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BS: Who else plays little # games?
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 19 Sep 13 - 11:44 AM Maybe it's a matter of notation although I think O.111... is a recognised abreviation for 0.1 recurring. Would O.(1) be better? I don't think it's an approximation. Pi written out to a trillion places of decimals would be an approximation but 0.(1) doesn't mean 0.111... written out to a trillion places of decimals, it means written out to an infinite number of decimal places. Infinity doesn't just mean a very big number. As far as I can see, 1/9 is exactly equal to 0.(1). Just because you can't write it out in full doesn't mean it isn't true. Others can argue this better than me - http://en.wikipedia.org/wiki/0.999%E2%80%A6 |
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Subject: RE: BS: Who else plays little # games? From: Nigel Parsons Date: 16 Sep 13 - 12:52 PM From: TheSnail - PM I'm a bit puzzled, though, by "until you have calculated all other transactions". What other transactions? The process of dividing by 9 leaving remainder 1 isn't going to change. The 'other transactions' are that you started by showing 1/9= 0.111.. (which is only an approximation) and then showed: 9 * 1/9 = 9 * 0.1111.... 1 = 0.9999.... It is in this second stage that you have overcomplicated things by replacing 1/9 by a close approximation of 1/9. 9* 1/9 = 9/1 * 1/9 = 9/9 =1 All you have shown is that an approximation of one ninth is very close to the value of One. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 09 Sep 13 - 06:39 AM Come now, Michael, don't underestimate yourself. As has been said, the maths involved is school level, the sort of stuff you were doing 65 years and more ago. The maths is just used to express ideas in a form that can then be manipulated mathematically. It could all be done in plain English but at considerably more length. Go on, give it a try. You might find you enjoy it. |
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Subject: RE: BS: Who else plays little # games? From: MGM·Lion Date: 09 Sep 13 - 06:03 AM Eeeeeeeeeeeeeeekkkk! I've opened Pandora's Box. Honest, Zeus, I never meant ~~~~~ Aaaaarrrggggghhhhh. It's off-down-the-garden-to-eat-worms-time again... |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 09 Sep 13 - 05:04 AM MichaelTheGM, may I remind you of your OP Do any other squares do that? And does anyone else find such things of as much interest ...You consider those who try to answer your questions "hijacking spoilsports"? |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 09 Sep 13 - 05:00 AM Come off it MtheGM. Do you seriuosly expect me to believe that you didn't realise that your question would unleash a hoard of geeks, nerds and anoraks? (Well, four or five of us.) You asked the question. Stop trying to evade your responsibility. IT'S ALL YOUR FAULT. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 09 Sep 13 - 04:35 AM DMcG has answered better than I could but, as I said, I think my "proof" works well enough in the context of little # games. I'm a bit puzzled, though, by "until you have calculated all other transactions". What other transactions? The process of dividing by 9 leaving remainder 1 isn't going to change. |
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Subject: RE: BS: Who else plays little # games? From: DMcG Date: 09 Sep 13 - 02:47 AM You are right, Nigel, but as I said earlier, to convert the method into a mathematical proof needs a few more powerful tools that would obscure the heart of idea, which is easy to grasp for a non-mathematician. This is essentially the same problem that crops up again and again in science and mathematics (and, indeed history and many other subjects): to teach something you often have to simplify it so much that you know is not strictly true. Alternatively, you can stick to truth (or I should say our best current understanding), and present something that is so convoluted and complex it is, quite literally, incomprehensible. You may remember that there was a TV show a looooong time ago called 'That was the week that was'. The intro varied each week, but on one occasion it contained the couplet "One eye open wide, one eye closed. And between the two the picture gets composed'. I've often thought that is one of the wisest remarks I've ever heard from a television program. And it applies here. |
