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#1




Here's an old argument between myself and a friend:
Does .9999... (nine repeating forever) really equal 1? I've had people swear it does, and try to prove it w/ equations, but the equations just boiled down to saying ".999 =1 is true because 1 = .999". Can anyone offer any decent proof either way? 
#2




This has been discussed before. .9999 is shorthand for "The limit as n goes to infinity of .9*10^(1)+.9*10^(2)+...10^(n)" which is indeed one.

#3




Rounding. Odds are you learned about it in first grade. No, .999... does not equal one. It equals .999... . Oh, you said "decent proof". Well, in that case, I can't answer your question. But I'm guessing that it equals .999... . Anyway, if it really did equal one, then 1.999... would really equal two. So we wouldn't even have decimal points or places.

#4




Well damn. I didn't even see Ryan's post. Okay, if you want to get all technical. Pay no attention to my previous post.



#5




This is often misunderstood because people forget the infinite repeating part.
Here goes the equations. Remember that a number with a line over it represents a repeating pattern. Code:
Let: _ N = 0.9 then _ 10N = 9.9 Since there's an infinite number of nines after the decimal: _ _ _ 10N  N = 9.9  0.9 = 9.0 10N  N = 9.0 9N = 9 N = 1 Code:
_ N = 0.06 _ 10N = 0.66 _ 100N = 6.66 _ _ 100N  10N = 6.66  0.66 _ 90N = 6.0 = 6 6 2 1 N =  =  =  90 30 15 N = 1/15
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#6




Well, how do you distinguish between two points (in this case one and .999...)? Well, if there's a point between them, then they're separate points, otherwise they're the same point, right?
Okay, no equations, just the simple fact that there is not a single point between .999... and 1. Not a one. As for the equations: 10 * .999... = 9.999... 9.999...  .999... = 9 Now, 10 * .999... can be written as 10x and as 9.999... so 10x = 9.999... and .999... would be x so 10x  x = 9x or 9.999...  .999... = 9 * .999... HOWEVER we noted that 9.999...  .999... = 9 so if 9x = 9, what does x equal? (the last step is left to the reader)
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I sold my soul to Satan for a dollar. I got it in the mail. 
#7




Oh, and slvr's right, 1.999... does indeed equal 2. However, she's wrong in that this does not invalidate the decimal system. 1.888... is still distinct from every number other than 1.888... Just think of it in terms of "is there a point in between A and B?" What about 1.888... and 1.9? 1.89
Nice and simple, this is not a contradiction of the rest of mathematics, but an inevitable result.
__________________
I sold my soul to Satan for a dollar. I got it in the mail. 
#8




Just out of curiosity, where is the thread that discussed this? I've searched but I apparently can't seem to hit on the right keyword or something... BTW, 0.999... is the same as 1. The algebraic proof shows that well enough. However, calculus works a bit differently. In calculus you can meaningfully talk of two numbers with an infinitely small difference between them. Imagine this: You take Zeno's paradox and express it in a program that calculates it infinitely. That is, the program adds 1/2 to 1/4 to 1/8 to ... without end. In theory, the program can be said to count in a fashion that takes it infinitely close to 1 but never lets it reach 1. In fact, the program does reach 1 because the system can only hold so many decimal places before it begins to round. This concept finds practical use in relativistic physics, where the acceleration of a particle can be plotted along a curve that acts the same way that Zeno's curve works, replacing 1 with c, the speed of light in a vacuum. In theory and in practice the particle's speed can get very close to c but can never quite reach it. Interesting, in an odd sort of way.
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"Ridicule is the only weapon that can be used against unintelligible propositions. Ideas must be distinct before reason can act upon them." If you don't stop to analyze the snot spray, you are missing that which is best in life.  Miller I'm not sure why this is, but I actually find this idea grosser than cannibalism.  Excalibre, after reading one of my surefire millionseller business plans. 
#9




Well, shit, didn't even notice that by the time I got around to answering, it'd already been answered better.
__________________
I sold my soul to Satan for a dollar. I got it in the mail. 


