I have self-taught myself undergraduate Real Analysis using S. Abbott's Understanding Analysis. I am now moving on to learning "graduate-level" Real Analysis using Stein & Shakarchi.
One thing, that I have seen come up a couple times now - not just in Stein & Shakarchi, but in some papers as well, is to create "base-n expansions" of numbers less than 1. Here is an example of doing this in base-3 (ternary) in an excercise.
And a similar "binary" expansion for numbers less than 1 in a paper I was reading:
I think I am just a bit confused on why these expansions are defined the way they are, and I'm even more confused by the fact that there are "ambiguities" - i.e ways to represent the same number with different expansions? I understand, mostly (although am not that comfortable with, honestly) creating base-n representations for natural numbers, but I am confused here. Can anyone explain intuitively why these numbers are defined the way they are?