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#151
08-04-2012, 08:41 PM
 President Johnny Gentle Guest Join Date: May 2007
Quote:
 Originally Posted by erik150x I know I can be irritatingly stubborn. I once had a 2 week long discussion of the chicken or egg question with a biologist. We were both arguing from an evolutionary perspective, but yet could not agree. He said chicken, I egg. It was an interesting discussion and in the end I think we agreed the question is not well defined enough to say exactly what is a chicken and what is an egg to answer. It is interesting to me to see how these things often come down to semantics or how you define things. Which in many cases can be somewhat arbitrary.
I missed this in my last post. This is almost exactly the point - in mathematics, definitions are everything. It's not so much an issue of semantics as it is being impossible to pose some mathematical questions without a firm definition.

Mathematically, it's not so arbitrary as asking whether your definition behaves in all the expected ways. So a working definition of an infinite sum should be associative and commutative, if at all possible. Also, it would be nice if the definition agrees with other accepted cases. As I mentioned earlier, we don't need limits to deal with rational numbers. However, once we have come up with a workable definition for an infinite decimal expansion, it had better give the expected results when applied to rational numbers.

Now, there might be multiple ways to do this. In that case, mathematicians would explore any such way which is discovered. In this event, it could be possible to come up with multiple extensions of the same system, which don't necessarily give the same mathematical object. For example, Euclid developed a consistent geometry using five postulates. However, if we replace the parallel postulate with something else, we can get a consistent geometry which differs from Euclidean geometry in interesting ways. For instance, in hyperbolic geometry, the area of a triangle entirely depends on the measures of its angles (and can never exceed p in the most common models).

Last edited by President Johnny Gentle; 08-04-2012 at 08:45 PM.
#152
08-04-2012, 09:15 PM
 erik150x Guest Join Date: Aug 2012
Indeed... mathematics is the last place I would like to see any arbitrary definitions. I may not be a mathematician. But I hold it very dear to me. It seems so pure an uncontaminated by the messiness of the "real" world, it lies outside of it in a Plutonian sense I think. It in theory it could exist on it's own whatever that would mean exactly... but the rest of the world could not exist without it. It dictates the framework in which everything must occur. At least this is the way I view it.

In this regard, Limits may be the best we have to build on... yet something about their definition does not yet seem elegant to me enough. As pointed out to me earlier, (unnecessarily I might add), I know we can't always trust our intuitions on these matters, but something about it just doesn't sit right with me and never has. It seems a little bit forced. It's like our "normal" rules of math break down at the Limit, and we can intuitively see what the answer should be, but have to add in these extra rules regarding limits to arrive at an answer. To me this strikes of something still missing in our understanding, rather than Limits being a fundamental Plutonian construct of math. Maybe its just me.

Last edited by erik150x; 08-04-2012 at 09:17 PM.
#153
08-04-2012, 09:18 PM
 Trinopus Member Join Date: Dec 2002 Location: San Diego, CA Posts: 5,069
Going back a bit to the idea of a "proof," I sometimes view it as a legal argument.

Suppose that .9999... < 1. Well, what is the difference? Where is the difference? Explicitly, it isn't in the first four decimal places. Is it in the fifth decimal place? No, because I can easily extend the description to .99999... Okay, is it in the 50th decimal place? No, because (with more effort than I care to expend) I can extend the description to 50 places.

Any time you say, it is in the "nth" decimal place, I refute this by extending the description to n+1 places.

In "legalistic" terms, I've got you beat. You can't describe the difference, because any time you try to state it, I can eliminate it.

This has led some people to "constructive mathematics," where a number isn't actually "a number" unless it can be constructed. Infinite expansions cannot be constructed (because I don't have "forever" in which to write down more 9's.) There is a counting number n1 and a counting number n2 such that n1/n2 is close enough to pi for any useful purpose. (Can any real-world engineer possibly require knowing pi to more than 20 decimal places?)
#154
08-04-2012, 09:27 PM
 zombywoof Guest Join Date: Jul 2009
Quote:
 Originally Posted by erik150x it lies outside of it in a Plutonian sense I think.
Quote:
 Originally Posted by erik150x Limits being a fundamental Plutonian construct of math.
Hmm, perhaps the word you're going for is platonic?

Last edited by zombywoof; 08-04-2012 at 09:29 PM.
#155
08-04-2012, 09:39 PM
 erik150x Guest Join Date: Aug 2012
[quote=Trinopus;15349776]Going back a bit to the idea of a "proof," I sometimes view it as a legal argument.

...

Any time you say, it is in the "nth" decimal place, I refute this by extending the description to n+1 places.

In "legalistic" terms, I've got you beat. You can't describe the difference, because any time you try to state it, I can eliminate it.

...

People will hate me for this..., but okay what if I want to say there is no "space" between them, they are literally "touching", while not occupying the same space/place on the number line.

An obvious response to this would be, well if there is no difference between A and B, then A - B = 0 and therefore A = B.

But what if I were to say well, actually A - B = 1/infinity? Perhaps there is a subtle distinction between say there is no space between 2 things yet they are not in the same space. But points have no dimension, right? So that leads us back to they must be in the same space again. What if I argue points do have a dimension? Say 1/infinity? Perhaps this is a number we don't fully understand (a bit of a planks constant of the math world?). I don't know, probably leads to contradictions. I'm sure many great mathematicians have already been down this road and reported the awful conditions. But I haven't and I find it interesting to toy with these ideas.
#156
08-04-2012, 09:41 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by zombywoof Hmm, perhaps the word you're going for is platonic?
Yes, thank you. LOL don't know why I said plutonian.
#157
08-04-2012, 10:02 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by DrCube I should mention that you have your work cut out for you. A system of math WAS developed that defines and uses "infinity" and "1/infinity", called hyperreals, or "non-standard analysis". In that system 0.999... STILL equals 1, so you're going to have to come up with something new. Basically, you're going to have to redefine what decimal numbers mean and hope what you come up with is constent.
That depends on how you choose to interpret decimal notation into the hyperreals. You could interpret a.bcd... as the hyperreal corresponding to <a, a.b, a.bc, a.bcd, ...> (i.e., as "a.bcd... truncated to the omega-th decimal place", where omega is the canonical hypernatural <0, 1, 2, ...> which generates the whole structure). Then 0.999... would be different from 1; their difference would be 1/10^omega (i.e., "0.000...1, with omega many 0s before the 1"). And this is probably a fair formalization of the intuition which erik150x is, in their own way, attempting to express and explore.