#10




The proof that I always like to give goes like this:
1/3 = .333333... (1/3)*3 = 1 .33333..... * 3 = .999999999999 Therefore, .9999 = 1 It usually shuts them up. 
#11




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I will represent Professor McWheely's Cough Elixir and Tonic. 
#12




mouthbreather
Thank you. It's like a trip down Memory Lane. There were at least three such treads at that time. Here is another for your education/amusement.
__________________
That's not a tau neutrino in my pocket; I've got a hadron. 
#13




Another way to put it, I guess is, if .9999 doesn't equal 1, then Zeno's paradoxes are true. Which throws the concept of Calculus, and algebra for that matter, out the window. One of them was, a man standing in a room cannot touch the other wall, because in order to do so he would have to walk half the distance, then half again, and half again, etc. You have to say the limit of the sequence a[n]/10^n is the full distance of the room (1) as n approaches infinity. Of course nowadays, one can get nitpicky and say he never actually touches the wall because the atoms of his finger never actually contact the atoms of the wall (unless a nuclear bomb drops on him and splatters him all over the wall..).
To argue that point just use the second paradox as a arguement, of course we know Archilles actually passes the tortoise, because his arguement was that in order to actually pass the tortoise he would have to first be at the same point he is  then overtake him. In other words, in order for you to actually say he passes him you have to say he actually gets to that point, in essence you say that .999.. converges to 1, THEN you can get numbers greater than 1 again, and at precisely that point, 1, he is right beside him, and then runs past him. This is more of a "logic"type and popular arguement settler for this question. 
#14




Quote:



#15




Actually, Ryan, I was looking for an obscenely old thread I remember from before I registered. I haven't been able to find that *exact* thread, but I've found ones that are essentially clones of it. BTW, links to threads created in the UBB system don't work. Ferexample, follow this link: http://boards.straightdope.com/ubb/F.../0074962.html You won't get to the thread (not by just pointing and clicking, anyway) but you'll instead be routed to http://boards.straightdope.com/sdmb/ which is the front page to all the several SDMB boards. I'm almost certain this has to do with the '/ubb/' in the addy, which was dropped when we switched from UBB to vB. It's kind of annoying to those with so little to do we read old threads.
__________________
"Ridicule is the only weapon that can be used against unintelligible propositions. Ideas must be distinct before reason can act upon them." If you don't stop to analyze the snot spray, you are missing that which is best in life.  Miller I'm not sure why this is, but I actually find this idea grosser than cannibalism.  Excalibre, after reading one of my surefire millionseller business plans. 
#16




all assumptions
All arguments about .999... = 1 in my mind boil down to what do you believe 1/infinity to be. If you say 0 then they are equal. If you say no, its a really really really... Small number, then they are NOT equal. Some people don't believe in infinity, some people don't believe in 1/infinity (other than saying it is zero, which is just saying it doesn't exist).
People use informal proofs like: 1/3 = .333... And since 1/3 x 3 = 1 Then 3 x .333... = .999.... = 1 The assumption here is that 1/3 in fact exactly = .333... But if you do the math for your self you can see after every (3) in the .333... There is always a remainder. What we are supposed to believe is some how after infinitely many 3s this remainder vanishes. And why should that be? I would say it's still there, you could say its remainder is in fact 1/(3 * infinity). So lets look at the informal proof again: 1/3 = .333... + 1/(3 * infinty) 3 * (.333... + 1/(3 * infinity)) = .999... + 1/infinity = 1 This is not the same as saying .999... = 1 ! All algebraic proofs use this trick to make you forget about the 1/infinity. .99 * 10 is not 9.99 but rather 9.90 Yet people will say matter of factly .999... * 10 = 9.999...(9) that extra 9/infinity just gets convenirntly thrown in there. Why not = 9.999...(0)? Okay so that number is hard to think about and so is 1/infinity. But the proof is none the less... A cheat. There are various calculus based proofs that are supposedly more rigorous, but built into the foundation of calculus is the concept of limits which allow us to say 1/infinity is arbitrarily close to zero, so lets just call it zero. For practical purposes, this makes sense. Lets face it if .999... Is not equal to 1, it's damn close. We all agree. But that's different than saying it IS EXACTLY 1. Let me ask you this. Is there a number bigger than infinity? How many rational numbers are? Infinity? Yes. How many real numbers are there? Infinity, yes, but also provably more infinite than the number of rationals. How can a number be bigger than infinity? I don't know but clearly there can be. So when people try to prove by contadiction that 1  .999... Can't exist because it would be the smallest number > zero, yet certainly that number divided by 2 would be a contradiction. Its like arguing infinity can't exist because than what's infinity + 1, or infinity * 2. I have never seen a proof for .999... = 1 that at its basis doesn't assume 1/infinity = 0. And we all know 1/infinity is undefined, so at best this question is undefined. Every time I begin to think I am the smartest person in the world, some udiot comes along to prove me even smarter. ;p 
#17




