D'Nalor Posted July 22, 2008 Posted July 22, 2008 Is it at all possible to have the number 0.0(reoccuring)1? or for that matter any other number at the end? my idea is that the number is 0.0 with an uneding string of 0s on the end, but if it did have an end, it would end in 1. Is this possible?
YT2095 Posted July 22, 2008 Posted July 22, 2008 do you mean as in a very small number like a Yoctosecond for instance 10^-24? or that "Zero" must resolve to something other than total Nothingness?
Air Posted July 22, 2008 Posted July 22, 2008 Is it at all possible to have the number 0.0(reoccuring)1? or for that matter any other number at the end? my idea is that the number is 0.0 with an uneding string of 0s on the end, but if it did have an end, it would end in 1. Is this possible? If zero's are recurring, at which point would you be able to attach a zero at the end?
Country Boy Posted July 22, 2008 Posted July 22, 2008 Is it at all possible to have the number 0.0(reoccuring)1? or for that matter any other number at the end? my idea is that the number is 0.0 with an uneding string of 0s on the end, but if it did have an end, it would end in 1. Is this possible? "but if it did have an end"? And you just said "unending string of 0s"? Surely you see the logical falacy! The nearest I can come to matching that is to consider the sequence of finite decimals: 0.1, 0.01, 0.001, 0.0001, 0.00001, etc. Its limit could be considered to be "an unending string of 0s with a 1 on the end". Of course, it is easy to see that the limit is simply 0. In fact, the limit of the sequence 0.a, 0.0a, 0.00a, 0.000a, 0.0000a, etc. is 0 no matter what digit 0 is. It really is just an unending string of 0s!
John Cuthber Posted July 22, 2008 Posted July 22, 2008 0.0recurring1 may be a bit meaningless, but it's the difference between 0.9recurring and 1 Since 0.9recurring is equal to 1, 0.0recurring1 is equal to zero.
ajb Posted July 22, 2008 Posted July 22, 2008 I wonder if we are heading towards infinitesimals? These are not real numbers, but "numbers infinitesimally close to zero". We use them all the time in calculus, differential geometry, deformation theory... the list goes on.... You really have to deal with them quite formally as "external parameters". For example, [math]dx[/math] is really a formal object.
D'Nalor Posted July 23, 2008 Author Posted July 23, 2008 0.0recurring1 may be a bit meaningless, but it's the difference between 0.9recurring and 1Since 0.9recurring is equal to 1, 0.0recurring1 is equal to zero. 0.9recurring is not equal to 1, it is just a tiny bit smaller than 1, but the difference is so small it is just considered to be 1.
Kyrisch Posted July 23, 2008 Posted July 23, 2008 0.9recurring is not equal to 1, it is just a tiny bit smaller than 1, but the difference is so small it is just considered to be 1. That's actually not the case... [math]0.\overline{9}[/math] is actually equal to 1. There are plenty of proofs for this.
Country Boy Posted July 23, 2008 Posted July 23, 2008 0.9recurring is not equal to 1, it is just a tiny bit smaller than 1, but the difference is so small it is just considered to be 1. 0.99999999... is, by definition, the limit of the sequence 0.9, 0.99, 0.999, 0.999, etc. That is a geometric sequence and it is easy to show that its limit (and so the value of 0.9999... ) is exactly 0.11/(1- 0.9)= 0.1/0.1= 1.
John Cuthber Posted July 23, 2008 Posted July 23, 2008 ".9recurring is not equal to 1, it is just a tiny bit smaller than 1, but the difference is so small it is just considered to be 1." Er, no. The difference is exactly 0.0recurring1 However as the difference is also zero then it follows that 0.0recurring1 is also zero. If you don't like limits of series try this X= 0.9... Multiply both sides by 10 10X= 9.9... Subtract X (ie 0.9...) from both sides 10X-X =9 Simplify 9X=9 leaving you with X=1
Bignose Posted July 23, 2008 Posted July 23, 2008 0.9recurring is not equal to 1, it is just a tiny bit smaller than 1, but the difference is so small it is just considered to be 1. Wonderful, we're going to start this again. Please answer this question which cuts right to the heart of the matter. If 0.99999... (infinitely repeating 9's) is NOT equal to 1, what is the number that comes between them? If there is no number that comes between them, then they are equal. If they are unequal, then there must be a number between them. So... since you say they aren't equal, what is the number between them? Same thing with your 0.0000 (infinitely repeating 0's)1 number. First of all, there really isn't any such number. If there are infinitely repeating 0's, how can there be a 1 on the "end"? It can't, there are infinitely many 0's. But, even if it were possible? What number comes between 0.00000 ... etc. and 0? Again, there isn't any such number, so they are equal.
ajb Posted July 24, 2008 Posted July 24, 2008 Same thing with your 0.0000 (infinitely repeating 0's)1 number. First of all, there really isn't any such number. If there are infinitely repeating 0's, how can there be a 1 on the "end"? It can't, there are infinitely many 0's. But, even if it were possible? What number comes between 0.00000 ... etc. and 0? Again, there isn't any such number, so they are equal. Yes, it I such more like what I said. In a rough sense you need to "add" such "numbers" to the real line. This is the theory of infinitesimals.
