The most intuitive explanation I've read is that if two numbers are distinct, then you can find a real number inbetween them. No such number exists for 0.999... and 1.
The answer isn't intuitive in any language other than the language of math. It can be debated, because the solution cannot be seen with the eyes. Even when it is shown, it is not believed.
Math is beautiful like that. You have to trust it and you have to trust your teachers.
The answer isn't intuitive in any language other than the language of math
Sorry for I might have not expressed myself clearly. "There is no number you can name between these numbers shown in this notation" is in English and appeals to intuition, it does not show that there is indeed no such number. Food for thought: http://scienceblogs.com/goodmath/2009/05/you_cant_write_that...
You have to trust it and you have to trust your teachers.
No. Mathematics is a virtue of explanation (i.e. proving). No trust involved.
=> proof there's a # between .9... and 1, assuming of course .9... < 1.
What's cute about this is it's employing a similar conceptual error: the above proof is literally circular -- it's proving something equivalent to '.9... < 1' (that there exists a distinct # between .9... and 1) by assuming '.9... < 1'; it's just different-looking enough to hide the circularity / equivalence if you don't want to see it (the analogy to the 'debate' as to whether .9... == 1 should be apparent).
And if you did 'solve' X = (.9... + 1)/2 using the standard 'algorithms' for doing it by hand you will wind up with X = .9... + .0...1 (more on this in a sec); you get there by noticing 1.9... / 2 looks like it ought to be .9..., but "of course" 2 x .9... == .9...8, ergo you can deduce you have .9... + a remainder of .0...1.
If you don't believe .0...1 exists (infinite 0s followed by a 1) then this ought to convince you that .9... == 1 (as (x+y)/2 == x => x == y) but since we're (hypothetically) arguing with someone who believes .9... != 1 it's quite plausible that the person is also going to believe in .0...1 existing, also.
What I suspect is actually going on (I've also met otherwise-savvy people who will debate this issue also) is that the human numerical-cognition system uses something like a crude form of nonstandard analysis internally; and, under this theory, you see this internal system (and its intuitions) leak out in some people smart enough to get that .9... really does have 'infinitely many' 9s but who also for whatever reason are both undereducated on this point of mathematics and generally self-assured.
Briefly: you can extend the real #s with a new entity (call it delta) such that:
- delta > 0
- if r is a nonzero # then delta < |r|
- (for now) delta^2 = 0
(in effect: delta is an infinitesimal, it's nonzero but smaller than any other positive #, and delta^2 is 0 is mainly an algebraic convenience).
Once you've extended things by including delta you can write extended #s in the form r + s x delta (eg: 1 - delta).
If we re-define ".9..." to be 1 - delta, then the following all drop out of that:
- .9... != 1 (1 - .9... == delta != 0)
- .9... < 1, (1 - .9... == delta > 0)
- and we even get: (1 + .9...)/2 = 1 - (delta/2) ==> (1 + .9...)/2 > .9...
And more generally in my experience most of the spinning-in-circles around people with beliefs about .9... != 1 is that the people who vociferously believe .9... != 1 don't even know about non-standard analysis (or infinitesimals, etc.) and thus can't articulate what their intuition is telling them, but what their intuition is using is basically (a nonrigorous form of) analysis with infinitesimals.
As to your general point of what constitutes "internal representation", someone below tries to convince people that definitions involved with Reals are arbitrary and can be made differently thus forming Hyperreals. But no. One first have to construct Reals anyways, then one arrives at infinity, which is not a number, hyperreal or otherwise, but a consequence of numbering per se. Nothing ad hoc here. Hyperreals are that or another notation, but the main thinking is still the same, just paradoxes pop in different places, the only gain is increase in confusion as hyperreals, tuns out, do not conform to that "internal representation" in many more places than the poor 0.(9) .
I'm not trying to claim the "internal representation" is somehow more correct or even prior-to the real #s, etc.; I do think (though admit I haven't proven) that a great many common examples of innumeracy can be explained by assuming the innumerate are using a "reals + infinitesimal" representation...which doesn't make the innumerate claims right, but possibly helps get at why they make those particular mistakes and not others.
