Prove that 0.99999….=1

[Proud90sKid]

I am devoting this thread to discussing or creating various proofs that 0.99999… repeating is in fact equal to 1. Or maybe you dont think that it is and want to present an alternative argument.

[¯|_(ツ)_|¯]

How good at math does one have to be to do this

[Proud90sKid]

Originally Posted by ¯|_(ツ)_|¯

How good at math does one have to be to do this

Not that good. There are a lot of different(and short) ways to prove this, so thought it would be a good topic for this section.

Can you think of a number in between 0.99999…. and 1? (Probably not, unless you construct some weird number system unlike those we use in real life).
What is 1-0.9999999…? If you can’t think of it, can you think of a nonzero number less than 1-0.99999……? I cant think of one.

There is even a wikipedia article on the 0.9999….. thing. And I didnt even have to look to know one almost certainly exists, although I checked anyways.

Edit: wow, never thought Id see an ad for reddit (another forum system) allowed on this site. I remember the days when mentioning another forum site was against the rules. Anyways, speaking of reddit: there is an entire reddit community devoted literally to the topic of 0.99999…=1
r/infinitenines

About the weird infinitesimal number system- a prominent member (maybe a mod) of that community is convinced 0.9999… is actually less than 1 because they seem to be working with an atypical number system that allows infinitely small non-zero quantities. They seem super convinced of it. Havent looked at their arguments thoroughly to know if they are really valid though. I suspect not. A lot of it sounds more like misconception than a rigorous construction of a new number system where these quantities arent equal. But, possibly could be depending on what their idea of a number is and if it is logically consistent. But in the real number system,.99999…=1

Different people have different weird ideas and sometimes these ideas are logically consistent and can be deemed “right”. For example, you are free to stop accepting the existance of irrational numbers if you make certain assumptions(don’t we pretty much just use an approximation like 3.14 whenever we need to use pi in real life?). Why not just see pi as a quantity that can be quantified to any precision we need but not a number in itself? If some ancient religious/extraterrestrial text wanted to tell us pi, they will have to stop the precision at some point and the text would technically be somewhat off. You literally cannot precisely quantify it in words. Only how to calculate it ourselves to any arbitrary precision.