The Plutonium Integers thread reminds me to a similar game, with which I played some years ago, and it made me to go through again - just to understand, whether I caught the idea correctly or not.

OK, I didn't know about adics - I just puzzeled with this matter of infinite divergent series which are here used as plutonium integers(PI) . Please ignore all this posting, if I am wrong in this point.

I understand that a PI is generally a infinite series written down like a number, where the digits d[i] are the coefficients of a series like

... d[2]*10^2 + d[1]*10^1 + d[0]*10^0 , for example

... 8*10^3 + 8 *10^2 + 8 *10^1 + 8 = ... 8888.

A specific numerical value certainly cannot be assigned to these series, as they are divergent, and so I am absolutely suspicios against claims, which use words of or recur to associations, which are commonly used in evaluation of the numerical value, like "this PI is greater than this PI", or "that PI has that value".

My initial interests were questions on other properties, like divisibility, for instance to what extend it may be allowed to cancel out similar divergent series, when used as nominator and denominator of a fraction.

What came out was, that PI's could be related to negative rationals between [0,-1]; they share properties with them. In this thread, the term of "mapping" came into discussion, and I want to summarize my opinion using that terminology.

I noted the posting of Professor Abian, who provided an analytical system to operate with these "infinite naturals" ( AP), or "Plutonium Integers", but where discrepances occur concerning the properties and the handling of these series. See the footnote of this posting for details.

It made me deciding between three types of PI's:

* periodic plutonium integers (PPI),

* periodic with an addition of a finite integer (composite PI=CPI),

* and infinite aperiodic (API).

While the (periodic) PPI's seem to be mappable to rationals of the range [0,-1] (if the basic PI (of basis 10) #I=...9999 is defined to map to -1), the composite CPI's are mappable to the whole range of rationals in [-inf,+inf]. I don't have an answer, to what the API's map, as I did not play around with them and have no ideas of their properties.

These observations/assumptions allow already to state, that

a) the product of two PPI's cannot map to +1 except #I*#I, as each mapping is of absolute value <1. So if the example in a previous posting concerns two PPI's, this equation is false:

> > > tail-ends of two nonunit ap-numbers which are multiplicative inverses of

> > > each other:

> > > (5) ....2403 x .....3067 = ..0001

> > Really? Persuade me by specifying the "...." and the "....."

If the two PI's are meant as nonperiodic CPI's or API's, then the solution could only be verified, if the "..." were explicated.

The following example is then also false, as one PI maps to a positive integer and the other to a negative - hence their product must map to a negative integer, and not to +1:

>

> How about a simpler example, instead? Try ...000003 and ...999997

> (all the missing digits of the first number are 0, and all the missing

> digits of the second number are 9).

Tried by multiplication or with direct PI-algebra the result does not map to 1, but to -9, which is expressible in PI-notation as ....9991.

b) Not all rationals in [0,-1] are mapped. The mapping 0.5 or -0.5 cannot be found in basis 10-PI's. There are two solutions to map each rational in [0,-1]:

- take a PI of approriate basis, for example 3, then -1 is mapped by #I=...2222, and -0.5 is mapped by ...1111

- introduce the decimal point and create plutonium reals (PR): of basis-10, PR's allow the notation ...9999.5 with the mapping to (-1 + 0.5)=-0.5 and generally any PR with a decimal fraction, as a composite PI with a normal PI plus a real instead of an integer.

The latter method possibly creates inconsistencies, because different PR's can map to the same value: all ...000.000..., ..111.111.., ...222.222.. etc map to 0 for example. I even didn't check, whether this notation makes any sense at all.

When I played around with these constructions, I did not find any use for them. The only interesting question for the future for me could be the exploration of the API's - but I don't have a door to step in.

Gottfried.

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Footnote:

Let the notation for any PI be "#x" with a small letter; let the notation for periodic PI be "#X" with capital letters, and specifically let the notation for the PI "...9999" be "#I" as a unit, or a base entity for the whole PI-analysis.

Let also be the equal-sign be used, although there is no actual value to compare, but only the analytical convertibility.

I really don't know, whether PI-arithmetics make any sense at all, but at least one can observe, that if we relate or "map" some PI's to real numbers, then the PI's, which are modified by algebraic operations, are related to reals, which were modified in the same way.

Let me use the notation {...9999} = -1, if I want to say, that the PI-unit #I maps to the real -1, or {#x} = 4, if I want to express, that the PI #x maps to the real value 4.

For mixed formulae, where PI's and real numbers are connected, let a capital letter be used for integer numbers [..-2,-1,0,1,2,...], any small letter be used for rational numbers exept r,s,t for reals.

