PDA

View Full Version : 0.999... again

Solfe
2013-Jun-03, 10:43 PM
We aren't going to have a 1=.999999.... show up are we?

caveman1917
2013-Jun-04, 04:01 AM
We aren't going to have a 1=.999999.... show up are we?

We do now... :p

BigDon
2013-Jun-04, 05:39 AM
We aren't going to have a 1=.999999.... show up are we?

I hope not, because it doesn't.

1 = 1 and ".999999 to infinity" is a non-statement because "infinity" is a human construct with no representative in reality.

grapes
2013-Jun-04, 08:26 AM
The first three posts were removed from this ATM thread:

http://cosmoquest.org/forum/showthread.php?144528-Hilbert-s-Second-Problem

Solfe
2013-Jun-04, 10:22 AM
All right mathletes, it wasn't my intention to start a 1 vs. .99999.... thread, but my question is, can you have a conjecture, theorem, or other math rule that can be negated by substituting .99999.... for one? It would seem to me that the answer is no, for the practical purpose that by whatever logic used to introduce the .99999.... value could be applied to every value in the conjecture, theorem, etc. and it would still have to be reasonably true.

Is that correct or is my logic busted? Or is the reverse true, that you prove all sorts of things by creating a problem that requires the answer not to be 1 and injecting a .99999.... value to make it work.

In the thread where this came from, it seemed to me that a requirement of "not 1" could be met by "is .99999...."

HenrikOlsen
2013-Jun-04, 10:30 AM
If you have something that's negated by 1=0.999... then it's not even false, it's simply not math, since 1 and 0.999... are two different decimal representations of the same member of the set of reals.

Strange
2013-Jun-04, 10:42 AM
Vi Hart does some great math videos. For example: YouTube: Why Every Proof that .999... = 1 is Wrong (http://www.youtube.com/watch?v=wsOXvQn3JuE)

(Note the date, before getting too wound up by it!)

Solfe
2013-Jun-04, 12:00 PM
If you have something that's negated by 1=0.999... then it's not even false, it's simply not math, since 1 and 0.999... are two different decimal representations of the same member of the set of reals.

The other thread was ATM...

The set of reals is something that I have heard of, I think I need to read up on that and watch the video.

I have this suspicion that people confuse the impossibility of writing out the value of .99999.... as a process instead of a representation a singular concept. I'd like to do it myself, but at least I understand that it is wrong.

Strange
2013-Jun-04, 12:23 PM
The set of reals is something that I have heard of, I think I need to read up on that and watch the video.

make sure you watch her "other" video on the subject as well (or instead) otherwise you might end up very confused ...

I have this suspicion that people confuse the impossibility of writing out the value of .99999.... as a process instead of a representation a singular concept.

I think you are right. And once again, math trumps "common sense". (Sorry, Don!)

Ivan Viehoff
2013-Jun-04, 01:00 PM
All right mathletes, it wasn't my intention to start a 1 vs. .99999.... thread, but my question is, can you have a conjecture, theorem, or other math rule that can be negated by substituting .99999.... for one? It would seem to me that the answer is no, for the practical purpose that by whatever logic used to introduce the .99999.... value could be applied to every value in the conjecture, theorem, etc. and it would still have to be reasonably true.

Is that correct or is my logic busted? Or is the reverse true, that you prove all sorts of things by creating a problem that requires the answer not to be 1 and injecting a .99999.... value to make it work.

In the thread where this came from, it seemed to me that a requirement of "not 1" could be met by "is .99999...."
You can't have a sensible discussion in mathematics unless you agree what you are talking about. In the case of whether 1=0.9999... you need to agree what kind of numbers you are talking about and what notations you use for those numbers. Once we agree we are talking about the real numbers, and agree that those are defined by a standard construction, eg Dedekind cuts, then that is a context in which 1=0.9999..., they are simply equivalent notations for the same thing within the real numbers. But if you refuse to agree on your terms of debate, your notation is ill-defined, and you could be talking about any kind of number system. It is straightforward to devise consistent definitions and notations where 1 and 0.9999... are different, but it won't be a number system of much applicability to any maths anyone is interested in.

This basically is the sophistry that the contrarians employ: they refuse to agree to a defined number system. And obviously without defined terms and notations, anything goes.

