9/11 = 0.818181818181818181…
0.81818181818181… is not 9/11 because there is endlessly a ‘81’. Endlessly…
I don’t think so.
St.James: Perhaps 20 trillion computer differentiations or so later, maybe even a googleplex later, (or above),if we survive, the ‘should’ finally trump the ‘is’? God does not play dice with the universe?
This is more conceivable than not, based on
the rate of increase of in computer technology. Why?
Because, the way structural patterns re-occur, in sub atomic, molecular, cellular, astrologically prepondent circular patterns.
Showing finity as sub-set of patterns of greater and greater aggregates , tending toward the greatest
aggregate, which in some extraordinary manner is/becomes
simultaneously the least. All the angels can dance on the head of the pin. Inconceivable, yes, but it has been described by way of the simple notion of the möbius. In this schema, there are not many
universes, only appearently, in fact it is one universe with it’s self bounding it.
The appearent end(s), become virtually the ‘real’ ones, whereas the ‘Real’ Ones, re-appear as Virtual.
Then, they are both: real and virtually different integral spatial/temporal effects.
This is a semantic disagreement. You’re using “reals” to refer to a construction of the real numbers that includes hyperreal numbers. That is not the only construction of the reals, and the common practice seems to be to refer to what you call the “standard reals” as “reals”, and the hyperreals as “hyperreals”. When I use the word “reals”, I’m referring to what you call the standard reals. I’ve tried, and will continue to try, to adopt your convention for ease of communication, but to be clear, I will never use the term “reals” to refer to the hyperreals.
This is incoherent. There is no last digit in an infinite string of digits. Especially because .1 can be written as .1000…, any “last” digit could always accept an infinite string of 0s on the end without changing the value, but then that wouldn’t be the “last” digit at all.
You’re suggesting that there’s both a number at the end (i.e., the “last” digit), and that there’s no end (“because ‘infinitely’ MEANS ‘ENDLESSLY’[sic]”). Your argument seems to be premised on a contradiction in terms.
But if the value you posit as being the difference between 1 and .999… cannot be coherently defined in the standard reals, that entails that, in standard reals, they are the same number.
There is zero distinction between I and 0.999 … because of the infinite number of 9s after the decimal point and so in order for there to be a
distinction there would have to be a finite number of 9s instead. Now I can be expressed infinitely as I.000 … so then I.000 … = 0.999 … And
the base ten system can have a number with more than one decimal representation but they are still of equal value however they are written
Oh my!
Ok, there’s conflation going on here!
Yes, 0 does equal 0.0… Because it is defined differently than non zero numbers.
BUT!!
.9
Is never equal to zero…
Doesn’t matter if the chain is infinite regress…
You see, when zero repeats, it never gets smaller, when non zero numbers repeat, such as nine, they get smaller forever !!
Never converge to zero!!
I don’t even want to bother to give a 20 page discourse on this shit…
Suffice it to say… James is right on the big points.
As has been said before, .999… is a fixed, static value, it isn’t ‘getting’ anything.
A 1 followed by a decimal point followed by any finite number of 0s and then any other digit, is greater that 1. There must be infinite 0s for it to be exactly equal to the integer 1.
Similarly, a 0 followed by a decimal point followed by any finite number of 9s and then any other digit, is less then 1. There must be an infinite number of 9s for it to be exactly equal to the integer 1.
The constructions are parallel, it’s just the intuitions that are different.
Because the symbols represent different quantities, looking different is all it takes.
So no. You are wrong again. It is not the same as “one differs from 1”.
Carleas, Phyllo, Is, and others trying to support Wiki and the public authority proclamation: I guarantee that if Wiki and public authority had stated the opposite, without any proof concerning it, you would each being trying desperately to support that opposite proclamation. You guys are the make of fundamentalist religions and propaganda momentum.
You should each seriously look at yourselves and imagine being on the opposite side just for a moment. What would your argument be?
No. It isn’t and for the exact same reason. Any time a number ends with “…”, it is telling you that the ratio producing the limit cannot be represented in decimal form (just like the Pi series can’t). It is saying that “this series cannot end such as to reach the limit”.
That isn’t why. It isn’t the number that is equal to the limit because the very definition of “0.999…” forbids it to ever, ever be equal to “1.000…”
You would merely be wrong again. There are very many numbers that cannot be represented in decimal form. There might be even more that cannot be than can be.
Maybe that is because that is how THEY define the words and categories.
Reals = “nonstandard reals” (aka “hyperreals” and any other nonstandards) plus the “standard reals”
We have been trough this. It is THEIR category definition, not mine.
BS. You show me precisely what is incoherent about that very simple series.
The fact that there is no end to the series is exactly what demands that the “1” ALWAYS be present no matter how far out it is taken.
THERE IS NO POINT where that “1” could ever suddenly become a “0”.
Carleas,
SHOW ME how that trailing 1 magically changes into a zero “at infinity”.
When does this series:
[list]0.1
0.01
0.001
0.0001
.
.
.
Ever reach exact zero?[/list:u]
False assertion. The fact that every finite string in the infinite list ends with a 9 tells you that there is no point where the 9 magically becomes a 0. How does that trailing 9 in the series suddenly become a 0? There is no “at infinity” so as to declare a reason for change. The trailing 9 can never change.
Precisely.
My argument would be that, as I’ve already acknowledged, there are number systems in which .999… != 1. The hyperreals are arguably more complete that the standard reals, in the same way that the reals are more complete than the integers.
But I encourage you to take your own advice, I agree that it is a good way to ward off dogmatic adherence to beliefs.
