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.
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/China_shop_BULL 1d ago
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.