John Cuthber Posted July 24, 2008 Posted July 24, 2008 "Please answer this question which cuts right to the heart of the matter. If 0.99999... (infinitely repeating 9's) is NOT equal to 1, what is the number that comes between them? " OK. Within the bizarre context of this thread the number 0.9recurring5 exists and meets that criterion.
ajb Posted July 24, 2008 Posted July 24, 2008 0.(9) is greater than 0.99999, which is in turn less than 1. (If I have read your reply right) The basic premise is that you can't find a real number that is closer to 1 than 0.(9). In fact, you can show that 1 and 0.(9) are actually just different representations of the same number. You can even see a proof of this on wiki.
John Cuthber Posted July 27, 2008 Posted July 27, 2008 "(If I have read your reply right)" No, it seems you have not. Have a look at what this whole thread is about. It's about the weird world of numbers that have an infinite number of, for example, 9s after the deimal point then another number. "In fact, you can show that 1 and 0.(9) are actually just different representations of the same number. You can even see a proof of this on wiki." Nevermind wiki, I posted a proof of that earlier in this thread (did you read the thread btw?). However if you accept that "zero point nine recurring with a five on the end " has a meaning, then it falls between 0.9recurring and 1 If you don't accept that the term has a meaning (and anywhere other than this thread I would go allong with that) then the thread is meaningless anyway.
ajb Posted July 27, 2008 Posted July 27, 2008 (edited) 0.(9)5 has no meaning. Full stop. So I have no idea what you are talking about! Edited July 27, 2008 by ajb
John Cuthber Posted July 28, 2008 Posted July 28, 2008 Admitting that you don't understand something while claiming it's meaningless isn't logical
ajb Posted July 28, 2008 Posted July 28, 2008 Whatever Cuthbert. I have no wish to converse with you any further.
Country Boy Posted July 29, 2008 Posted July 29, 2008 Admitting that you don't understand something while claiming it's meaningless isn't logical ??? Actually, it would be illogical to say you DO understand something while claiming it is meaningless.
John Cuthber Posted July 29, 2008 Posted July 29, 2008 True, but it doesn't detract from my point. If he doesn't understand it, he cannot judge whether or not it's meaningless.
Kyrisch Posted July 30, 2008 Posted July 30, 2008 [math]0.\bar 95[/math] is meaningless. Since it is meaningless, it cannot be compared to [math]0.\bar 9[/math], an actual, rational number and evaluated as either greater or less than.
D'Nalor Posted July 30, 2008 Author Posted July 30, 2008 (edited) If 0.9(recurring)5 is meaningless, then why do I understand what it means? I know that although the acctual number is impossible(or improbable), isn't the number i (the square root of -1) in physics also an impossible(or improbable) number? (by the way, Kyrisch, nice paradox) Edited July 30, 2008 by D'Nalor
Mr Skeptic Posted July 30, 2008 Posted July 30, 2008 If 0.9(recurring)5 is meaningless, then why do I understand what it means? Because you don't? The number itself is a contradiction. You have an endless repetition of 9's, and then at the end, a 5. But there is no end, so there can't be a 5 at the end.
John Cuthber Posted July 30, 2008 Posted July 30, 2008 The original question was about 0.0recurring1. That has a meaning of sorts. It's the difference between 0.9recurring and 1 It's the limit (though not, I think, in the strict mathematical sense) of 1-.9=0.1 1-.99=0.01 1-.999=0.001 and so on to 1-.9recurring= "0.0recurring with a 1 tagged on the end". From that point of view it's a perfectly reasonable concept. It happens to be another way of writing zero. Once you accept that you can label some of these weird (and very near useless)figments of imagination as having certain properties you can look at those properties. For example, in the same way that 0.95 is between 0.9 and 1 you can see that 0.9Recurring5 is between 0.9recurring and 1 It's not physically meaningful, but (as has been pointed out) nor is i. That doesn't stop mathematicians playing with it.
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