The number of cranks willing to deny long and strongly established mathematics is extraordinary. Prominent maths professors receive numerous unsolicited papers purporting to prove things contrary to long and established mathematics, where the proofs are in no doubt. I read recently a claim that the most popular subject for those cranks is Cantor's famous "diagonalisation proof" that the reals cannot be enumerated. Which is actually of the essence in the case of what we are talking about here, since it depends upon much the same thing.

caveman1917
2013-Jun-04, 01:17 PM
It is straightforward to devise consistent definitions and notations where 1 and 0.9999... are different, but it won't be a number system of much applicability to any maths anyone is interested in.

The hyperreals are one such number system, and they have quite a significance in mathematics - they allow the construction of non-standard analysis. Just because it's not the real numbers doesn't mean that it hasn't much applicability to any maths anyone is interested in. In fact, if you ever used any calculus at all you could have used the hyperreals as a basis rather than the \epsilon, \delta-formalism (see non-standard calculus).

Strange
2013-Jun-04, 01:26 PM
Here is an interesting article on mathematical cranks, and how to respond to them: http://web.mst.edu/~lmhall/WhatToDoWhenTrisectorComes.pdf

Grey
2013-Jun-04, 04:36 PM
Here is an interesting article on mathematical cranks, and how to respond to them: http://web.mst.edu/~lmhall/WhatToDoWhenTrisectorComes.pdfI find it entertaining that, since the paper was written before Wiles's proof of Fermat's Theorem, he lists proofs of Fermat's Conjecture as an example of the sort of thing he receives from cranks.

Of course, I have no doubt that mathematicians do receive erroneous proofs of Fermat's Theorem, even today (and indeed, Wiles's proof is so fantastically complex, using mathematical techniques developed long after Fermat lived, that it seems almost certain that whatever Fermat himself had in mind as a proof was probably flawed). Even if something is true, it's still possible to have an incorrect proof of it.

Strange
2013-Jun-04, 04:40 PM
Here is another article that points out that just occasionally it is worth reading the letter: http://www.nytimes.com/1999/02/09/science/genius-or-gibberish-the-strange-world-of-the-math-crank.html

Trebuchet
2013-Jun-04, 05:02 PM
Here is another article that points out that just occasionally it is worth reading the letter: http://www.nytimes.com/1999/02/09/science/genius-or-gibberish-the-strange-world-of-the-math-crank.html

That's a great article. I had a college friend who had a method of trisecting an angle, which worked as nearly as we could measure it with a plastic protractor marked out in whole degrees. That, of course, is probably about as accurate as the method was. I wish I still remembered how to do it though.

grapes
2013-Jun-04, 05:28 PM
Here's an exact way to trisect an angle. :)

http://mentock.home.mindspring.com/trisect.htm

ETA: https://web.archive.org/web/20010513173745/http://mentock.home.mindspring.com/trisect.htm

Solfe
2013-Jun-04, 08:42 PM
I find it entertaining that, since the paper was written before Wiles's proof of Fermat's Theorem, he lists proofs of Fermat's Conjecture as an example of the sort of thing he receives from cranks.

Of course, I have no doubt that mathematicians do receive erroneous proofs of Fermat's Theorem, even today (and indeed, Wiles's proof is so fantastically complex, using mathematical techniques developed long after Fermat lived, that it seems almost certain that whatever Fermat himself had in mind as a proof was probably flawed). Even if something is true, it's still possible to have an incorrect proof of it.

I seriously love the story behind Fermat's Conjecture. I often wonder if he was expressing something completely non-informational such as his amazement and love of math instead of proposing a real proof. Wouldn't that be funny?

Strange
2013-Jun-04, 09:51 PM
This is neat; "origami geometry" (or should that be origami kikagaku?) extends Euclid's axioms and means you can trisect an angle:
http://www.sciencenews.org/view/generic/id/8567/description/Trisecting_an_Angle_with_Origami

Nowhere Man
2013-Jun-04, 10:06 PM
Sure you can trisect an angle, given the correct tools. But you can't trisect an angle with only a compass and an unmarked straightedge.

Fred

grapes
2013-Jun-04, 10:35 PM
Sure you can trisect an angle, given the correct tools. But you can't trisect an angle with only a compass and an unmarked straightedge.

