Yes, I get it now, thanks Michael!

0.6666659999999999999999999999999... = 0.666666

--Michael

Nobody else has had the poor taste to muddy the waters further by introducing non-standard arithmetic. So I shan't either. WFL On 1/28/13, Dan Asimov <dasimov@earthlink.net> wrote:

To Eric Angelini:

Besides what Whit and Michael said,

The only way to understand an infinite decimal representation N.abcde... is to understand

a) That it represents the limit of the sequence S = N.a, N.ab, N.abc, N.abcd, N.abcde, ... .

b) What it means for a sequence of real numbers to have a limit, and that when this is true, the sequence has only one limit.

c) That the sequence S must in fact have a limit.

So e.g. 0.99999... means the limit of the sequence of real numbers

S_1 = 0.9, 0.99, 0.999, 0.9999, 0.99999, ... .

This must be 1, because no matter how close you may want the sequence to get to the limit (say within the number e > 0 of the limit), then after a certain number of terms of the sequence, the rest of the terms all lie within e of 1. (In other words, they are all between 1-e and 1+e.)

Let's *check* this claim. If we pick a sequence of closenesses e_1 > e_2 > e_3 > ... > e_n > ... > 0 such that this sequence contains arbitrarily small numbers, and we verify the above sentence for e = e_n no matter what n we choose, then we have verified that 0.99999... = 1.

So, let's choose e_n = 1/10^n for each n = 1,2,3,... . Clearly these get arbitrarily small.

Then for any e_n = 1/10^n, we can look at all the terms of the sequence S_1 *after the nth term*. Now, the nth term has n 9's after the decimal point, so the terms *after that* have at least n+1 9's after the decimal point. It's easy to check that this means they all lie between 1-(1/10^n) and 1+(1/10^n). This proves that the limit of the sequence S_1 is 1. This means that 0.99999... = 1.

--Dan

="Dan Asimov" <dasimov@earthlink.net> The only way to understand an infinite decimal representation ... ="Fred lunnon" <fred.lunnon@gmail.com> Nobody else has had the poor taste to muddy the waters further by introducing non-standard arithmetic.

Hear hear! And here we go 'round the infinite decimal discussion, again... Look, I've nothing 'gainst the Standard Theory of the Real Numbers (STotRN); an' I too'll testify t' y'all, it surely do come in plenty handy, of a time. So (pacem, please, concerns of arithmetical apostasy) what bugs me is that darn word "only", and the vertiginous semi-circular reasoning it obscures. Surely there are many ways to "understand" infinite decimal representations? Of COURSE when we START by implicitly binding the strings 0.9, 0.99, 0.999 and so on to a convergent sequence of real numbers then 1.0 is where our interpretation of the limit of the sequence of strings MUST inevitably lead. But strings and numbers are truly different things, and different routes to the infinite can and do afford divergent perspectives. For example we've discussed the mirror world of 9.0, 99.0, 999.0 and so on. We can also model these nicely using the SAME closed form for the finite sums of a geometric series. Except now, in passing to infinity, we get a sequence of INCREASING POSITIVE values whose "algebraic limit" is demonstrably -1! The quandary is, if we favor the algebra we break with STotRN convergence. But once string ordering no longer entails strong ordering there's no longer an air-tight argument that 0.999... HAS to NAME the same THING as 1.000... And of course when it doesn't we can get "perfectly good" but "non-standard" arithmetical systems that are internally consistent (in some symbolic sense) but royally suck at modeling our conventional magnitudinous quantities. I'm totally OK with explaining to folks why STotRN PLUS the usual implicit interpretation of numeral strings leads to the identification of 0.999... with 1.000... Just, please, at least, can we somehow cool it with glibly calling 0.999... a "number"?

I may be confused, but I know of no theories, however exotic, in which 0.999... has a well-defined meaning that is not 1. In the surreals, the sequence 0.9, 0.99, 0.999, ... has no single well-defined limit. It converges on the *set* of numbers within an infinitesimal distance of 1. I would love to hear of any theory in which it makes sense to say 0.999... != 1. On Tue, Jan 29, 2013 at 3:46 PM, Dan Asimov <dasimov@earthlink.net> wrote:

Mark, I respectfully could hardly disagree more strongly.

The question of why .99999... = 1 that was initially brought up by Eric Angelini was clearly about the ordinary real numbers.

Now, I have nothing against exotic theories of numbers. (In fact, I love the surreal numbers.) But when people want to know whether, and if so why, .99999... = 1, they are in almost all cases wanting to learn or be reminded of the conventional meaning of that statement.

