I actually had this post deleted on other forums as they banned me for sexual selection theory and deleted all my posts after the bannings. Anyone who’s into math should enjoy this!!!
I discovered this method when I realized that if you mirror all of the counting numbers, you end up with every possible decimal expansion, particularly when it converges with infinity, however I have not found out how to make them all correspond with counting numbers, and am working on a proof or disproof that all the reals can be counted with infinite lists in infinite dimensions per list, something Cantors diagonal argument doesn’t address and a method which makes Cantors diagonal argument meaningless.
But what I will show you from this method is a way to count all of the rational numbers, and it is a new way!!!
First you start by listing them and their negatives…
0
1, -1
2, -2
3, -3
4, -4
5, -5
6, -6
7, -7
8, -8
9, -9
Now at this point, when you hit base, or 10 in this instance, the rules change somewhat, this is where you start mirroring. 10 mirrors as 01, so the sequence for 10 is:
10, -10, 0.1, -0.1, 0.(1 repeating), -0.(1 repeating)
11, -11, 1.1, -1.1, 1.(1 repeating), -1.(1 repeating)
etc…
when you hit 100 and beyond a new rule needs to be used, and this is the very last rule you need to use in order to correspond all of the rational numbers with the counting numbers in sequence!!!
The rule is: Anytime a number that’s about to be mirrored ends in a zero, only move the decimal point in one place!!! For example: The mirror of 100 is 001, which leaves us with 0.01. If you move it in one more place after that ONLY if it ends in a zero before being mirrored, then you will have a number like 00.1, which will cause INFINITE overlap, because 0.1 was already the mirror of 10, and will occur an infinite amount of times as the zero’s keep expanding as the numbers get larger… I’ll show you 100-102, so you can get the picture…
100, -100, 0.01, -0.01, 0.0(1 repeating), -0.0(1 repeating), 0.(01 repeating), -0.(01 repeating)
101, -101, 1.01, -1.01, 1.0(1 repeating), -1.0(1 repeating), 1.(01 repeating), -1.(01 repeating), 10.1, -10.1, 10.(1 repeating), -10.(1 repeating)
102, -102, 2.01, -2.01, 2.0(1 repeating), -2.0(1 repeating), 2.(01 repeating), -2.(01 repeating), 20.1, -20.1, 20.(1 repeating), -20.(1 repeating)
If you use this method, you will have all rational numbers correspond to the counting numbers without any overlap!!!
What I’m trying to prove right now is whether the reals can all be listed or not using infinite dimensions per list using infinite lists.
I hope you enjoyed this!