Then the following formulae -in my notation- can be stated, seemingly without contradiction.

a) In these examples the basis for the expansion of a PI into a series-notation is 10; a PI of a basis 10 can be converted in a PI of another natural-number-basis>1.

b.1) There are PI's with periodic digits-sequence. They will be called PPI, and they map to the rationals in the range [0,-1]. For instance the PPI's #X=...3333 maps to -3/9 =-1/3 , or #X=...212121 maps to -21/99 = -7/33.

b.2) there are PI's which have an aperiodic sequence of digits of finite length at the tail, and a infinite periodic sequence at the head. For instance #x=...33334 or #x=...2121123, or #x=...99990.

b.3) there are PI's with a complete aperiodic sequence of digits, which are described or generated by specific functions, for instance the PI with the inverse digit-sequence of sqrt(2). My article does not deal with these entities, possibly with the help of such constructions some contradictions or paradoxa can be generated. If they are specifically referred then I use the notation #~x

c) Two basic mappings are defined

1) { 0# } = 0

2) { 9# } = {#I} = -1.

As a consequence of the application of the arithmetic symbols to the PIs any #0n maps to n:

{ 0#n } = n

c.1) If a mapping of a PI is found for i.e. the rational number q, then all arithmetic including PI's and integers is mapping-consistent; that means: if I apply the same operation to the mapped number, then the resulting PI maps to the resulting number. For instance:

{....99999} = -1,

....99990 = ...999*...0010

{....99990} = {...9999*...0010} = {9#}*10 = -1*10 = -10

....99990 = ....9999- ...0009

{....99990} = {...9999- ...0009} = {..9999}- 9 = -1- 9 = -10

{..9999-...99990}=(-1) - (-10)=9 .

or in a better formal notation

{9#*0#n} = {9#}*{0#n} = -1*n = -n

c.1.1) {#X-N} = {#X} - N and also {#X+N} = {#X} + N .

For instance ...9999-9 = ...9990 and the mapping

{...9999-9} = {...9990} = {...9999} -9 = -1 -9 = -10

c.1.2 {#X*N} = {#X}*N and {#X/N} = {#X}/N

For instance ...999/3 = ...333 and the mapping

{...999/3} = {...333} = {...999} / 3 = -1 /3 = -0.333..

c.2) If a mapping of a PI is found for example the rational number q, then all arithmetic including two PI's is mapping-consistent; that means: if I apply the same operation to the mapped number, then the resulting PI maps to the resulting number. For instance:

c.2.1) {#X-#Y} = {#X} - {#Y} and also {#X+#Y} = {#X} + {#Y} .

For instance ...9999-...1111 = ...8888 and the mapping

{...9999-...1111} = {...8888} = -8/9

(-1) - (-1/9) = -8/9

c.2.2) {#X*#Y} = {#X}*{#Y} and {#X/#Y} = {#X}/{#Y}

For instance ...999/...333 = ...003 and the mapping

{...999/..333} = {...999} / {...333} = -1 / (-1/3) = 3

What is stated here, is that the mapping of the result of a PI-arithmetic operation is equal to the same operation performed on their maps, if basically #I is mapped to -1.

A result is, that if p-adics map to [0,-1], then the PI's (possibly except the irrational) map to all rationals, as they can be composite of a PPI and a natural number and/or a rational multiplicator, so #I-9 maps to -10, which is outside [0,-1].

A conjecture, that may follow, is, that all periodic PI's map to the rationals in [0,-1], and all composite PI's map to rationals execept [0,-1].

Another conjecture, which results from the observation, that with the PI's of basis 10 a mapping to -0.5 cannot be found, but easily with another basis, could be: all rationals can be mapped to composite PI's, if all naturals are allowed as basis. For instance, if the basis 3 is used, then #I is ...222 and maps to -1, and ...111 maps then to -0.5 .

Another conjecture could be, that the product of two periodic PI's cannot be periodic, as the mapping will be a positive rational number, which cannot be mapped by a periodic PI, since periodic PI's only map to rationals in [0,-1].

This allows to reflect a previous posting:

> > > Also, contrary to the familiar Arithmetic of Natural numbers, a nonunit

> > > ap-number may have a multiplicative inverse. The following shows the

> > > tail-ends of two nonunit ap-numbers which are multiplicative inverses of

> > > each other:

> > > (5) ....2403 x .....3067 = ..0001

> > Really? Persuade me by specifying the "...." and the "....."

> >

For me this example is problematic for the same reason. If the dots represent periodic digits, then there would be no {#A*#B} =1 except {#I*#I}. If the example recurs on infinite aperiodic PI's, then I would like to see the generation formula for the digits.

>

> How about a simpler example, instead? Try ...000003 and ...999997

> (all the missing digits of the first number are 0, and all the missing

> digits of the second number are 9).

It was meant, that the result would be ...00001. I don't think so. In this example we have the conjecture {#A*#B} = 1, which has to be consistent with {#A} * {#B} =1.

...003 is #I+4, ...9997 is #I-2. The mapping may be expressed retrieving from :

{(#I+4)*(#I-2)}
= {#I+4}*{#I-2}
= {#I*#I} +(4-2)*{#I} - 8
= (1) + (-2) -8
= -9 which is

= {#I-8}

= {...99991} (which is not equal 1)

If you perform the multiplication digit per digit, then you come to the same result:

....999997 * 0003
21
27
27
-------------
...9999991

A further conjecture could be: the inverse of a periodic PI must be composite, because a periodic PI maps to rationals in [0,-1] , the inverse of that must be a rational outside this range, and all rationals outside [0,-1] are composite PI's.