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%3 N_20180202212840_CDW    By CDW 2018/02/02 @ 21:28:40 -------------------------------- Yes. In base 2 we say that zero point 1 recurring is equal to 1, and in base 8 we say that zero point 7 recurring is equal to 1, and so on.        (select only this node)     N_20180201174515_CDW    By CDW 2018/02/01 @ 17:45:15 -------------------------------- This discussion is intended to explain why in modern mathematics we say that "zero point nine recurring" (ZP9R) is said to have the value 1.        (select only this node)     N_20180201175120_CDW    By CDW 2018/02/01 @ 17:51:20 -------------------------------- So the first question is: Why do we think ZP9R should have any meaning at all?        (select only this node)     N_20180201174515_CDW->N_20180201175120_CDW N_20180202213433_CDW    By CDW 2018/02/02 @ 21:34:33 -------------------------------- Consider: if we compute 1/3 as a decimal then we get zero point three recurring. We can write that as ZP3R.  That clearly has no end, so it makes sense to say that the syntactic construction ZP3R has a meaning, and has value 1/3.        (select only this node)     N_20180202213517_CDW    By CDW 2018/02/02 @ 21:35:17 -------------------------------- If we compute 2/9 as a decimal then we get zero point two recurring. That clearly has no end, so it makes sense to say that the syntactic construction ZP2R has a meaning, and has value 2/9.        (select only this node)     N_20180202213433_CDW->N_20180202213517_CDW N_20180201175211_CDW    By CDW 2018/02/01 @ 17:52:11 -------------------------------- I'm going to ignore that question because people seem to think that it should have a meaning, and it should have a value.  Feel free to reply to the node above if you want to explore this question.        (select only this node)     N_20180201175120_CDW->N_20180201175211_CDW N_20180201175120_CDW->N_20180202213433_CDW N_20180202213553_CDW    By CDW 2018/02/02 @ 21:35:53 -------------------------------- So ZP3R is 1/3. Let's multiply that by 3, and we get the answer 1.        (select only this node)     N_20180202213517_CDW->N_20180202213553_CDW N_20180201175425_CDW    By CDW 2018/02/01 @ 17:54:25 -------------------------------- The next question is: If it have a "value", what value should it be?        (select only this node)     N_20180201175211_CDW->N_20180201175425_CDW N_20180202213632_CDW    By CDW 2018/02/02 @ 21:36:32 -------------------------------- But if we write out ZP3R we have 0.3333333333333333... and if we multiply that by 3 we get 0.999999999999... which is ZP9R.        (select only this node)     N_20180202213553_CDW->N_20180202213632_CDW N_20180201175450_CDW    By CDW 2018/02/01 @ 17:54:50 -------------------------------- If it has a value then we kind-of assume that the value will be a number of the sort we're used to.        (select only this node)     N_20180201175425_CDW->N_20180201175450_CDW N_20180202213731_CDW    By CDW 2018/02/02 @ 21:37:31 -------------------------------- So start with 1, divide by three, multiply by three, we should get 1. But if we do that with decimals we get ZP9R. Doesn't that mean that ZP9R has to equal 1?        (select only this node)     N_20180202213632_CDW->N_20180202213731_CDW N_20180202170231_Anon    By Anon 2018/02/02 @ 17:02:31 -------------------------------- This is a big assuption that I'm not clear why we need to make. Maths is full of numbers I'm not used to.        (select only this node)     N_20180201175450_CDW->N_20180202170231_Anon N_20180201175509_CDW    By CDW 2018/02/01 @ 17:55:09 -------------------------------- Yes, there are other sorts of numbers - ask if you want to know more.        (select only this node)     N_20180201175450_CDW->N_20180201175509_CDW N_20180201175544_CDW    By CDW 2018/02/01 @ 17:55:44 -------------------------------- Whatever the number will be, whatever value we ascribe to ZP9R, it's going to be bigger than 0.        (select only this node)     N_20180201175450_CDW->N_20180201175544_CDW N_20180202170359_Anon    By Anon 2018/02/02 @ 17:03:59 -------------------------------- At some point I'd like to know more but for now I'd observe that all the sorts of numbers have been created in order to deal with what mathematicians want to do, no?        (select only this node)     N_20180201175509_CDW->N_20180202170359_Anon N_20180201175603_CDW    By CDW 2018/02/01 @ 17:56:03 -------------------------------- It's going to be bigger than 0.8.        (select only this node)     N_20180201175544_CDW->N_20180201175603_CDW N_20180201175631_CDW    By CDW 2018/02/01 @ 17:56:31 -------------------------------- It's going to be bigger than 0.95, and you can probably see where I'm going with this.        (select only this node)     N_20180201175603_CDW->N_20180201175631_CDW N_20180201175718_CDW    By CDW 2018/02/01 @ 17:57:18 -------------------------------- So whatever value we give to ZP9R, it's going to be bigger than 0.9999999, which is 1 minus one-ten-millionth.        (select only this node)     N_20180201175631_CDW->N_20180201175718_CDW N_20180202171041_CDW    By CDW 2018/02/02 @ 17:10:41 -------------------------------- So whatever value we assign to ZP9R it has to be bigger than 0.999...9 where there are a specific, finite number of 9's.        (select only this node)     N_20180201175718_CDW->N_20180202171041_CDW N_20180202170049_Anon    By Anon 2018/02/02 @ 17:00:49 -------------------------------- OK so we're headed for a row of 9s that we can literally never get to the end of. Never ever. For all eternity those 9s will carry on without changing the digit before th dot to a 1. So never a 1.        (select only this node)     N_20180201175718_CDW->N_20180202170049_Anon N_20180202203628_CDW    By CDW 2018/02/02 @ 20:36:28 -------------------------------- We can show for that series that if you pick any number you like, taking enough terms of that series produces a sum that's bigger. Since the infinite sum (whatever that means) must be bigger than just an initial finite portion, the series as a whole has no sensible value.        (select only this node)     N_20180202171514_CDW    By CDW 2018/02/02 @ 17:15:14 -------------------------------- You can choose to say that ZP9R is not a number that we're accustomed to, that it doesn't live on the number line, and it doesn't represent a quantity in the same way that, say, 3/5 does.        (select only this node)     N_20180202170231_Anon->N_20180202171514_CDW N_20180202203511_CDW    By CDW 2018/02/02 @ 20:35:11 -------------------------------- An example of that is 1/2+1/3+1/4+1/5+1/6+1/7+...        (select only this node)     N_20180202203511_CDW->N_20180202203628_CDW N_20180202171649_CDW    By CDW 2018/02/02 @ 17:16:49 -------------------------------- Sometimes "numbers" have been created purely because mathematicians have been playing around and said: "Oh look, these things behave like numbers"        (select only this node)     N_20180202170359_Anon->N_20180202171649_CDW N_20180202203720_CDW    By CDW 2018/02/02 @ 20:37:20 -------------------------------- For the series 1/2+1/4+1/8+1/16+... has the property that it's strictly larger than anything less than 1, and strictly less than anything bigger than 1.        (select only this node)     N_20180202203810_CDW    By CDW 2018/02/02 @ 20:38:10 -------------------------------- As such we have two things. Firstly, at can't be anything other than 1, and secondly, assigning that infinite sum the value of 1 makes sense and never goes wrong.        (select only this node)     N_20180202203720_CDW->N_20180202203810_CDW N_20180202174033_Anon    By Anon 2018/02/02 @ 17:40:33 -------------------------------- I would propose it is a notation that expressly does not represent a quantity, despite working within calculations just as if it were 1.        (select only this node)     N_20180202181329_CDW    By CDW 2018/02/02 @ 18:13:29 -------------------------------- So if it behaves exactly as if it is 1, why is that not the same as saying that it's just another way of writing 1?        (select only this node)     N_20180202174033_Anon->N_20180202181329_CDW N_20180202205204_CDW    By CDW 2018/02/02 @ 20:52:04 -------------------------------- So are you suggesting that ZP9R is just a "thing" and it's not a number, and has no value?        (select only this node)     N_20180202174033_Anon->N_20180202205204_CDW N_20180202174202_Anon    By Anon 2018/02/02 @ 17:42:02 -------------------------------- Can you clarify what you mean there, as is it obviously NOT less than +0.8 for example.        (select only this node)     N_20180202181159_CDW    By CDW 2018/02/02 @ 18:11:59 -------------------------------- Ah, it's not ZP9R that's smaller than all positive numbers. If you declare that ZP9R is *not* equal to 1 then we can consider 1 minus ZP9R and show that it can't be zero (because ZP9R is not 1) and yet it must be smaller than every positive number.        (select only this node)     N_20180202174202_Anon->N_20180202181159_CDW N_20180202171919_CDW    By CDW 2018/02/02 @ 17:19:19 -------------------------------- So you choose how many 9's you want, write down a "0." followed by that many 9's, and whatever value we assign to ZP9R, it must be larger than what you've written down.        (select only this node)     N_20180202171041_CDW->N_20180202171919_CDW N_20180202171150_CDW    By CDW 2018/02/02 @ 17:11:50 -------------------------------- Yes, we're headed for a row of 9's that never ends, and has a "0." at the beginning.        (select only this node)     N_20180202171301_CDW    By CDW 2018/02/02 @ 17:13:01 -------------------------------- But it's simply that representation of ZP9R that doesn't have a 1 at the front. There is nothing to say that it can't still have the value 1.        (select only this node)     N_20180202171150_CDW->N_20180202171301_CDW N_20180202171409_CDW    By CDW 2018/02/02 @ 17:14:09 -------------------------------- The fraction 12/15 has the same value as the fraction 16/20, they are different representations of the same value.        (select only this node)     N_20180202171301_CDW->N_20180202171409_CDW N_20180202172817_CDW    By CDW 2018/02/02 @ 17:28:17 -------------------------------- The modern understanding in mathematics is that ZP9R is simply another representation of the value "1".        (select only this node)     N_20180202171409_CDW->N_20180202172817_CDW N_20180202171514_CDW->N_20180202174033_Anon N_20180202171557_CDW    By CDW 2018/02/02 @ 17:15:57 -------------------------------- But that then has other consequences that you might not like. It implies the existence of a positive quantity that's bigger than zero, but less than every positive number.        (select only this node)     N_20180202171514_CDW->N_20180202171557_CDW N_20180202171557_CDW->N_20180202174202_Anon N_20180202174259_Anon    By Anon 2018/02/02 @ 17:42:59 -------------------------------- Yes, that's what I thought, which I suppose is fine!        (select only this node)     N_20180202171649_CDW->N_20180202174259_Anon N_20180202172030_CDW    By CDW 2018/02/02 @ 17:20:30 -------------------------------- So we know that ZP9R is larger than 0.999999999 and all its ilk. They are lower bounds. What about upper bounds?        (select only this node)     N_20180202171919_CDW->N_20180202172030_CDW N_20180202172110_CDW    By CDW 2018/02/02 @ 17:21:10 -------------------------------- I think we can all agree that whatever value we ascribe to ZP9R, it will be smaller than 723. So 723 is an upper bound.        (select only this node)     N_20180202172030_CDW->N_20180202172110_CDW N_20180202172145_CDW    By CDW 2018/02/02 @ 17:21:45 -------------------------------- It's not a very good upper bound, though, 257 is a better upper bound.        (select only this node)     N_20180202172110_CDW->N_20180202172145_CDW N_20180202172238_CDW    By CDW 2018/02/02 @ 17:22:38 -------------------------------- And 9.7 is better still, and 3, and 2, and 1.13496. I think everyone would agree that these are all upper bounds for the possible value of ZP9R.        (select only this node)     N_20180202172145_CDW->N_20180202172238_CDW N_20180202202620_CDW    By CDW 2018/02/02 @ 20:26:20 -------------------------------- So 1.001 is an upper bound for the value of ZP9R, and 1.0001 is an upper bound, and 1.00001 is an upper bound, and so on.        (select only this node)     N_20180202172238_CDW->N_20180202202620_CDW N_20180202172855_CDW    By CDW 2018/02/02 @ 17:28:55 -------------------------------- This isn't an arbitrary choice, randomly selected. It's consistent with all the behaviour that we would want of a value represented by ZP9R.        (select only this node)     N_20180202172817_CDW->N_20180202172855_CDW N_20180202212501_CDW    By CDW 2018/02/02 @ 21:25:01 -------------------------------- If it's a number on the regular number line then you can deduce that it has to equal 1.        (select only this node)     N_20180202191359_Anon    By Anon 2018/02/02 @ 19:13:59 -------------------------------- 1. Simply because it has the zero there, which asserts that as long as the row of 9s is it's still not 1. I haven't claimed that it works in calcs just like a 1, maybe the only reason it might , could be that mathematician choose to treat it so.        (select only this node)     N_20180202191532_Anon    By Anon 2018/02/02 @ 19:15:32 -------------------------------- 2. Doesn't there NEED to be be a way of notating an infinite sequence which is infinite BY VERY VIRTUE of the fact that it never reaches that 1. That would be the function of the 0.9 recurring as distinct from 1.        (select only this node)     N_20180202191359_Anon->N_20180202191532_Anon N_20180202201849_CDW    By CDW 2018/02/02 @ 20:18:49 -------------------------------- This is the part of your line of reasoning that I don't understand. Two things that have the same value can have different representations.        (select only this node)     N_20180202191359_Anon->N_20180202201849_CDW N_20180202203337_CDW    By CDW 2018/02/02 @ 20:33:37 -------------------------------- When we have some sort of infinite syntactic form we try to understand what value it has.        (select only this node)     N_20180202191532_Anon->N_20180202203337_CDW N_20180202202344_CDW    By CDW 2018/02/02 @ 20:23:44 -------------------------------- You are simply asserting repeatedly that it can't equal 1 simply because it doesn't look like 1. Maths doesn't work like that.        (select only this node)     N_20180202191532_Anon->N_20180202202344_CDW N_20180202181329_CDW->N_20180202191359_Anon N_20180202212415_CDW    By CDW 2018/02/02 @ 21:24:15 -------------------------------- Part of the problem is that you're not trying to say what you think ZP9R is. As soon as you start being precise about what it is, one of two things happens.        (select only this node)     N_20180202212554_CDW    By CDW 2018/02/02 @ 21:25:54 -------------------------------- If you insist that it's not equal to 1, then you take 1 minus ZP9R and think about that. Then you can deduce that the result is bigger than zero, but smaller than every positive number.        (select only this node)     N_20180202212415_CDW->N_20180202212554_CDW N_20180202212415_CDW->N_20180202212501_CDW N_20180202170049_Anon->N_20180202171150_CDW N_20180202211220_Anon    By Anon 2018/02/02 @ 21:12:20 -------------------------------- And it might not have a value. It might be a written expression expression that has no definable value. If it did, it wouldn't be recurring, surely?        (select only this node)     N_20180202211624_CDW    By CDW 2018/02/02 @ 21:16:24 -------------------------------- Most people are happy that 1/7 is 0.142857142857... which is infinite, recurring, and a definite value.        (select only this node)     N_20180202211220_Anon->N_20180202211624_CDW N_20180202201929_CDW    By CDW 2018/02/02 @ 20:19:29 -------------------------------- Just because this representation has a 0 at the front doesn't imply that at can't have the value 1.        (select only this node)     N_20180202201849_CDW->N_20180202201929_CDW N_20180202202215_CDW    By CDW 2018/02/02 @ 20:22:15 -------------------------------- And in fact it behaves exactly like 1, and that's why mathematicians say that it has the value 1.        (select only this node)     N_20180202201929_CDW->N_20180202202215_CDW N_20180202202442_CDW    By CDW 2018/02/02 @ 20:24:42 -------------------------------- When the form "zero point nine recurring" is studied it is found to have the value 1. The line of reasoning elsewhere is trying to explain why that is so.        (select only this node)     N_20180202202344_CDW->N_20180202202442_CDW N_20180202211630_Anon    By Anon 2018/02/02 @ 21:16:30 -------------------------------- No I didn't. The reason I'm asserting that 0.9 recurring can't equal 1 is not about the look of the symbols, but to do with the concept of 1 and the concept of infinity being unreachable.        (select only this node)     N_20180202202344_CDW->N_20180202211630_Anon N_20180202202725_CDW    By CDW 2018/02/02 @ 20:27:25 -------------------------------- So we choose how many 9's we want, and how many zeroes. Whatever value ZP9R has, it must be between 0.9999...9 and 1.000...01, where those can be any (finite) length you want.        (select only this node)     N_20180202202620_CDW->N_20180202202725_CDW N_20180202202751_CDW    By CDW 2018/02/02 @ 20:27:51 -------------------------------- There is only one number that satisfies those bounds, and that's the number 1.        (select only this node)     N_20180202202725_CDW->N_20180202202751_CDW N_20180202202838_CDW    By CDW 2018/02/02 @ 20:28:38 -------------------------------- So if we give any value to the syntactic form "zero point nine recurring" we find that the only value it can have that's consistent with standard mathematics is the value 1.        (select only this node)     N_20180202202751_CDW->N_20180202202838_CDW N_20180202203404_CDW    By CDW 2018/02/02 @ 20:34:04 -------------------------------- For some, there is no consistent value that can be assigned to it.        (select only this node)     N_20180202203337_CDW->N_20180202203404_CDW N_20180202203337_CDW->N_20180202211220_Anon N_20180202203429_CDW    By CDW 2018/02/02 @ 20:34:29 -------------------------------- For some there is a single consistent value the expression can be given.        (select only this node)     N_20180202203337_CDW->N_20180202203429_CDW N_20180202203404_CDW->N_20180202203511_CDW N_20180202203429_CDW->N_20180202203720_CDW N_20180202211736_Anon    By Anon 2018/02/02 @ 21:17:36 -------------------------------- And by the way, I was wondering does it make any difference whether we're in base 10 or base 12 or base 2 or whatever?        (select only this node)     N_20180202211630_Anon->N_20180202211736_Anon N_20180202212758_CDW    By CDW 2018/02/02 @ 21:27:58 -------------------------------- If you say that infinity isn't reachable then perhaps you simply have to say that we can recite the words "zero point 9 recurring" but they are meaningless.        (select only this node)     N_20180202211630_Anon->N_20180202212758_CDW N_20180202211630_Anon->N_20180202212415_CDW N_20180202211728_CDW    By CDW 2018/02/02 @ 21:17:28 -------------------------------- But what you say is covered by my comment at 2018/02/02 @ 20:34:04 where I say that a syntactic expression might not have a value.        (select only this node)     N_20180202211624_CDW->N_20180202211728_CDW N_20180202211736_Anon->N_20180202212840_CDW N_20180202211820_CDW    By CDW 2018/02/02 @ 21:18:20 -------------------------------- So do you want it to have a value? Or not? You seem now to be arguing that ZP9R is not a number or value at all.        (select only this node)     N_20180202211728_CDW->N_20180202211820_CDW