<snip>Mentor note: Off-topic post and reply have been excised.
DanKnaD said:
couldn't any real number made using the construct by adding 1 to the nth digit of the nth number be turned into a natural "version " by removing the 0. And writing the number out backwards.
How does this method work for two real numbers with the same digits after the decimal but different digits before? For example, 1.2345 and 2.2345. Do these turn out to be 54321 and 54322? If so, then you run into an issue where multiple numbers are mapped to the same number. Both 2.2345 and 0.22345 map to 54322 in this case. That seems like a problem, as you lose injection and bijection.
If you keep the decimal after the swap then you still run into the issue of trying to make a number with infinitely many digits after the decimal turn into a number with infinitely many digits before the decimal. Both 0.3333... and 0.4444... become infinitely large if you try to reverse their digits. This is a problem. A number like 0.333... is, well, a
number. It has a unique representation and a unique value. A change to any digit makes it a different number. But neither ...333 and ...444 (where the ... means the digits continue on to the left without end) are numbers as far as I can tell. They have no unique value; both ...333 and ...444 are infinitely large. I can change ...333 to ...332 and the 'value' if you want to call it that is still infinite.
This issue occurs because a number with infinite digits after the decimal converges on a value as the number of digits increases. That is, going from 0.3 to 0.33 to 0.333 and so forth makes sense. There is a single value or quantity that this process will never exceed (more accurately, it is the smallest value that this process of adding digits will not exceed), which we define as the final value of this process and is represented by 1/3 or 0.333...
But going the other way the number diverges, never reaching a unique value. 5 to 55 to 555 and so forth results in increasingly large values and there is no value that this process will fail to exceed. In other words, pick a finite number of any size and this process of adding digits will eventually exceed it, no matter how large the number you pick. So I'm not even sure these qualify as numbers.