Of course, this wouldn't be the usual way of interpreting infinite decimal notation. But it's a not unsensible one, for that context and erik150x's purposes.

Last edited by Indistinguishable; 08-04-2012 at 10:06 PM.
#158
08-04-2012, 10:03 PM
 bobot Member Join Date: Jan 2009 Location: Chicago-ish Posts: 816
My 2 cents, and 2 tenths..
I work as a machinist. If I'm machining a feature, say a slot width, and the print
calls for the slot to be .999" +/- .0002, and I make the slot 1 inch wide, guess where
the part goes? In the scrap bin. Why? Because .999 does not equal 1.
#159
08-04-2012, 10:05 PM
 President Johnny Gentle Guest Join Date: May 2007
Quote:
 Originally Posted by erik150x In this regard, Limits may be the best we have to build on... yet something about their definition does not yet seem elegant to me enough. As pointed out to me earlier, (unnecessarily I might add), I know we can't always trust our intuitions on these matters, but something about it just doesn't sit right with me and never has. It seems a little bit forced. It's like our "normal" rules of math break down at the Limit, and we can intuitively see what the answer should be, but have to add in these extra rules regarding limits to arrive at an answer. To me this strikes of something still missing in our understanding, rather than Limits being a fundamental Plutonian construct of math. Maybe its just me.
It's not just you. Some of the greatest mathematicians of the 19th century felt the same way.

And you have a very good point. There's a reason that infinitesimals stuck around for so long, despite their lack of rigor. They captured the intuition of the mathematicians using them. However, they proved to be stifling in the long run.

Believe it or not, the limit definition makes intuitive sense for working mathematicians. It does an excellent job of encapsulating everything that ought to be true about the real numbers. But more importantly, it generalizes to other situations in a very simple fashion. Limits make sense wherever we have a geometric situation with a concept of "nearness." In fact, in the most abstract situation, the definition is actually much simpler. It's only because we add so much more to the geometry in specializing to the real number system that the definition becomes more complex. But since the abstract definition is so intuitively nice, there's not much in the way of worry about the general concept of limits. There is worry about teaching limits however, since the formal e-d definition of a limit is extremely daunting to a freshman calculus student.
#160
08-04-2012, 10:07 PM
 President Johnny Gentle Guest Join Date: May 2007
Quote:
 Originally Posted by bobot My 2 cents, and 2 tenths.. I work as a machinist. If I'm machining a feature, say a slot width, and the print calls for the slot to be .999" +/- .0002, and I make the slot 1 inch wide, guess where the part goes? In the scrap bin. Why? Because .999 does not equal 1.
The title of the thread is unfortunate here. No one is arguing that 0.999 = 1, but that 0.999999999..... = 1, where the sequence of 9s never stops.
#161
08-04-2012, 10:31 PM
 bobot Member Join Date: Jan 2009 Location: Chicago-ish Posts: 816
Quote:
 Originally Posted by President Johnny Gentle The title of the thread is unfortunate here. No one is arguing that 0.999 = 1, but that 0.999999999..... = 1, where the sequence of 9s never stops.
I see, so my .0002 are very large by comparison.
I've never studied mathamatics, other than algebra and trig., ...
but in my ignorance I contend that .999999999... however far it's carried out is not 1.
And I am not able to defend this.
And I will not be convinced otherwise.
Unless someone pays me, then it's ok with me.
What's a few .000000000000000000000000000000000000000000....1 between friends?
#162
08-04-2012, 10:36 PM
 friedo Charter Member Join Date: May 2000 Location: Brooklyn Posts: 19,333
Quote:
 Originally Posted by bobot What's a few .000000000000000000000000000000000000000000....1 between friends?
How many zeroes are there between the last one you wrote and the 1?
#163
08-04-2012, 10:39 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by bobot I see, so my .0002 are very large by comparison. I've never studied mathamatics, other than algebra and trig., ... but in my ignorance I contend that .999999999... however far it's carried out is not 1. And I am not able to defend this. And I will not be convinced otherwise. Unless someone pays me, then it's ok with me. What's a few .000000000000000000000000000000000000000000....1 between friends?
Bless you, Bobot.

I will be sending you roughly a trillion infinitesimal dollars in the mail shortly to not be swayed!
#164
08-04-2012, 10:47 PM
 bobot Member Join Date: Jan 2009 Location: Chicago-ish Posts: 816
Quote:
 Originally Posted by erik150x Bless you, Bobot. I will be sending you roughly a trillion infinitesimal dollars in the mail shortly to not be swayed!
Could you please put that in a very large box, like a refrigerator box, so I know it's from you?

Peace
#165
08-04-2012, 10:49 PM
 Indistinguishable Guest Join Date: Apr 2007
bobot: There's nothing wrong with using "0.999..." to mean something different from 1. It's just not how mathematicians standardly use that notation. The way mathematicians standardly use infinite decimal notation has the result that "0.999..." is defined to be equal to 1. There's not much to argue about, since it becomes, for them, true by definition.

Everyone else: There's nothing wrong with using "0.999..." to mean something different from 1, except that it is nonstandard. There's nothing wrong with saying the difference is 0.000000...1, with infinitely many zeros before the 1. This is perfectly formalized by the interpretation of infinite decimal notation as representing the hyperrational given by truncation to a fixed infinite number of decimal places, which probably captures quite well what bobot would end up sketching out, in their own amateur, inchoate way, were they pressed to spell out their intuitions.

