Mathematical
Induction postulate. The P-adics do have a largest member as
demonstrated by ...999. The number ...999. is the last of all possible
digit arrangements. There exists no other maximal possible digit
arrangements. The Naturals are Whole Numbers and the P-adics are Whole
numbers. Since the P-adics have a largest member and the Naturals do
not then the P-adics are a proper subset of the Naturals. When
infinite
sets are shown to be proper subsets of each other then they are set
theoretic equals.

((2)) Consider factorials. Notice that as you take the factorial of
every consecutive Natural number starting with 1 that 5! has the last
digit 0. That 10! has the last two digits 0's, and 15! has more last
digits 0 and so on. As the numbers become larger and larger the
factorial of these larger numbers continues to have more 0's as the
last digits. In the limit, n! is the number ...000. which is 1 larger
than the largest P-adic ...999.

The third argument addresses the conception that the P-adics are
layered or stepped with gaps. The P-adics terminating in endless 0's
are our Natural numbers. Leaving all those P-adics terminating in
endless 1's,2's,3's,4's,5's,6's,7's,8's,9's (those terminating in
endless 9's are our Negative numbers) other steps which Mathematical
Induction cannot bridge the gap. Here is a proof that the P-adics are
countable and that the Peano Axioms yield P-adics.

(((3))) The Peano Axioms make it such that it makes no difference on
how we represent numbers. That is, the theorems which talk about
numbers are true regardless of number representation. No theorem of
mathematics, unless the theorem is about number representation
specifically, changes when using a different number system. For
example, Unique Prime Factorization Theorem is the same true theorem
in
base 2 (binary number system) as it is in our familiar base 10 number
system. Now taking the definition of a Whole Number given above we
notice that the Natural numbers are a subset where the leftward
strings
go on endlessly in 0's. Let us deal with the binary system and apply
it
to this definition of Whole Number. Notice that in the application of
the Peano Axioms to the definition of Whole Number in binary system
that Mathematical Induction counts every one of those binary infinite
leftward strings. Notice that the binary system makes the
string...1111. equal to the string ...999. of base 10. Notice that in
binary there are no gaps of levels. Thus, this proof argument says one
of two things. Either numbers are different in different base
representations, or the Peano's Axioms yield infinite integers. QED
(Quantum Electrodynamics)

END

Notice also in the above post that I was very close to discovering
FrontView and
BackView which every number must possess, as shown in my remarks about
the number ....99999. So unconsciously or subconsciously I had the
idea of
FrontView and Backview but it would actually require 13 years later in
2007 to
discover FrontView and BackView so that it was at the Front of my
brain (sorry
for the metaphor).

Now I believe I am the first person in the history of mathematics to
claim the
discovery that the Reals represent All Possible Digit Arrangements.
That many
mathematicians came across the idea of All Possible Digit Arrangements
in association
with probability theory or some other mathematics but not with a link
to the Reals
themselves.

I believe I am the first person to say that the Reals are All Possible
Digit Arrangements.

The reason I say this is because in order for anyone to claim All
Possible Digit Arrangements
for the Reals means they have to throw-out other concepts of
mathematics that are contradictory
to All Possible Digit Arrangements such as these contradictory
concepts:

(a) transcendental numbers contradict Reals as All Possible Digit
Arrangements
(b) Reals uncountable contradict All Possible Digit Arrangements
(c) 1/3 = 0.333..... contradicts Reals as All Possible Digit
Arrangements
(d) 1 = 0.999.... contradicts All Possible Digit Arrangements
(e) betweenness on Reals where there is an infinite supply of Reals no
matter which two
Reals are chosen contradicts All Possible Digit Arrangements

So one can quickly get the picture. That if anyone discovers Reals as
All Possible
Digit Arrangements, could not have discovered it without realizing the
faults and flaws
of the Old Reals.

And below is a post to sci.math in 2007 where Dik Winter claims that
Cantor had
known of the concept of All Possible Digit Arrangements. I did not
accept Winter's
assessment. I am not fluent in German, and do not know what the German
words
would be for the concept of All Possible Digit Arrangements. I suspect
that Cantor
was not cognizant of All Possible Digit Arrangements. I suspect that
Cantor never
entertained the diagonal method on 00, 01, 10, 11, and that no other
mathematician
asked Cantor about that. I suspect that the diagonal method of Cantor
was like some
nightmarish fools trick played in math history, where so many weak
minds would
instantly cotton on and lapp it up, that it would become stuck in math
as perhaps the
most phony of phony math in recent times. Like the flat-earth people
after Copernicus
and Galileo. Cantor with his diagonal trickery would be the flat-earth
of mathematics.

