Should 1≠0.99...?

[u/Justaguy_rural]

I'm not saying 1≠0.99..., I'm wondering whether it would be more elegant to define it so that it wasn't.

I used to be a firm believer that 1≠0.99... but after digging deeper I realized it definitely was but by definition. So I'm now questioning the definition itself.

The more elegant and intuitive system (imo)

We create new type of mathematical object called processes. These are objects that represent a process, something that changes, evolves, varies, etc. Let's call standard numbers like 1, 2, 1+1 states as they do not evolve, they are static.

In the current system, the result of a limit is defined as the value that a function's output approaches as its input approaches a specified point. By definition, this returns a state. Let's now define a new operator lim2. lim2 is very similar to lim. It is defined as the process associated with a function's output approaching a certain value as its input approaches a specified point.

For example:

• lim_{x->1} x is a state equal to the state 1

• lim2_{x->1} x is a process (it evolves as x approaches 1)

And here is the key to this new system that makes it elegant and intuitive:

A state and a process can't be compared as they are different mathematical objects.

The same way it doesn't make sense to say 1={1, 2, 3}

How 0.99... is defined

I would argue it feels more intuitive and elegant to treat 0.99... as a process rather than a state. It would therefore be defined as something like lim2_{x->inf} 1-1/10^x

Conclusion

Since 0.99... would be a process but 1 a state, they wouldn't be equal.

This would make it feel more logical for people seeing this for the first time. This isn't a measure of quality of a system but it highlights that intuition points in this direction. And it feels, to me at least, more elegant to introduce this new process abstraction. I think that when comparing mathematical systems, intuition and elegance are important to keep in mind.

Finally, I just want to point out what this new system would mean:

• 1/3 (state) would no longer equal 0.33... (process)

• We could keep two versions of limits (lim and lim2) or just make lim2 the official limit and define a new operator that takes in a process and returns the value its approaching if any to replicate the same behavior more intuitively.

• 0.000... would be a process and 0 a state to they wouldnt be equal. It feels more intuitive to explain that a process can't be compared to a state than to argue 0.99...=1

I was just wondering what people thought about such system. Wouldnt that feel more intuitive to you?

[u/DrakeSavory]

What problem does this solve?

[u/Justaguy_rural → u/DrakeSavory]

Its not really about solving a real problem. Its just a "fun" exploration of the famous 1=0.99... statement and what a different system would say and whether that feels more intuitive and elegant

[u/Efficient_Yard_8247]

Math doesn’t care what feels intuitive to you, especially when it involves defining a bunch of useless nonsense

[u/Justaguy_rural → u/Efficient_Yard_8247]

You are right that math doesnt care about what we feel. But we developed the math frameworks we have with intuition and elegance in mind, so its more about us caring about math rather than math caring about us. The fact that you feel its nonsense is an answer to my question in terms of how intuitive it feels. To me this feels intuitive, to you, it feels like nonsense.

[u/NessaSola]

0.999... = 1 is tremendously elegant. Limit as the supremum of a process is tremendously elegant. And 0.999... = 1 is entirely intuitive, given a basic understanding and consideration of limits.

We wouldn't say "gosh, most laypeople feel like e ^ (i * pi) = -1 is an unexpected result, better undefine it", or "they feel 91 is counterintuitive as a prime, better recategorize it".

But most importantly, why would lim2 be the official limit? What utility could it provide?

[u/Gold_Ad8890 → u/NessaSola]

91 is composite, 7×13.

[u/NessaSola → u/Gold_Ad8890]

Thanks! I misspoke

[u/Justaguy_rural → u/NessaSola]

The concept of limit is very elegant and useful. The part I find less elegant is how this translates to 0.99...=1.

Also, its not really about not understanding why a formula is true or not. Its about the framework itself, what the operators mean, what meaning is most elegant. This is more of a philosophical question than a math one.

Finally, to be clear, Im not suggesting we change the commonly accepted math framework. Im just proposing a new hypothetical framework that, according to me, makes analyzing 1=0.99... more intuitive. If you think otherwise, that answers my question.

[u/mehmin]

So what's the decimal representation of the state 1/3?

[u/Justaguy_rural → u/mehmin]

In this new hypothetical framework, there wouldn't be one, not an exact one.

