Last night I remembered this thread around adics in this group. As I myself was interested on the theme (but in fact as a discussion of problems and specialities of divergent series like sum(a[i]*10^i)) I made some thoughts around that.
Now I found the solution and joined the community of mathematics, who started to trust on adics as natural numbers since 1993.
I found out some more nice and never documented properties of this numbers.
a).let â (always an adic) be ...99999. Multiply it by 10. you get ....999990. Now
â - â*10 = ...99999 - ...999990 = ....0000009
Funny: â has another representation than as an adic; it’s the well known
â *(1-10) = â *(-9) = 9 where follows that â =-1
So first theorem of G is
G1: an adic sometimes can be expressed in equivalence to a negative rational number.
other examples found on the same way:
....33333 - ....33333*10 = ...33333-...33330 = 3 what means that â - â *10 = 3 and â =-3/9 and â =-1/3
....77777 - ....77777*100 = ...77777-...77700 = 77 what means that â - â *100 = 77 and â =-77/99 and â =-7/9
Especially:
....99999 - ....99999*10 = ...99999-...99990 = 9 what means that â - â *10 = 9 and â =-9/9 and â =-1
which is equivalent to the highest positive infinite number, and what in elegance shows, that
G2: seen as an adic the HIGHER-THAN-INFINITY circles back to negative naturals.
But what happens if you add the Unity to the highest adic? All digits convert to 0 and it comes out that
...99999+1 = ...00000 = 0 ; the same as -1+1 = 0, which is a nice, short and elegant proof that the natural numbers exist in circularity, and the overdriving of infinity from the positive side makes you walk to the negatives and so on.
let â = ....99999 and ê = 10* â. In natural numbers ê is greater than â, but in adics, like in negatives is NOT:
...99999 * 10 is ...99990 which is SMALLER than ...99999 the same as -1 *10 is smaller than -1. So it comes out that:
G3: â * 10 < â , but not in every case â * 11 and so on
Now we see, that ...33333 = -1/3 that means ....33333 = -0.33333...., so „ADICS“ means „ELANOITAR“ or „ADICS“ = - „RATIONALE“ or at last „ZWRXH“
Now sum the ADIC and the -.ELATIONAR (or ZWRXH) then you get
....33333 + 0.3333.... = ....33333.33333.... = -1/3 + 1/3 =0
The same occurs with
....77777+0.77777 ... = ...77777.77777... = -1/7 + 1/7 = 0
which proves that
G4: each ADIC incremented by it’s ZWRXH is just disappearing or nothing.
There are many representations for zero:
0,...000.000...,...999.999...,...888.888... and so on
There are funny representations for some adics:
...666.333... = ...333.
...333.666... = 0.666..
Now, what is the adic - representation of -1/7?
1/7 = 0.142857...,
if â = ....142857 then â - 1000000*^a = 142857 where follows that -999999/1000000 = 142857 . So because 1000000*1/7 - 1/7 =142857 is the same as â it follows, that the adic ...142867 is equivalent to -1/7 ( and not, as an example ...758241).
Now because -1/7 - 6/7 = -7/7 = -1 we try to find the -6/7 representation in adics-format:
it is just (simple multiplication):
..142867142857 * 6 =
857142857142
which adds up with â to
..999999999999
which is just the same as -1
One can see, that the adics shown until here are equivalent to rationals between 0 and -1. Bigger negative equivalences are easily computed
-2 = ....999998
- 3=.....999997
-100 = ....999900
What about positive numbers?
1 = -1 +2 = ....999999 + 2 = ....00001
One can see, that the last 4 values are aperiodic adics, or periodic adics with an offset.
Next: after adding adics, what happens on multiplication?
....1111 * 2 = ....2222
....3333 * 6 = ...9998 (equivalence: -1/3*6 = -2)
adic multiplied by adic
...1111 * ....1111 = ......54321 where the decimals-resolution is aperiodic. Note that the square of an adic leads to an aperiodic sequence as well as the root of a rationale leads to a aperiodic sequence. The same number 1/81 is representable by -1/81 - -2*1/81
The sequence of magnitude of infinity multiplied by itself is aperiodic. Another representation in another numbers system (with base of infinity) shows the digits of this product as the sequence of natural numbers! That means that the simple square 1/81 is to be represented by a number of complete beautyness: the sequence of naturals. The third potence gives the sequence of sums of natural numbers and so on.
In base inf it is ..... 15/10/6/3/1 and in base ten it is 260631, and periodic with a length of 81 digits, the same as 1/729.
Will the root of an adic lead to a periodic sequence?
adics represent negatives, so the root of an adic is imaginary. What about the third root?