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Subject: RE: BS: Who else plays little # games? From: Nigel Parsons Date: 08 Sep 13 - 08:57 PM Snail: Grishka, I would have thought that my method was more accessible to other readers than the epsilon stuff. I reproduce it here - 1/9 = 0.1111.... 9 * 1/9 = 9 * 0.1111.... 1 = 0.9999.... I may have been a bit sloppy with the notation for recurring. It could have been - 1/9 = 0.(1) 9 * 1/9 = 9 * 0.(1) 1 = 0.(9) but I think people knew what I meant. I'm still unclear as to what is specifically dodgy about it. Your problem is in the first line: 1/9 = 0.1111.... (Assuming the multiple dots mean 'recurring') NO 1/9 approximates to 0.111(recurring). No matter how many decimal places you add, you will never get a valid value of 1/9 in decimal notation. 1/9=3/27, or 4/36 etc. To work with 1/9 mathematically, you do not replace it with its decimal approximation until you have calculated all other transactions. To do so is to bring in inaccuracies which may be multiplied by other transactions. |
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Subject: RE: BS: Who else plays little # games? From: MGM·Lion Date: 08 Sep 13 - 04:20 PM No, I didn't. Don't tell me what "I must have known", please. I asked if anyone else played these little games. For all Snail's thinking, three threads back, that these great big suge ginormous non-games are "what this thread is about"-- It's my thread. And it isn't. So yah-sucks-boo! to all you hijacking spoilsports & party-poopers. Likewise Chiz-Chiz! And great big botty-boo-bums! ɷɷɷɷɷɷɷɷɷɷɷ |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 08 Sep 13 - 04:08 PM Grishka, nothing you have posted so far would have suggested that English wasn't your first language. if this were serious rather than a "game", I would have been more cautious and meticulous). Sorry, don't get that. Whether it's serious or a game, I can't see the point of doing it if you aren't going to do your best to get it right. I'm sorry if we'v scared off the other readers but MtheGM must have known that he was setting a chalenge. If he doesn't like the answer, he has only himself to blame. Maybe you or I should try and explain your friends proof in plain English. "10n-1 ≤ x²" doesn't make a lot of immediate sense to someone who isn't used to mathematical notation. |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 05 Sep 13 - 01:53 PM You know the answer, Snail, and you have read that my friend left the details to me (which well he could; if this were serious rather than a "game", I would have been more cautious and meticulous). Remains the question whether we enjoy such "games". I do to some degree, but only once in a while. This one will last for the rest of this year. In fact I must confess generally that a part of my personality enjoys discussing details, whereas the larger part strongly objects to the negligence of more important problems. I cannot really complain about lack of genuine challenges for my mental muscles; posting to Mudcat may amount to escapism altogether. One of my excuses is that it is a good training for my proficiency in English. Some training in mathematics and logic has a great practical value as well: it helps us to be not so quickly convinced by apparent "common sense", neither by others nor by ourselves. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 05 Sep 13 - 01:12 PM Snail, I hope you are just trying to tease me. Not at all. Just flexing my creaking mental muscles on an interesting problem. I thought that was what this thread was about. If you didn't want your friends "proof" discussed, why did you post it? Hint: if the inequality holds for some n (> 1), it will also hold for n+1... Yes, the assertion is true for all values of n greater than or equal to 5. No problem. n can in fact be either 1, or 2, or 3, or 4, by that argument. This leaves us with checking all values of x from 1 to 99 - a "finite" task. That falls short of your friends claim he had a proof of MtheGM's conjecture that 1 and 81 were the only numbers with this property. It does not rule out the possibility that there could be 3 and 4 digit numbers that share the porperty. OK, there aren't but that isn't the point. Have you got back to your friend about this? (All right, I might be teasing a bit.) |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 05 Sep 13 - 11:25 AM Instead of "nonnegative" read "positive" - pedants! |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 05 Sep 13 - 11:16 AM Snail, I hope you are just trying to tease me. I'll join the game one more time, though I fear that readers are already bored. Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned. Sorry, not proved by your friends argument. As I wrote before, it was my error; n can in fact be either 1, or 2, or 3, or 4, by that argument. This leaves us with checking all values of x from 1 to 99 - a "finite" task. ("The details are left to the reader.") There's a challenge. (I only claimed to have "a bit more than school maths".) It does not seem too difficult to me, just boring. Hint: if the inequality holds for some n (> 1), it will also hold for n+1, since the left side is nonnegative and increases by a factor 10, whereas the right side only increases by a factor (1 + 2/n + 1/n²), which is less than 4. By induction, the assertion is proved. Wrong again? Requiring non-school maths? |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 05 Sep 13 - 09:58 AM ("The details are left to the reader.") There's a challenge. Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned. Sorry, not proved by your friends argument. (I only claimed to have "a bit more than school maths".) |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 05 Sep 13 - 07:53 AM Snail, you are the one who claims maths proficiency. n=5 works, the cases of n up to 4 can be tested manually or by a little computer script. My friend was not too specific in his email. Little # games? Questions of the type "is this true for all integers?" are often easy to ask and difficult to answer with a proof. This one turned out to be of medium difficulty, others like Fermat's theorem took centuries to be answered by professional mathematical researchers. Football is a little ball game, but we might not enjoy playing it against Bayern München or Real Madrid if we seriously want to win. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 05 Sep 13 - 06:02 AM You mean "10n-1 ≤ x²"? Fine, but now if n=3... |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 04 Sep 13 - 12:58 PM Snail, you are right, it must be n-1 (all my own error when reformulating and trying to simplify). You can fix the proof though, I am sure, and the assertion stays the same. |
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Subject: RE: BS: Who else plays little # games? From: DMcG Date: 04 Sep 13 - 10:55 AM Yes, it was I. My phone doesn't like holding cookies so I often forget to check whether it still remembers who I am. Going back to the original question: I haven't spent the ten minutes investigating this because it is a much harder problem than it first appears. The difficulty is that it is a mixture of a properties about numbers and properties about notation. In particular when you get a carry' it becomes really difficult to express 'the sum of the digits'. On the other hand, the 81 is part of a more obsure set that is quite interesting: suppose you write a number in base N rather than base 10. Then, for every N there is a number that has the same property of the sum of the digits matching the square root. So for example 41 in base 6 is 25 in base ten, which is the square of 4+1. This doesn't need more than school algebra to work out what the number is, but even so it might take a few minutes. The Eureka! Moment when you see how to solve it might make the exercise worth it. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 04 Sep 13 - 09:17 AM Anonymous GUEST (is that you, DMcG?) But nothing fundamentally wrong, no. O, good. I think it fits well with little # games His proof (sketched): assume the number to have n decimal places, and x being its square root (both positive integers), then 10n ≤ x². I've got a bit more than school maths but you've lost me already. |
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Subject: RE: BS: Who else plays little # games? From: MGM·Lion Date: 04 Sep 13 - 07:33 AM _All school maths, nothing esoteric. _ Oh goody. How consoling... |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 04 Sep 13 - 07:13 AM My friend has taken the challenge of the OP and asserts that 1 and 81 are the only positive integers with that property. (Admitting 0 makes for an additional solution.) His proof (sketched): assume the number to have n decimal places, and x being its square root (both positive integers), then 10n ≤ x². Since x is the sum of those decimal digits, x ≤ 9n. Everything being positive, we can square this to get x² ≤ 81n². Together, we get 10n ≤ 81n². This is not true for n=3 (1000 being greater than 729), and it cannot be true for any higher n, roughly because the left side grows exponentially. ("The details are left to the reader.") Thus n can only be either 1 or 2, easily seen to yield no more than the two solutions mentioned. All school maths, nothing esoteric. |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 03 Sep 13 - 06:29 PM That anonymous guest was not I, but I agree. |
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Subject: RE: BS: Who else plays little # games? From: MGM·Lion Date: 03 Sep 13 - 05:44 PM DMcG -- Indeed. But I can keep up with Tolstoy as well as Lear -- whereas... Ah well... ~M~ |