Last edited by Indistinguishable; 08-04-2012 at 10:53 PM.
#166
08-04-2012, 10:58 PM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by bobot Could you please put that in a very large box, like a refrigerator box, so I know it's from you? Peace

Actually Bobot, They don't take up too much space, I will be sending in a small envelope. Just in case you still can't see them there will be two marks inside, labeled .999... and 1. You will find them taped right between those two marks.

Last edited by erik150x; 08-04-2012 at 11:02 PM.
#167
08-05-2012, 05:16 AM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 11,583
Quote:
 Originally Posted by Indistinguishable ... There's nothing wrong with using "0.999..." to mean something different from 1. ...
Agreed. It's just that:

(1) It's not the generally-agreed "standard" definition of real numbers. If you want to argue about it with mathematicians, and not be laughed at, you need to make this clear, e.g., by saying you are talking about a kind of non-standard analysis.

(2) Even if you do talk about a system using infinitesimal numbers like .000...1 (i.e., the difference between .999... and 1), you need to make sure that your axioms (a technical word that logicians and mathematicians use for assumptions) and definitions build a consistent mathematical system. If it's inconsistent (e.g., because you can prove that 1=0), then your system is useless.

Last edited by Giles; 08-05-2012 at 05:17 AM.
#168
08-05-2012, 09:27 AM
 jtgain Guest Join Date: Jul 2007
Quote:
 Originally Posted by Indistinguishable There's nothing wrong with saying the difference is 0.000000...1, with infinitely many zeros before the 1.
Ok, I thought I had it understood until here. How can you say that there would be infinitely many zeros BEFORE the 1? Wouldn't the 1 never show up because the zeros just keep coming? And then 1 at the end implies a definite termination at some point, which is what .999...... does not do. It never terminates.
#169
08-05-2012, 10:39 AM
 Malacandra Guest Join Date: Jan 2003
I'm not going to spend too long getting mired in this one, but I don't know if anyone mentioned that if you take 0.9999.... and multiply it by 10 to get 9.9999..., they both have the same number of 9s after the decimal point - an infinite number. You'd like to say that 9.9999.... has one fewer, but that's not how infinity works; there is no such thing as "infinity minus one" because infinity is not a number you can count up to.

Therefore 9.9999.... - 0.9999.... equals exactly 9 and not an infinitesimal more or less (because every 9 after the decimal point on one side has a corresponding such 9 on the other side), and we're off to the races.

I'm not expecting that this will settle a damn thing, but hey, at least I've had my 2 x 0.999.... cents' worth.

Last edited by Malacandra; 08-05-2012 at 10:40 AM.
#170
08-05-2012, 11:04 AM
 leahcim Member Join Date: Dec 2010 Location: New York Posts: 1,238
Quote:
 Originally Posted by jtgain Ok, I thought I had it understood until here. How can you say that there would be infinitely many zeros BEFORE the 1? Wouldn't the 1 never show up because the zeros just keep coming? And then 1 at the end implies a definite termination at some point, which is what .999...... does not do. It never terminates.
It doesn't make sense in standard arithmetic, yes, but there's no particular reason not to use the notation in an alternative math system. The notation is, after all, just some digits, some dots and another digit -- perfectly finite. Note that even in standard arithmetic, 0.9999999.... does not mean the 9s "keep coming". The 9s are already there, there's just an infinite number of them already there.

As a somewhat orthogonal example, what if I wanted to use an alternative ordering (using << instead of <) of the natural numbers where:
• a << b means the same thing as a < b if a and b are both even or both odd.
• a << b is always true of a is even and b is odd.
• a << b is always false of a is odd and b is even.

Then we could write the natural numbers in order as:

0,2,4,6,8,...,1,3,5,7...

And the ellipsis in the middle doesn't prevent us from considering the ordering to the right of it.
#171
08-05-2012, 11:28 AM
 TriPolar Member Join Date: Oct 2007 Location: rhode island Posts: 20,163
Assuming that definitions are the ones people commonly use, is there any reason to use the notation .999... for anything other than showing that the result of an arithmetic operation produces an infinite series of 9s? Is there a reason at all? After all the value is 1. Or is it 'one'?
#172
08-05-2012, 11:33 AM
 jtgain Guest Join Date: Jul 2007
Quote:
 Originally Posted by leahcim It doesn't make sense in standard arithmetic, yes, but there's no particular reason not to use the notation in an alternative math system. The notation is, after all, just some digits, some dots and another digit -- perfectly finite. Note that even in standard arithmetic, 0.9999999.... does not mean the 9s "keep coming". The 9s are already there, there's just an infinite number of them already there.
Okay, please try to keep it to the idiot level of conversation so I can understand. How does it ever make sense to say .000000....1? There is an infinite number of zeros. By definition it can never terminate (even under an alternate system or any system). You literally never, ever, ever get to the point where the 1 appears. Because if it did appear, you would have a terminating, fixed and defined number. Which can't happen because the zeros are infinite.
#173
08-05-2012, 11:45 AM
 KarlGauss An old man in a dry month Charter Member Join Date: Mar 2000 Location: Between pole and tropic Posts: 5,967
Quote:
 Originally Posted by jtgain You literally never, ever, ever get to the point where the 1 appears.
The ellipsis means it's 'already happened', an infinite number of 0's have 'already passed', you do "get there", by definition.