Just more proof that zombies are bad at math.

#18




I believe that .999... equals 1 and that 1/infinity is not equal to zero. What now?

#19




Please don't laugh at me, but can you have a repeating decimal when measuring time?
edit: If I say that I got there in 10.33... seconds and Jamie got there in 10.34 seconds, did we arrive at the same time? Last edited by Farmer Jane; 08032012 at 11:54 PM. 


#20




No, it's not the same time. 10.33... is equal to 10 and 1/3, which is a different number than 10.34.

#21




U then believe in a contradiction. If you accept a contradiction to be true then you prove anything to ne true and not true. It is a meaningless belief.

#22




erik150x, I've learned things here on the SDMB. One thing I learned was that there are no other numbers between .999... and 1. That means they are the same number. Try looking for other threads on this subject.

#23




Quote:
A. All mammals are animals. B. Carrots are mammals. C. Therefore, carrots are animals. Unassailable, sound logic, isn't it? A syllogistic thing of beauty. My point is that you don't understand how infinities work, or what infinity means. 
#24




Quote:
If you would like to point me to the exact proof you mention, perhaps I too could be enlightened. Thanks. 


#25




Quote:

#26




1/infinity isn't defined because infinity isn't a number (at least not in the same way that the reals are numbers). However, it is undeniably true that the limit of 1/x as x approaches infinity is 0. It is also true that the limit of the sum of 9/10^x with x going from 1 to infinity is 1.0. 0.999... is just shorthand for that limit.

#27




Let me add I am trying to say that I don't believe .999... = 1. And i also believe the question can be rephrased as what is 1/infinity, (zero or not). Your state they are not essentially the same. Ok. I think your wrong. I have explained my view and why. I'd be interested to here your view and why they are not the same questions. You say I don't understand infinity or its workings. Please explain what leads you to this conclusion?

#28




Quote:
Right. And a limit is basically saying if one can come arbitrarily close to some nimber, we will just say it is that number. For practocal purposes this is fine, but this is just an accepted assumption in calculus. It is not an absute truth. Show me the proof that the limit of a function is exactly the same as the result of the function? 
#29




[QUOTE=erik150x;15347336]
Quote:



#30




Quote:
11=0. It's an identity function. If there are no other numbers between .999... and 1, then they are the same. If there are other numbers between, then 1.999... does not equal 0.1 If you add any number to .999..., you will get something greater than one. It's just an anomaly of notation that there can be more than one representation of the same number. Now if you have some new definition for a numeric system that allows them to be different numbers without breaking down elsewhere, good for you. Nobody else is using it, and it wasn't presumed to be used in the question here. So go open a thread on your new topic, or find one that is more on the topic of your new system. The mathematicians on this board will sense the shift in the force that results and you will be able to participate in a thorough and detailed analysis of your theory. 
#31




Quote:
When you can name a number between 0.999... and 1.0, you'll have a case. 
#32




Quote:
Just for fun though... (1999....)/2. 
#33




Sorry that should be (1.999...)/2 + 1  .999...

