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
In anticipation of your response, I had updated my post to include “number or pattern” before you posted that. It just depends on which kind of list you are making whether it is a single number pattern or a more complicated pattern. Either way, the pattern doesn’t change such as to become so small that a part of the pattern must become different.
Again, I’m not saying that anything loses a 1. Rather, finite strings just aren’t relevant here, because we’re talking about an infinite string that by definition cannot have any digit “at the end”.
The difference between .999… and 1 is not a member of that set. That is the number we’re talking about.
At some point, the infinitesimal reaches a critical point, where it transforms infinity into finity. That’s got to be it. But it has to be a very high n to the X power.
The null space in an eigenvector function can be summed in linear systems, by linear algebra. - is proof of this.
I am only showing it this way, so as to desribe the irresistible truth in Plato’s Meno-s slave,as a way to reach the anti derivative. Kind of like showing the wisdom of zeno’s Paradox, and it’ solution being solvable backwards.
Plato, it seems left it to the future to come up with an explanation of this idea, or of ideas in general.
A “converging series” specifies a LIMIT for a summation. The LIMIT is equal to xxx. You can have two obviously different summation series with the same limit. Note that the summation must REACH INFINITY before it can equal its limit. And there is no “at infinity” to be reached.
And even a higher truth is that even if the summation could actually reach infinity, it still would not equal its limit because its definition forbids it from EVER summing up the last tiny bit.
That is irrelevant and Zeno’s can be solved easier than that via relative infinitesimals, “calculus”.
That is what must happen in order to make 1.0 = 0.999…, but the problem is that such can never happen.
Strange! Before James and I posted one could read the following text below Jerkey’s last post, although he was the last one who posted (before James and me):
“Last edited by jerkey on Wed Jun 29, 2016 5:24 pm, edited 2 times in total.”
Yes. Therefore all this operations have to do with the infinitesimal calculus (inveted by Gottfried Wilhelm Leibniz). So James is right with his answer.