ETA: i.e. one defines ". . . " to mean an infinite number of them have 'already passed'

Last edited by KarlGauss; 08-05-2012 at 11:47 AM.
#174
08-05-2012, 11:48 AM
 jtgain Guest Join Date: Jul 2007
Quote:
 Originally Posted by KarlGauss ETA: i.e. one defines ". . . " to mean an infinite number of them have 'already passed'
Then you have a contradictory definition. Infinity never passes.
#175
08-05-2012, 11:54 AM
 KarlGauss An old man in a dry month Charter Member Join Date: Mar 2000 Location: Between pole and tropic Posts: 5,967
Quote:
 Originally Posted by jtgain Then you have a contradictory definition. Infinity never passes.
To say it's a contradiction, you have to show that defining things that way leads to a contradiction in that system, in that notation.

You can define whatever you want. Then you see if it leads to a contradiction within the system you've defined it. I think the reason you believe the definition itself is contradictory is because you're considering it within the 'usual' rules of numbers which don't necessarily apply to infinities, or at least can be defined not to.
#176
08-05-2012, 11:57 AM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by jtgain Okay, please try to keep it to the idiot level of conversation so I can understand. How does it ever make sense to say .000000....1? There is an infinite number of zeros. By definition it can never terminate (even under an alternate system or any system). You literally never, ever, ever get to the point where the 1 appears. Because if it did appear, you would have a terminating, fixed and defined number. Which can't happen because the zeros are infinite.
It's all about finding a suitable interpretation.

We can choose to interpret the word "infinity" like so: to say a property definitively holds of "infinity" is just to say that it holds of all natural numbers beyond some point. And to speak of any function of "infinity" is just to construct a term which you might later use in speaking of some property of "infinity".

This is often not the way you would want to interpret that word, but it is also often a useful way to interpret that word. Different strokes for different contexts.

(Note that, on this particular interpretation of "infinity", we will have that "infinity" is definitively greater than 0, greater than 1, greater than 2, etc. But "infinity" + 1 will be even greater than "infinity", and "infinity"^2 will be greater than those, and so on. [We will also have that "infinity" fails to be definitively even and fails to be definitively odd, but is very definitively (either even or odd). If this last part makes you unhappy, we can go ahead and ascribe "infinity" properties at random beyond (but consistent with) the basic ones we've already settled, till there's no uncertainty left.])

Now having done that, we can talk about 1/10^"infinity" and its decimal expansion.

In general, 1/10^n has a decimal expansion of the form 0.000...1 with n many zeros before the 1.

And thus, we can definitively claim, on this interpretation of our mathematical language, that 1/10^"infinity" has a decimal expansion of the form 0.000...1 with "infinity" many zeros before the 1.

If you multiply it by 10, you get a decimal expansion with "infinity" - 1 (which is smaller, but still infinite in the sense of being greater than 0, greater than 1, greater than 2, etc.) many zeros before the 1. If you divide it by 10, you get a decimal expansion with "infinity" + 1 many zeros. If you square it, you get a decimal expansion with 2 * "infinity" many zeros.

All of this is perfectly consistent. It's all given by our simple rule (to say something holds of "infinity" is to say that it holds of all sufficiently large natural numbers).

Last edited by Indistinguishable; 08-05-2012 at 12:00 PM.
#177
08-05-2012, 11:57 AM
 Frylock Guest Join Date: Jun 2001
It's no different than if we'd notated it as a pair of numbers, for ecample: (0.999..., 1) "After" is not central to the concept, I think.
#178
08-05-2012, 12:59 PM
 Senegoid Guest Join Date: Sep 2011
Quote:
 Originally Posted by jtgain Okay, please try to keep it to the idiot level of conversation so I can understand. How does it ever make sense to say .000000....1? There is an infinite number of zeros. By definition it can never terminate (even under an alternate system or any system). You literally never, ever, ever get to the point where the 1 appears. Because if it did appear, you would have a terminating, fixed and defined number. Which can't happen because the zeros are infinite.
Aha! The game is a-foot! This four-page (and counting...), multi-zombified thread, having beaten to death the discussion of what 0.999999..... means (a discussion, by the way, that will not die) . . .

We are now moving into new territory! Now, let's beat to death what 0.000000000000......1 means! What fun!

Okay, I can see a way to give a sensible interpretation of what 0.000000......1 would mean, analagous to 0.999999....

As with 0.9999999, we consider a sequence of numbers, each with one more digit:
0.1
0.01
0.001
0.0001
...
0.0000000000000000000001
...
0.00000000000000000000000000000.....00000000000000000000000000.....1

Now, as with 0.99999...., we stand back and look at this sequence, and ask: Does this sequence approach a limit? If it does, then define the notation 0.000000000000.....1 to be the value thus approached.

The result: Yes, it approaches a limit. The limit thus approached is 0. Thus, the notation
0.00000000000000......1, by this definition, is exactly zero. Not infinitesimally near zero. Exactly zero. And this approach, essentially the same way we got the meaning of 0.999999..... gives us a consistent result too: 1 - 0.9999..... sure enough gives us 0.000000......1

In other words, 1 - 1 = 0

Hey, I knew that!
#179
08-05-2012, 01:11 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by Senegoid The result: Yes, it approaches a limit. The limit thus approached is 0. Thus, the notation 0.00000000000000......1, by this definition, is exactly zero. Not infinitesimally near zero. Exactly zero. And this approach, essentially the same way we got the meaning of 0.999999..... gives us a consistent result too: 1 - 0.9999..... sure enough gives us 0.000000......1
Of course, the only reason you can say it is "exactly" zero, rather than infinitesimally near zero, is because you've baked into your (or, rather, we've baked into our conventional) definition of "limit" that infinitesimals are zero.

That is, we can define another notion of "limit", where the limit of a function f(n) as n approaches infinity is the value f("infinity"), interpreted as in my last post. Then the limit will be 1/10^"infinity", which will not be zero. It will only be infinitesimally close to zero.

Now, if you happen to decide that you no longer care to distinguish between values whose difference is infinitesimal, you will recover the conventional definition of limit. In some sense, that's the motivation of the conventional definitions. Very often, one has the tools to show that two values are infinitesimally close, and does not care about infinitesimal differences, so one might as well identify them.