#34




Quote:
9/10. + 9/100 +9/1000 ... Just a sum of an infinite sequence, no limit involved. 


#35




Quote:
I find it best, if only "for old time's sake" , to appeal to the Axiom of Archimedes, which can be paraphrased as For any positive ɛ there is a finite integer N such that Nɛ > 1 Notice that for any N, you can make N(1  0.999...) < 1 by providing sufficiently many 9's. Last edited by septimus; 08042012 at 01:55 AM. 
#36




It's not about 1/infinity or limits or sums or what have you. It's all about notational convention.
Here's my standard reply to this debate: Keep in mind that one must distinguish between notation and what that notation represents. Different notation can represent the same entity, as in, for example, the equality of "1/3" and "2/6": they are not equal as notation, but the fractions they denote are equal. Now, does "0.9999..." denote the same thing as "1"? Well... first off, a disclaimer: of course, one could invent an interpretation of this notation on which they denoted different things, just as one could invent an interpretation of notation on which "1/3" and "2/6" denoted different things (for example, they denote different dates...). But I'm not going to talk about that sort of thing right now. Instead, I'm going to talk about the standard, conventional interpretation of infinite decimal notation, the one that mathematicians mean when they use this notation, and the one which justifies the claim that "0.9999..." denotes 1. When a mathematician gives an infinite decimal as notation for a number, what they mean by it is this: the* number which is >= the rounding downs of the infinite decimal at each decimal place, and <= the rounding ups of the infinite decimal at each decimal place. This is the definition of what infinite decimal notation means; it's true because we say it is, just as the three letter word "dog" refers to a particular variety of fourlegged animal because we say it does. So, for example, when a mathematician says "0.166666...", what they mean, by definition, is "The number which is >= 0, and also >= 0.1, and also >= 0.16, and also >= 0.166, and so on, AND also <= 1, and also <= 0.2, and also <= 0.17, and also <= 0.167, and so on." What number satisfies all these properties? 1/6 satisfies all these properties. Thus, when a mathematician says "0.16666...", what they mean, by this definition, is 1/6. Similarly, when a mathematician says "0.9999...", what they mean, by that same definition, is "The number which is >= 0, and also >= 0.9, and also >= 0.99, and also >= 0.999, and so on, AND also <= 1, and also <= 1.0, and also <= 1.00, and also <= 1.000, and so on." What number satisfies all these properties? 1 satisfies all these properties. Thus, when a mathematician says "0.9999...", what they mean, by definition, is 1. [*: Of course, when one says a thing like "THE number which is...", this may be taken to involve an implicit claim that there is a unique such number. So when mathematicians use infinite decimal notation, they also generally have a very particular numbersystem in mind in which these uniqueness claims are all justified. But, there are many other numbersystems (just as useful ones, or even more useful ones, for many purposes; the world is diverse and our mathematical analyses needn't be shoehorned into "one size fits all" form) in which there may be no number or many different numbers satisfying such systems of constraints; in such contexts, infinite decimal notation is generally less useful as a way to denote numbers, though it can still be used in essentially the same way to denote certain intervals instead.] Last edited by Indistinguishable; 08042012 at 02:32 AM. Reason: Having said my piece, I now depart 
#37