And that requires understanding a) that an infinite decimal represents an infinite sequence, and b) that this sequence has a limit, and c) that having a limit has a very specific meaning that can be verified in this case.

I have no idea what vertiginous or any other kind of semi-circular reasoning the word "only" obscures. But let me remind you that the fundamental group of a semi-circle is trivial, as compared to that of a circle, which is the integers.

Of course, there's always this famous quote: << "When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."

...

...

--Dan

On 2013-01-28, at 11:42 PM, Marc LeBrun wrote:

<< "Dan Asimov" <dasimov@earthlink.net> << The only way to understand an infinite decimal representation . . .

...

...

Hear hear! And here we go 'round the infinite decimal discussion, again...

Look, I've nothing 'gainst the Standard Theory of the Real Numbers (STotRN); an' I too'll testify t' y'all, it surely do come in plenty handy, of a time.

So (pacem, please, concerns of arithmetical apostasy) what bugs me is that darn word "only", and the vertiginous semi-circular reasoning it obscures.

Surely there are many ways to "understand" infinite decimal representations? . . .

...

...

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On the subject of what "0.999..." _means_ (which we need to agree on before we can discuss what number it denotes), let me suggest that instead of thinking of it as the sum of an infinite series, one can think of it as an address on the number line. The two points of view can be shown to be equivalent, but pedagogically they're different. (Students don't need to know what an infinite series is or how limits are defined in order to appreciate that 3.1415... means a number that's between 3 and 4, and also between 3.1 and 3.2, etc.) Maybe someone who knows more of the history of mathematics than I do can comment on which of these two notions is closer to the meaning intended by the people who first used infinite decimals? Note that my use of the word "between" in the preceding paragraph is ambiguous vis-a-vis the handling of the endpoints. If 3.1415... means the unique number in all the intervals [3,4], [3.1,3.2], ..., then 0.999... means the unique number in all the intervals [0,1], [0.9,1.0], ..., namely 1. But one can argue that 3.1415... means the unique number in all the intervals [3,4), [3.1,3.2), ..., in which case 0.999... means the unique number in all the intervals [0,1), [0.9,1.0), ... --- but there is no such number! Jim Propp Jim On Wed, Jan 30, 2013 at 12:44 AM, James Propp <jamespropp@gmail.com> wrote:

Allan (and others),

A theory of formal infinite decimals in which 0.999... is not the same as 1.000... is described in the talk "Why 0.999... is greater than 1.000...", presented by my student Linda Zayas-Palmer at the ninth Gathering for Gardner (slides at http://jamespropp.org/g4g9-slides.pdf).

As long as you do arithmetic with just addition and multiplication (not subtraction or division), the distinction between 0.999... and 1.000... can be sustained. But once you bring subtraction into the picture, you basically have to identify the two expressions with one another. (Or, as I was taught to say forty or fifty years ago, these two _numerals_ represent the same _number_.)

There are connections with the Faltin-Metropolis-Ross-Rota construction of the real numbers as a wreath product, and a somewhat more arcane connection with the theory of chip-firing (aka the abelian sandpile model).

Jim Propp

On Tue, Jan 29, 2013 at 5:29 PM, Allan Wechsler <acwacw@gmail.com> wrote:

...

I may be confused, but I know of no theories, however exotic, in which 0.999... has a well-defined meaning that is not 1. In the surreals, the sequence 0.9, 0.99, 0.999, ... has no single well-defined limit. It converges on the *set* of numbers within an infinitesimal distance of 1. I would love to hear of any theory in which it makes sense to say 0.999... != 1.

On Tue, Jan 29, 2013 at 3:46 PM, Dan Asimov <dasimov@earthlink.net> wrote:

...

Mark, I respectfully could hardly disagree more strongly.

The question of why .99999... = 1 that was initially brought up by Eric Angelini was clearly about the ordinary real numbers.

Now, I have nothing against exotic theories of numbers. (In fact, I love the surreal numbers.) But when people want to know whether, and if so why, .99999... = 1, they are in almost all cases wanting to learn or be reminded of the conventional meaning of that statement.

And that requires understanding a) that an infinite decimal represents an infinite sequence, and b) that this sequence has a limit, and c) that having a limit has a very specific meaning that can be verified in this case.

I have no idea what vertiginous or any other kind of semi-circular reasoning the word "only" obscures. But let me remind you that the fundamental group of a semi-circle is trivial, as compared to that of a circle, which is the integers.

Of course, there's always this famous quote: << "When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."

...

...

--Dan

On 2013-01-28, at 11:42 PM, Marc LeBrun wrote:

<< "Dan Asimov" <dasimov@earthlink.net> << The only way to understand an infinite decimal representation . . .

