I wonder where you get your idea of correct from. That everyone does it, doesn't make it correct, read my example above. You support mob rule? — Metaphysician Undercover
Clearly "three dozen" does not represent a quantity of three, just like "four score" does not represent a quantity of four, and "twenty six" does not represent a quantity of twenty. — Metaphysician Undercover
I've explained very thoroughly why 1/2 is not a number. — Metaphysician Undercover
I think I got it (incidentally, c=k here, right?) — InPitzotl
Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity — EnPassant
But I note that your OED definition talks about values referring to the same particular quantity. And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value.I said these do not represent any particular quantity, and ought not be considered as numbers. — Metaphysician Undercover
Your problem is your problem though, not mine.It is the belief that they are numbers which is what I consider to be a problem.
No, they define .999... in such a way that it has the same value; it's not a different value that's close enough, it's the same value. But .999... having that value comes from the definition assigned to it. Like I said at first, this is a language barrier issue. You don't speak the same language.The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. — Metaphysician Undercover
You can do whatever you wish, but I'm under no obligation to take you seriously, especially at your word.But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim.
That's not an argument, it's a claim.My argument, if you've read what I posted, is that .999... does not represent a particular quantity. — Metaphysician Undercover
Again, you didn't make an argument (it was just a claim) and, until you do, I can dismiss your claims with Hitchen's razor. Where we left off is your claim that .999... does not represent a "particular value" despite it being equal to 1, which does. I repeated the inconsistencies I pointed out last post in this post for you.I suggest that you come back when you've got an argument to make. — Metaphysician Undercover
That's true for all finite x. But you need it to be true for an infinite x. To see the problem, here's a "troll proof" that infinity is finite. 1. 1 is finite. 2. For all x where x is finite, x+1 is finite. 3. By 1 and 2, and infinite recursion, infinity is finite, QED.As the number of terms taken increases 1/x decreases but never becomes zero. — EnPassant
You missed the point of the example, as is your habit. — Banno
Well, no; what you did was explain how you use the word "number" in a rather eccentric fashion. You told us nothing about numbers. — Banno
But I note that your OED definition talks about values referring to the same particular quantity. — InPitzotl
And I note that you've chosen of your own will in this post to not actually argue the relevant point... which was that .999...=1 is equivalence under equality, and under equality equivalence implies having the same value. — InPitzotl
Until you do, there's nothing to argue against. You have no point to make, just a problematic claim. And by Hitchen's razor, I can dismiss that without argument. — InPitzotl
No, they define .999... in such a way that it has the same value; it's not a different value that's close enough, it's the same value. But .999... having that value comes from the definition assigned to it. Like I said at first, this is a language barrier issue. You don't speak the same language. — InPitzotl
Again, you didn't make an argument (it was just a claim) and, until you do, I can dismiss your claims with Hitchen's razor. Where we left off is your claim that .999... does not represent a "particular value" despite it being equal to 1, which does. I repeated the inconsistencies I pointed out last post in this post for you. — InPitzotl
(a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1. — InPitzotl
I do not argue against the fact that mathematicians believe that .999..., and 1 refer to the same value. The difference between these two is a difference which does not make a difference, for them, so they say that it is the same value. But that doesn't prevent me from arguing that the claim that there is a difference which doesn't make a difference is a contradictory claim. — Metaphysician Undercover
MU, you're pretending here to be making an argument about .999... = 1:I don't deny that in some cases different symbols represent the same quantity. The op does not provide one of those instances. — Metaphysician Undercover
A, therefore B, where A is .999... = 1, and B is some rambling about equivalence. But here we don't merely have equivalence, we have equality. Because we have equality, they do represent the same value. I've never heard of someone so far gone as to commit an amphiboly by changing the word. But in this post, and here?:As I've explained to fishfry already, that two things are equivalent does not mean that they are the same thing. Therefore what is on the left side of the "=" (which indicates equivalent) does not provide a definition of what is on the right side. — Metaphysician Undercover
...you're still not talking about the relevant point. You're still not advancing any reason why you think equality represents different values, or why you think the same value representing a particular quality can actually wind up representing different qualities. Nor are you even trying to make this point; you're just, instead, playing hide-the-ball.I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity. — Metaphysician Undercover
I haven't seen that definition. care to provide it? — Metaphysician Undercover
That definition was already discussed in the thread. And that definition is used in the pdf provided by the op in section 1. By that definition, .999... = 1 exactly.