No, that's sufficient (see link above). There's one more restriction. :)

Grey
2013-Jun-04, 10:47 PM
Sure you can trisect an angle, given the correct tools. But you can't trisect an angle with only a compass and an unmarked straightedge.And you have to follow all the rules. The method grapes uses only involves a compass and unmarked straightedge, but he (very subtly) breaks the rules for how to use the compass. It's a very nice example of how you can break the rules in a way that most people might not notice (many people might not be clear about exactly what the rules are).

Edit to add: And grapes himself makes the same point, slightly faster than me. ;)

grapes
2013-Jun-04, 10:59 PM
Oops, I shoulda let you handle it, better. :)

Solfe
2013-Jun-05, 02:08 AM
There is a thread about constructions with only compasses and straight edges from about two years ago. I dropped in to mentioned that my drafting teacher had those the final exam.

It turns out that the final exam contained drawings of a trisecting an angel, doubling a cube and squaring a circle. The actual point was duplication of an image, correctly labelling and a few other items of interest in drafting. It had nothing to do with the actual construction with restricted tools and must have been a joke on the teacher's part. I didn't get it at all until I posted here.

Nowhere Man
2013-Jun-05, 02:35 AM
Anything to keep this thread from becoming another long argument over whether or not 0.999... = 1.000...

Fred

Paul Beardsley
2013-Jun-05, 05:47 AM
I thought the previous thread was closed shortly after the claim was comprehensively proven. Something like:

1 divided by 3 is 0.333...

0.333... times 3 is 0.999...

End of story, surely?

NEOWatcher
2013-Jun-05, 12:47 PM
It turns out that the final exam contained drawings of a trisecting an angel
That sounds bloody.

Nick Theodorakis
2013-Jun-05, 01:30 PM
I thought the previous thread was closed shortly after the claim was comprehensively proven. Something like:

1 divided by 3 is 0.333...

0.333... times 3 is 0.999...

End of story, surely?

That only works if one accepts that 0.333... is actually 1/3.

The better one is

Let x = 0.999...
multiply both sides by 10:
10x = 9.999...
so
10x = 9 + 0.999...
but since x=0.999...
then 10x = 9 + x
9x = 9
x =1
QED

This is basically the way to convert repeating decimals into fractions.

caveman1917
2013-Jun-05, 01:31 PM
That sounds bloody.

And makes one wonder, how many thirds of an angel fit on a pinhead?

ipsniffer
2013-Jun-05, 01:45 PM
All right mathletes, it wasn't my intention to start a 1 vs. .99999.... thread, but my question is, can you have a conjecture, theorem, or other math rule that can be negated by substituting .99999.... for one? It would seem to me that the answer is no, for the practical purpose that by whatever logic used to introduce the .99999.... value could be applied to every value in the conjecture, theorem, etc. and it would still have to be reasonably true.

Is that correct or is my logic busted? Or is the reverse true, that you prove all sorts of things by creating a problem that requires the answer not to be 1 and injecting a .99999.... value to make it work.

In the thread where this came from, it seemed to me that a requirement of "not 1" could be met by "is .99999...."

I am a bit surprised that at a science-themed board, this would be controversial.

There are some references to another thread - was that at this board? If so, where is it? (I went looking, without success, but I didn't really know a whole lot about what I was looking for.)

I don't know what "reasonably true" in a maths setting is; we would usually talk about statements being true or false, not degrees of truth. But, if a statement about the number "1" is true, then it remains true if you replace "1" by "6/6" (provided the statement is about values, not about particular ways of representing the value). It will also remain true if "1" is replaced by "0.999...", since these are different ways of writing the same number.

I find the complaint that 0.999... is a human construct to be particularly strange. Of course it is a human construct, like all numbers. Was the number "6" discovered by an archeologist in a cave somewhere? Numbers are human constructs. And since humans constructed the number 0.999... to have the same meaning as "one", it has all the same representations in reality that "one" has.

Edited to add - I guess I should make it clear that the last paragraph is responding to someone else's post, not Solfe's.

Grey
2013-Jun-05, 02:43 PM
It had nothing to do with the actual construction with restricted tools and must have been a joke on the teacher's part.That's awesome.

I thought the previous thread was closed shortly after the claim was comprehensively proven. Something like:

1 divided by 3 is 0.333...