Arrrrgghh! Not this old stuff again! I am reminded of the Petunias in HitchHiker's Guide to the Galaxy saying "Oh no, not again!" Wasn't this settled in a much earlier thread, like in the days of Archimedes?
Okay, I'll make an attempt to add something to the discussion. I think I see a problem with many people's understanding of .99999..... like erik150x that I don't think I've seen mentioned yet. I'll mention it, but I don't feel much like expounding in detail. What does an infinitely long decimal fraction mean in the very first place, anyway? Consider a finitely long fraction, like .357  In our decimal number system, that means: 3 x 10^{1} + 5 x 10^{2} + 7 x 10^{3} Everybody agree so far? It's the sum of a finite number of terms, one term for each decimal digit. So how do we define the meaning of an infinitely long decimal fraction? Well, it's the sum of an infinite number of terms! Kaboom! We have a problem right there, Houston. Adding up an infinite number of terms is NOT like normal addition, and the very concept of a sum of infinitely many terms needs to be carefully defined right from the start. It's defined in terms of convergence to a limit. You create a sequence of sums of finitely many terms. That is, you create this sequence of sums:  Sum of just the first term.  Sum of the first two terms.  Sum of the first three terms.  Sum of the first four terms. and so on. Now you stand back and look at your sequence of sums, and ask yourself: Does this sequence of partial sums seem to be approaching a limit? (And if so, can you prove it? That's even better!) If the sequence of partial sums approaches a limit, then the sum of your infinite series is defined to be that limit. And that is exactly how we define any meaning at all for infinitely long decimal fractions. Once you get your mind wrapped around that, you can begin to deal with things like .99999.... Your sequence of partial sums is none other than:  .9  .99  .999  .9999  .99999 and so on. Does this sequence approach a limit? Yes, it approaches a limit. It approaches 1 as a limit. That's why, by definition, .999999..... equals 1. It is reassuring to notice that there is consistency in the mathematical world. You've seen all those proofs that .99999..... = 1, for example by shifting the digits and subtracting, and tricks like that. Those are all tricks, but it's comforting to know that they all give answers consistent with our definition of infinitely long decimal fractions in the first place. There. How about that. I got started and ended up giving the full lecture after all. So there. 
#38




Quote:
My point here has been and remains that I have yet to see a convincing proof that .999... = 1 Every proof I have seen has hidden somewhere in the assumptions that 1/infinity = 0. When you use limits for purposes of proving .999... = 1 whether you are aware or not you are making an assumption that for the purpose of this question, 1/infinty = 0. Which in itself leads to contradivtions. Is 0 * infinity = 1? Another way of looking at it limit use here without saying 1/infinity = 0, is that the limit of the sum of the series 9/10^x as x aproaches infinity is 1. However a limit is an inuitive assumption here, not a proof. It is not a mathematical identity like 1  1 = 0. I'm honestly not trying to be stubborn here (though I can be and may seem to be). I really just haven't seen a proof that convinces me. I have shown the holes in the algebraic proofs and the assumptions made in the limit process. Are the no definitive proofs? Again you can say that the limit of 1/x as x aproaches infinity is 0, .but that is very different than saying 1/infinity is zero. Please helo a lost soul sonewhere between exit 10 and exit 9.999... On the number highway. 
#39




Quote:
Now, 1/10 = .1, agreed? And 1/100 = .01 = .1^{2}, agreed? So .99999... = 9/10 + 9/100 + 9/1000... = 9 * .1 + 9 * .01 + 9 * .001... = 9 * .1 + 9 * .1^{2} + 9 * .1^{3}... = Code:
∞ Σ { .9 * .1^{n} } n=0 Code:
∞ Σ { a * b^{n} } n=0 a/(1b) So .9/(1.1) = .9/.9 = 1. 


#40




19th century mathematicians, especially Dedekind and Weierstrass, defined real numbers and then proved statements about limits. Those suggesting OP is ignorant might overlook that things were less clear before 19thcentury rigorous definitions.
In particular, as Indistinguishable points out, there are number systems where 1 and (1  ɛ) are distinct, where ɛ is infinitesimal. It happens that this is not the system formalized by Dedekind and Weierstrass. Quote:
I still think that best is to simply postulate The Axiom of Archimedes. If it was good enough for the greatest genius who ever lived, then it should be good enough for us. (ETA: Indistinguishable writes "by definition" so we arrive at the same result, the difference being that between "Do as you're told" and "Here's why.") Last edited by septimus; 08042012 at 03:23 AM. 
#41