As you have defined it in this thread, in all caps and bold, infinite means endless. So you have posited an endless string of 0s with a 1 at the end… That is incoherent, there is no end at which to place the 1.
The series clearly goes to zero, and it seems it should have a value of zero in the standard reals. 0.000… and 0.000…1 will have the exact same digit in every place; the 1 “at the end” is meaningless when appended to a string of zeros that is endless by definition.
It doesn’t matter (as I have already explained) whether you use merely the standards or whether you use the entire real spectrum. The result is the same.
Every single argument that you have presented, I have shown you where you went wrong in your reasoning. But every argument that I make, you don’t show where it is wrong, you merely present your own argument in its place.
That is the difference. I can show you the detailed flaws in your reasoning. You cannot do the same in return.
So no. We are not equals with merely opposite opinions.
That is what I have shown throughout this thread, usually in caps.
And like your other strawmen, that is NOT what I said or showed.
I showed you an infinite series of finite strings, each with a 1 at the end. And I asked for you to show me how “at infinity” or wherever you think, that 1 suddenly becomes a 0.
“Clearly goes to zero”???
Then show us how that happens, because it is certainly not clear to me.
The 1 isn’t “appended to” the infinite string of 0s. The 1 is at the end of EVERY finite string in the ENTIRE series.
So the question is, what happened to it such that it somehow became a 0 and when/where??
And this too is another example of me showing your detailed mistake (in this case a strawman). And it is also another example of you not addressing my argument, but rather merely inventing another argument of your own (in this case, the strawman and your response to it).
What do finite strings get us? We’re talking about infinite strings. I don’t disagree that any finite string of 9s will be less than 1. But .999… isn’t a finite string. Are you using your series to conjure an infinitely long finite string?
This,
“1.000… - 0.999…” doesn’t show what an infinitely long string would look like, thus we are debating about it.
This;
"0.1
0.01
0.001
0.0001
.
.
.
"
Doesn’t show us what that exact same infinite line looks like either. But it DOES show us something that the other symbol didn’t show. It shows us that every single line in the ENTIRE series ends with a 1. And that there is no excuse for that 1 to ever not be there no matter how infinite the series is.
So now, how about you tell us how that 1 disappeared from the most infinite line in the series, the same line as the original would be that you say is “0.000…”
But you’re just baking in a contradiction in terms: “every single line in the ENTIRE series ends with a 1”, but an infinite string of 0s has no end. That 1 is not there when the end is not there.
Each line has an end. There is not one single line in the entire infinite set without an end. And at the end of each and every line is a 1.
Except you don’t believe that (despite it being true by definition). You want to tell us that eventually, “at infinity”, the 1 that always appears at the end of each and every other line has disappeared into merely a 0.
I am asking, “How did the 1 become a 0?”
And if you are going to say again that the end isn’t there (with which I agree), realize that the end can’t be a 0 either. But for the series to be equal to 1.000… EVERY digit must be a 0. So where did the 1 go? How did it become a 0? In effect, you are saying that the pattern changed.
James, if your argument were true in terms of counting finite strings, then all the reals would have been counted thousands of years ago by rationals alone !! This is what Carleas is trying to say!!
Carleas, the logic in this post clearly disputes any thesis that .9… Is 1
This also links to finite element simulations and infinitesimally small elements.
With finite elements, the elements can be as small as you want, the result will always be an approximation at the end, even if a very good approximation at that.
With infinitesimally small elements the result is not an approximation, it is exact.
0.000…1, where the ‘…’ represents an infinite number of zeros, is not a member of the series you constructed. It is incoherent in the standard reals to have a number represented by an infinite string of digits with two ends. It is internally contradictory, there’s just no consistent way to make meaning of it (again, within the context of the standard reals).
That’s my rejection of your argument from the series. Not that the “1 become[s] a 0”, but that the question is incoherent, it bakes in incompatible requirements.
For .999… to equal 1, every digit of their difference must be a zero, I agree. And we know that this is in fact the case, because we know that it is an infinite string of zeros. The string is bound on the left by the decimal point (i.e., it is an infinite string of zeros after the decimal point), and cannot be bound on the right by any other digit (that would be incoherent). So every digit is 0.
Again, that is YOUR construct and contradiction. I said nothing of an standard infinite string of 0s followed by a 1. In the standard reals, you can’t even have an “infinite”, certainly nothing following it. Again, a strawman.
I defined an infinite set of reals, each having a 1 at the end of a FINITE list of 0s. Each and every member of the set is a real (much like the infinite set of integers). Again, I am asking how any of a set of numbers that have a 1 at the end loses the 1. What possible logic could say that despite being defined as only numbers that have a 1 at their end, one of that exact same set actually doesn’t have a 1 at the end.
Well, this point carleas made earlier is somewhat unresolved… That .9… Is it’s own entity without sequence! He is right however that it’s not in the list you made James !
The only way we even build these types of numbers is 1 at a time, so I don’t think they are entities out of sequence as carleas said, but what you call infinitesimal 1 James, cannot be derived from your algorithm… It’s dimensional flooding on a single line!!
If your technique actually worked James, all the reals would be listed by the rationals!!
There is no “technique” involved, Ec…
It is merely a question of the definition of the terms involved.
“…” means that the preceding number or pattern continues eternally without change.
Well, decimals that are non zero regress, that is change… What happens with those, is that we find a pattern in regression, infinite regression to be exact.
But to say they don’t change is to say they don’t continue… Just to articulate the issue