can you explain the logic you think is being failed? The alternative (that 1/3 is exactly 0.3333...) seems equally arbitrary to me.
It appears to me that the people on this thread arguing that 1!=0.9999... have come up with a fully consistent, alternative way of defining equality when involving infinite fractions that is no worse than the conventional definition.
You say "It appears to me that the people on this thread arguing that 1!=0.9999... have come up with a fully consistent, alternative way of defining equality..." but if you look, there are absolutely no mathematical proofs outside of the people showing that 1=0.9r. Instead, there are people blankly asserting that there's a 0.0...1 remainder, which as many have pointed out, is meaningless without a precise mathematical definition and perhaps a proof of existence within the set of reals. And the reason it's meaningless is because it's ill-defined and misunderstands infinite series, etc...
I'm no math whiz, but the reasoning on display here isn't just sloppy, it's totally perpendicular to mathematical reasoning. Arguing against it is like trying to argue with someone who asserts that "the sunset tastes exuberant." The statement may look a lot like a meaningful phrase, but it's just nonsense.
well the suggestion being made is that an infinite series can be terminated - and with a different digit. That's finite right? :)
Irrespective of 1/3 = 0.333 (which can be proven mathematically) terminating an infinite series is impossible.
The same fallacy persists in all his other points. For example when he does the 10x business and points out there are still only 5 9's then his series is clearly finite and not infinite - and therefore no longer 0.9 recurring.
The main issue with people "disbelieving" 0.9 recurring equals 1 is a misunderstanding of what infinity actually is :)
EDIT: in reply to your edit - you've fallen for the misunderstanding of infinite. No one can dispute that 0.3<lots of 3's>3 + 0.0<losts of 0's>1 = 1. But in an infinite series you never get to that 1. The number is 0 (this can be mathematically proven I am told; but it is beyond my ability to do so)
> Kind of an interesting idea, actually. (Of course, that's after only looking at it a couple minutes... the idea might fall apart under more scrutiny)
Yes, it is an interesting idea, and yes, it falls apart :) (As I pointed out lower in the thread, it's hard to define 10*0.00...1)
For a more rigorous version of infinitesimals, try:
"can you explain the logic you think is being failed?"
You can not have an infinite series of something, followed by something else. One definition of "infinite" is that it never ends, which is also why it isn't a number (a number is intrinsically something that would have a definite end). So to have "an infinite series of something, followed by a 1" is to have "a series of something that never ends, followed by a 1 after the end". I can type those words without crashing English, but it has no meaning.
"The alternative (that 1/3 is exactly 0.3333...) seems equally arbitrary to me."
No. I'm not even sure what else to say. If you need convincing, start the long division on paper and keep going until you're convinced.
No, seriously, keep doing it until you're convinced. Right now. Don't reply with a counterargument until you've done that. There's nothing arbitrary about it; to prove that there is, you need to show some point where you came to a choice and you chose to add the next 3, rather than some other hypothetical alternative.
"have come up with a fully consistent, alternative way of defining equality when involving infinite fractions that is no worse than the conventional definition."
Well, no worse, other than also defining an internally-contradictory definition of "infinity" for the sole purpose of winning an internet argument. Other than that, no worse, no.
Recall that introducing one contradictory premise into a logical system allows you to prove any statement. If you are not bothered by a "small" contradiction to prove a dubious point, you don't understand math. Thus, making up definitions to prove a point must be analyzed in the full context of a mathematical system, not just analyzed on some other abstract measurement system (that doesn't matter). Breaking infinity to win an argument doesn't work.
...
Most people go about proving this the wrong way. Proving something about the string "1" vs. ".999..." is wrong. In order to show the two are distinct entities, you need to show a situation in which they behave differently. No such situation exists. Until you do that, "1" = ".9999..." is no more surprising that "2/2" = "1". Or (ahem) "9/9" = "1". Numbers are what they do, they have no existence beyond what they do.
On that note, when teaching I think the point that "=" means "is bidirectionally fully substitutable by" is not taught properly. "2 + 2 = [underline]" is a flawed question; oh, we all know what it means, but "2 + 2 = 2 + 2" is a perfectly correct answer to the question as written. We ought to define a separate operator for what we really mean, "simplification", so we can do something like put "2 + 2 => [underline]" on a test, and save the equality operator for actual equality.
That is also why, for instance, the definition of the square root function must include the stipulation that it takes the positive root, because otherwise "sqrt(4) = 2" is not a true statement; if sqrt(4) means "both" roots somehow it is not fully substitutable with 2. (You can't say that both "sqrt(4) = 2" and "sqrt(4) = -2", either, because then "2 = sqrt(4) = -2". Oops.)
That's the background for my statement that .9999... = 1. It means that there's no way to tease the two apart with any (valid!) mathematical operation. Here's another way to say it: On the real number line, if you have real number A and real number B, and there are no points between A and B, then A and B must be equal. To show they are two different real numbers, you must show a number C between 1 and .9999.... You can not do this... well, again, you can't do this validly.
