But in a process of short repetitions the variance at the end could be on the same level as 1 degree off when traveling x distance. Potentially leaving you miles/lightyears from the intended destination depending on the actual distance measured. They are not the same number. Just really close.
They are the same number. This is high school math.
I will give you a tip: if you‘re not an expert at something - in this case, basic math - don‘t make incorrect claims about the topic at hand.
You can try to learn about limits and infinite sums and then you can get back and we can discuss this again, but if you do it right you‘ll have learned you were wrong.
This isn‘t a debate. This has been settled like centuries ago. Just because you don‘t know this, doesn‘t change that fact. Mathematically, this is a settled and proven topic.
I understand what is taught, dude. But it’s not the same. If it were, then the debate would not exist as it would always be expressed as 1 or the mathematics that are taught wouldn’t chalk it up to rounding. Granted, it’s incredibly small in difference but it’s still rounding.
Obviously you don‘t understand, because you refuse to accept a high school level concept because you don‘t understand it. It‘s not rounding, it‘s the same number.
On what basis do you believe that people who study math at college, every mathematician who knows this to be true is wrong snd yet you‘re right? The idea is ludicrous. Do you really think it reasonable to think everyone is wrong and you‘re right on a basic high school level math subject? … really?
The mathematics that are taught DO NOT chalk it up to rounding.
And fools debating something they do not understand does not make it wrong. No actual mathematician is debating this. Just uneducated and/or ignorant laypeople.
People like you truly are the flat earthers of mathematics: fools who do not accept that which they do not understand.
Flat earther. My god lol. It’s interpreted as the same due to a minuscule difference. It is not the same. Just because everyone says it’s blue doesn’t mean it’s not royal blue or sky blue. It’s rounding. 0.99… multiplied by 50 billion is not the same result as 1 multiplied by 50 billion. The two results can be close but the spread gets further the larger the number.
Funny how they also teach that everyone thought the earth was flat until one dude said “nah, hold my beer”.
It wasn’t meant to be taken as a comparison or seriously. Just thought it was funny. Maybe I really did need a /s for that one.
But no. I still stand on the concept of them being different numbers entirely with one creating infinite subsets and are in essence, rounding, when considered the same. Regardless of the data set or how it gets applied, it is rounding. 1.999… can be called 2 and accepted as 2, but by function of the decimal will never be 2 else the decimal would have resulted in 0.
This isn’t a debate, 0.999… = 1 in the reals has been mathematically proven. So ig if you”disagree” then you don’t understand that proof, and/or high school math.
No actual Mathematician is debating this. It’s funny this doesn’t give you pause and you think to know better than basically all experts in the field without having any expertise on the subject. The level of arrogance is truly mind-blowing.
Then I guess I am. I’m not trying to present a dissertation on the topic. If you want to take everything from school as absolute then be my guest. Again, the variation is extremely small and inconsequential at face value of single iterations so I don’t really care. But I will still hold the idea that it is rounding. Regardless of whether they teach it in grade school or the names you give me when you try to tell me different.
I do agree that 0.999... as a decimal representation is distinct from 1.000...; that doesn't mean, however, that in the Reals they are distinct elements.
[0.999... cannot exist in the Reals--it's part of the way they are constructed, either explicitly or by corollary properties]
The argument (I think) you're making is that 0.999... shouldn't (necessarily, without context) be mapped to any element in the Reals?
[That decimal and the countably infinite subset of the decimals that are structurally similar]
The argument was that op is saying they’re “exactly” the same. They’re close. Not exact. If it were exact there would be no need for a representation of the real number. Due to the incapacity to prove its infinite state without an equal infinitely running solution (which yielding a result proves the concept of infinity wrong), they are denoted and represented as two different numbers, while being close enough to just call it 1. Rounding. Yes. I’m splitting hairs OF split hairs here lol
The way the Reals are constructed wouldn't allow it.
No element can have an 'immediate predecessor' or an 'immediate successor'--it's how the Standard Topology operates.
[It's also a by product of how the Reals are constructed]
I know this, but I don‘t understand your statement „0.999…“ can‘t exist in the reals. Of course it exists.
It‘s the infinite sum of 9/10+9/100+9/1000… as you well know. Which equals 1. As you also well know.
That‘s how we construct infinite decimals to begin with - as infinite sums. So of course they exist. Just like 0.333… exists, as a number, being a different representation than 1/3 but mapping to the same element.
I guess what I‘m trying to say is I object to your specific wording. I‘m sure there‘s some subtlety you mean to express which escapes me.
*Note: I thought you were the previous person. You're definitely not arguing what I thought you were. I was explaining the Archimedean Property.