While everything you say is true, I don't think the OP is making that point. That is a very advanced point to make. If somebody said "well, it's trivial to define a space where these numbers are distinct" most people who have ever done higher level math will probably nod and say "ayup."
I don't think it's fair to disparage the OP, but we HAVE been using the aforementioned rigorously defined real numbers as the de facto standard for a while now. I mean, I guess it's possible he's making the very advanced mistake of misapplying late 18th century mathematics to modern number systems, but I think it's more likely that he's just misunderstanding the modern common real number system with no middleman involved. Last edited by Jragon; 08042012 at 03:26 AM. 
#42




Many thanks to Indistinguishable and Senegoid for well thought out responses.
Re: "Similarly, when a mathematician says "0.9999...", what they mean, by that same definition, is "The number which is >= 0, and also >= 0.9, and also >= 0.99, and also >= 0.999, and so on, AND also <= 1, and also <= 1.0, and also <= 1.00, and also <= 1.000, and so on." What number satisfies all these properties? 1 satisfies all these properties. Thus, when a mathematician says "0.9999...", what they mean, by definition, is 1" Pretty stupid notation, why not just say "1". Seriously is the some Mathematical Authority which has proclaimed this definition? What if I just want to talk about the geometric series 9/(10^x ) for x = 1 to Infinity and I do not want any assumptions that say well intuitively if we get arbitrarily close to some number we'll just call it that number. I'm not to say that this is not reasonable to make such assumptions, but they are assumptions not proofs. What is the smallest number less than 1? If I say it is .999... as defined by the geometric series geometric series 9/(10^x ) for x = 1, what proof can be offered I am wrong with out the assumptions made by a limit process? I don't know why any one would find that mathematical trickery like saying 10 x .999... is 9.999... where one must add this mysterious 9/infinity to the end is comforting. I would argue it is the opposite, a trick is something to fool you into believing something other than the actual reality. I will repeat once more that when you use the Limit process you are making an assumption that x is close enough to n, that we will just call it n. Perhaps it is n, but it is not a proof is it? I don't need a proof that 1 1 = 0. I can accept that a given. But to me it seems many have taken the result of a limits as a mathematical practicality to the conclusion that they are in fact some undeniable truth. Okay so you can say by the axioms of the real number system, .999.... must be 1. Fine but the real number system is not perfect is it? Does it describe (1)^(1/2)? Does it describe 1/infinity? I can accept that by the rules proclaimed by the axiom of Real Numbers or whatever .999... must be 1. But can some one tell me what 1/infinity is or why it is not logical to assume 1  .999... would be 1/infinity? 
#43




The real number system IS perfect. In fact, so is this number system I just made up, comprised of the set {Dog,Cat,Fred} where Dog+Cat=Fred and FredCat=Fred are the only valid forms of manipulating the set.
We get results that work in the real world with the Real Number system. Trying to philosophically suss out why 1.999... =/= 1/infinity or what the square root of 1 is in a system that doesn't deal with it is madness. There's a rather recent famous proof that you can take a sphere and, doing a bunch of slices and rotations, clone it into five spheres, completely identical (even in size and volume!) to the first. It's not useful, but it's kind of cool. It's the same with trying to do what you're doing. If you WANT to define a number system where 1.999...=Fish, by all means. If you can do math that makes science or real world applications easier, then definitely go for it! The mistake you're making is that you're asking a question that basically has no meaning. It's the math version of "Does a dog have a Buddha nature?" No, in the real numbers sqrt(1) doesn't mean anything, that doesn't mean it's imperfect, it just means that we don't particularly care about the result. There's no point trying to criticize it. The point is, yes, mathematical systems are rather arbitrarily defined to have certain properties. Yes, some of them lead to counterintuitive results like .999... = 1 or don't cover all cases like deciding sqrt(1) doesn't make sense. However, despite their arbitrariness we get RESULTS with them. Like cell phones. And the internet. If you want to invent a system that deals with all the crap that doesn't make intuitive sense, and can still help us make cell phones, great! If not, then you have to accept that given standard conventions, 1/infinity has no meaning and it's wrong to say 1  .999... = 1/infinity because the standards and conventions of the number system say so. And that same convention invented cell phones! And the internet! Even if you find it hard to swallow, you can't really argue with what it produces. Last edited by Jragon; 08042012 at 03:41 AM. 
#44