In the previous post I was hoping to make (only) the following claims(s):
- in my experience, the theory "people who claim .9... != 1 are (unknowingly) equating .9... to 1 - delta" has explanatory power (in that the results of doing calculations with deltas are aligned with what these people claim)
- I suspect that the internal human numerical representation is often a cruder, less rigorous form of nonstandard analysis, and that the reason you encounter otherwise-smart people who won't budge on .9... != 1 is that for them the question goes straight to that internal representation
The just-so story for why I think the internal numerical representation is approximately real #s + some infinitesimal would go like this:
- small distances aren't reliably perceivable under natural conditions: if I have two identical 8L pots and one has 4L and the other has 4.001L in it I probably can't tell which is which visually; if I'm not in very controlled conditions artifacts of ambient lighting or perspective or variations in the pot material will overrule any perceivable visual difference between the two
- actions with nonzero actual effect often have no perceivable effect: if i have an 8L pot with 4L in it and I insert 1 ml from a pipette I probably can't see a difference even after the ripples settle down, but of course I will know that there's more water in the pot after I put some in than before
...and thus it'd make sense (at least for items with a continuous scale of size) to keep track of both:
- what size do I perceive this to be? (the real component)
- if I know extra information (eg: that I added a .001L) about the size does that information say that the actual size is going to be > or < the size I perceive it to be (the "infinitesimal" component)
...and it's not unreasonable that for ad-hoc, intuitive reasoning the mind would (roughly speaking) make the "correct" adjustments when combining size measurements (eg: (x + delta) + y > x + y, (x + delta) + (y + delta) > x + y + delta, (x + delta) + (y - delta) 'is a wash between' x + y, etc.)...after all even babies and animals apparently can do simple sums intuitively/automatically.
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.
People are still talking about this? I remember being convinced of the right answer ("yes") back in elementary school. If 0.9 repeating is not equal to one, name the number between them.
I was fortunate enough to have a very patient math professor in high school who argued with me for days about this topic and eventually I agreed that .9999... = 1
Not everyone learns this very odd mathematical issue at the same age, some will never hear of it in their entire lives and if they do, will never believe it.
Math is abstract, you can't see it to believe it and that's the problem here, you can't see the end of .9999...
I think it's interesting to see it asked here. Probably the vast majority will believe the proof - but of course some wont. And I would imagine the level of counter argument here to be not only high but potentially approaching it from obscure and intriguing new angles :)
Sigh. Here we go, a definition of the real numbers. I'm only doing this once.
Let's look at the set of all cauchy sequences of rational numbers. A sequence of rational numbers is just a map N->Q. We write a_0, a_1, a_2, ... for the sequence which maps 0|->a_0, 1|->a_1, 2|->a_2, ... Such a sequence is called "cauchy" if it has the property that:
For all epsilon > 0 there exists an N such that n,m > N => |a_n-a_m|<epsilon.
Basically, this means that the tail of the sequence bunches together.
Now, we define an equivalence relation ~ on the space of cauchy sequences. We say:
a_0, a_1, a_2... ~ b_0, b_1, b_2... if:
a_0-b_0, a_1-b_1 ,a_2-b_2... converges to zero
(A sequence a_0,a_1,... converges to zero if, for every epsilon > 0, there exists N such that n>N => a_n < epsilon.)
Now, the real numbers are the set of equivalence classes of rational cauchy sequences under ~, as defined above.
Clearly, .999... is (the equivalence class of):
0.9, 0.99, 0.999, ...
And 1 is the equivalence class of:
1, 1, 1, ...
So are these the same? Well, we have to check if our representatives are equivalent. Does this sequence converge to 0?
It still doesn't convince my colleague, because he says that the sequence never gets to zero, but now I know where I stand. He can't deal with the formalisms of convergence and limits.
It's also interesting to try to prove this via the Dedekind Cut construction, rather than the equivalence classes of Cauchy Sequences. I'll have a go at that later tonight.
That's why it's an interesting post to me, since this guy came up with a clever device (and I think internally self-consistent device) that lets him argue to the contrary.
And to answer your question, a number between 0.9 repeating and one is 0.999...5 (yes this is 9 recurring with a five at the end)
The naive perspective on this issue is because people intuitively perceive the hyperreal numbers. In the hyperreal number system, .9999999999999999999999... and 1 are different. But in almost all phrasings of this question it is implied to be talking about the real number system.
Specifying it gets rid of all ambiguity. The real numbers .999... and 1 are equal. The hyperreal numbers .999... and 1 are not.
I find these threads deeply ironic (the 400 page ones, anyways) because both sides arguing are ultimately wrong because both perspectives are correct. People with the naive perspective aren't wrong for not specifying they aren't talking about the hyper real numbers, because the other side doesn't specify they are talking about real numbers.
At a philosophical level, the answer is that .999999999999999999999999999... is different from 1. The difference is real enough to let you rigorously define calculus.
This is one of those subjects that makes an online community go insane (with 400 page threads), because it's possible for the math people and the philosophical people to go on and on without figuring out the right answer.
