It has been many years since I noticed something bizarre in our number system. It has caught my eye since then. In the beginning, I discarded it as something irrelevant, obvious maybe and something that was a mere illusion.
Start with a number, any four digit number will explain best and easiest. For example $6173$ (any random number). First remove the alternate digits from right side, i.e. the number becomes $67$ (removing $3 $and $1$). Write$ 6173$ in front of it again. It becomes $617367$. Repeat the steps. $$ 1. 6173$$ $$67$$ $$2. 617367$$ $$676$$ $$3. 6173676$$ $$ 137$$ $$4. 6173137$$ $$ 133$$ $$5. 6173133$$ $$133$$
In four steps, we got a number 133 which will be repeated forever. I like to call this 'purest form of a number'. This is applicable to any digit number, any number till infinity.
Further, I observed that single digit numbers cannot be made any more 'pure'. They reach their purest form in one step. 2 digit numbers reach their purest form in 2 steps, 3 digit numbers in 4 steps, 4 digit numbers also in 4 steps. 5 digit ones in 5, 6 in 6, 7 in 6...
Writing them up in a series: $$1, 2, 4, 4, 5, 6, 6...$$
Alternate odd digit numbers don't purify in as many number of steps.
Now,1 digit numbers have 1 digit pure number, 2 have 1, 3 have 2, 4 have 3, 5 have 4, 6 have 5, 7 have 6 and so on.
This pattern becomes: $$1, 1, 2, 3, 4, 5, 6...$$
I studied removing alternate digits from the left side too. It is almost the same. Also, in the second last step of reaching the pure forms, the numbers get very very close in any instance. In the above example, 137 and 133.
My question is:
(a) Is this actually a good observation or just an obvious fact? (b) In any case, why are we sticking to a number which doesn't change after a few steps?