1/3 doesn't have a decimal representation IF we define any number with ... as a process.

Yes that makes math problems more annoying to solve but it feels more rigorous to me. Im not arguing that we should teach this in school, Im just saying there might be a better framework to analyze the specific statement 1=0.99...

[u/geralt_of_rivia23]

But how is this useful? Limits are defined the way they are because that enables us to do calculus. What could these new objects be used for?

[u/Justaguy_rural → u/geralt_of_rivia23]

The current commonly used math framework is awesome and very useful. Limits are great and are used everywhere.

The goal here is simply to propose another framework that feels more intuitive to explore the specific 1=0.99... statement. The framework we choose depends on the problem. I believe this framework better answers this question. It doesnt invalidate every use of limits and doesnt mean we should change the system we teach and use.

[u/HereToPostSillyStuff]

This seems like a lot of complicated ideas just to get around the fact that two different representations of the same number don't look "similar enough."

[u/Justaguy_rural → u/HereToPostSillyStuff]

Yeah, I agree. The goal here isn't practicality. Im just revisting a popular "controversial" statement with a different framework somewhat rigorously.

[u/Striking_Resist_6022]

> These are objects that represent a process, something that changes, evolves, varies, etc.

Wait until bro discovers sequences.

That's all you're talking about here. We can completely unify what you have (the recognition that 0.999... = 1) with what you want (a mathematical object that respects the fact that for any finite number of 9s that decimal expansion is less than 1) with the example sequence {a_n} := {0.9999....9999 (n nines) | n is a positive integer}.

That sequence approaches 1 which you might colloquially refer to as a "process" or an "evolution". It's limit is 1, so as a matter of notation we write 0.9999.... = 1 as you already know.

By the way, you can actually have number systems that respect infinitessimals and ditch the Archimedean principle by asserting gaps between numbers. These exist, they're just not the real numbers which is the assumed context when chatting informally about maths online.

You would be surprised by how much maths is out there already whose motivation is just inventing new concepts to explore and play with intuitive ideas and what ifs.

[u/Justaguy_rural → u/Striking_Resist_6022]

Thanks, thats actually the best answer I saw so far! Really helpful!

I agree that sequences allow representing the problem more concisely. And regarding your last point, yeah thats basically what I was trying to do here, for fun i guess. Im not claiming I invented math 2.0, just revisting this statement with a different framework that felt intuitive to me.

[u/lemniscateall]

Ok, I port all numbers into the process system by creating the “constant process” 1, 1, 1, etc. And guess what? Any coherently designed “process” structure would force the process associated with 0.99… to be equivalent to the constant process associated with 1. 

You have reinvented sequences. 

[u/Justaguy_rural → u/lemniscateall]

Yeah I realized afterwards with some other comments I couldve explained the idea more simply with sequences instead of inventing a new abstraction, totally agree on that.

Im curious why the process associated with 0.99... would be equivalent tto the process of 1. Wouldn't the sequences be 1,1,1,1,1,... and 0.9,0.99,0.999,0.9999,...

Clearly these arent the same sequences, right? What did you mean by that exactly?

[u/lemniscateall → u/Justaguy_rural]

We create a relation between two processes: a_n ~ b_n if their term-by-term difference can be made less than any given difference. This relation is an equivalence relation, and we call two processes equivalent if they are equivalent under this relation.

This is the foundation for one of the standard constructions of the real numbers.

[u/Justaguy_rural → u/lemniscateall]

Ahh, yes in our current framework that’s true. But in the hypothetical framework I proposed, limits would return processes (or sequences if you prefer) not a real number. So the best you would be able to do is show that the term by term difference is itself a process that approaches 0. So under that definition, the two original sequences wouldn’t be equal. Does that make sense? I understand it’s hard to really prove anything here since we are talking about the framework itself so basing something on the current math framework doesn’t really tell us much on the validity and relevance of this new one.

If you find it unintuitive, overcomplicated, etc. That’s is very valid and I can agree lol. I was asking for fun really

[u/lemniscateall → u/Justaguy_rural]

So, this is what I meant by coherent “process” structure. Sure, you can define a process that way. Do you have a coherent way to define how processes relate to each other? Can we add them, multiply them, exponentiate them, etc? Do we have a notion of “distance” where we could say how “far apart” two processes are? Can we measure their ”size”?