Quote:
Your proof of the convergence uses limits does it not? Does it not at its foundation say well "it gets as close as we want it to get and therefore we will assume by time we get to infinity that the difference is 1/infinity and let's just call that difference zero?" Does it not do that? If except limits only as a practicality for say calculating the amount of fuel needed for a rocket is only going to be off by 1/infinity, good enough. But if your looking to prove the universe has no dirt under it's fingernails... maybe you need to do better? 


#45




Quote:
Again, if you want to define a number system that makes us cell phones that makes all the weird counterintuitive stuff disappear, I'm sure we'll all thank you. But pretending the real number system should be defined as something other than it is, or insisting it's "hiding" properties it doesn't have is a bit fruitless. 
#46




Quote:
Why say 4/6 when we can say 2/3? Numbers can have lots of different names. Assuming you meant "the largest number less than 1", the answer is that there isn't one. That is a fundamental difference between the reals and the integers. 
#47




Earlier, I gave meaning to .9999.... by defining it as a limit, and suggested that it's best if you can prove your alleged limit  But I didn't define what a limit is or how you compute the actual value of one, or prove anything about. I'm not sure if anyone else in this thread has, either.
If any of us actually felt it necessary to do that, we'd just by typing our Calculus I textbooks into this thread. Can we assume that all the parties to this discussion at least know those basics of Calculus? I'm not getting the impression that erik150x does. And I'm not seeing that anyone else has tried much to explain that, as that is exactly what I would mean by typing in our Calculus I textbooks. And no, I'm not going to do that here. I'll just point out the problem  more clearly, I hope, than has been done so far. Forget all that horsemanure about 1/infinity and stuff like that. At the end of that path lies madness, not meaningful results. And skip all the nonsense about finding the From the days that Newton and Leibniz developed Calculus until sometime in the 19th century, nobody knew what limits were or how they worked, and mathematicians couldn't do much formal proofs or logic dealing with them. They just had a bunch of formulas for derivatives and integrals that seemed to work, and they built up Calculus from there. A formal development of Calculus was not possible until somebody (sorry, I'm not enough into math history to know the names) came up with the epsilondelta definition of a limit, and the corresponding epsilondelta method of proving theorems about them. In order for this entire thread to make any sense, everybody needs to wrap their minds around the epsilondelta definition of a limit. There's no infinities involved. There's no 1/infinities involved. There's nothing about finding just the right number that's infinitesimally close to another number. Before you can deal with any abstractions like those, you have to know how to work with limits first. Last edited by Senegoid; 08042012 at 04:13 AM. 
#48