...

...

Hear hear! And here we go 'round the infinite decimal discussion, again...

Look, I've nothing 'gainst the Standard Theory of the Real Numbers (STotRN); an' I too'll testify t' y'all, it surely do come in plenty handy, of a time.

So (pacem, please, concerns of arithmetical apostasy) what bugs me is that darn word "only", and the vertiginous semi-circular reasoning it obscures.

Surely there are many ways to "understand" infinite decimal representations? . . .

...

...

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Let me be more precise about what I mean by equivalence here: in the presence of the ordered field axioms, the least upper bound property is equivalent to the property that every infinite decimal corresponds to an element of the field (here "corresponding" means "belonging to all the 10-adic intervals that are tacit in the decimal expansion"), and any ordered field satisfying either of these "completeness properties" is isomorphic to the real numbers. (Technical caveat: in addition to the ordered field axioms, you also need some axioms about the natural numbers, since the natural numbers are used to index the successive digits.). See jamespropp.org/reverse.pdf for details. Jim On Wednesday, January 30, 2013, Dan Asimov wrote:

"The two points of view can be shown to be equivalent" is very hard to believe, without a great deal more detail about just what the "address on the number line" concept means.

--Dan

<< On the subject of what "0.999..." _means_ (which we need to agree on before we can discuss what number it denotes), let me suggest that instead of thinking of it as the sum of an infinite series, one can think of it as an address on the number line. The two points of view can be shown to be equivalent, but pedagogically they're different.

...

...

---

math-fun mailing list math-fun@mailman.xmission.com <javascript:;> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

Well, since f(x) = {-1 if x is even; 1 if x is odd} works, I guess you could use f(n) = -1 * cos(pi*n) But that's needlessly complicated. The constant function f(n)=-1 works just fine for all n; or you could use f(n)=2 for all n. Maybe you could explain what you mean by "asymptotic". On 1/30/13, Dan Asimov <dasimov@earthlink.net> wrote:

While studying something apparently unrelated, I came upon a function

f: N_0 -> N_0

(where N_0 denotes {0,1,2,3,...})

with this curious property:

f(2n) + 2 = f(n)^2.

Puzzle: Find a closed-form asymptotic expression for f(n).

-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

On 1/31/13, Dan Asimov <dasimov@earthlink.net> wrote:

Parts of what Robert wrote don't use the specified domain and range.

Oh yes! /-: thanks!

But he's right that the constant function f(n) == 2 satisfies all conditions.

OK, let me add that f is *nonconstant*. I hope this makes the asymptotic expression unique. But I don't know that and haven't checked it out

Okay, then it's MCS1733378 : A[0] = 2; A[1] = 3; A[N] = 3 A[N-1] - A[N-2] 2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, 39603, 103682, 271443, 710647, 1860498, 4870847, 12752043, 33385282, 87403803, 228826127, 599074578, ... Way I solved the problem: Make a spreadsheet with N_0 (natural numbers starting at 0) in the left column. Next to N we'll put f(n). Next to 0 put 2, because that's the only value it can have satisfying f(0)^2 = f(0)+2 Next to 1 put "a", which represents an unknown variable. f(1)=a By the f(2n) + 2 = f(n)^2 rule, f(2) must be a^2-2, and f(40 must be (a^2-2)^-2, etc. The first row without an f(n) yet is row 3, so put "b" (another unknown) next to 3. In this way we get a table like this: 0 2 1 a 2 a^2-2 3 b 4 (a^2-2)^2-2 5 c 6 b^2-2 7 8 ((a^2-2)^2-2)^2-2 9 10 c^2-2 Next I choose an a. It can be anything, but I'll just use the smallest valid choice, which is 3. Then we get: 0 2 2 1 a 3 2 a^2-2 7 3 b ? 4 (a^2-2)^2-2 47 5 c ? ... 8 ((a^2-2)^2-2)^2-2 2207 Noting that when we double n, the value of f(n) approximately squares, that implies that b should be close to the geometric mean of 7 and 47. sqrt(7*47 is a little over 18, so I used 18. Now I have 5 terms: 2, 3, 7, 18, 47, ... I looked this up with my recurrence-relation search MCS [1] and found the above solution. You can use a calculator and verify that for the terms shown satisfy A(2*N) = A[N}^2 - 2. - Robert [1] http://mrob.com/pub/math/MCS.html

...

On 1/30/13, Dan Asimov <dasimov@earthlink.net> wrote:

...