0.333... times 3 is 0.999...

End of story, surely?You'd think so, but there's this thread (http://cosmoquest.org/forum/showthread.php?14593-Do-you-think-0-9999999-1-that-is-infinite-9s), and this thread (http://cosmoquest.org/forum/showthread.php?16961-Two-reasons-why-999-1), and this one (http://cosmoquest.org/forum/showthread.php?16806-Where-do-you-stand-now), and this one (http://cosmoquest.org/forum/showthread.php?18175-Bad-Math). Notice how long some of those were. There may be others hiding somewhere. ;)

That only works if one accepts that 0.333... is actually 1/3.

The better one is...Actually, form seeing lots of conversations about this, it appears that Paul's method is usually more convincing to people that don't initially accept the equality. I'm not sure why; maybe the method you're using (which is more commonly used, I think) seems to some people like there's some trickery in one of the steps somewhere.

ipsniffer
2013-Jun-05, 03:10 PM
You'd think so, but there's this thread (http://cosmoquest.org/forum/showthread.php?14593-Do-you-think-0-9999999-1-that-is-infinite-9s), and this thread (http://cosmoquest.org/forum/showthread.php?16961-Two-reasons-why-999-1), and this one (http://cosmoquest.org/forum/showthread.php?16806-Where-do-you-stand-now), and this one (http://cosmoquest.org/forum/showthread.php?18175-Bad-Math). Notice how long some of those were. There may be others hiding somewhere. ;)

OMG, my eyes are bleeding now :(

Actually, form seeing lots of conversations about this, it appears that Paul's method is usually more convincing to people that don't initially accept the equality. I'm not sure why; maybe the method you're using (which is more commonly used, I think) seems to some people like there's some trickery in one of the steps somewhere.

The scary thing to me is that some of the pro-equality posts are just as bad as the anti-equality posts. Quite a lot of the "proofs" aren't proofs at all. Most of them rely on hidden assumptions which (although true, given the usual definitions) are not proved in-thread, and some of them are just wrong, even if they reach the right conclusion.

I guess that which persuades is different than that which is correct.

caveman1917
2013-Jun-05, 03:49 PM
I guess that which persuades is different than that which is correct.

And if people use appeal to intuition to state that the equality is false, what good does it do to appeal to that same sense of intuition to convince them that 0.9... = 1? Sure they will now parrot the correct answer but one has not addressed the underlying problem, which is that this person uses appeals to intuition to approach mathematical questions. Rather, one has actually reinforced that underlying error.

An actual proof would be noticing that real numbers are equivalence classes of sequences of rational numbers that have the same limit, ie \sim: \lim x_n = \lim y_n, and then showing that both 0.9... and 1.0... are both sequences of rational numbers with the same limit, thus in the same equivalence class, and thus being the same real number.

Grey
2013-Jun-05, 04:57 PM
And if people use appeal to intuition to state that the equality is false, what good does it do to appeal to that same sense of intuition to convince them that 0.9... = 1? Sure they will now parrot the correct answer but one has not addressed the underlying problem, which is that this person uses appeals to intuition to approach mathematical questions. Rather, one has actually reinforced that underlying error.

An actual proof would be noticing that real numbers are equivalence classes of sequences of rational numbers that have the same limit, ie \sim: \lim x_n = \lim y_n, and then showing that both 0.9... and 1.0... are both sequences of rational numbers with the same limit, thus in the same equivalence class, and thus being the same real number.The difficulty, of course, is that for most people (particularly those who do not accept the equality), ideas like limits, a rigorous method of defining the real numbers, the very notion that the real numbers are distinct from the rational numbers, and pretty much all the other mathematical concepts required to address this question with rigor are completely over their heads. So you're left with the choice of just ignoring the conversation entirely (allowing them to persist in an incorrect belief about the way real numbers work), insisting that 0.999... does equal 1 but without offering a proof (suggesting that they should just accept the word of someone who understands mathematics better than they do), trying to teach them everything they need to know to approach the question (only possible if everyone involved has sufficient time and interest to cover some pretty lengthy ground; I know people who would probably take months of work to really understand the concepts involved), or trying to simplify the explanation to something that they will understand and accept, but which tries to minimize the use of mathematical concepts with which they are not familiar. To me the latter seems like a good realistic compromise (although there are plenty of times when the first choice is perhaps the best way to go...).

agingjb
2013-Jun-05, 09:36 PM
Indeed, if people don't accept, or more usually don't bother to learn, the properties of the reals, then they are not going to get very far with invented alternatives.