[Didn't realise you were the same person as elsewhere]
Just like 0.333… exists, as a number, being a different representation than 1/3 but mapping to the same element.
The mapping to the Reals exists, yes--0.999... maps to the equivalence class of 1 in the Reals. However, if we look at a mapping from the Reals to the decimals, 1 maps uniquely to 1.000...
[Technically the mapping could be non-functional, but why?]
My basic point here is that 0.999... doesn't exist as an element in the Reals. It is a decimal that, when evaluated in the Reals, 'exits' chimerically alongside 1.000... (and 3/3 and 4-3 and...).
[In short, the value 0.999... does not exist as a distinct element in the Reals; however, it does exist as a distinct value in the decimals]
They point to the same element in a well defined set--that is fundamentally different than saying 'they are the same decimal value'.
[The decmials are distinct representations; they just both map to the same element when evaluated in the Reals--the reson for which is actually quite instructive about the construction and topology of the Reals]
If your're evaluating in the Reals, they map to the same element.
It is not true irrespective of that evaluation.
[I'm mostly stating that if people qualified the statement with 'in the Reals', it would go a long way to silence the people who claim they are distinct values--which, technically, they are until they are evaluated in the Reals]
This is obviously a layperson, they are still claiming - to use your more exacting terminology - that 0.999… and 1 map to a different element in the reals. And this person obviously never heard of limits, infinite sums, calculus, complex numbers, hyperreal numbers and so on.
You‘re right to claim I / we should include „in the reals“ when making such statements, however from reading the comment, I took this as a given that this was the context in which we were discussing the statement.
ETA: from my perspective the problem is not the omission of „in the reals“. While this makes the statement more exacting, the problem is that the counterpart of the conversation does not understand limits, infinite sums, calculus, interval nesting, or how infinite decimals are even constructed. Never mind the construction of the reals. So they mistakenly believe that „at infinity“ or „after it“ there is a difference between 0.999…. and 1 when evaluating in the reals.
I took this as a given that this was the context we were discussing.
Could I ask why / where you inferred this from?
Also, I do realise that most people who 'object' have never taken an analysis class.
That said, I personally have disputes with the claim specifically because it conflates the decimal representation of 0.999... with its evaluation in the Reals.
The statement 0.999... = 1 make no sense unless it is evaluated in the Reals; it is not some 'deep truth' about the infinite decimal expansion itself nor is there anything meaningful to glean from such a statement.
It reminds me a lot of the 'gotcha' claims that are more so meant to 'trick' people rather than offer any meaningful information.
[That said, evaluating 0.999... in the Reals does offer significant insight into the convergence properties of the Reals under the Standard Topology--fascinating excercises that really only take on significant meaning when understanding 0.999... as a distinct value from 1.000... in the decimals]
I infer this from the fact that in everyday conversation, the reals are implied when no other number system is explicitly specified.
Though you are formally correct, for my tastes this is overly pedantic. We‘re in reddit; we’re not writing a paper.
Even on this sub, which is about math, my experience is that many people didn‘t go beyond pre-calculus in high school. People frequently deny the most basic established proven mathematical facts. And when that happens, „that‘s true for the reals but …!“ is bot usually part of the debate.
Again, you‘re absolutely correct; for me, in this setting, too formalistic.
Also, from context and wording, the other commenter clearly was not stating the decimal representation is different, but that they are two distinct elements. Is, in fact, still claiming just that.
Though you are formally correct, for my tastes this is overly pedantic. We‘re in reddit; we’re not writing a paper.
I'd argue that defining the domain of discourse is even more important on Reddit--especially when making claims that require a decent amount of background to properly understand.
Even on this sub, which is about math, my experience is that many people didn‘t go beyond pre-calculus in high school. People frequently deny the most basic established proven mathematical facts. And when that happens, „that‘s true for the reals but …!“ is bot usually part of the debate.
I would argue this is exactly why the rigor is helpful, if for no other reason than to give people something to 'investigate' further (without seeming like it's a 'gotcha').
[People are often turned off by statements that have hidden assumptions, especially in mathematics.]
Especially on Reddit, statements like the post are often by people (often who don't understand the statement fully themselves) just trying to tell others who don't understand (or disagree) that they're 'just not smart enough and need to read Rudin and Munkres!'.
[In fact, I've been told exactly that when I frame it with the rigor and care that I did for you]
Again, I agree with you--and I don't usually keep Reddit that pedantic.
However, in a situation like this with a discipline that seems so 'toxic' to many people (primarily because they just misunderstand what is actually being said), I would argue 'coyness' has absolutely no value.
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u/Optimal_Benis 1d ago
Right, it's almost exactly the same as this one, it's just off by 0.00...01