Show me: “You have insisted that someone solve this problem without limits (which has been done, though you ignore it)”
RE:" it’s an artifact of decimal notation." - I’ll do you better than that. It’s pretty much a definition of the decimal system. There is no argument about what .999… = 1 within the rules and definitions of the real number system. I would not argue that, it would be pointless. The question is deeper than that. The real number system does not necessarily give us some profound insight into the matter of what
9/10 + 9/100 + 9/1000 + …
ACTUALLY sums to. The limit tells us that it = 1 and it also has the limit of .999…
This is no big deal as the real number system defines them to be the same. No where in the real number system is this ever proven to be the case. And frankly, I am not concerned with what the real number system says precisely because it starts out with the assumption they are the same.
Oh my this isn’t the secret council of mathematicians who control the world of mathematical definitions? :smack:
I like to debate. And apparently people like to debate with me. I have no intent (immediately anyway) of changing the definition real numbers. There is I BELIEVE a deeper question here than just what the real number system defines .999… to be. And I have explained this numerous times. That is what I am discussing and arguing here. And this is something the real number system tells us nothing about.
The standard way of adding infinite terms uses limits. If you want to use another method called the “actually sums” method, you need to define that method. And maybe you’ll come up with a cool way to do it.
The actual sums method as defined by Erik150x: “Add just like you did in grade school.”
Yes, I know that is quite impossible to do with an infinite series.
But taking the Limit does not really address the question at hand. If you understood the definition of a Limit you would understand that by it’s very definition .999… = 1. The Limit process specifically is designed to avoid infinitesimal differences. Now I have no way to actually add this infinite series, but simple logic tells me that no matter how many 9s you stretch out there, even to infinity, it will never precisely equal 1.
I’m going to emphasize again for late comers who may not have been up to reading all 666 posts so far. I am not discussing this strictly in terms of real numbers. Since the real number system very much defines .999… = 1 without question, that would be pointless. I would also point out that the real number system defines them to be the same because it would mess things up terribly if it didn’t. We can’t have 2 different real numbers whose difference is zero (at least in real numbers). However neither the Limit definition nor definition of real numbers proves in some absolute sense that .999… = 1. Understanding this distinction would be key to this discussion. There is no point in debating anything here if you simply wish to discuss strictly in terms of real numbers.
I guess what I have been using this notation for would be described like this. If a number is given as such “.aaa…” then a is assumed to stretch out to the infinity (of the countable type).
If a number is given as such “.aaa…b” then a is presumed to stretch out to just short of infinity and at infinity the decimal place is = b. I have been tentatively referring to this as the infinite’th decimal place. Go ahead add that to your dictionary
I can see this needs further refining (assuming your eyes have not already rolled into the back of your head). So let us further say that “.aaa…bc” mean a stretches out to the infinite’th decimal place - 2. I.e. c would be in the infinite’th decimal place and b the infinite’th decimal place -1.
Lastly, I would describe any decimal places beyond the infnite’th as such: “.aaa…bc.de” where “.aaa…bc” are exactly as described above and d is the infnite’th place +1 and e is the infnite’th place + 2.
But nothing can provide that absolute sense you’re talking about.
Anyways,
You say it’s “intuitive” to think of some finite digit after the 3 dots “…”
Why is that “intuitive”? The 3 dots signifies infinite digits expansion so appending a finite digit on the end contradicts the entire notation and acts as if the 3 dots “…” was never there at all.
In your “alternative” notation which you’re trying to reconcile with “intuition” what does the following mean?
2.7…5…3
Hmmm… 2dot7infinitedigits5infinitedigits3
I don’t see any value-added “intuition” by think of “…” as a finite textual substitution.
Is 0.0…05 the same number as 0.0…5? Yes.
Is 0.0…5 the same number as 0.0…50? No. The difference here would be 5/infinity.
This should be clear by my notation. But alas sometimes we need examples to clarify.
