#380 a 21st century better definition of Irrational, Rational, and Transcendental ; new book 2nd edition: New True Mathematics
Archimedes Plutonium
plutonium....@gmail.com wrote:
> lwal...@lausd.net wrote:
>
>>On Apr 6, 3:31 pm, David R Tribble <da...@tribble.com> wrote:
>>
>>>Archimedes Plutonium wrote:
>>>
>>>>As I said, both L and 999...99999 have the same infinite digit place
>>>>values. Both are located as a point on the Greenwich Longitude between
>>>>0 and the South Pole. 999...9999 is one unit short of SP, whereas, L
>>>>is further away from the South Pole
>>>
>>>Does that mean that L has fewer digits than 999...999?
>>>Or if they have the same number of digits, does that mean
>>>that the leftmost string of 9's in L is not really the same as
>>>the string of 9's in 999...999?
>>
>>It means that the number of 9's in both 999...999 and L is
>>_Dedekind_ infinite. That means that there can be exactly
>>as many digits in L and in 999...999, _and_ the leftmost
>>string of L is exactly the same as the string of 9's in
>>999...999 (i.e., SP-1).
>>
>>When a set is Dedekind infinite, there is a bijection
>>between the set and its proper subset. The part can be
>>equal to the whole. Therefore, there can be exactly as
>>many 9's in the leftmost string of L as there are total
>>digits in the entire string L.
>
>
> That maybe one of the few things Dedekind got correct in
> his career in mathematics for his Dedekind Postulate for the
> "cut on Reals" will likely disappear in the near future.
>
> But anyway, Tribble, in a serendipity (look up serendipity
> discoveries) post, happened to put two seemingly unrelated
> questions in the same post.
>
> He asked me about Champernowne type numbers in AP-adics
> 99999....99999....9999....999989999.......11109876543210
>
> which is a string that contains all the sequentual numbers all
> contained inside that string. Call them sequentual, call them
> all of Peano Natural Numbers where they include infinite integers.
> And what troubles people about this number although it contains
> all the numbers, even the largest at 9999...9999, yet is smaller
> than the largest. Well, that is infinity for you. I call them Hologram
> numbers because like a Hologram, cut a corner away and the
> same picture appears on the cutaway as the larger piece.
>
> But in a previous Tribble post he asked me about that
> Champernowne number and the sqrt2 = 1.414..... for me
> to elaborate in AP-Reals.
>
> Well, that was fortunate because I will now combine the
> two, Champernowne numbers and Irrationals of Old Reals
>
> In AP-Reals, the concept of Irrational versus Rational no longer
> has any credence. The best that the Old Reals could differentiate
> Rational from Irrational was to say that a decimal expansion
> has a "block repeating out to infinity" if Rational and no repeating
> if Irrational. So that 2/3 is Rational because 0.6666...... the 6's
> repeat or that a number like this 0.12381238..... is Rational because
> of the repeating block 1238.
>
> But then the Old Reals never had BackView and can never have
> backview so the Old Reals are like a corruption curtain or a
> crime post where they sweep under the rug.
>
> Because numbers like this in AP-Reals 1.414.....999999
> never come up in Old Reals because they had no BackView
> and as Champernowne's number shows us, that one of the
> infinite-substrings can repeat such as the 999's whereas the
> FrontView substring which is infinite also of 414 is nonrepeating.
>
> So as to the question of whether the AP-Reals can elaborate
> more and better as to what is the sqrt2, well, the AP-Reals
> can do that.
>
> Consider that all we need for a clear understanding of the full
> sqrt2 as a number is for some number N such that N x N = 2
> and this 2 is 2d0000...000000 with nothing but 0s to the rightward
>
> So how do we get nothing but 0s out of an N such that
> NxN = 2d000000....0000
>
> Well, we need not have to have 2d0000....00000 but can
> have 1d99999....999999 since the second-decimal-point
> L such as 2/3 = 0d6666....66666L where the L storages
> the remainder carryover of a "2" in the endless division process
> of 20 divided by 6 is 3 for 18, carryover the 2
>
> So that a 1d9999....99999 from a N equal to a number which
> has a L second decimal point restores that carryover of a "1"
> and thus elevates the 1d9999....9999 answer to a 2d000...000
> answer.
>
> So now I ask myself in finite approximations in taking the square
> root of numbers, what number string delivers me the most 9s ????
>
> And I have a limited calculator but the most 9s in the first four
> digits
> is the number 9999 in AP-adics and 0d9999 in AP-Reals
>
> sqrt 9999 = 99d994999
>
> sqrt0d9999 = 0d99994999
>
> So it is safe to say that although five 99999 breaks the pattern
> but that somewhere out there another even string of solid
> 9s when taking the square root delivers the very most 9s
>
> So that what is sought for as the sqrt2 in AP-Reals
> is a infinite string with at least two infinite-substrings
> like this:
>
> 1d414........9999L
>
> and where the L carryover of 1 is thus forcing the 9s to act as
> a bridge. or this:
>
> 1d414.....99994L and where the L carryover is a 6 and thus the
> chain reaction of forcing all the 9s to revert to a 0 digit
>
> P.S. also, I must include the fact that in AP-Reals we can easily
> slide over to the AP-adics for help in conceptualization because
> there is a similar number in AP-adics of a square root of
> not of 0000....0002 since the AP-adics are finite in radix but
> for a square root of the North Pole as 2H000....0000 and that
> square root is the number 1H414......r and is a point located about
> 41% past the South Pole enroute to the North Pole.
> As in a previous recent post of mine where I compute the
> multiplicative
> inverse of 757575.....7575 as 1H32......r
>
> So unlike the Old Reals where they never had help from a symmetrical
> other number system. Any questions posed about AP-Reals can easily
> transfer over the same question to the AP-adics which may come
> up with the correct answer easier than having just the one Old Reals
> or AP-Reals
>
The above gives me a great new idea. How to make a better definition of
Rational, Irrational and Transcendental Real Numbers, of course they
have to be AP-Reals.
So notice above where I outline how to get the Irrational string for
sqrt2 that I use two infinite-substrings.
Notice in the Champernowne's number that you have an infinity of
subinfinities.
So now a modern day Rational number repeats in blocks. Fair enough.
But when they repeat in only one infinite string such as
the string 1/3 = 0d333...3333L that there is no infinite substring
different from the repeating 3s
But suppose we had a AP-Real such as sqrt2 as 1d414.....99994L
Now I am not sure yet whether that in fact is sqrt2 in BackView
as posted earlier today above. But notice that it has at
least two infinite substrings whereas Champernowne's number
what I like to call Hologram numbers, have an infinitude of
infinite substrings which makes for the case that we can define
Rational as repeating blocks of one solo infinite substring.
We can define Irrational as a greater than 1 of infinite substrings
but a finite number of infinite substrings. And finally we can
define a Transcendental Number as that number in which it has
an infinitude of infinite-substrings and which therefor leaves
Champernowne Numbers or Hologram Numbers as transcendental numbers.
And so my earlier thoughts in bygone years where I declared the world
had only two transcendental numbers is revised.
Also, since the discussion pivoted recently around the question of how
much of a influence does Physics have on mathematics and most readers
know my position that math is a tiny subset of physics, but I want
to comment on those Hologram Numbers in that if Physics has a
recognizable feature of the World in that a big hologram with a picture
and a corner cutout of that same hologram leaves the same picture on
both the cutout and the bigger piece, since Physics has that feature,
then mathematics must have some feature of that same characteristics
within its geometry and its Number Systems.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies