RE: Euler.
Did he have a method of building up the real numbers? I can’t think of any, but I could be mistaken. At any rate, my main point is we don’t really need to appeal to his authority to build up the reals. That can be done in several different ways with pretty much the same end result.
Building up the reals does not “depend” on infinite sum theory (whatever that is).
I presume you mean “For what N”, and the answer is “N = 0”. But alright, it seems you intend always to interpret an infinite decimal as meaning the infinite series given by truncating it at successive decimal digits, starting from a fixed position. Which is perfectly fine; I just wanted to make sure you intended this.
The rest, of course, turns on how you interpret these series. My first two questions were to help me understand that (I suspect I know how you would want to treat these series, but I wanted to clarify certain issues). Alas, you did not respond to those two questions. That’s ok; I should spend my time on other things.
Oh what a jolly caper this all is!
Cantor lived a full century after Euler. So why are you not mentioning him and his contributions?
I have given many proofs that 1 ≠ 0.99999…
It is amazing that you are ignoring them and always return to the same old way of
thinking.
I am tired to repeat what I said many times.
Tell me, why number 1 should have another decimal representation besides 1.000000…?
Solve the equation:
-1 + x = 0
the only solution is x = 1
x=0.9999…is not the solution.
Furthermore, these two identities cannot be both true at the same time:
…999999 = -1
1=0.99999…
It is as clear as why these two identities cannot both be true at the same time:
1/3 = 0.33333333…
1/3 ≈ 0.33333333…
It is very hard for me to understand why it is taking so long time for you understand
these proofs. The only explanation is that you are ignoring them, although it
does bring any benefit to you or anyone else.
Ah, nevermind
What is 1 - 0.999… equal to?
That equation has one solution, but there are many ways of writing that solution.
x = 1
x = 2/2
x = cos(0)
x = 100%
x = 13 - 12
So iff 0.9999… = 1, then x = 0.9999… is yet another way of writing the solution to this equation. This “argument” brings nothing new to the discussion.
Why shouldn’t it?
No, seriously.
Other than an appeal to some notion of personal aesthetics, why can it not have another decimal representation? Do you have a proof of the uniqueness of decimal representations? Is it inherent to the real numbers and what is your evidence? There’s no axiom of the real numbers that requires such uniqueness.
This reminds me a lot of false proofs of the Parallel Postulate. Many of them assumed that some notion was so ridiculous as to require no additional comment. They were wrong. Believing that uniqueness must exist is similar. It is a difficult pill to swallow, but there’s no rule that says it must be so. It’s “common sense”, but “common sense” is often neither.
Yes, “arbitrary” is not a number, just as “infinity” is not a real number. When a mathematician says that an infinite sequence, such as the difference between the partial sum of a sequence and its limit, becomes “arbitrarily small”, it means:
For the sequence S1, S2, S3, …, Sn, …,
given any positive number d, no matter how small,
there exists an number m such that
for every number n > m
the absolute value /Sn/ < d.
That is, no matter how small a number you give me, I can prove that all members of the sequence are smaller than that past some point in he sequence. The word “arbitrarily small” refers to the choice of the number d.
So what it really comes down to is that you have a problem with the notation “…” You don’t want 0.999… to notationally mean a limit. You are welcome to do that, and to some extent I agree with you since it’s not that rigorously defined. For example: if I wrote 3.14159… I could be talking about Pi or I could be talking about 355/113. But for the most part mathematicians understand that what is meant by 0.999… is the limit of the infinite series that results from continuing to add 9’s on to the end. If you have another definition you want to attach to it and can convince your audience to go along with it, more power to you, but most of the mathematics community will give you the same skepticism that you would get if you told them that 2+2=3 for very large values of 3.
A while ago I pointed out that 0.999… is shorthand for an infinite series:
910[sup]-1[/sup] + 910[sup]-2[/sup] + 910[sup]-3[/sup] + 910[sup]-4[/sup] + …
and you seems quite happy with this. But you keep coming back to the idea of “decimal representation” of a number. So which is it? Is it a series, or a decimal representation of a number?
If it is a series, this constant worrying about a unique representation for a number isn’t part of the question. If it is a decimal representation of a number, you might like to begin by defining which kind of number you consider it to be a valid to representation of. (Hint, clearly it isn’t a Natural or an Integer.)
Correct. …999999 = -1 is not true.
And again, already pointed out. “Approximately equal to” has a clear and well understood definition that does not exclude also being exactly equal to. To claim it does exclude equality is not mathematics but just playing with words.
It is amazing that you are ignoring them and always return to the same old way of thinking.
We understand exactly what you’re saying; you’re just wrong.
Huh. Just figured out why that doesn’t work. This is why I’m not a mathematician.
(Oops, posted this before your follow-up post.)
Okay. So shout out to **Itself **for not calling me an idiot! :)’
Here’s the best I can come up with, but now I think I understand the problem*:
Prisoner 1 looks at Prisoner’s 3 hat. If it’s the same color as Prisoner’s 2’s hat, he says that color. If it’s not, he says the opposite color. Prisoner 2 now knows what color his hat his. So does Prisoner 3 (based on Prisoner 2’s answer). They both get it right. The sequence starts over. So, at a minimum we have 2/3 of the prisoners getting the right answer. BUT, Prisoner 1 (and Prisoner 1+3x) have a 50% chance of being right themselves, so the number of prisoners that die is:
Infinity-((2/3)Infinity)+(.5(Infinity/3))
Which is not a finite number.
Do I get an A? 
*FWIW, I understand the .9999…=1 thing.
I like to phrase the prisoners problem in an even more outlandish way: instead of prisoners having two possible hat colors (and thus, we might say, 50-50 odds of guessing their hat color correctly), let’s suppose each prisoner actually has an arbitrarily long essay written on their hat, and they have to announce the contents of this essay perfectly (letter-for-letter, comma-for-comma). Or, if you like, a real number between 0 and 1, or any other such thing, is written on their hat.
Show that there’s still a strategy that guarantees all but finitely many of them will go free.
For any number of prisoners, N, I know the solution that saves all but the first one, but I don’t see how to extend it to an infinite number of prisoners. Something for me to ponder…
But I didn’t show that finitely many could go free if I understand the problem correctly.
I’d be interested to hear it. That’s what I thought I posted the first time and then deleted after I thought about it.
It’s not possible to guarantee the freedom of any particular prisoner. All the hat-declarations are done simultaneously and without any communcation.