But if one did care about infinitesimals, for whatever reason (even no reason other than to explore the idea of infinitesimals), you could easily use a suitable notion of "limit" which would make everything go through quite similarly, but end up with "infinitesimally near zero" and not "exactly zero".

Last edited by Indistinguishable; 08-05-2012 at 01:14 PM.
#180
08-05-2012, 04:05 PM
 erik150x Guest Join Date: Aug 2012
So a lot people mocked me and made fun of me... and perhaps to some degree I deserved it because I was not well prepared enough to defend my position. touche.

But now we have come full circle... if you look back at my original post: #35http://boards.straightdope.com/sdmb/...3&postcount=16

You'll see I was basically saying what we have all mostly concluded here.

Built in to the definition of calculus with limits is the very notion that .999... = 1

The theory of Limits asks us to simply accept this as fact. Although one could argue it doesn't actually ever say this is some absolute truth, but rather if you want to do math without a lot of headaches, we suggest you make the assumption that .999... = 1, although it may not actually be true. Others may say that because it works so well, it must actually be true, a reasonable position for some. Yet Limit theory offers us no proof!

Great Antibob refuted my claim that there is no evidence outside of using limits, that 10 x .999... = 9.999...

I re-post here:
---------------------------------------------------------------------------------------------------
We're not just "accepting" that there is a 9 there.

Here's an example of one of the "contexts" where infinity is actually defined.

Write 0.9999..... in a different form:

sum(i = 1 to "infinity") [9/10^i]

Note that the "infinity" in the index of the summation just tells us not to stop adding more terms ever.

Now, multiply this by 10:

10*sum(i = 1 to "infinity") [9/10^i]

We can bring the 10 "inside" the sum:

sum(i = 1 to "infinity") [10*9/10^i]

Now simplify:

sum(i = 1 to "infinity") [9/10^(i-1)]

Written in a more 'standard' form, this is 9.99999........

There is no '0' at the end at all. Nor are we "adding" any digits at all. We're just multiplying by 10.
-----------------------------------------------------------------------

I didn't like it then and still don't. This proof uses Limits. Without Limits you can't do this, it's what allows you to sneak in the extra 9 at the end [9/10^infinity]. Therefore again outside of assuming .999... = 1 via Limits, it is no proof at all.

I don't reject Limit theory at all. In fact as I have always interpreted it, it is a useful tool when you don't really care about 1/infinity or other such situations. But it is and never has been in my mind some absolute truth.

It also true that 99.999...% it probably makes no earth;y difference in any practical way whether they are equal or not. But some of us form a purely mathematical metaphysical point of view would like to know if they REALLY do = each other. Is that so wrong? Is it so wrong to believe they don't? After all tell me how you ever really get to 1 from .999...?

I keep hearing they are different notations for the same thing (well with limits indeed they are), but Limits just make the assumption they are. For practical purposes this makes a TON of sense, but when your asking the question, is .999... really equal to 1? It makes no sense to say of course they are because I assume they are.

I want to you all to know I have done the math, just finished a while ago and my hand is really cramped. But the answer I got to .999... was .999...

Last edited by erik150x; 08-05-2012 at 04:09 PM.
#181
08-05-2012, 04:59 PM
 leahcim Member Join Date: Dec 2010 Location: New York Posts: 1,238
Sorry to be coming into this late, but:

Quote:
 Originally Posted by erik150x sum(i = 1 to "infinity") [9/10^i] Note that the "infinity" in the index of the summation just tells us not to stop adding more terms ever.
Actually, this is not a conventional formal definition of an infinite sum. If we were forced to use it, we would have to conclude that the value of any infinite sum is unknowable, since one can't finish ever.

The more conventional definition of that notation is:

sum(i = 1 to "infinity") [9/10^i] := lim (N->infinity) sum(i = 1 to N) [9/10^i]

which permits you to get a value for this sum in finite time, but forces you to use limits.

Quote:
 I didn't like it then and still don't. This proof uses Limits. Without Limits you can't do this, it's what allows you to sneak in the extra 9 at the end [9/10^infinity]. Therefore again outside of assuming .999... = 1 via Limits, it is no proof at all.
I'd take you one step further, and suggest that without limits, the notation .9999... doesn't actually mean anything at all. Of course, using notation that doesn't make sense without limits, and then rejecting the concept of limits after the fact, can lead to inconsistencies.
#182
08-05-2012, 05:52 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by erik150x So a lot people mocked me and made fun of me... and perhaps to some degree I deserved it because I was not well prepared enough to defend my position. touche.
erik150x, I never mocked or made fun of you. The thing you need to know is that people using the conventional definitions of limits and "real numbers" aren't wrong; they're just using definitions that are in tune with what they are interested in. They are absolutely correct, according to their definitions, which is as correct as it gets in mathematics.

The only thing they are guilty of is persisting to use mathematical language in accordance with definitions which you are not that interested in. But this is alright; this is their prerogative. (And there are good reasons for them to be interested in those particular natural definitions; they present a convenient and useful system of calculation. But you seem to already accept this, which is great.)

On the other hand, what they need to understand is that you aren't really wrong, either. For the most part, you've seemed to me not particularly crankish (there are those who are a lot worse). You've been focusing on a rather natural and useful idea, albeit you, as a non-mathematician, may have struggled to express or formalize it. You are also absolutely correct, according to your definitions, which is as correct as it gets in mathematics.

The only thing you are guilty of is persisting to use mathematical language in a non-conventional way. But this is alright; this is your prerogative. (And this is also often how mathematics progresses). So long as you are willing to acknowledge the non-conventionality of your use of the language and understand the conventional use and how it relates to yours, all is well.

This thread is full of people saying things that aren't incorrect, except to the extent that they flatly proclaim that other people are incorrect.

