[gus_massa]

You mapped the numbers between 0 and 1 that have a finite number of decimals, like 0.123123425736289

But there are many more real numbers between 0 and 1 that have an infinite number of decimals, like 0.1231234736876578632786278365782635879281987087420...

As jgreen says in a sibling comment, which number do you assign to 1/3 = 0.333333333333333333333333333333333333333333... ?

The result you got is well known but interesting, because it's unintuitive. Moreover with some tricks you can try to extend your method to all fractions, like 1/3, 5/7, 27/127, ...

The problem is that there are irrational numbers like pi, sqrt(2), ... and many other numbers that we have assigned no friendly name. Those "unfriendly" numbers are the actual problem.

[smac__]

Again, excellent point! These numbers clearly show why* my reasoning is missing a rather large portion of this set.

[edit] - fixed typo

[gus_massa]

Too late, but I hope you see it. I only want to add a highlight.

One you fix a finite set of symbols, let's say ascii [1], the amount of numbers you can describe with it with strings of finite length is different from the amount you can describe if you allow also infinite length.

The difference between finite length and infinite length in this problem is very important.

[1] I'm not sure about unicode /s