Mathematical Girls - Gödel's Incompleteness Theorems - Vol. 1 Ch. 5

[MangaDex]

 

[Simpleton]

Another way to explain the 0.999... = 1 phenomenon is to go through fractional math.

Most people learn that 1/3 = 0.333... and can accept this

Most people then learn or understand that 2/3 = 2*1/3 = 2*0.333... = 0.666...

Continuing down that path, 3/3 = 3*1/3 ? 3*0.333... = 0.999...

And since early fractional math teaches you that 3/3 = 1 this leads us to the conclusion that 1 = 0.999...

 

[syketuri]

Surprisingly they didn’t use geometric series for this. We can write the funny number “0.9…” as an infinite sum \sum_{k=1}^{\inf} [9 * 10^{-k}]. As the common ratio r = (1/10) is in (0,1) we can equate the sum with the formula a/(1-r) where a is the first (k = 1) term and we have already defined r. Hence 0.999… = (9/10) / (1 - (1/10)) = (9/10) / (9/10) = 1. QED

 

[phosphatprostrat]

Surprisingly they didn’t use geometric series for this. We can write the funny number “0.9…” as an infinite sum \sum_{k=1}^{\inf} [9 * 10^{-k}]. As the common ratio r = (1/10) is in (0,1) we can equate the sum with the formula a/(1-r) where a is the first (k = 1) term and we have already defined r. Hence 0.999… = (9/10) / (1 - (1/10)) = (9/10) / (9/10) = 1. QED

The reason they did not do this is because to define infinite sums, one must define limits in the first place. The limit of partial sums is what makes your expression well defined. So if you go around explaining this first, then you might as well be more direct like them.

I appreciate the attention of this work to using as basic of notions as possible. It builds a lot of things from ground up, without relying solely on intuition, while still giving intuition with their axioms afterwards.

 

[syketuri]

The reason they did not do this is because to define infinite sums, one must define limits in the first place. The limit of partial sums is what makes your expression well defined.

Fair enough, I am implicitly using preliminary results. Though after re reading the chapter, they are kind of informally defining limits with the “getting closer to” notion. They could also probably get the class to agree that, with S=0.999… (or heart as suggested in the chapter):
S/10 = 0.999…/10 = 0.0999…, and so
S/10 = S - 9/10 => S = 1