It may be worth saying that the hyperreals are a side issue here. They don't provide numbers between two identical reals with differing notations. The hyperreals are an extension of the reals and, if used, require additional notations. My view is that unless the properties of the reals are accepted and familiar the hyperreals are best left till later. They are interesting but hardly essential, and perhaps rather misleading.

caveman1917
2013-Jun-05, 10:35 PM
It may be worth saying that the hyperreals are a side issue here. They don't provide numbers between two identical reals with differing notations.

I disagree. Whilst notation can be defined however one wants, it seems natural to consider 0.9... as the sequence 0.9, 0.99, 0.999, ... and likewise 1 as 1, 1, 1, ... like we would in an actual proof that as real numbers they are equal. Interpreting these in terms of hyperreals does give two different numbers where 0.9... < 1, because while the hyperreals (just like the reals) are (or rather can be) defined as equivalence classes of sequences, the equivalence relation is different. Using the ultrapower construction one can immediately see this because the set of indices where these two sequences coincide is the empty set, which is by definition not a member of a filter.

ETA: Maybe it doesn't seem as natural to consider 0.9... as that sequence, but we usually do see the interpretation 0.9... = \sum_{n = 1}^\infty 9\frac{1}{10^n} and given that an infinite sum is the limit of the sequence of partial sums that does implicitly use the (limit of the) sequence above.

Solfe
2013-Jun-05, 11:15 PM
Let us all remember that this thread comes from a foolish comment I made in ATM and that post was used to create this thread. I didn't actually name it or create another .99999....= 1.0000 thread on purpose. I fear the day blame comes to town. :)

This thread could be more appropriately named "What to say when someone (such as Solfe) enjoys math, but doesn't really get math".

ipsniffer
2013-Jun-06, 03:50 AM
The difficulty, of course, is that for most people (particularly those who do not accept the equality), ideas like limits, a rigorous method of defining the real numbers, the very notion that the real numbers are distinct from the rational numbers, and pretty much all the other mathematical concepts required to address this question with rigor are completely over their heads. So you're left with the choice of just ignoring the conversation entirely (allowing them to persist in an incorrect belief about the way real numbers work), insisting that 0.999... does equal 1 but without offering a proof (suggesting that they should just accept the word of someone who understands mathematics better than they do), trying to teach them everything they need to know to approach the question (only possible if everyone involved has sufficient time and interest to cover some pretty lengthy ground; I know people who would probably take months of work to really understand the concepts involved), or trying to simplify the explanation to something that they will understand and accept, but which tries to minimize the use of mathematical concepts with which they are not familiar. To me the latter seems like a good realistic compromise (although there are plenty of times when the first choice is perhaps the best way to go...).

I think I have to agree that "bad argument leading to right conclusion" instead of "bad argument leading to wrong conclusion" is probably the best that can be done realistically in many cases. But judging from the other threads, the issue is not just lack of ability or time to learn in many cases; it's lack of willingness. Some of the people seem fresh from conspiracy theory summer camp.

Ivan Viehoff
2013-Jun-06, 09:49 AM
The hyperreals are one such number system, and they have quite a significance in mathematics - they allow the construction of non-standard analysis. Just because it's not the real numbers doesn't mean that it hasn't much applicability to any maths anyone is interested in. In fact, if you ever used any calculus at all you could have used the hyperreals as a basis rather than the \epsilon, \delta-formalism (see non-standard calculus).
That is why I was careful to say "not of much applicability". I managed to get through a maths degree without studying hyperreal or surreal numbers - despite doing the set theory option - but I am aware of them. If the typical contrarian had in mind hyper-reals or the like, then they wouldn't be so stupid as to deny the equivalence in the reals or rationals.

HenrikOlsen
2013-Jun-06, 08:15 PM
An actual proof would be noticing that decimal representations of real numbers are equivalence classes of sequences of rational numbers that have the same limit, ie \sim: \lim x_n = \lim y_n, and then showing that both 0.9... and 1.0... are both sequences of rational numbers with the same limit, thus in the same equivalence class, and thus being the same real number.
Red is my addition.
Part of the problem is when people's understanding of limits is not rigorous, another is when their intuition about real numbers is based on the subset of the rational numbers computer science calls reals.