[I would also note that arithmetic ignoring infinitesimals and arithmetic paying attention to infinitesimals are not disconnected, such that there is any sense in falling into one "camp" or the other; they are as closely related as modular and integer arithmetic, or the studies of linear and analytic functions. Both these systems shed light on each other. Both can be used to help analyze the other. The better we understand either, the better we understand both. There's no reason either one should be ignored in favor of the other, except the vagaries of personal interests.

For example, epsilon-delta arguments are all about the passage between the world where we don't and the world where we do ignore infinitesimal differences, even if they are not often presented this way. And understanding this fact will give you a better understanding of the foundations of calculus, make you sharper at recognizing how it relates to other topics in mathematics, etc. In mathematics, there is a tendency for knowledge to interconnect. It's all good. You ignore this at your peril.]

Quote:
 Originally Posted by erik150x I keep hearing they are different notations for the same thing (well with limits indeed they are), but Limits just make the assumption they are. For practical purposes this makes a TON of sense, but when your asking the question, is .999... really equal to 1? It makes no sense to say of course they are because I assume they are.
Keep in mind that that's what mathematics is. The only way anything is mathematically true is by assumption (either directly or indirectly). Every fact in mathematics is like the fact that bishops move diagonally: it's true because, and only because, you made it a rule of the game, and having done so it is unquestionable, except insofar as you can use different rules if you're interested in looking at some other game.

Last edited by Indistinguishable; 08-05-2012 at 05:55 PM.
#183
08-05-2012, 08:24 PM
 Trinopus Member Join Date: Dec 2002 Location: San Diego, CA Posts: 5,069
Quote:
 Originally Posted by erik150x . . . People will hate me for this...
Nah; there is no hatred in mathematics. (Okay, yeah, we all hate "word problems." But...)

Quote:
 An obvious response to this would be, well if there is no difference between A and B, then A - B = 0 and therefore A = B. But what if I were to say well, actually A - B = 1/infinity? Perhaps there is a subtle distinction between say there is no space between 2 things yet they are not in the same space. But points have no dimension, right? So that leads us back to they must be in the same space again. What if I argue points do have a dimension? Say 1/infinity? Perhaps this is a number we don't fully understand (a bit of a planks constant of the math world?). . . .
Seems to me you're trying to define a number that has only some of the properties of "zero" but not all of its properties. It has no width...but it has a *little* width.

But, as I argued, from a legalistic standpoint, you can never "win" in describing this number, because I just shrug and add another "0" in front of it. No matter how you describe it, I can always trump it.

I have a trivial mechanism by which I can "construct" counter-examples, but you have no means by which you can "construct" the number you're looking for.

Quote:
 Originally Posted by Indistinguishable . . . And thus, we can definitively claim, on this interpretation of our mathematical language, that 1/10^"infinity" has a decimal expansion of the form 0.000...1 with "infinity" many zeros before the 1.
And, as I've been suggesting, if someone disagrees, you just ask them to point to the exact decimal number where this differs from "zero." The tenth decimal place? The millionth? The googleplex-to-the-google-squared place? I simply smile, and say, "Plus one..."

Quote:
 Originally Posted by Indistinguishable . . . For example, epsilon-delta arguments . . .
Oh, right: we hate those, too!
#184
08-05-2012, 08:43 PM
 Indistinguishable Guest Join Date: Apr 2007
I'm not sure I understand what you are getting at, Trinopus, but I will say, on the interpretation I gave above, 1 - 0.999... will differ from 0 in the "infinity"th place after the decimal point. And, yes, dividing this by 10 will produce an even smaller number, differing only in the "infinity" +1th place. And squaring that will produce an even smaller number, differing only in the "infinity"^2 + 2"infinity" + 1th place.

That's alright. That's just how this interpretation runs. It's not the conventional interpretation of either the notation or the ambient numeric system, but it's another one. It takes 0.999... as running to "infinity" many 9s, but not "infinity" + 1 many 9s. We can make sense of that, if that's something we wish to make sense of.

Last edited by Indistinguishable; 08-05-2012 at 08:45 PM.
#185
08-06-2012, 12:15 AM
 Monimonika Guest Join Date: Aug 2012
Quote:
 Originally Posted by erik150x ... it's what allows you to sneak in the extra 9 at the end [9/10^infinity].
Am I correct to assume that you still think there should be a 0 added to the "end" of the infinite string of 9s for the answer to 10x0.999...?

Or, if 0.999... is multiplied by 10, the infinite string of 9s will have one less 9, or at least a different "infinite" number of 9s to the left of the decimal point and thus cannot be used for later calculations like 9.999...-0.999... since the decimal places do not match and/or the "end" digit is uncertain?

If so, I would like to ask you if you can find the flaw(s) in the following (in which I try to see if 10x0.999... really does result in having a 0 at the end):

I start off assuming three premises:

1) When two numbers have the exact equal infinite number of decimals places to the right of the decimal point (hereon denoted as [A], with [A-1] being "infinite decimal places-1" or some such), the two numbers can be used in calculations together since all the decimal places match.

2) When two numbers with exact equal infinite number of decimal places to the right of the decimal point are added together, the result will also have the same infinite number of decimal places, since addition does not shift the decimal point in any way.
x[A] + y[A] = z[A]

3) A number multiplied by 10 is equal to ten of the number being added together.
10x = x+x+x+x+x+x+x+x+x+x

((The above is non-rigorous on the definition of "number", but please be kind in assuming what I meant.))

x= 0.9999...[A]

I calculate x+x, which I can do thanks to Premise 1.

Calculation is as follows (the * are for spacing purposes):

*0.9999...[A]
+0.9999...[A]
___________
*0.
*1.8
*0.18
*0.018
*0.0018
*0.00018
+ etc.
___________
*1.9999...[A]

Premise 2 applies on the result.

I now calculate (x+x)+x, also known as x+x+x.

Using the same calculation method above, the result is 2.9999...[A].

I repeat this until I have added up 9 of the x with the result of 8.9999...[A].