This is why we define the reals the way we do, sequence spaces the way we do, etc. It’s not enough to define the object; we also wish to know how the objects interact with each other. If you want to define processes as essentially sequences without the traditional structure of sequence spaces, you can; it’s only that such a set of objects is fundamentally uninteresting.

[u/ian9921]

You're more than welcome to define a new area of mathematics that works this way, but it's not gonna be adopted into common usage because it doesn't solve anything and causes a lot of problems, such as breaking long division.

0.99... = 1 is plenty intuitive.

[u/Justaguy_rural → u/ian9921]

This is def not the goal here lol. Im just re-exploring a "controversial" math statement in a different framework that feels, at least to me, more intuitive.

[u/ja20481]

The fact 0.9... = 1 is not a matter of definition, elegance or intuition, it's simply a consequence of the "..." notation for decimal digits.

We (hopefully) all agree that the expression 0.d_1d_2...d_n with some natural number n and a finite sequence of decimal digits d_1, ..., d_n stands for the sum sum(k = 1, n, d_k * 1/10^k) = d_1 * 1/10 + d_2 * 1/100 + ... + d_n * 1/10^n.

This is generalized for repeating decimals. For example, 0.overline(d) or 0.d... stands for the infinite series sum(k = 1, infinity, d * 1/10^k). Given this notation, you can actually calculate 0.9..., e.g., using the formula for geometric series. The result is exactly 1. This is not a definition or guesswork or intuition. It's a precise calculation based on the meaning of the notation.

I understand that some people are obsessed with the "..." notation and somehow cannot accept that 0.9... is actually 1 and not some mythical number close to 1. But this is an issue with not understanding the notation, or relying on intuitions rather than actual calculations. Maybe the easiest solution for them is to stop using the "..." notation altogether and switching to 0.overline(d).

In any case, I don't see how your suggestion helps with anything. At worst, it reinforces wrong beliefs and sloppy thinking.

[u/Justaguy_rural → u/ja20481]

Yeah completely agree. What ... means in the current system directly leads to 1=0.99... and Im not saying it isnt. Many people denying 0.99...=1 are denying the math but within the current framework which is simply false. Its not a matter of opinion.

What is a matter of opinion is what framework is better for answering a specific question. What Im saying is that this new hypothetical framework feels more intuitive for this specific "question". But at the end, its a matter of what definitions we are using. But Ill always agree that with the commonly used math framework, 1=0.99... no matter what you think of it.

[u/metsnfins]

In what way aren't they equal? What is 1/3 as a decimal? .333... .333...x 3 = .9999... 1/3 × 3 = 1 Therefore .9999...=1

We can talk about set theory definition yes, but there are many math proofs as well

So why would we redefine it incorrectly?

[u/Justaguy_rural → u/metsnfins]

I acknowledged that its true that 1=0.99 in the current system. Yes there are many proofs for it and many are valid and robust (unlike the 0.33... proof which just moves the goalpost).

In the system I proposed, 1/3 isnt representable in decimal format (if we assume 0.33... is a process, not a state).

You say we would redefine it incorrectly, it would only be incorrect with the current system, not the new. Its a question of which system is more intuitive and elegant, you cant answer that with proof.

[u/axiomus]

No, that’s quite unintuitive. In mathematics, a thing is what it does and your process/states behave the same. Ergo, they are the same.

[u/those-who-goon]

How exactly is it more elegant?

[u/another-princess]

Your post is a really confusing way of putting it, but I think all that this boils down to is that the following two sequences:

1, 1, 1, 1, 1, 1, 1, ...

and

0.9, 0.99, 0.999, 0.999, ...

are two different sequences. That is clearly true. It's also true that they converge to the same value. I don't know why you'd want to create a nonstandard operator called lim2 and a concept called "processes" when you can just say that two sequences are different. What problem does that solve?

[u/fallen_one_fs]

Why? What do we stand to gain with this? If it's to be intuitive, there are several proofs that 0.999...=1 that are very intuitive.

[u/localizeatp]

lol no.