(Also, per RiderOfGiraffe's point, yes, there's some simplification here, but I think it's fair to run on the theory that intrinsic to this argument is that we're on the real number line. People who know enough to talk about transfinite ordinals don't usually get into these arguments; by then, they've learned the secret of math lies in the definitions chosen and understand the ultimately contingent nature of any answer. :) )
> You can not have an infinite series of something, followed by something else.
Actually, in the more general sense, you can. The transfinite ordinal numbers do exactly that. Using w to represent the first infinite ordinal we have w=1+w, but w<w+1. The transfinite ordinal w+1 is modelled as the natural numbers, in order, with an extra element stuck "on the end."
However, your point is valid in the context of the decimal representation of real numbers.
"can you explain the logic you think is being failed?"
You can not have an infinite series of something, followed by something else
Colonel Oates: Get down and give me infinity.
[Bill and Ted drop]
Colonel Oates: You stupid, pathetic, craven little cretins.
Colonel Oates: You petty, base, bully-bullocked bugger billies!
You're not strong! You're silky-boys...
Dead Bill: Dude, there's no way I can possibly do infinity push-ups.
Dead Ted: Maybe if he lets us do them girly-style.
Perhaps this is the key to the confusion. If we can reason about infinities, and "put" them in mathematical proofs, why can't we just "put" an infinity of zeros followed by a 1 somewhere?
Again, it's the paucity of English as a language precise enough.
Again, I don't think this is necessarily that fruitful a discussion, since the whole point of the guy's argument, as I see it, is that you can slightly adjust the meaning of "equality" as it relates to infinite fractions and maintain a completely self-consistent mathematical system.
> You can not have an infinite series of something,
> followed by something else.
Why is that? How about the number 19...91? why are you denying me that number? :)
> If you need convincing, start the long division
> on paper and keep going until you're convinced.
When I do that, I constantly am left with a remainder. If I accept that 1/3=0.3333..., as you're arguing, then the remainder disappears, for unexplained reasons. An alternative argument is that the remainder persists in the form 0.0...1 and then you don't have to "hand wave away" the remainder.
> Recall that introducing one contradictory premise
> into a logical system allows you to prove any statement
I agree with that completely, but I haven't seen the "contradictory premise" yet... all I see (at this point) is some alternative premises that seem fully internally consistent.
> In order to show the two are distinct entities, you
> need to show a situation in which they behave
> differently.
I think that's actually the best argument you've put forward so far. Basically, an argument in favor of choosing the more pragmatic way of handling these situations.
"When I do that, I constantly am left with a remainder. If I accept that 1/3=0.3333..., as you're arguing, then the remainder disappears, for unexplained reasons."
No, it doesn't. You stopped finitely soon into an infinite process. You apparently didn't take it far enough.
No disrespect intended, but a HN comment is not a place to post a full description of infinity. Hie thee hence to a textbook. And read it like a textbook, not an internet post. You know, where you go over it with a fine-tooth comb looking for opportunities to leap up and call your opponent a Nazi. Yeah, I know that style. Without that, counterarguments are getting mangled on their way into your brain, because you don't actually understand them in Math. That can't be corrected by people batting down various ill-posed English-based objections.
(If it could, I would have seen it happen by now.)
0.0...1 is a very interesting number. Normally, one indexes digits by integers (there's a first digit, a second digit,...) But here you have digits seemingly indexed by ordinals. The omega-th digit is 1. I don't think you can make a consistent number system this way. Here's the problem: what is 10*0.00...1? It can't have omega-th digit "10" because 10 isn't a digit. It can't have omega-th digit 1 and omega+1-th digit 0, because that should be the same as 0.00...1, under any logic I can think of. So that number system doesn't work.
The real numbers, by contrast, have the advantage of working (so far as we know.)
OK, that's the most convincing argument I've heard so far. The omega+1-th digit would need to be 1, for your example.
I'm not entirely sure that it has to be the same as 0.00...1, as you argue. However, you're definitely showing that this way of handling fraction would probably get to be pretty ridiculous at some point.
ok some key points about infinity ot address here :)
>then the remainder disappears, for unexplained reasons.
Actually no - it doesnt disappear. That's why it works. Dont forget the definition of infinity is "forever". The remainder is always there - and it always reduces further (edit: if you continue completing the sum. As you can see below the sum is already completed for infinite length anyway - your just manually bound checking it at the beginning)
In your counter argument where the 0.0....1 acts as a remainder. The set of 0's before the 1 can only be finite. One reason people fall for this is because they imagine the infinite set growing by adding a 0 on the end each time. Or perhaps pushing on one the front (which is what would have to happen for your requirement). This is *wrong( entirely - the infinite set exists, in entirety for the whole of it's infinity.
It appears to me that the people on this thread arguing that 1!=0.9999... have come up with a fully consistent, alternative way of defining equality when involving infinite fractions that is no worse than the conventional definition.
To expand his argument a bit: The argument is:
...seems reasonable to me (again, it requires a slightly different definition as to what "equality" truly means)Kind of an interesting idea, actually. (Of course, that's after only looking at it a couple minutes... the idea might fall apart under more scrutiny)