I don’t agree that 0.999… = 9/10 + 9/100 + … = 1 as an infinite sum, (since this is impossible and oxymoronic to say “infinite sum is a finite sum”).
But yes, I do agree with the *limit *of said series being 1.
The reason is that endless strings of digits to not become a finite number.
PhD’s have said that […] means “and so on”, or “for infinity”. Infinity is endless, so if you are stating endless 9’s, ie: 0.999…, then that is what you have, and not 1.
However if everyone agreed that […] means “limit”, which I have a hard time believing, then we would be getting somewhere I suppose.
True. You construct the number 1, from a sequence that converges to it.
The number ONE is output as the constructed number. What you are doing next is attempting to write the sequence of partial sums as “the number”, and claiming it is equal to the number which was constructed.
Dr. Wildberger has a lot of very interesting videos on youtube, and he points out some very interesting topics. Since Cauchy Sequences was recent;y brought up, have a look at MF94.
I think there is an issue here. Infinity does have a very clear definition, and it is likely that not using it does cause some difficulty. Words like “endless”, whilst not wrong, are not precise enough, and have the unfortunate habit of becoming confused with notions of time. Lets be quite clear. In this discussion -
Infinity is the cardinality of the Natural numbers.
(It also happens to be the cardinality of the Integers and the Rationals, but we can leave that for the moment.)
Any other notion of Infinity is going to get you into trouble in one form or another. The numerous attempts through this thread to construct some form of finite infinity are one, and notions of “never completed processes” another.
The only useful precise definition of 0.999… is as discussed. It is an infinite series. Sidetracks into how it is “written down” or otherwise miss the point. You can have any number of imprecise colloquial ideas about what it means, but if you want to argue the fine truth, you need to stick to the precise definitions. That of course includes sticking to the well understood precise definitions for all the concomitant operations that are applied.
Whoa there. I thought you were arguing that 0.999… is less than one. Now you are saying it is not even a finite number.
Precision in the use of words and symbols is a mathematician’s friend. Either you have claimed something brand new here (which I have no time for) or you have been imprecise.
It is exactly that kind of imprecision in definitions that leads to fuzzy thinking and ultimately to erroneous conclusions. If you are at odds with a bunch of professionals (you really should read their stuff some time) and if you have demonstrated at least one instance of fuzzy thinking then you really should entertain the possibility of your being wrong.
Does not follow. ≈ means that a number is either equal to or almost equal to another number. 1/3 ≈ 0.33… does not exclude the possibility of 1/3 = 0.33…
However, assuming your usage of the symbol, which apparently means “close to, but not equal to”:
This last step does not follow. If the series (2.9, 2.99, 2.999, …) is convergent towards 3, then the limit of the row is equal to 3. You cannot simply insert the limit of that series and expect to get the same result. Things behave differently at the limit.
We don’t have to assume that, though. We’ve proven it. Multiple times. And everything past this point is just unintelligible gibberish. You are not explaining yourself well, even when ignoring the errors.
Since it is endless, there is always another 9 and therefore, the difference is always 90% less than the iteration before. That is why I said something along the lines “0.(9) is less than 1 by an unidentifiable amount”.
That is because endless 9’s: 0.999… is also an unidentifiable amount, since we do not know how many 9’s are really there, and using induction, for every (n) amount of 9’s, there are n+1 9’s which is greater than the previous. There are no amount, even infinity 9’s that suddenly change 0.(9) to 1. That is what an infinite sum is saying. But as discussed, if […] means limit, then of course the solution is 1.
This is all basic analysis. I can understand what you want me to understand, as well as professionals. Except, if you discuss these things in real life, you are quick to be shunned, and labeled “crank”, etc etc… so you end up just jumping through hoops, pretending you “agree” with everything you are reading and doing, etc, etc. Again, if I may refer you to Dr Wildberger, for example, MF94: Problems with Limits and Cauchy Sequences, he is a professional who is not afraid to bring up issues, but as you said, because he doesn’t agree with his co-mathematicians on everything, “you (he) really should entertain the possibility of your being wrong.”
I don’t think that is the case at all. I believe mathematics should be advancing and evolving. By being close minded and thinking everything that was said hundreds of years ago are the only facts, and refusing to entertain any new thinking, is pretty selfish.
Very early on in the video he tells me that sqrt(2) is a fictitious number and does not really exist. Are you really sure that you want to submit such an outrageous statement as supporting evidence for your cause. Because if you do, I anticipate a systematic and decisive shredding. Dr or no, this is so far at odds with the way that mathematics uses the Real numbers as to be unbelievable.
Limit is a voluntary concept. Endless amount of digits is easily understandable as a concept of endless digits. You are making limits “automatic” and doing analysis voluntarily on an object.
It seems like this is the case:
I give you: Σ 9/10ⁿ (n ∈ N)
Then you start constructing it: 9/10 + 9/100 + (and so on)
You get tired of writing 9’s, and eventually realize you can never finish writing them, then decide you are getting closer and closer to 1 (while really, you are still infinitely far from reaching it, since for every (n), you are still infinitely far from ∞).
I watched his (very boring) 40 minute video, and his claim that √2 does not exist boils down to the fact that we cannot calculate its decimal representation completely.
At the very end he makes one half-amusing point, he says that “we” all accept that
√2 = 1.41421356237309504880168872420969807856967187537694807317667…
But do we need all those decimals? How about
√2 = 1.4142135623730950…
Or
√2 = 1.4142…
But could we go further?
√2 = 1. …
And why not just simply
√2 = …
Which is reminiscent of the THEN A MIRACLE OCCURScartoon, so I smiled. But he is wrong in the thrust of his argument, no-one believes that the ellipsis “…” indicates “and then ALL the other digits to infinity that we have calculated(or could conceivably calculate)” – there is no miracle that is being glossed over, it is just an informal shorthand.
[referring to post 1977]But everyone else here is using the word infinite to refer to the cardinality of the set of natural numbers and you have agreed with this. Only you are using this word to refer to the amount of ink or time required to write a number down.
You are not being consistent and you are definitely employing terminology in an unorthodox fashion which leads to your conclusions being incorrect. Strinka put it really clearly for you.
It’s a good question. You are pretty much on your own with your interpretation of terms and symbols. I am not sure what you gain by persisting with it.
And with that, I retire. It is past bedtime where I live.
I have already explained why I accept …99999 = -1 as true. I got the idea
from reading James Tanton’s document which I mentioned.
If you have carefully considered everything I have said, you should also yourself
be able to arrive at the same conclusions.
On the other hand , if you don’t take me seriously, then it is, I think, impossible to
understand what I say. Tell me, what is, for example, …99999.99999…,
what is it equal to, is it infinity etc?
Can you answer? Can you answer if you think that what I write is nonsense?