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Subject: RE: BS: Who else plays little # games? From: GUEST Date: 03 Sep 13 - 05:36 PM Not dodgy; just to make a formal mathematical proof out of it you have to demonstrate that the power series that the notation is shorthand for is well behaved and that would take a few more lines and need a few more powerful tools. But nothing fundamentally wrong, no. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 03 Sep 13 - 05:20 PM Grishka, I would have thought that my method was more accessible to other readers than the epsilon stuff. I reproduce it here - 1/9 = 0.1111.... 9 * 1/9 = 9 * 0.1111.... 1 = 0.9999.... I may have been a bit sloppy with the notation for recurring. It could have been - 1/9 = 0.(1) 9 * 1/9 = 9 * 0.(1) 1 = 0.(9) but I think people knew what I meant. I'm still unclear as to what is specfically dodgy about it. |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 03 Sep 13 - 03:22 PM DMcG and TheSnail, I do not think it is of interest to other readers. (If you are interested privately, a hint, courtesy of my high school teacher: the epsilon thing is about sequences or series. See Wikipedia, also at Decimal.) |
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Subject: RE: BS: Who else plays little # games? From: DMcG Date: 03 Sep 13 - 02:50 PM It works as a demonstration fine. The problem from the point of view of pure mathematics is that it is an infinite process. Now, when you represent a number like, oh, 0.7687326 to one less decimal place the convention normally used is that you round the penultimate digit up if it is 5 or more, and down if it less than five (at least, that's the normal convention - there are weirdnesses like banker's rounding, but we will ignore that!), this would be0.768733. Mow, if we have 0.99999999... to any *finite* number of places, and you wish to stop writing the places out, then the rounding rules take you to 1.00000000.... which does equal 1, but not in an interesting way. Conversely, if you use the other common convention and truncate, you have 1=0.9999 (say), which isn't true. Either way, you don't really fully express the concept. And if you want to 'go on forever' you are on treacherous ground: infinite series are vicious beasts. By way of illustration, and for a spot of light-hearted maths (!!!!), think about the series S = 1-1+1-1+1... What does it add up to? Well, lets bracket things this way S = (1-1) + (1-1) + (1-1) ... ===> S = 0 But noting that a-b is the same as a+(-b), a different bracketing gives S = 1 + (-1+1) + (-1+1) + (-1+1) ... ===> S = 1 Or even S = 1 - (1-1+1-1+1-1..) ... ===> S = 1 - S ===> S = 0.5 It's problems like that which made mathematicians very wary of infinite series and the need to treat 'em with kid gloves. |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 03 Sep 13 - 02:28 PM I wouldn't mind a bit of feedback on my contribution of 31 Aug 13 - 05:54 AM. Does that work as a proof and if not, why not. No axe to grind; I'm here to learn. |
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Subject: RE: BS: Who else plays little # games? From: DMcG Date: 03 Sep 13 - 01:25 PM Grisha - I reckon we should leave this one, as we risk completely alienating the MtheGM's of this world. I'll just say I used the term "disproof" not be way of an accurate definition, but because that's how it had been labelled earlier. I've reread what I wrote and stand by it as long as we include the caveat I gave in my 03 Sep 13 - 09:09 post (epsilon bigger than zero). |
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Subject: RE: BS: Who else plays little # games? From: gnu Date: 03 Sep 13 - 12:38 PM Laff and learn. This guy is gooood. |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 03 Sep 13 - 12:30 PM DMcG, sometimes I enjoy playing the temporary pedant myself, but not this time. Mysha (who is male, as my private directory of Mudcatters tells me) did not provide a disproof - how could he - but a "disproof", i.e. as precise as your "proof", and therefore pointing at that lack of precision. No need though to mend it, though. It is basically high school maths, not requiring higher academic studies. Still, regarding Airymouse of 31 Aug 13 - 10:41 AM, it is noteworthy (though not "puzzling") that all numbers that have a finite decimal representation also have another, infinite one (ending with an infinitely many 9s), whereas all other numbers only have one representation. In contrast, MtheGM's original question, which properly reads "Are there infinitely many numbers with that property, and if not, which ones or how many or which is the largest one?", does not look easy to me at all. A friend of mine who minored in maths gave up after ten minutes. I would not bet that every PhD mathematician performs better. |
