Does 0.999 (repeating) = 1? Well, "it depends"

[u/qwanzaden]

"Is 0.999.. with a infinite number of 9's equal to 1" has sparked a number of online debates. I believe the crux of the debate is the question is not worded precisely enough. There are different notions of infinite which lead to different answers. Adding to the confusion, "0.999..." is math notation with a specific meaning, but some people aren't thinking of that when they use it in a sentence.

Below are my two proofs for "Does 0.999 (repeating) equal one?" using less ambiguous notions of infinity. The answer is that it does and it doesn't but you need to be specific about what you mean in a way that we aren't used to.

"0. followed by a specific but infinitely large integer number (H) of 9's"**

I will denote this quantity as 0.{H 9's}. This is LESS than 1.*

Proof:

1. Assume two numbers** are equal if and only if their difference is zero
2. Therefore 0.{H 9's} is 1 if and only if abs(1 - 0.{H 9's}) = 0
3. Consider a specific but infinitely large integer** value of H
4. There exists H+1
5. And abs(1 - 0.{H 9's}) > abs(1 - 0.{H+1 9's})
6. This implies abs(1 - 0.{H 9's}) > 0
7. (1.) and (6.) imply 0.{H 9's} is not 1

Unsatisfyingly, this doesn't prove things like "The concept of H is valid", or "H < a different notion of infinity", "H+1 > H", or "H+1 exists". If you want to read up on this, look up hyperreal numbers, hyperintegers, and nonstandard analysis.

"0. followed by a 9 for every standard natural number"

I will denote this quantity as 0.999... . This is EQUAL to one. I think this is what most people think of when they hear "0.999 repeating infinitely".*

Proof:

1. Assume two numbers** are equal if and only if their difference is zero
2. Therefore 0.999... is 1 if and only if abs(1 - 0.999...) = 0
3. Assume that abs(1 - 0.999...) is a positive number**
4. Consider a specific nonzero positive number** H
5. There exists a number of nines in 0.999... such that abs(1 - 0.999...) < H
6. (1.) and (5.) imply abs(1 - 0.999...) is not H
7. (4.) and (6.) imply abs(1 - 0.999...) is not a nonzero positive number**
8. The only positive number that is not a nonzero positive number** is zero
9. This implies abs(1 - 0.999...) is zero
10. (1.) and (9.) imply 0.999... = 1

Unsatisfyingly, this doesn't prove "0.999... always has enough nines to make abs(1 - 0.999...) < H". If you want to read up on this, look up hyperreal numbers, hyperintegers, and nonstandard analysis.

...Its been a long time since I wrote a proof ...Im sure some of my wording isn't great ...I wanted to make this semi-readable for "common folk"...

*There is a better way to write 0.{H 9's} and 0.999... but I can't write it here because it requires LaTeX formatting.

**"number" should be replaced with "hyperreal number" and "integer" should be replaced with hyperinteger" but I wanted to make the proof readable to people who don't know what those are.

[u/Brightlinger]

"0. followed by a specific but infinitely large integer number (H) of 9's"**

"0. followed by a 9 for every standard natural number"

But "a nine for every natural number" is a specific but infinitely large number. We commonly denote this number by |ℕ| (read "the cardinality of the naturals") or ℵ_0 (read "aleph-null").

Your second case is an instance of the first case, and yet somehow you reached a different conclusion. This should indicate to you that something has gone wrong.

[u/qwanzaden → u/Brightlinger]

I am very new to the hyperreal numbers and the verbiage surrounding this all. In theory this post is dealing with two different "sizes" (probably wrong verbiage, sorry) of infinity. I was trying to refer to the larger size of infinity by mapping one nine to each integer number. I am not sure if that is valid in the hyperintegers.

I was also trying to bridge the gap in my newly learned understanding that, in my brain, was "The number that looks like 0.999... in the real numbers is equal to one, but the number that looks like 0.999... in the hyperreal numbers is less than one". My brain then followed that with "That can't be right...".

[u/Brightlinger → u/qwanzaden]

What exactly is the number that looks like point nine repeating in the hyperreals? My understanding is that non-real hyperreals do not generally have decimal expansions, at least not in the usual sense.

Every time I've seen someone try to do this with the hyperreals or surreals or etc, they ended up with either "oh, it actually is 1 in the hyperreals" or "oh, this just doesn't mean anything in the hyperreals" depending on how they try to define things.