PLUTONIUM INTEGERS (Algebraic structure)
Alexander Abian
This is the Algebraic structure of Plutonium Integers. For their
Topological (Analytic) Structure see a previous posting
of mine "Topology for Plutonium Integers")
I have defined a Plutonium integer (based on Plutonium's earlier ideas
and postings) according to the following:
DEFINITION. A Plutonium integer is a beginningless sequence of digits
0, 1, 2, ..., 9
Examples: ..........643098002, ...........361351351
.......99999999999, .... 0100001000100101
They are introduced for the purpose and objective of having
many, many (in fact continuum many) distinct infinite integers
enriching the theory of finite integers and avoiding considering
each one of them as, say, the same infinite countable ordinal "omega"
I have emphasized that Plutonium integers have nothing to do with
p-adic or m-adic numbers. I again restate that Plutonium
integers are infinite integers and must be considered as
enriching the notion of arithmetic by introduction of many
nuances in the concept of infinite integers. They must not
be reduced to inverse limits of anything .
Their arithmetic is defined as follows:
ADDITION. If A and B are ap-integers (short for Plutonium
integer) their sum A + B is defined as a beginningless
sequence which at the n-th decimal place has the digit
which appears at the n-th decimal place of the sum
(in the usual sense) of the n-th final segments of A
and B
For instance if A = ..........53879296 and B = .......87502968
Then the 1st digit in A+B will be 4 since 6+8 = 14
the 2-nd digit in A+B will be 6 since 96+ 68 = 164
the 3-rd digit in A+B will be 2 since 296+968 = 1264
the 4-th digit in A+B will be 2 since 9296+2968 = 12264
...........................................
So, A + B = ............2264
MULTIPLICATION. If A and B are ap-integers (short for Plutonium
integer) their product AB is defined as a beginningless
sequence which at the n-th decimal place has the digit
which appears at the n-th decimal place of the product
(in the usual sense) of the n-th final segments of A
and B
For instance if A = ..........53879296 and B = .......87502968
Then the 1st digit in AB will be 8 since 6x8 = 48
the 2-nd digit in AB will be 2 since 96x68 = 6528
the 3-rd digit in AB will be 5 since 296x968 = 286528
.............................
So, AB = ................528
REMARK Addition and multiplication are commutative, associative
and the usual distributivity law holds.
The additive inverse of P is the unique Q such that
P + Q = 0
THERE ARE NO NEGATIVE PLUTONIUM INTEGERS and (same as that there
are no negative natural integers)
SUBTRACTION M - K is defined as the unique ap-integer S
such that M = K + S
Thus .....000003 - ....000005 = .....99999998
The equation a + x = b always has a unique solution
There are Divisors of Zero in Plutonium arithmetic, i.e.,
ED = 0 does not imply A = 0 or B = 0
The equation ax = b is not always solvable and if solvable
need not have a unique solution.
All the above is a summary of what was posted before.
PARTIAL ORDER IN PLUTONIUM ARITHMETIC. ( <= read "less than or equal")
DEFINITION. For ap-integers A and B we define
(1) A <= B iff from somewhere on all the final segments of A
are less than or equal to the similar (lengthwise) final segments
of B.
We paraphrase the above by saying that A is eventually less
than or equal to B.
Clearly <= is a partial order since for every A, B, C
A <= A (reflexivity)
if A <= B and B <= A then A = B (antisymmetry)
if A <= B and B <= C then A <= C (transitivity)
However <= is not a linear order since
........121212121212 and ....... 212121212121
are not comparable w.r.t <=
So, it is in the spirit of the above exposition that I meant
"there is no connection of Plutonium integers with p-adic
integers". I insist that infiniteness of ap-numbers and
the variety of infinite ap-numbers should play the central role.
If A. Plutonium desires to see the relation with p-adic numbers,
then he should consult Professor C.N. Giffen's extremely able
description of the said relationship in his 6-9-99 sci math post
#329464 subject: "Plutonium Integers".
So, Archie, Prof. C.N. Giffen has answered your specific question.
(as mentioned above for the Analytic structure of Plutonium integers
see a previous posting of mine "Topology for Plutonium Integers"
--
-------------------------------------------------------------------------
ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND EPIDEMICS.
JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR SYSTEM.
REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN EARTH(1990)
THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF SPACE
Archimedes Plutonium
ab...@iastate.edu (Alexander Abian) writes:
AA This is the Algebraic structure of Plutonium Integers. For their
AA Topological (Analytic) Structure see a previous posting
AA of mine "Topology for Plutonium Integers")
AA
AA I have defined a Plutonium integer (based on Plutonium's earlier
ideas
AA and postings) according to the following:
AA
AA
AA DEFINITION. A Plutonium integer is a beginningless sequence of
digits
AA 0, 1, 2, ..., 9
AA
AA Examples: ..........643098002, ...........361351351
AA .......99999999999, .... 0100001000100101
AA
AA They are introduced for the purpose and objective of having
AA many, many (in fact continuum many) distinct infinite integers
AA enriching the theory of finite integers and avoiding considering
AA each one of them as, say, the same infinite countable ordinal
"omega"
AA
AA I have emphasized that Plutonium integers have nothing to do with
AA p-adic or m-adic numbers. I again restate that Plutonium
AA integers are infinite integers and must be considered as
AA enriching the notion of arithmetic by introduction of many
AA nuances in the concept of infinite integers. They must not
AA be reduced to inverse limits of anything .
AA
AA Their arithmetic is defined as follows:
AA
AA ADDITION. If A and B are ap-integers (short for Plutonium
AA integer) their sum A + B is defined as a beginningless
AA sequence which at the n-th decimal place has the digit
AA which appears at the n-th decimal place of the sum
AA (in the usual sense) of the n-th final segments of A
AA and B
AA
AA
AA For instance if A = ..........53879296 and B = .......87502968
AA
AA Then the 1st digit in A+B will be 4 since 6+8 = 14
AA the 2-nd digit in A+B will be 6 since 96+ 68 = 164
AA the 3-rd digit in A+B will be 2 since 296+968 = 1264
AA the 4-th digit in A+B will be 2 since 9296+2968 =
12264
AA ...........................................
AA So, A + B = ............2264
AA
AA MULTIPLICATION. If A and B are ap-integers (short for
Plutonium
AA integer) their product AB is defined as a beginningless
AA sequence which at the n-th decimal place has the digit
AA which appears at the n-th decimal place of the product
AA (in the usual sense) of the n-th final segments of A
AA and B
AA
AA For instance if A = ..........53879296 and B = .......87502968
AA
AA Then the 1st digit in AB will be 8 since 6x8 = 48
AA the 2-nd digit in AB will be 2 since 96x68 = 6528
AA the 3-rd digit in AB will be 5 since 296x968 = 286528
AA .............................
AA So, AB = ................528
AA
AA
AA REMARK Addition and multiplication are commutative, associative
AA and the usual distributivity law holds.
AA The additive inverse of P is the unique Q such that
AA P + Q = 0
AA
AA THERE ARE NO NEGATIVE PLUTONIUM INTEGERS and (same as that there
AA are no negative natural integers)
AA
AA SUBTRACTION M - K is defined as the unique ap-integer S
AA such that M = K + S
AA
AA Thus .....000003 - ....000005 = .....99999998
AA
AA The equation a + x = b always has a unique solution
AA
AA There are Divisors of Zero in Plutonium arithmetic, i.e.,
AA ED = 0 does not imply A = 0 or B = 0
AA
AA The equation ax = b is not always solvable and if solvable
AA need not have a unique solution.
AA
AA All the above is a summary of what was posted before.
AA
AA
AA PARTIAL ORDER IN PLUTONIUM ARITHMETIC. ( <= read "less than or
equal")
AA
AA DEFINITION. For ap-integers A and B we define
AA
AA (1) A <= B iff from somewhere on all the final segments of A
AA are less than or equal to the similar (lengthwise) final
segments
AA of B.
AA
AA We paraphrase the above by saying that A is eventually
less
AA than or equal to B.
AA
AA Clearly <= is a partial order since for every A, B, C
AA
AA A <= A (reflexivity)
AA if A <= B and B <= A then A = B (antisymmetry)
AA if A <= B and B <= C then A <= C (transitivity)
AA
AA
AA However <= is not a linear order since
AA
AA ........121212121212 and ....... 212121212121
AA
AA are not comparable w.r.t <=
AA
AA
AA So, it is in the spirit of the above exposition that I meant
AA "there is no connection of Plutonium integers with p-adic
AA integers". I insist that infiniteness of ap-numbers and
AA the variety of infinite ap-numbers should play the central role.