I now add the tenth x:

*8.9999...[A]
+0.9999...[A]
___________
*8.
*1.8
*0.18
*0.018
*0.0018
*0.00018
+ etc.
___________
*9.9999...[A]

Given Premise 3, 10x0.9999...[A] equals 9.9999...[A].

Wait, there has been no shift of the decimal point during any step, thus nowhere for the 0 to be added, and nowhere to add an "extra" 9 either. Where have I gone wrong?
#186
08-06-2012, 12:50 AM
 erik150x Guest Join Date: Aug 2012
Response to Monimonika:

.999...9
+ .999...9
-----------
1.999...8
+ .999...9
-----------
2.999...7
+ .999...9
-----------
3.999...6
+ .999...9
-----------
4.999...5
+ .999...9
-----------
5.999...4
+ .999...9
-----------
6.999...3
+ .999...9
-----------
7.999...2
+ .999...9
-----------
8.999...1
+ .999...9
-----------
9.999...0

Last edited by erik150x; 08-06-2012 at 12:50 AM.
#187
08-06-2012, 12:55 AM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 11,583
Quote:
 Originally Posted by erik150x Response to Monimonika: .999...9 + .999...9 ----------- ...
But there is no last 9 in .999... (as usually defined). After every 9 in that decimal fraction there is still an infinite number of 9s to go.
#188
08-06-2012, 12:56 AM
 erik150x Guest Join Date: Aug 2012
Not sure why you think you can ingnore what happens at the "end" of this infinite string of 9s?

Granted even I have some uneasiness about either your version or my version. But in my mind, mine feels more correct to me. There is no getting around the fact we are talking about infintesimals here, and I have no perfect theory on them, maybe not even a good one. I remain to see any clear reason to say they don't exists or except some ulitmate truth based on Limits whcih purposeful ignore them (with good reason), that .999... is really the same thing as 1 from our mast basic understaning of the idea...

9/10 + 9/100 + 9/1000 +... + 9/infinity
#189
08-06-2012, 01:00 AM
 Trinopus Member Join Date: Dec 2002 Location: San Diego, CA Posts: 5,069
Quote:
 Originally Posted by Indistinguishable I'm not sure I understand what you are getting at, Trinopus . . .
Doubtless I'm phrasing it poorly, but I'm taking my cue from epsilon-delta proofs, which begin, "For any epsilon > 0 there exists a delta > 0 such that..." I always viewed those as a "burden of proof" argument. If you give me an epsilon, I will construct a delta, which fits the criterion. As if in a court of law, you say, "Is it within one part in a million?" and I can say, "Yes it is, and here's the delta that works." "Is it within one part in a trillion?" "Yes, and here is the new delta that turns the trick."

I never have to prove that it works "for all epsilon." I only have to show that it works "for any given epsilon." I have an algorithm that I use to construct my new delta from the epsilon someone else gives me.

Thus, with .9999... = 1. Tell me where you think the difference might be, and I will, upon the instant, show that, by simply putting in a few more 9's, that the sum is closer to 1 than the difference you proffered.

Like two lawyers arguing in court.

Monimonika: I like it! By adding the numbers, you appear to have nicely gotten around the matter of "the extra decimal place."

Last edited by Trinopus; 08-06-2012 at 01:01 AM. Reason: invented a new Greek alphabetic letter epsiolon!
#190
08-06-2012, 01:05 AM
 erik150x Guest Join Date: Aug 2012
Giles your point is well taken. Niether my nor Monimonika's version makes perfect sense. It is not a simple issue to deal with. But if the proof was as easy as Monimonika makes it out to be, it would have been made long ago. Poeple accused me of thinking I am smarter than all the great mathematicians in history. I certainly do not even compair my self.. There is no proof I know of in a very elemantry manner describing a proof for .999... = 1. If there was it would be all over the interent. Everyone i have seen uses Limit based calculus. If your going to use limit based calculus then you have already made the assumption they are equal, so what's the point of even considering the question.

Now... for me the problem with using limit based calculus as answer, is that doesn't really say anything about HOW or WHY .999... = 1. Nothing to support it's conclusion, other than it let's avoid the very quesiton of does .999... = 1 and if so how? If not what is 1 - .999...? It simply avoids the question altogether.
#191
08-06-2012, 01:08 AM
 erik150x Guest Join Date: Aug 2012
Trinopus,

What you are saying is you can prove .999... is as close to 1 as you wish to prove, but you cannot prove it EQUALS 1.

If there was a proof... it would have been long ago given. Instead we are asked to accept the definition of a limit, which says if you can prove a number is close enough to another number with arbitrary accuracy (just short of infinity), then let's just call it that number. It is in effect a very miniscule rounding off at the infinity decmial place.
#192
08-06-2012, 01:12 AM
 Derleth Guest Join Date: Apr 2000
Quote:
 Originally Posted by erik150x Trinopus, What you are saying is you can prove .999... is as close to 1 as you wish to prove, but you cannot prove it EQUALS 1.
We can and have. You refuse to accept such proofs, which says a lot about you and nothing about the proofs.
#193
08-06-2012, 01:12 AM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 11,583
Quote:
 Originally Posted by erik150x Not sure why you think you can ingnore what happens at the "end" of this infinite string of 9s?
It's not being ignored because it does not exist: there is no end to the infinite string of 9s. If you start counting the natural numbers 1,2,3,... you can go on forever: there is no last natural number. Similarly, an infinite decimal fraction has no last digit.
#194
08-06-2012, 01:22 AM
 erik150x Guest Join Date: Aug 2012
[quote=Indistinguishable;15351997]erik150x, I never mocked or made fun of you. The thing you need to know is that people using the conventional definitions of limits and "real numbers" aren't wrong; they're just using definitions that are in tune with what they are interested in. They are absolutely correct, according to their definitions, which is as correct as it gets in mathematics.