If they only learned about limits as a vague description of a process of sequential iteration, that's when you get the "but it never gets there" counter arguments.

Solfe
2013-Jun-06, 08:32 PM
Er... I shouldn’t have said "reasonably true" back in that post. I should have said "is true".

I have a rather complex definition of "reasonably true", but it has to do with the lack of rigour. It is reasonably true that you can figure out the area under a curve without calculus and some people intuitively try such things because it their approach makes some obvious assumptions that can be true. It is a lousy way of doing things because the assumptions can create really wrong answers, and there is a whole field of math that provides excellent answers using an easier method. "Reasonable", yes, best no.

HenrikOlsen
2013-Jun-06, 09:17 PM
In Danish some of us have a punnish way of describing two ways of arguing about a mathematical theorem. A proof is a "bevis" in Danish, and to convince is to "overbevise" which can be nouned to make an "overbevis".

If you're trying to convince someone who is fairly wall grounded in mathematics and already knows how to connect dots, it's often enough to just provide enough of the intervening dots that they can see they can all be connected without having to work out exactly how in a rigorous manner.

Paul's "proof" is really a "convincion" as it uses the same notation that some people aren't clear about, but does it in a way that bypasses the bit they normally get hung up on.

caveman1917
2013-Jun-07, 02:42 AM
Red is my addition.

And it is wrong, i'm not talking about the representation (decimal or otherwise), i'm talking about the real numbers themselves. A decimal representation of a real number is a certain sequence of rational numbers, and as such uniquely defines a real number, but a real number itself is an equivalence class of such sequences - not just a sequence itself. At least, that is one of the ways to define the real numbers, which seems the most appropriate in this case.

caveman1917
2013-Jun-07, 08:42 AM
If they only learned about limits as a vague description of a process of sequential iteration, that's when you get the "but it never gets there" counter arguments.

On the other hand, it doesn't matter if it "gets there" or not, since the real numbers are not limits of sequences of rational numbers, or at least you can't define them that way (since before you introduce the real numbers you don't have any number system in which the limit belongs), though after you introduced the real numbers you can certainly identify them with those limits. That's why i stressed that they are equivalence classes of sequences, not limits of sequences. Strictly speaking equivalence classes of cauchy sequences.

If the difference between 1, 1, 1, ... and 0.9, 0.99, 0.999, ... gets arbitrarily close to zero then we define the sequences as equivalent and thus the same real number. The "it never gets there" counter argument is a true statement, and we don't really need any other exposition of the concept of a limit other than the statement in the previous sentence (you don't even have to use the word limit), all you have to get them to agree to is that the difference between those sequences gets arbitrarily close to zero.

HenrikOlsen
2013-Jun-07, 09:45 AM
But then arbitrarily close to zero is one of the ways of defining limits which bypasses the process idea.:)

I think this is yet another case of agreement which sounds like disagreement because of different phrasing, I'm definitely not going to treat it as anything other than that.

caveman1917
2013-Jun-07, 10:10 AM
But then arbitrarily close to zero is one of the ways of defining limits which bypasses the process idea.:)

Sure, but the difference is that if you define the real number as the limit you have to state something about the result of the process, which opens you up to "but there is no result because it never ends" etc. If, on the other hand, you define the number by the process itself (without reference to any sort of "result" or limit) in saying that the process is one where you can get as close to zero as you want, you won't get that "but it never ends" and, i assume, are more likely to get agreement that it is indeed such a process. An added bonus is that this way is, strictly speaking, the correct one :)

I think this is yet another case of agreement which sounds like disagreement because of different phrasing, I'm definitely not going to treat it as anything other than that.

We certainly don't disagree about what a limit is, it just seems a minor discussion about possible strategies of persuading people of the equality.

Solfe
2013-Jun-07, 11:03 AM
Speaking of which, is there a method of attack in calculus that would make this equality easier to understand? I only ask because I bought a calculus book that started with limits, then the next edition was early transcendentals and the current version is back to limits.

I fail to see how calculus teaching methodology changed in three years to require 3 books by one author with two different starting points.