What’s (0.0…1) / 3? = 0.0…0.333… + 1/3(infinity-aleph-1)
0.0…3…? meaningless
What’s (0.0…3…) * 10? meaningless
0.0…3…0? meaningless
What’s actually kind of fun is that Indistinguishable pointed the way to an approach to these, upthread. I don’t know how to access a Greek alphabet here, but, for the moment, pretend that “i” is an iota…
And that this iota is defined (however) as an infinitesimal number, “smaller than any real number.” (It is NOT the same “i” as in complex numbers, the square root of negative one.)
0.000…5 is, then, 0 + 5i
0.000…50 is 0 + 50i.
As was also noted, you don’t get closure if you multiply them, so that you, instead, end up with polynomials in i.
Another way to model it is to declare that “…” means “one trillion decimal places.” It’s so vast as to have no physical real-world meaning…but it does actually give you a system you can work with, formally and rigorously.
All of this is completely made-up, a neo-definition, and, furthermore, one that contradicts the standard definitions of numbers. It breaks set closure, for instance. You can now have two numbers, which, multiplied by one another, results in a product which is not a number!
So, our correspondent can have it his way – if he’s willing to take the effort to create rigorous definitions, and to pay the price of the consequences.
I don’t believe he is well-enough educated to do this. He is at the stage of having nothing to offer but denial. He won’t accept what he is told, but lacks the expertise to work with the implications of his intransigence. It took a much better mathematician – a damn sight better than I will ever be! – to provide a way of justifying these declarations in a rigorous fashion.
I’m sure an inventive mind, backed with some solid education, could come up with other rigorous definitions that would offer some abstract functionality.
I once had a lengthy phone call from a stranger who had devised an entirely new mathematics (or arithmetic) simply by doing away with the number “nine.” He talked enthusiastically to me for an hour, but never quite managed to say anything cogent. It was clear in his mind, but he had no way to communicate it to anyone else.
RE: But nothing can provide that absolute sense you’re talking about.
Your opinion. I respect that. But do not necessarily agree.
RE: Why is that “intuitive”? The 3 dots signifies infinite digits expansion so appending a finite digit on the end contradicts the entire notation and acts as if the 3 dots “…” was never there at all.
I wouldn’t say its exactly intuitive at all. And to some degree this leads us into the discussion I was having with Frylock and others about the existence of an infnite’th decimal place. I have constructed an informal proof that one must exists. And I am certain that is debatable. I think if perhaps you go back and read some of this you will be kind enough to not make me re-hash it all and probably answer the same questions that have been asked already. A good starting point maybe post 574.
RE:"What’s actually kind of fun is that Indistinguishable pointed the way to an approach to these, upthread. I don’t know how to access a Greek alphabet here, but, for the moment, pretend that “i” is an iota…
And that this iota is defined (however) as an infinitesimal number, “smaller than any real number.” (It is NOT the same “i” as in complex numbers, the square root of negative one.)
0.000…5 is, then, 0 + 5i
0.000…50 is 0 + 50i.
As was also noted, you don’t get closure if you multiply them, so that you, instead, end up with polynomials in i."
That would be yours and/or Indistinguishable’s notation. not mine.
RE: “All of this is completely made-up, a neo-definition, and, furthermore, one that contradicts the standard definitions of numbers. It breaks set closure, for instance. You can now have two numbers, which, multiplied by one another, results in a product which is not a number!”
**Of course it’s all made up. And being that you don’t even understand the notation I have given I don’t see how you can make such claims. **
RE: “… It took a much better mathematician – a damn sight better than I will ever be! – to provide a way of justifying these declarations in a rigorous fashion.”
Indeed, I dare say that’s true. And what does the fact that a damn sight better mathematician took the time to do so, tell you about the idea of doing so?
RE: “I’m sure an inventive mind, backed with some solid education, could come up with other rigorous definitions that would offer some abstract functionality.”
**He already has, their called Hyperreal numbers which extend the real number system to include infinite and infinitesimals. But I would think someone like you would already know that?