ME: My comments regarding being mocked were not directed towards everyone here. I have enjoyed very much the ones who have given me at least some respect. In fact you have given me the most, and I appreciate it mostly becuase you have had the most respectable resposes. (Some) on here were very foolishly spouting of about my ignorance without even understanding the very basis for their own belief in the matter of .999... = 1.
#195
08-06-2012, 01:24 AM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Derleth We can and have. You refuse to accept such proofs, which says a lot about you and nothing about the proofs.
Where is the proof with out using limit based calculus?

You do understand that limit based calcuslus starts out with the assumption, with out proof, that they are equal? It never does make any claim to such a proof.

#196
08-06-2012, 01:26 AM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Giles It's not being ignored because it does not exist: there is no end to the infinite string of 9s. If you start counting the natural numbers 1,2,3,... you can go on forever: there is no last natural number. Similarly, an infinite decimal fraction has no last digit.

Mine always starts at the last decimal place. Where does yours start? ;-)
#197
08-06-2012, 01:45 AM
 Giles Charter Member Join Date: Apr 2004 Location: Newcastle NSW Posts: 11,583
You start at the beginning, but recognise that every step is a partial sum.

The number .999... is the limit of the series .9, .99, .999, .9999, ..., so to add it to itself, you get:

.9 + .9 = 1.8
.99 + .99 = 1.98
.999 + .999 = 1.998
.9999 + .9999 = 1.9998
...
.999...999 + .999...999 = 1.999...998
...

So, given enough steps, you can have as many 9s as you like after the decimal point, and the value of the final 8 can be as small as you like. With limits, "as many as you like" is an infinite number, and "as small as you like" is zero. The limit of the sum is 1.999... (i.e., 1 followed by an infinite number of 9s), which is equal to 2, because if you go far enough you can make it as close as you like to 2.
#198
08-06-2012, 02:01 AM
 erik150x Guest Join Date: Aug 2012
I wish to express that I do not reject the idea of using limits. They are a spectacular creation allowing us to progress mathematics in many ways. However there is at least 2 different ways I think you can interpret limits:

1) Limits express the ultimate truth in the real number system about the issue of .999... = 1 and that truth is that they are indeed they are precicely the same.

(It would seem from what I gather on here, at least many are being tought this principle)

At some point, not sure when this occurred, I don't ever remember being tought it, that we now incorporate this Limit definition right into the very definition of numbers. Wow, that blew me away. Either I am just old, or things have chnaged recently, or maybe I skipped class that day. Very possible.

2) You can view Limits as a tool, a very very powerful wonderful tool, but none the less just a tool which helps us avoid these troublesome questions about things like does .999... = 1?

I think within most applications of math it doesn't really matter which view you take. I don't really know what practical matter really involves the quesiton of .999... = 1? There could be, I don't know them is all. But for me I do not see how they are equal from a fundemtnal standpoint of the sequence:

9/10 + 9/100 + 9/1000 + ... + 9/infinity

At the end of the sequence all I see is .999... I don't know how you get to 1. Just saying it so by deinfition of limit which basically just says trust us... they are they same.

They could be equal... but I have not seen anythign to make me think they are and I have never seen a proof. Of course I won't get one from Limit based calculus because they just assume they are equal. What right is there to do so, other than it works quite nicely for the vast majority of mathematics.

If you believe Limit based calculus express some ultimate truth in this matter, ok. When someone asks you if .999... = 1, just tell them in the standard analysis or real numbers we assume them to be true is all, and in fact we deinfe the numbers that way. I leave it to you to tell them why. But don't give them any proof, becuase you have none.

Last edited by erik150x; 08-06-2012 at 02:03 AM.
#199
08-06-2012, 02:06 AM
 erik150x Guest Join Date: Aug 2012
Quote:
 Originally Posted by Giles You start at the beginning, but recognise that every step is a partial sum. The number .999... is the limit of the series .9, .99, .999, .9999, ..., so to add it to itself, you get: .9 + .9 = 1.8 .99 + .99 = 1.98 .999 + .999 = 1.998 .9999 + .9999 = 1.9998 ... .999...999 + .999...999 = 1.999...998 ... So, given enough steps, you can have as many 9s as you like after the decimal point, and the value of the final 8 can be as small as you like. With limits, "as many as you like" is an infinite number, and "as small as you like" is zero. The limit of the sum is 1.999... (i.e., 1 followed by an infinite number of 9s), which is equal to 2, because if you go far enough you can make it as close as you like to 2.
Very good. Now do that same analysis with 10 (.999...998), you are only adding 2 of them you need 10 to simulate mulitplication by 10. Notice how you have an 8 at the end now? What do you think you will have in that exact decmial place after adding 10 of them?

Last edited by erik150x; 08-06-2012 at 02:07 AM.
#200
08-06-2012, 02:12 AM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by erik150x At some point, not sure when this occurred, I don't ever remember being tought it, that we now incorporate this Limit definition right into the very definition of numbers. Wow, that blew me away. Either I am just old, or things have chnaged recently, or maybe I skipped class that day. Very possible.
It's not that you're old or things have changed recently; these definitions have been standard since the 1800s.

Rather, the problem is that most people are never taught anything about them. The typical math curriculum for anyone who isn't a math major never (or just barely, as in "It gets half a page in the part of the textbook we'll never be looking at") actually explicitly defines "real numbers" and the interpretation of infinite decimal notation, even while expecting students to be comfortable with these.

This is maybe not the worst thing in the world. You can get a lot done with "real numbers" without being too formal about them; indeed, for many purposes, it would be premature to fix their formalization.

But it does lead to people (quite naturally) making up their own meanings for infinite decimal notation, and then being surprised and argumentative when their particular reconstructed definitions turns out to conflict with others' or the never mentioned standard.

Last edited by Indistinguishable; 08-06-2012 at 02:16 AM. Reason: "0.999... + 0.999... = 1.999...8 < 2, with N digits after the decimal point in each, even if N = 'infinity'"

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