HenrikOlsen
2013-Jun-07, 11:25 AM
As I said, we're in agreement.

The definition of limits I've learned is the one where \lim_{n \to \infty} f(n)=1 if, for any real \epsilon>0 there exists a positive integer s such that for any n>s we can show that \left\vert{} {f(n)-1} \right\vert{}<\epsilon (which is basically the explicit mathematical definition of "can get arbitrarily close"), so proving the equality is normally done by constructing a function for s such that it is clearly true.

Note that there is no process that'll never end in this definition, there are only static statements that are shown to be true.

ipsniffer
2013-Jun-07, 11:56 AM
As I said, we're in agreement.

The definition of limits I've learned is the one where \lim_{n \to \infty} f(n)=1 if, for any real \epsilon>0 there exists a positive integer s such that for any n>s we can show that \left\vert{} {f(n)-1} \right\vert{}<\epsilon (which is basically the explicit mathematical definition of "can get arbitrarily close"), so proving the equality is normally done by constructing a function for s such that it is clearly true.

Fair enough, the issue in the construction of the real numbers from the rationals is, a lot of sequences of rational numbers (for example, 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, etc.) do not have limits that exist as rational numbers. The limit (in this case, pi) is the thing we are trying to define. So rather than referring to the limit of the sequence, we talk about it converging, in that for any \epsilon, there is an s such that for all m,n>s, we have \left\vert f\left(m\right)-f\left(n\right)\right\vert<\epsilon. So we have a notion of convergence, with no regard to what the sequence is converging to (and the limit may or may not exist as a rational number).

HenrikOlsen
2013-Jun-07, 12:13 PM
But that's just an existence proof which shows that the sequence converges, which is interesting if that's what you're trying to show, but it isn't really useful in this case where the question quite definitely is "What does it converge to?".

I should have mentioned that if we define f(n)=1-10^{-n} (which is the 0.9, 0.99, 0.999 ... sequence) it's easy to see that we can always pick a positive integer s>-log_{10}(\epsilon) and this s fulfills the requirement, so we have clearly shown that specific convergence.

ipsniffer
2013-Jun-07, 12:21 PM
Yep, I just used 1 because that's the special case we were examining in this thread's OP.

Yes, I guess we could address the issue in the thread title strictly within the realm of the rationals. We would only need to introduce the reals for non-repeating decimals.

HenrikOlsen
2013-Jun-07, 12:35 PM
I would conjecture that the subset of reals that have multiple decimal representations is also a subset of the rationals.

Grey
2013-Jun-07, 12:41 PM
I fail to see how calculus teaching methodology changed in three years to require 3 books by one author with two different starting points.My guess would be that a revision schedule that frequent is motivated by the author's desire for additional income, rather than a real need to change the methodology. ;)

HenrikOlsen
2013-Jun-07, 01:13 PM
Or by political control on the schools' curriculum, which means it gets changed at the whim of the politicians, which is what happens too often in Denmark.

Ivan Viehoff
2013-Jun-07, 03:30 PM
Fwiw, I see that this comes as #2 in the 12 most disputed facts in mathematics as listed here. http://www.businessinsider.com/the-most-controversial-math-problems-2013-3
Though I doubt any contrarian would be happy with the explanation given. I also think the presentation of the Monty Hall problem there, at #1, is incomplete - in that case, it really is more complicated than that, as they say. (But I'm not going to participate in a discussion of it, you can find better discussions of it elsewhere.)

grapes
2013-Jun-07, 04:10 PM
It's hard to believe that the perfect parallelogram is in dispute. Even, in discussion. :)

SeanF
2013-Jun-07, 04:42 PM
I have a problem with #6, the Broken Water Heater Problem. It's reasonable to assume that everybody in the "accountant-and-plumber" group is capable of fixing a broken water heater, but not everybody in the "accountant-but-not-plumber" group is. Since the water heater was fixed, we know that the person is in the "capable-of-fixing-a-broken-water-heater" group, and need to take that into account.

Don't we?

grapes
2013-Jun-07, 06:42 PM
I have a problem with #6, the Broken Water Heater Problem. It's reasonable to assume that everybody in the "accountant-and-plumber" group is capable of fixing a broken water heater, but not everybody in the "accountant-but-not-plumber" group is. Since the water heater was fixed, we know that the person is in the "capable-of-fixing-a-broken-water-heater" group, and need to take that into account.