I have no intention of creating my own number system, and why would I when hyperreals have already been created. I am simply playing what if as far as defining notation here. You asked me to which is fair enough. But you don’t even understand the simple notation I have given?
Furthermore, I am not sure so much rigor is required here to discuss the matters at hand. You simply seek to confuse the issue, (if you even understand it), by asking me to further refine some new number system. That’s my take anyway.
I mean your statement here:**
"Another way to model it is to declare that “…” means “one trillion decimal places.”
**is about as ridiculous as I can fathom. Why on earth would I want that. **
RE: Even **1+1=2 **cannot be proven in the absolute sense. You realize that right?
Touche. It is true. Yet, I am not sure anyone in there right mind would argue that. But perhaps I am wrong? The idea that .999… might not exactly = 1 in a a deeper sense than reals can offer, is not exactly crazy to my mind. Absolute truth is probably not the best way to say it. A deeper truth perhaps is better.
RE: But nothing I saw in your post #574 renders that notation meaningless.
No not in that post, and rightly so someone asked me here to clarify that notation which I did. I think the key here again is accepting the idea of infnite’th decimal place. It’s a bit extreme I grant you. And I am simply playing with the idea to express my ideas on .999… not equaling one in some system that would extend or modify the reals. And no I have neither the inclination or eduction to do so. Plus its already been done. There is no harm in playing the what if game here, and as needed I am happy to try to clarify my ideas. I’d be especially happy to discuss where my ideas my be faulty if presented in a reasonable and non-condescending manner.
You’ve been trying to argue with me that there must be a last member of the infinite series 1/2, 3/4, 7/8, … But your argument has relied completely on questions asked about what happens when you traverse the real number line. If you’re talking about some model other than the real number line then you need to explain what model that is. As long as you’re talking (as you have been, in your arguments) about how to traverse the real number line, you are by definition incorrect in your claim that there must be a last member of the infinite series denoted above.
Augh. I’m sorry, but augh. (Hey, you said above we were “killing you” so a little expression of emotion is warranted.) You’re right that no one would “argue it,” but what someone *would * do in their right mind is build a model in which 1+1=/=2 and see where they can go with it. It’s not a question of whether the statement is true. It’s a question of what the system it is embedded in can model.
Similarly, you seem to think you’re talking to us about whether it’s “really really true” that 0.999… = 1. You admit that it’s true by definition in one system, but then you go on to try to argue that it’s not “really” true or at least, not “really really” true in some sense. But this is a nonstarter. None of us is concerned about whether it’s true in some sense independent of any mathematical system you might care to build. There’s literally no answer to that question–it’s actually a senseless question that cannot have an answer. The only thing you could say that could be interesting would be to show us a mathematical system in which the two notations do not refer to the same value, even though in some sense they refer similarly to the way they refer in “normal math.” For this to be worthwhile, said mathematical system has some interesting properties that make it worth studying. As you’ve seen, someone has already built such a system–the hyperreals. Go learn about it! See if it gets you the kind of result you’re looking for. If not, go to school! Get a PhD! Build your own system! Everything else is pretty useless for your (or anyone else’s) purpose.
There are no “deeper” truths about what equals what. There are just different models in which different statements are true. If there are “deeper” truths they concern metamathematical matters about relationships between mathematical systems, the expressive power of different systems, and so on. But identity is quite literally a shallow truth, simply stipulated by fiat in each mathematical system. There are no deeper truths about which expressions should have an equal sign placed between them. Different systems are different, and that is all.
I will also stand by my basic assertion that the real number system offers us nothing in the way of deciding whether .999… = 1. Would the hyperreals do so? I kind of hope so, but don’t know them well at all. I think it is very convenient to define .999… = 1 in the real number system. I am not sure it’s an absolute necessity. Perhaps it is. Or perhaps there is just no need within the context of reals to really care about it. Surely there must be some need for further refinement or else why then are some calculus text books being re-written in terms of infinitesimals then?