So if anyone knows German really well and can research Cantor, I
suspect the outcome
of that research is that Cantor never involved himself with the
concept of
"All Possible Digit Arrangements" and if he somehow did, he never
understood the concept
for it would immediately defeat the diagonal as phony baloney.

Newsgroups: sci.math, sci.physics, sci.edu
From: a_plutonium <a_pluton...@hotmail.com>
Date: Tue, 13 Nov 2007 23:23:38 -0800
Local: Wed, Nov 14 2007 1:23 am
Subject: #289 Cantor's diagonal is all wet; new textbook: Mathematical
Physics (Reals & Counting Numbers/AP-adics Primer) for age 6 years
onward

Dik T. Winter wrote:
> In article <1194931892.063146.142...@d55g2000hsg.googlegroups.com> a_plutonium <a_pluton...@hotmail.com> writes:
> > Dik T. Winter wrote:

> > Can you understand that this
> > list is immune to the diagonal:
> > 00
> > 01
> > 10
> > 11
> > Apparently, you are unable to see or understand that the two place
> > value is immune to the Diagonal.

> Apparently you did not read what I wrote. This is a clear proof. In the
> material I wrote, and that you quoted, but did not read, I gave precisely
> this one to show that it would not work with the diagonal argument.

I do not remember that. So did you run a full race or did you quit the
race?
By that I mean, do you see where that leads to a proof that the Reals
as
All Possible Digit Arrangements cannot have a Diagonal applied?

Do you agree that

Since 2 place value is impossible to apply diagonal, and since 3, and
4 and 5
ad infinitum, Thus infinity place value of Reals is immune to the
Cantor diagonal.

Do you agree?

> > You do not understand Cantor's argument, Dik. You do not understand
> > that Cantor can not have a list of All Possible Sequences, and you
> > made that up. You cannot understand that Cantor never had All Possible
> > Digit Arrangements, because he still applied a Diagonal.

> "Sind nämlich m und n irgend zwei einander ausschliessende Charaktere, so
> betrachten wir einen Inbegriff M von elementen
> E = (x1, x2, ..., xv, ...)
> welche von unendliche vielen Koordinaten x1, x2, ..., xv, ... abhängen,
> wo jede dieser Koordinaten entweder m oder w ist. M sei die Gesamtheit
> aller Elemente E."
> And if you wish, in translation:
> "Namely, let m and n be two different characters, and consider a set
> M of elements
> E = (x1, x2, ..., xv, ...)
> which depend on infinitely many coordinates x1, x2, ..., xv, ..., and
> where each of the coordinates is either m or w. Let M be the totality
> of all elements E."
> In what way do you think that here M is *not* the set of all possible
> arrangements?

Alright, then I think I can pinpoint the trouble. That Cantor wanted
to use,
and did use,
All Possible Digit Arrangements and was striving for that in his
narrative,
but that Cantor was blind and not consciously aware of the fact that a
diagonal would not work on such a list.

Cantor was not consciously aware to the fact that the diagonal does
not work on
All Possible Digit Arrangements.

And this is very much understandable, that many people, nay, all
people have
experiences in life where they expect something to work, yet not
consciously aware
that their actions do not work as expected. Some of us have two or
three such
experiences almost every day. Just today, I was expecting to work on a
project
and had the weather in my favor in that there was no rain forecasted
however, it
was all spoiled because we had voracious winds which cut short my
project.

So that what happened to Cantor in the later half of the 19th century
is that he
expected and envisioned that a Diagonal would solve all his problems
and cough up
a new number not in his list and Cantor did not realize that All
Possible Digit Arrangements
destroys the diagonal. So he was blinded by the diagonal and never
thought to
experiment with it on two place value or three to see if it works.

So that if Cantor had played around with 00, 01, 10, 11 first and
played around with
that set and realized the Diagonal could not work, then saw that 3
place value
and 4 and 5 ad infinitum, that none of them can apply the Diagonal and
make the Diagonal
work. Well that led Cantor astray.

And this blindness led every follower of Cantor astray. That they
picture a list and where
the diagonal changes one digit in every number in that list. And they
walk away thinking they
have produced a new number not on the list, when in fact, they
repeated a number already
buried inside the list.

So Cantor and the people who think Cantor had something, all were
fooled by the Diagonal.
And even you Dik was fooled by the diagonal.

> And what he actually *does* show that each list of such things is incomplete.

No he does not. For Cantor did not comprehend that when a list is All
Possible Digit
Arrangements, that the diagonal crumbles apart and only repeats a
number already
within the list. Cantor never fetched a new number, but only repeated
a number already contained.