You can reprise another thread that did the issue to death http://boards.straightdope.com/sdmb/...d.php?t=545282
My posting was this: I'll repeat (with a few edits to improve it) it since I think it is hard to dispute. Another way of looking at the issue, is to go back to the question: why do these infinite length representations exist anyway? The answer lies in the unique prime factorisation theorem. Sometimes known as the fundamental theory of arithmetic. Simply put, for any natural number you can select, there is only one set of factors of that number that are all prime. So 20 = 2*2*5 42 = 3*3*7 and most importantly for this discussion, 10 = 2*5 and 3 = 3. It is never possible to come up with an alternate set of prime factors for any natural number. If you want to express a fraction (or rational number) in decimal form one of two cases are possible. Either the denominator will have prime factors which are only 2s and/or 5s, or there will be some other prime in the denominator's factors. If the only prime factors are 2 and 5, you can represent the number as a terminating sequence in decimal notation. Any other number than 2 and 5 and the representation is not terminating. Any. So, 1/4 = 1/(2*2) = 0.25 Note that this is exact. 6/32 = (3*3)/(2*2*2*2*2) = 0.1875 and again, this is exact. But, say, 4/7 = 2*2/7 = 0.571428571428571428571428571428.... and note the recurring sequence 571428. Another thread pointed out the notation 0.(571428) to denote this. (When I was at school we used a superscript dot.) The parentheses representation is much easier to read, so I'll stick with that. Now, note that 2 and 5 are important only because they are the factors of the base we are using for the representation (decimal). We could easily represent the number in a different base. In base 2, aka binary notation, we only have one prime factor  2. The only fractions that can be represented in a terminating form are those where the denominator is a power of 2. So, in base 2: 1/4_{10} = 1/(2*2)_{10} = 0.01_{2} Exactly. 1/8_{10} = 1/(2*2*2)_{10} = 0.001_{2}, exactly and 3/8_{10} = 3/(2*2*2)_{10} = 0.011_{2}. also exactly. But 1/5_{10} is 0.001100110011001100110011001100110011001100110011001101.... or 0.(0011) in base 2. Which is only exact if you sum the infinite series: 1/8 + 1/16 + 1/128 + 1/256 + 1/2048 + 1/4096 ....... Yet in base 10, 1/5 was terminating and thus exact without recourse to a series. How about trying base 3 then? 1/3_{10} = 0.1_{3} Exactly. 2/3_{10} = 0.2_{3} Also exactly. 1/3_{10} + 2/3_{10} = 0.1_{3} + 0.2_{3} = 1 Exactly. 1/2_{10} in base 3 doesn't terminate  since 3 has only one prime factor: 3. So, 1/2_{10} in base 3 is 0.111111111111111111111111111111111....... = 0.(1) Now how about that!!! Guess what is going to happen? 0.111111111111111111111111111111111...._{3} + 0.111111111111111111111111111111111...._{3} = 0.(1)_{3} + 0.(1)_{3} = 0.(2)_{3} = 1/2 + 1/2 = 0.2222222222222222222222222......_{3} = 1 in base 3. Look familiar? It is also worth noting that in base 2: 0.111111111... = 0.(1) = 1. Which is identical to the expression for the sum of the infinite series 1/2 + 1/4 + 1/8 ...... In base 10 1/3 = 0.33333... = 0.(3)_{10} 2/3 = 0.66666... = 0.(6)_{10} 1/3 + 2/3 = 0.(3)_{10} + 0.(6)_{10} = 0.(9)_{10} So 1/3 + 2/3 = What? Which brings us full circle. 0.9999999..... = 0.(9) is not a process, it does not exist in any definition that uses a finite number of terms. It is a notation for a sum of an infinite series. No different to any of the other infinite series above. If you mess about changing bases you can make any fraction drop in and out of having a terminating representation. You didn't change a notation into a process or back again. (Not unless you define 1.(0) as a process too. Which means you define all numbers as an infinite process. Which we don't. Or to put in the obverse: 0.999999... is just as much a process as 1.00000....  neither are.) 
#49




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(Numbers don't stop at quaternions, necessarily, but after that I find mathematicians tend to refer to them more as matrices or vectors than as 'numbers' as such.) There are even numbers that represent concepts you can't really apply to cocoa beans directly, and while most people think those ideas don't amount to a hill of beans, they can still be quite interesting. My point is, one, ignore the posts I made in this thread back in 2000, and, two, the concept of 'number' is complex enough to extend in many dimensions well beyond the reals you're used to. (Also, the naming scheme we have is idiotic. The 'reals' are no more real than any rotation matrix, the 'imaginary' numbers no more imaginary than any integer, and the 'complex' numbers no more complicated than watching a compass spin around.) Last edited by Derleth; 08042012 at 04:22 AM. 


#50




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