When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia
In other words, 65 pennies, a dime and a quarter is not worth a dollar because pennies are 1/100th of a dollar and that's not a particular quantity of money. I mean, sure, some pennies are smaller than other pennies slightly; but some dollar coins are also smaller than other dollar coins. But apparently the pennies being smaller implies that pennies aren't a particular value, whereas the dollar coins being smaller does not indicate such a thing. Such is the tomfoolery I've heard from you so far. That's a garbage argument that can be ignored just on its merits.I deny that .999..., as presented in the op, represents a particular quantity, because there is no quantitative value given for 1/9. Therefore I deny that .999...which in this instance does not represent a particular quantity is equivalent to 1 which does represent a particular quantity. — Metaphysician Undercover
Yes, but it's all gibberish nonsense.I made my point — Metaphysician Undercover
Sure they do. 1/2 represents one half. As you said, one of anything represents a particular quantity. The quantity that half represents is very clear... that is the multiplicative inverse of two. It takes two halves to make the quantity one.I made my point, symbols such as 1/2, 1/3, 1/9, are representative of ratios between quantities, they do not represent any particular quantitative value. — Metaphysician Undercover
What argument?The argument is very clear in my discussion with Banno. — Metaphysician Undercover
Your "premise" isn't a premise... it's a pointless language game. It shows you cannot speak the language (or at the very least, refuse to). You invented some niche and uninteresting alternate meaning for "particular quantity" that mathematics speakers do not use. The way mathematics speakers use the term "particular quantity", 1/9 is indeed one of those things. So you're not really advancing a "view" of quantities, you're promoting a language that's uninteresting. Therefore, your real burden is to show what's wrong with the language of math; you can't just say, "I don't 'believe' 1/9 is a particular quantity"... you have to say, "saying '1/9' is a particular quantity leads to the following problem" and say what that problem is.You just cannot grasp the first premise, that 1/9 does not represent any particular quantity, and therefore it is not a number.
.999... is an infinite string of 9's. There's no problem with that per se.Ok, but can't this be also said for 0.999...? — EnPassant
We don't have to say "at infinity"... it's an infinite string of 9's. We add one 9 for all finite numbers, there are an infinite number of finite numbers, therefore there are an infinite number of 9's. But I think you misunderstand what the problem is...Adding terms and then saying 'at infinity'. — EnPassant
In the infinite string .999...., I can say that the first digit after the decimal is 9. That's how I construct that infinite string. It's done homogeneously... it's always there at all steps.You can't have (b)
An extremely charitable remark.There's nothing here but Meta's queer usage. — Banno
And this, some time ago, s/h/b the last word in this thread by a participant qualified to make it.We have something that at first makes no sense, then we MAKE it make sense with a clever definition. What exactly are you objecting to? — fishfry
When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1. — InPitzotl
No.Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? — EnPassant
Sorry, what is c here and how does that relate to 0.999...?Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1 — EnPassant
Yes, by using a different definition for a divergent infinite sum. I toyed with that here:Ramanujan summed the natural numbers and got -1/12. — EnPassant
...since ...999=-1 in 10-adics.So does this mean ...999.999... = 0? — InPitzotl
Sorry, what is c here and how does that relate to 0.999...? — InPitzotl
So you're talking about 0.111...? Then @Pfhorrest's post applies:It's the same idea 0.999.. - 1/10 + 1/100 + ... — EnPassant
That doesn’t sum to 1, that sums to 1/9. — Pfhorrest
There is no "at infinity" here though. Every term here is a finite number; there's just an infinite number of finite numbers. Think of it intuitively this way... imagine the set you're trying to picture... it has "at infinity" in it, and maybe some other things. Remove every infinite-th step from this; we only care about finite steps. But we do want all of the finite steps. Now you still have an infinite set, but it only has finite terms in it. That is the thing we're describing.What seems to be happening here is that 1/x = 0 at infinity. — EnPassant
If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined.So 0 + 0 + ... = 1 after an infinity of terms.
But most mathematicians probable would not accept this. — EnPassant
But you could do the same thing with a segment of length 2, 50, 0, and -7. So that infinite sum could also add up to 2, 50, 0, or -7, or any other value. This is what undefined refers to.But they add up to 1 unit. How do da? — EnPassant
No, they aren't. The unit position is not defined in terms of infinite additions of 0. That would be useless, since infinite additions of 0 is undefined.Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. — EnPassant
That doesn’t sum to 1, that sums to 1/9. — Pfhorrest
If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined. — InPitzotl
But you could do the same thing with a segment of length 2, 50, 0, and -7. — InPitzotl
It's undefined! :wink:Yes, because it can't be defined in terms of calculus but the question remains, what is it? — EnPassant
No, it's not arbitrary. It's just infinitely non-specific. That sum genuinely is sometimes 1 inch, sometimes a light year, sometimes 0, sometimes negative. So it's undefined. You can't reduce the sum to 1 inch if it could be negative or a light year. So it's useless to ponder whether it "truly" is 1 inch or "truly" is a light here, because your infinite sum doesn't give you the information to distinguish any length from any other.Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit can be 1 inch or 1 light year. — EnPassant
Okay, so that sum is 50 units.Say 0 + 0 + 0 + ... = 50 units. — EnPassant
You can only say that if you're literally talking about that 50 unit thing, because:(0 + 0 + 0 + ...)/50 = 1 — EnPassant
Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. To recap, that sum is undefined.But 0/50 = 0. — EnPassant
You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever. — InPitzotl
There's nothing here but Meta's queer usage.
— Banno
An extremely charitable remark. — tim wood
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