Don't we?
That's the problem I have with their explanation. :)

They could have taken that tack in their Venn diagrams, and the explanation would've been a whole lot easier to swallow.

In other words, draw a circle of your "accountant and capable-of-fixing-a-broken-water-heater" group, then draw the *smaller* circle inside of that of the "accountant and plumber" group. The conclusion is still the same, but it seems to make it more obvious--they're still more likely to be an accountant than to be an accountant *and* a plumber.

caveman1917
2013-Jun-07, 06:45 PM
Fair enough, the issue in the construction of the real numbers from the rationals is, a lot of sequences of rational numbers (for example, 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, etc.) do not have limits that exist as rational numbers. The limit (in this case, pi) is the thing we are trying to define. So rather than referring to the limit of the sequence, we talk about it converging, in that for any \epsilon, there is an s such that for all m,n>s, we have \left\vert f\left(m\right)-f\left(n\right)\right\vert<\epsilon. So we have a notion of convergence, with no regard to what the sequence is converging to (and the limit may or may not exist as a rational number).

Strictly speaking the notion is introduced by the difference between (cauchy) sequences, we can define our equivalence such as \sim: (x)_\mathbb{N} \sim (y)_{\mathbb{N}} \Leftrightarrow \lim (x_n - y_n) = 0, so all we need is convergence towards zero, which is a rational number and thus an accessible notion before defining the real numbers. Once we then have the real numbers we can certainly identify them with the limits of those sequences (as i did previously and everyone naturally does), but this way we avoid the circular definition where we need the limits to define the real numbers and need the real numbers to define the limits - which is required for a rigorously correct definition.

But that's just an existence proof which shows that the sequence converges, which is interesting if that's what you're trying to show, but it isn't really useful in this case where the question quite definitely is "What does it converge to?".

The question is not "What does it converge to?", the question is "Does the difference converge to zero?". It's the difference between asking "what real number is this sequence" and "are the real numbers represented by these sequences the same (irrespective of what number they may be)?". We only need the latter question for this equality, and since it doesn't require identifying the number with the limit of 0.9... i think it would work better in convincing those that state things like "it never gets there".

caveman1917
2013-Jun-07, 06:54 PM
I would conjecture that the subset of reals that have multiple decimal representations is also a subset of the rationals.

You could even prove it rather than just conjecture it ;). It's a straightforward proof.

Grey
2013-Jun-07, 07:08 PM
In other words, draw a circle of your "accountant and capable-of-fixing-a-broken-water-heater" group, then draw the *smaller* circle inside of that of the "accountant and plumber" group. The conclusion is still the same, but it seems to make it more obvious--they're still more likely to be an accountant than to be an accountant *and* a plumber.And of course the whole question is made deliberately confusing by leaving out what is probably the most likely choice (at least, given that the person fixed your water heater): that the person is a plumber but not an accountant. Heck, even the fourth possibility (which they also don't mention, that the person is neither a plumber nor an accountant) is probably more likely than either of the two possibilities that they do present, since there are more non-accountants than accountants, and there's nothing about the scenario that would suggest that accountants have been preferentially selected.

Edit to add: To be clear, their answer is correct. It's just that it's a question phrased in a way to make it as confusing as possible.

HenrikOlsen
2013-Jun-08, 08:32 AM
You could even prove it rather than just conjecture it ;). It's a straightforward proof.
Sure, but since I can't be bothered, it's still a conjecture for me. :)

Ivan Viehoff
2013-Jun-10, 10:30 AM
I have a problem with #6, the Broken Water Heater Problem. It's reasonable to assume that everybody in the "accountant-and-plumber" group is capable of fixing a broken water heater, but not everybody in the "accountant-but-not-plumber" group is. Since the water heater was fixed, we know that the person is in the "capable-of-fixing-a-broken-water-heater" group, and need to take that into account.

There is a bit of an issue with that whole presentation is that it doesn't fully explain all the conditions necessary for the arguments he makes to be true, though that would probably detract from the fun of it. For that particular one, it is necessary to assume that a person who can mend a water heater is not necessarily or by definition a plumber. There are various versions of that "paradox" and I don't think he chose the clearest version of it.