AA
AA If A. Plutonium desires to see the relation with p-adic numbers,
AA then he should consult Professor C.N. Giffen's extremely able
AA description of the said relationship in his 6-9-99 sci math post
AA #329464 subject: "Plutonium Integers".
AA
AA So, Archie, Prof. C.N. Giffen has answered your specific question.
AA
AA (as mentioned above for the Analytic structure of Plutonium integers
AA see a previous posting of mine "Topology for Plutonium Integers"
Thanks Dr. Abian. I would like to see a Schaum's type outline for the
p-adic numbers for smart High School students. And I think that you
methods are the easiest and best to start into the Infinite Integers.
I still think that there maybe a way to teach p-adics, not just
10-adics via some alteration of Dr. Abian's operation scheme for
addition and multiplication. It would be nice for the mind to shift
from Reals to 2-adics and then from Reals to 5-adics and from Reals to
10-adics as a for instance sake.
It would be nice to do this rather than rely on the series definition
and the Hensel development of Adics.
I still have no proof that the P-adics form cones of
conic-section-systems. I only have my intuitive guess. Hopefully
someone more expertise at p-adics will confirm that the p-adics form
cones of conic sections. Where is Karl Heuer? I miss him.
Archimedes Plutonium
> They are introduced for the purpose and objective of having
> many, many (in fact continuum many) distinct infinite integers
> enriching the theory of finite integers and avoiding considering
> each one of them as, say, the same infinite countable ordinal "omega"
I am going to take issue and to explore this flaw. I see it as a
flaw.
Dr. Abian could you please write out Occam's Razor mathematically? The
reason I ask is because (1) if one says that this set
{0,1,2,3,4,5,.....} merges into Infinite Integers (which you have
called Plutonium Integers), then, Cantor transfinites and omega
disappears altogether.
(1) Occams Razor implies take the simpliest explanation. Should the
larger interval of Reals {0,1,2,3,4,5,.....} be less numerous than the
tiny interval of Reals [0,1]. Occams Razor would say, no. That the
larger interval should be more numerous, or if worse comes to worse,
equinumerous with a smaller interval of the larger interval. How to
correct this paradox? Easy. Accept the idea that this set
{0,1,2,3,4,5,.....} merges into Infinite Integers. That the endless
adding of 1 is the same as the p-adics series definition which is equal
to endlessly adding 1.
(2) Does it logically make sense to say that the concept of
"endlessness" has many varieties of endlessness? Let us look at the
reverse concept for help. Does it make sense to say that "nothingness"
has many kinds and types of nothingness? My answer and most people with
good commonsense would be that there exists one and only one type of
endlessness and one and only one type of nothingness.
None of the above is anything new from my arguments back in 1993-1994
with
Natural Numbers = Infinite Integers = P-adics
However, I had a huge problem with that equation geometrically. For I
posited that p-adics were Riemannian Geometry. And I knew that the
Reals, the positive Whole Reals of this set {0,1,2,3,4,5,.....} were
Euclidean Geometry.
So, you can see my huge problem. If I say that Natural Numbers =
P-adics would be like saying that Euclidean space eventually bends and
folds itself back around like on the surface of a big sphere. This is
tantamount to saying
Euclidean 3-space = Riemannian 3-space.
But recently I have made progress to overcome that difficulty. I have
speculated that the P-adics form conic section systems.
So, picture either Euclidean 2-dimensional Cartesian coordinate
system or 3-dim, it makes no difference. In 2d, we have the 4 quadrants
of I, II, III, and IV. Thus in 2d Cartesian coordinates we have two
conic section systems. We have quadrants I & III, and we have II & IV.
What keeps the Reals going *straight* geometrically is the numbers
that fill in between each Whole Real. Thus, between the Whole Reals of
0 and 1 are an infinite number of other Reals and it is these that keep
the Whole Reals straight.
But in p-adics or Infinite Integers, there are no
fillers-in-between-contiguous as the Reals. This lack of makes the
P-adic Integers or Infinite Integers bend geometrically and bend
Riemannian. However, the Doubly Infinites do have fillers like the
Reals but the double infinites force the space to bend negatively
yielding Lobachevskian geometry.
More later...