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Subject: RE: BS: Who else plays little # games? From: GUEST,DMcG Date: 03 Sep 13 - 11:48 AM Oh, I hope not, MtheGM. There's a good case for all sorts of number games just for the fun of it, and many a deep theory is inspired by a simple game, exactly is it is in science or, analoguously, literature. Lear has his place, not just Tolstoy. Not everything needs the PhD, whether maths, english or music. "There is beauty in the bellow of the blast, There is grandeur in the growling of the gale ..." |
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Subject: RE: BS: Who else plays little # games? From: Keith A of Hertford Date: 03 Sep 13 - 10:28 AM Only our generation Mr Happy. The formula is now provided in GCSE papers. |
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Subject: RE: BS: Who else plays little # games? From: MGM·Lion Date: 03 Sep 13 - 09:58 AM This has now turned into BIG number games. The numerate have hijacked the thread, dammit, leaving all us poor old games-playing hammer-chooers out in the cold! Chiz-chiz! ~M~ |
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Subject: RE: BS: Who else plays little # games? From: GUEST,DMcG Date: 03 Sep 13 - 09:09 AM I should add to that, of course, that any attempt to translate maths into ordinary English risks ambiguity or omission. I did not say, though it was my intention, that epsilon should be bigger than zero, as that is a well understood convention is science generally. Clearly, an epsilon of -1000 would make a lot of numbers equal that we did not intend. If you want to include that as an example of sloppiness, feel free. While we are at it, I also assumed scalar quantities, not, for example, complex numbers. All such things would have been explicit had I used formal notation not English, but it would have limited the readership even more than tedious pedantry (and is painful to enter as html) |
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Subject: RE: BS: Who else plays little # games? From: GUEST Date: 03 Sep 13 - 08:54 AM Not a disproof or a revelation of my sloppiness either, Grisha, though I admit to quite a bit of sloppiness in my postings in general, and indeed put in a correction to the definition of equality on this thread precisely because of a sloppiness on my first attempt. It is difficult to avoid pendantry in maths precisely because being exact is important, so I do plead guilty with mitigating circumstances on that one. Rapparee pointed out with some subtlety why Misha's disproof wasn't - think about his post a little more |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Grishka Date: 03 Sep 13 - 07:33 AM I found Mysha's post quite to the point. Her or his first assertion gives one more solution to the OP's question, and the other one points to some sloppiness in DMcG's writing. Hypothesis: (1 - 0.9*) is smaller than any arbitrary epsilon. Mysha's "disproof": (1 - 0.9*)/2 for epsilon is even smaller. Pedants, however temporary, should take particular care to be exact and correct themselves. |
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Subject: RE: BS: Who else plays little # games? From: GUEST,DMcG Date: 02 Sep 13 - 02:18 PM That was why, Mysha, I had to change the phrase 'any epsilon' (which is ambigous in English, to 'every epsilon'. Having a single epsilon can't on its own, show equality (though a single could disprove equality) I don't go on like this in real life, honest! |
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Subject: RE: BS: Who else plays little # games? From: Rapparee Date: 02 Sep 13 - 12:18 PM Mysha, you'll softly and silently vanish away. Another thing to mess your mind up with is: If 1+1=2, then the sum of any number added to itself is the next number in the sequence: 2+2=3, 3+3=4, etc. This would mean that 1+2=1.5. Try not to get your pay calculated this way. |
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Subject: RE: BS: Who else plays little # games? From: Mysha Date: 02 Sep 13 - 12:37 AM Hi, Well, on the one hand: 1*1= 1; adding all the digits of 1 gets you 1. Yet, on the other hand: What if I choose (1 - 0.9*)/2 for epsilon? Bye Mysha |
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Subject: RE: BS: Who else plays little # games? From: TheSnail Date: 01 Sep 13 - 04:20 AM With you now Airymouse. I hadn't realised you were dealing with the six digits in isolation. If you were still looking at the recurring decimal, the 1 you add to the 6 in 1 14285 6 would be the carry from the next set of recurring digits 1.142856 + 0.000001142856 + 0.000000000001142856 ... = 1.142857142857142857... |