Mythical Tale When Cantor and Winter meet in the Alps and Diagonalize:

A tale of yore. There was Cantor and Winter in the Alps and they were
diagonalizing
lists of numbers. They came to the list of 00, 01, 10, 11 of two place
value and asked
to diagonalize and retrieve a number number not on the list and be two
place value.
Impossible they found out. Then they were given three place value list
of
000, 001, 010, 100, 110, 101, 011, 111 and told to diagonalize and
come up with a
new number not on that list yet still being three place value.
Impossible they said.
They did it with 4 place value and 5 , and 6 and 7 and 8 and on and on
they went to
infinity place value and every time the answer was the same, their
diagonal was impossible
to deliver a new number not already on the list. So what did they
learn?

They should have learned that the Reals are Countable since the
Counting Numbers are
also All Possible Digit Arrangements and where a diagonal does not
work.

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

END

######

FRONTVIEW & BACKVIEW history
(5) Frontview and Backview on all numbers, 2007

I am going to write a whole chapter on FrontView and BackView and use
the metaphor
of a house. A small house for a finite number and a large house with
alot of rooms for
a infinite number. So to say that every number has a FrontView and
BackView is similar
to recognizing each house has a frontdoor and a backdoor, or a
frontlawn and a backlawn.
And just because it is an infinite number does not mean that it has
only one View.

Newsgroups: sci.math, sci.physics, sci.edu
From: a_plutonium <a_pluton...@hotmail.com>
Date: Mon, 01 Oct 2007 20:19:22 -0700
Local: Mon, Oct 1 2007 10:19 pm
Subject: #41 Presenting Infinite Integers or P-adics as front view or
as rear view and a major new first in mathematics; new textbook;
"Mathematical-Physics (p-adic primer) for students of age 6 onwards"

Archimedes Plutonium wrote:
> So, examine this Infinite Integer or 10-adic:

> 5...000000000

> Now that is a Infinite Integer and not to mistake it for .....00000005

> Where 5...0000000 is one unit larger than 4.....999999999

> Now 9....99999 is approx 180 degrees and 1/2 of 9....999999 is
> 4....99999 and this
> number is angle approx 90 degrees.

I do not remember when I first posted that P-adics could be front
displayed. I do remember
there were replies of outburst of skepticism and even mockery. Their
disgust and complaint
is that since the digits are an infinite string, that you cannot
display the "point of infinity".

I can sympathize with their concerns for infinity is an "endless
process" so there is no end.
So how dare I say that .....99999999 is 9....9999999.

But I am pragmatic if nothing else. And the above has to be examined
as per pragmatism and as
per philosophy.

We all know that the digits of .....999999 are all 9s.

So here is the pragmatism. What is 1/2 of ....999999 and it is
4...9999999. An infinite string
which has all 9s except for one digit that is 4.

So the debate ends by saying that if I cannot display ....99999 as
9...99999 then there is no
midpoint of this number and that the number 4...999999 does not exist.

So I and my critics and skeptics have a choice. If 9...99999 does not
exist then its midpoint
of 4...99999 is also nonexistent. Which is the most repulsive? It is
more repulsive to think that
a arc has no midpoint.

Give me any Rational Infinite Integer or Rational P-adic and I can
display it front view and where
I can thence give you a midpoint of that number or give you 1/2 of
that number.

So pragmatism wins over any critic or skeptic.

I think I first introduced this front loaded display or presentation
around 1995. It is important because
it allows us so much more calculation ability and opens up a whole new
vista of mathematics to explore.

And also, it makes us aware that the 10-adic of ....999999 is not (-1)
but the largest 10-adic integer.

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

END

Newsgroups: sci.math, sci.physics
From: plutonium.archime...@gmail.com
Date: Sat, 12 Jul 2008 17:45:52 -0700 (PDT)
Local: Sat, Jul 12 2008 6:45 pm
Subject: #579 some history of Frontview on the Internet; new textbook:
Mathematical Physics (AP-adic Primer)
Reply to author | Forward | Print | Individual message | Show original
| Remove | Report this message | Find messages by this author
--- quoting #299 post in the series ---
Newsgroups: sci.math
From: a_plutonium <a_pluton...@hotmail.com>
Date: Wed, 14 Nov 2007 22:11:13 -0800 (PST)
Local: Thurs, Nov 15 2007 12:11 am
Subject: #299 All Possible Digit Arrangements gives order and pattern
to the Reals and makes them Countable; new textbook: Mathematical
Physics (Reals & Counting Numbers/AP-adics Primer) for age 6 years
onward

- Hide quoted text -
- Show quoted text -
Jesse F. Hughes wrote:
> David R Tribble <da...@tribble.com> writes:
> > David R Tribble writes:
> >>> I guess you're right - I don't understand what all that is
> >>> supposed to mean. I can see what those rightmost digits
> >>> mean:
> >>> ...00054321 =
> >>> 1x10^0 + 2x10^1 + 3x10^2 + 4x10^3 + 5x10^4
> >>> + 0x10^5 + 0x10^6 + 0x10^7 + ...
> >>> But I can't make any sense out of the leftmost digits.
> >>> Could you express the number 9876000...00054321 in
> >>> the form of a sum of digits and powers of 10?
> >>> Otherwise I can't make any sense of where those leftmost
> >>> digits are supposed to mean mathematically.
> > Jesse F. Hughes wrote:
> >> Golly, are you just slow or what?

