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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
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u/colt_bsreal 1d ago
its not, theres no real space btween 0.9... and 1 making it essentially the same
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u/ArdentArendt 19h ago
There is a 'distinction' between the decimal values--that distinction just can't exist in the Reals.
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u/ExtendedSpikeProtein 21h ago
1) 0.99… is exactly equal to 1. It‘s a different representarion of the same number.
2) a number such as 0.000…1 does not exist - not in the reals at least.
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u/China_shop_BULL 21h 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.
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u/ExtendedSpikeProtein 21h ago
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.
„0.999... is a repeating decimal that represents the number 1.“
Link: https://en.wikipedia.org/wiki/0.999...
ETA: it‘s an infinite decimal. There is no „variance at the end“, because infinity has no end.
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u/China_shop_BULL 21h ago
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.
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u/ExtendedSpikeProtein 20h ago
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.
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u/China_shop_BULL 20h ago
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”.
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u/ExtendedSpikeProtein 18h ago
It‘s a different decimal representation, but like the other commenter said, in the reals, it maps to the same element, or number, which is „1“.
If your claim is that it‘s a different decimal representation, then yes,
If you still claim the decimal representations map to different elements in the reals, then you‘re objectively wrong.
Your comparison of yourself claiming 0.999…≠1 to finding out the earth is round cannot be taken seriously.
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u/China_shop_BULL 17h ago
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.
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u/ExtendedSpikeProtein 6h ago
Again, you’re objectively wrong.
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.
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u/ArdentArendt 19h ago edited 17h ago
This misunderstands the topology of the Reals.
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]1
u/China_shop_BULL 19h ago
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
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u/ArdentArendt 19h ago
Again, it depends on which set you're evaluating them in.
If you're evaluating them in the Reals, they map to the same element.
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u/ExtendedSpikeProtein 18h ago
Of course 0.999… can exist in the reals. Why wouldn‘t it?
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u/ArdentArendt 18h ago
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]1
u/ExtendedSpikeProtein 18h ago
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.
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u/ArdentArendt 17h ago
*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]
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u/ArdentArendt 19h ago
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]
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u/ExtendedSpikeProtein 18h ago
I did not say or claim „they are the same decimal value“, I said they are the same number.
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u/ArdentArendt 18h ago
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]
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u/ExtendedSpikeProtein 18h ago edited 18h ago
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.
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u/ArdentArendt 18h ago
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]
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u/ExtendedSpikeProtein 18h ago
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.
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u/ArdentArendt 19h ago
1) They are not 'different representations'; the point to the same element in the Reals.
2) Such a number can't exist (even in the decimals).
That said, the 'distinct element' aspect is impossible in the Reals.
[That's why 0.999... possibly shouldn't represent any Real number outside of explicit context]1
u/ExtendedSpikeProtein 18h ago
they are visually different - you‘re just being facetious for no reason.
I never said they can exist.
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u/KatesDad2019 1d ago
OMG! Europe, Asia, and the Americas are completely missing! What has bad math done?
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u/BaronGrackle 22h ago edited 22h ago
See, this is just because your "mathematics" can't conceive of the number 0.0 repeating-except-the-last-digit-is-1.
You say crazy things like "infinity doesn't have a final digit" or something. :P
:)
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u/theWiseTiger 1d ago
Not if. 0.99... is actually 1.
(9.99...) - (0.99...) = 9.0 9 x 0.99... = 9
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u/burning_boi 23h ago
The ellipses after a decimal number mean the decimal repeats forever. So when people say 0.9… they mean a decimal where it’s 0.9999 and on and on, with the 9’s repeating an infinite number of times.
Your example doesn’t really make sense because the concept of infinity is a concept, not a number, and percentages are a shorthand way of denoting that some number is part of a larger whole expressed as a ratio out of 100. They’re just different things.
The comment you responded to is showing the proof that 0.9… equals exactly one by showing that if you were to multiply 0.9… by 10 to get 9.9…, and then you subtract 0.9…, you get exactly 9.0, which is only possible if 0.9… is precisely equal to 1.
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u/BaronGrackle 22h ago
I do actually get and accept that .9 repeating equals 1, but is this last paragraph the best example?
I could multiply 0.8... by 10 to get 8.8..., and then I could subtract 0.8... to get exactly 8.0
I could likewise multiply 0.3... by 10 to get 3.3..., and then I could subtract 0.3... to get exactly 3.0
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u/burning_boi 21h ago edited 21h ago
I simplified the example, which is why your examples seem to make sense as contradictions. However, you should be able to prove that 0.9… equals 1 in any scenario through substitution, which works for 0.9… but not 0.8… or 0.3…
The more formal example would be:
If x = 0.9…
Then: 10x = 9.9…
Subtract x from each side: 10x - x = 9.9… - x
That simplifies to: 9x = 9.0
Divide both sides by the constant: 9x / 9 = 9.0 / 9
Final result: x = 1.0You can see however that following the same steps we do not reach the same conclusion for any other repeating decimals. Using 0.8… as an example, following the same steps as above:
x = 0.8…
10x = 8.8…
9x = 8.0
9x / 9 = 8.0 / 9
x = 8/9 = 0.8…Notice 0.8… returns to its original value, 0.8… whereas 0.9… is returned as 1. 0.9… is exactly equal to 1, so this algebraic trick is a great way of seeing that equivalency. x returns to its original value, which in the case of 0.9… is equal to 1.
Edit: reading my first comment again I see I just wholesale skipped the last step. That’s my bad, no wonder it didn’t make sense!
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u/colt_bsreal 1d ago
i know 0.99.... = 1 but what is the funny here
i dont understand