451–460 of 647 posts
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I mean you could if you want to: 1/10 + -9/100 + -9/1000 + -9/10000 + ... But I just meant them as series, not necessarily as sums.
Or 1 - 9/10 - 9/100 - ... or 1 - ( 9/10 + 9/100 + ... ) So it becomes circular: 0.00...1 = 1 - 0.999... (In math a series is a sum: https://en.m.wikipedia.org/wiki/Series_(mathematics)
What if you have 0.9̅4? Can we say 0.9̅5 > 0.9̅4 > 0.9̅3? More on what happens if you allow this: https://mathwithbaddrawings.com/2013/08/13/the-kaufman-decim...
Well, you fundamentally can't. If the 9s go on for forever then you never reach a point where you can add the 4. The definition of infinity precludes anything after infinity, because it never ends so you can never get there.
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I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…
Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since…
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There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true. Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was alway…
.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.
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Can you explain what you mean with "real" in that sentence? Because in the context of maths, a real number is "a number in ℝ", which this absolutely qualifies for. Whereas in plain English the term doesn't really have a clear definition. You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long,…
Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is m…
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great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in…
Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems. It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.
Earlier quoted context omitted.
I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…
Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since…
I'll just chime in with my completely ignorant theory that 1 - 0.999... = the infinitely smallest number, but is still, in my mind, regardless of any logic, reason, or educated calculations, greater than 0. I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.
Philosophically you are right. Mathematically you are wrong. It is indeed true that if there was an entity that could reason beyond infinity 1-0.999... would be greater than 0.
This seems like a common thread in the comments on this article, and I don't quite understand it.
Humans can reason about infinity just fine. We have a hard time picturing it, but it's overall a pretty simple concept: It never ends.
So there's no such thing as being able to reason "beyond infinity", because beyond infinity doesn't exist. Its very existence is precluded by the definition of infinity.
I don't think that 0.999... = 1 is actually provable. I think this and all of calculus is actually axiomatic, which has the following axiom: Given ε = 1/∞ then: ε = 0 Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)
I think the mistake in your thinking is that infinity is a value or reachable destination. It's a concept and it behaves differently than a number. Also go read Generatingfunctionology and Concrete Mathematics