He is more concerned about finding me wrong on something than the
pursuit of
mathematics.

> >> The number is
> >> 1x10^0 + 2x10^1 + 3x10^2 + 4x10^3 + 5x10^4
> >> + 0x10^5 + 0x10^6 + 0x10^7 + ... + 5x10^{oo - 4}
> >> + 4x10^{oo - 3} + 3x10^{oo - 2} + 2x10^{oo - 1}
> >> + 1x10^oo
> >> Duh.

Pretty good, but we can replace the infinity symbol with the world's
largest integer 999....99999

> > I guess I'm just not up to big thinking like AP.
> > I guess it's that magical "..." part in the middle that bothers me.

My vote is that you remain in the conversation for although it is very
negative towards me, I still seem to thrive better in a negative
environment
than a yes-man environment of praise.

> > I know you and me and a lot of others went through all this
> > some time ago with Tony Orlow over his infinite "T-riffic"
> > numbers, and it's all starting to sound like the same monotonous
> > tune, like deja vu all over again.
> Except that AP discovered FrontView.

Well that is one nice feature of the Internet newsgroups is that it
can validate and
sort out priority of discovery and I knew that I had discovered
frontview first.

--- end quoting last years post in this book ---

So the Internet can sort out priority of discovery. I had discovered
FrontView sometime
in the early 1990s when I posted the largest Natural Number of
9999...99999 and although
I did not state FrontView, the mere fact that I posted 9999....99999
instead of ...999999
signifies that FrontView was borne when I posted 9999....9999. I did
not realize its
vast importance as a tool of mathematics until 2007

The logic is quite simple. Any given infinite entity can be said to
have a front section, a middle section
and a rear section. Think of a cat as a infinite number. Are we going
to spend all our time on cats talking
only about the hind quarters and tail. Or would it make more sense to
focus on the head region. To think
that in mathematics, because a number is infinite, that you can only
talk and work on the hind quarter
region of the number is rather ludicrous. Although it is infinite, we
can shift focus on the frontview of the
number.

A number like this ....999999 we can easily focus on its frontview as
9999.....9999. But a number
like this .....151413121110987654321 well, we have a hard time of
focusing on its frontview even though
we suspect it has alot of 9s digits.

What makes it very exciting is that FrontView has alot to say about
Reals. If I divide 3 into 1, is the
answer really 0.33333..... which implies 0.3333.....333333 Well,no,
because 3 divided into 1 constantly
leaves a remainder of a 1 carryover.

3 divided into 0.9999..... is precisely 0.3333....33333
but 3 divided into 1.000....... is not 0.33333..... and is not
0.333....33333
it is actually 0.333....33333r where the r signifies a residue
carryover of 1.

And that is the crux of the flaw of so called proofs that claim to
prove 0.9999.... is equal to 1.
that is a false proof.

So these two concepts open up brand new vistas into mathematics

(1) All Possible Digit Arrangements

(2) FrontView

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

#######

TWO DECIMAL POINTS FOR EVERY NUMBER

(6) Every number has two decimal points, a prefix and suffix decimal,
circa 2007
and fully bloomed 2008

I vaguely remember at the end of the first edition of this book back
in 2007 of blurting
out some words saying to the effect:

"this division behaves like an extra decimal point"

I have not located that post, and whether it is prudent to spend time
on tracking it down.
The important thing is that in 2007 the idea-seed was sown that
something behaved
like an extra decimal point. So that when I came into trouble in 2008
with 1/3, that I
would immediately apply that idea. That it would rescue and solve my
problem of
what is 1/3? It is 0.3333....3333c1 which has two decimal points and
the c1
indicates a remainder at the point of infinity. Now I should specify
what I mean by
point-at-infinity for the small scale it is 10^(-)999...9998 place
value and the point
at infinity on the large scale is the 10^999...9998 place value.

The concept of two decimal points solves the problem of completing the
Reals as an
algebraic field with respect to division, otherwise we have no number
to represent
1/3 as a decimal, because of the 1 remainder carryover. And this Two
Decimal
Point concept revises the limit concept in Calculus for we no longer
need Limit
in Calculus, perhaps in all of mathematics, since we always know what
the neighborhood
of a Real Number is.