While studying something apparently unrelated, I came upon a function

f: N_0 -> N_0

(where N_0 denotes {0,1,2,3,...})

with this curious property:

f(2n) + 2 = f(n)^2.

Puzzle: Find a closed-form asymptotic expression for f(n).

-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

="Dan Asimov" <dasimov@earthlink.net> Mark, I respectfully could hardly disagree more strongly.

Yes, and also for my part. Please know any florid vehemence in this disputation in no way diminishes my affection or respect for anyone.

The question of why .99999... = 1 that was initially brought up by Eric Angelini was clearly about the ordinary real numbers.

[...] But when people want to know whether, and if so why, .99999... = 1, they are in almost all cases wanting to learn or be reminded of the conventional meaning of that statement.

And that requires understanding a) that an infinite decimal represents an infinite sequence, and b) that this sequence has a limit, and c) that having a limit has a very specific meaning that can be verified in this case.

I guess my main concern is that in our enthusiasm to get on with b) and c) we insufficiently acknowledge that step a) is where some key conventional choice-making comes in (from which the rest follows so agreeably). I feel a tug of intellectual obligation to examine and acknowledge that critical step (much as I'd feel some duty to somehow at some point distinguish a line drawn with pen and ink from an abstract geometrical line, or to mention there are non-Eucildean options, etc). Glossing too glibly that .999... "is" a real number leaves unexamined some curious topics which I tried to allude to before (no doubt unintelligibly). For example someone might wonder: if the half-infinite .999... "is" a number, then is the doubly infinite ...000.999... also a number? Surely they seem equal, and hence both equal to 1. But then if doubly infinite ...000.999... is a number, couldn't doubly infinite ...999.000... also be a number? And indeed one can engage in all sorts of arithmetic manipulations with these strings and very consistently find that ...999.000... = -1. So we have developed a kind of "numeral theory" which seems consistent with the intuitive ideas we have of numbers. But if we now try to incorporate that a)b)c) there's a dissonance to resolve, because the increasing sequence 9.0, 99.0, 999.0 clearly has limit -1, which is less than any of them. This line of reasoning seems to lead to a serviceable interpretation of repeating decimals which moreover covers a larger domain but which doesn't seem compatible with the usual ordered real line model. Now awareness of this approach doesn't mean I reject the utility of the STotRN nor that it's mandatory to drag in non-standard exotica, I'm just uncomfortable when presenting the conventional practice with implying it is the ONLY way to think on these matters, nor that it covers the whole topic.

="Allan Wechsler" <acwacw@gmail.com> I would love to hear of any theory in which it makes sense to say 0.999... != 1.

Um, I'd say "string theory" but that seems to be taken, so "numeral theory"?

="James Propp" <jamespropp@gmail.com> (Or, as I was taught to say forty or fifty years ago, these two _numerals_ represent the same _number_.)

Yes. Perhaps my mind was simply disfigured by "New Math", but thank you for acknowledging the distinction, and for the wealth of interesting commentary.

="Dan Asimov" <dasimov@earthlink.net> Mark, I never said that .99999... "is" only one thing.

Well, from Monday, January 28, 2013 10:04 AM I have

The only way to understand an infinite decimal representation N.abcde... ^^^^

I see I was probably WAY too literal-minded, being a bit obsessed with there being more than one way to *understand* infinite strings of decimal digits. Of course if, as I now think you doubtless intended, the full phrase was implicitly "an infinite decimal representation N.abcde... OF A REAL NUMBER" then indeed our train is set on a smooth track with a foregone destination.

What I emphasized is what almost all people are asking when they inquire about the meaning of ".99999... = 1".

Heh. I grant that's reasonable...at least in 99.999...% of the cases.

=Jan Willem Nienhuys (wsadjw@rw7.urc.tue.nl) The natural numbers start with 2 (TWO). The name "counting numbers" make that clear. If there's only one item of a certain kind, one doesn't start counting.

Actually, I've heard that the ancient Greeks similarly didn't consider 1 a number, because a number was implicitly a possible number of things, and 1 thing was just that thing, not a number of them. Or something like that (I'd love to hear more authoritative opinions on this). Further, I've heard it alleged that because of this certain reasonings in, eg, Euclid, were broken out into the cases N=1 and N>1--not because, as it might seem to modern assumptives, to support any inductive pattern, but simply because they were viewed as intrinsically distinct cases.

I hope this simple solution will stop the silly discussions about 0 being a natural number. My hope will be in vain, I know.

Your Honor, the prosecution intends to show, beyond a reasonable doubt, that no things are surely not some things, and so zero is not a possible number for some things to be, hence manifestly zero is not a number of any sort, natural or unnatural.