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Subject: RE: BS: Who else plays little # games? From: GUEST,DMcG Date: 31 Aug 13 - 03:31 PM Sorry, that was me above and I got cut off mid sentence. The mistake is getting too hung up on the method of getting the answer rather than the answer itself. That way, instead of concluding that the terminating and recurring numbers are in some sense distinct, you start wondering if the problem lies in the method of adding the digits rather than the numbers themselves. Obviously enough, mathematicians don't handle infinite series by adding all the terms together explicitly since by definition that is an infinite process. Exactly how the do it varies but a common way is to prove the sum of the series must be X or more and simultaneously X or less. Hence you can work out the sum without actually doing the calculation. And to get back onto representation again: there's no need to stick to a base-10 notation. If you use a base 7, for example, a number like 2/7 that is recurring in base 10 'decimals' is not in base 7 'septimals'. So a suggestion there are distinct types of number depending on whether they recur of not is clearly an artifact of the chosen notation, not of the number |
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Subject: RE: BS: Who else plays little # games? From: GUEST Date: 31 Aug 13 - 02:24 PM I can't answer that one, Mr Happy, beyond saying it is amazing how much mental clutter we pick up along the way. Airymouses' comment is more subtle. It took mathematicians literally centuries to get a firm grip on infinite series and thinking of a decimal representation of a number as such a series is valid. But it is a mistake to think of the mechanical process of adding digits and hence conclude recurring decimal numbers are in some way in a different class to those we can add up in a finite number of stepps |
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Subject: RE: BS: Who else plays little # games? From: Mr Happy Date: 31 Aug 13 - 11:59 AM .............but why do I always remember x= -b +/- sq.root.b squared - 4ac 2a ?? When I've never ever had to use it in everyday life?? |
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Subject: RE: BS: Who else plays little # games? From: GUEST,Guest from Sanity Date: 31 Aug 13 - 11:04 AM Rapparee: "License plates are something that drives me nuts. For example, I'll see one with a number like, oh, 4679. Right off I think..." A few years back my wife and I had license plates on each of our cars....Mine: OI12EU ..Hers: OI12EU2. People who 'got it' thought it was pretty cool! Regards, Rap.. GfS |
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Subject: RE: BS: Who else plays little # games? From: Airymouse Date: 31 Aug 13 - 10:41 AM Snail. Sorry about my muddling. I meant 8 times 142857, the product to two whole numbers. The product is 1 14285 6. I isolated the first and last digits of this product, because they are the digits you have to add to get "142857". For some psychological reason that I don't understand, people seem happy with 1/3= .333333... and 2/3= .6666666... but when you add 1/3 + 2/3 they are puzzled by 1=.9999.... Of course infinite decimals really are something different from ordinary arithmetic. You can add only two numbers at a time, so if Lewis Carroll asks what's 1 and 1 and 1 and 1, Alice should be asking herself is this ((1+1)+1) +1) or is it (1+ (1+1)) +1 or is it .... But we all know that no matter which of the possibilities Lewis has in mind the answer is 4. Putting this quibble aside finite decimals make sense .385 means 3/10 + 8/100 + 5/1000. Since you can add only two numbers at a time, no matter what order you choose you will never be able to add up the infinite list 9/10 + 9/100 + 9/1000 etc. Thus, infinite decimals are a different animal than finite decimals, which are just the finite sum of fractions. |
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Subject: RE: BS: Who else plays little # games? From: Rapparee Date: 31 Aug 13 - 10:26 AM Spaw, teaching (from a transcript): Okay, you dipshits. Lissen up. You take the number 9, see? Only make it with a straight tail. Flop it over and you got either a 6 or a lower-case B -- that's a little B to you shitheads. Lay is down on the round part and it looks like that thing you see in your camera, right? Lay it down on the flat part and it looks like Whatsisname over there in the corner, only without the peter to play with like he's doin' now. Learn this, because it's important. Why is it so fuckin' important? Because I said so, you little.... It goes on like that for every single number up to a googleplex. |
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Subject: RE: BS: Who else plays little # games? From: catspaw49 Date: 31 Aug 13 - 09:52 AM When I taught school I did a math segment in AutoTech to my usually non interested teen males. Most were very poor in math and had learned to hate it. I started out with number games and especially a few nine games which always fascinated them, caught their interest, and at least made them more easily taught. You have to be more entertaining than MTV.......... Spaw |
