Was just trying to confuse some people who are new to the thread and can't be bothered to read all of it
Was just trying to confuse some people who are new to the thread and can't be bothered to read all of it
DrPizza
Administrator Elite Member Goat Whisperer
DrPizza
Administrator Elite Member Goat Whisperer
Geeez, that's about as close as you can get to saying "I've been a member here for a long time"
Banned before???
DrPizza
Administrator Elite Member Goat Whisperer
Originally posted by: Kalvin00
ehhh, even if he was, I just did a search of his recent posts... he's not obnoxious in those posts, so...
Welcome Back!
Perhaps we should clear it up rather than continuing to be so condescending to the poll majority.
The content of the link looks like a school of Mathmatical thought related to Donald Knuth, whose work has been brought up here before. Building up a number system allowing for the existence of an infinitessimal real, or as Knuth called it, the "dark number".
Cliffnotes: kick out a few Axioms of conventional mathematics, and rebuild a number system using different axioms, and you can have situations where it is impossible to prove that 1=0.9999....
From Webster:
Axiom: [n] (Logic & Math.) A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted;
Even more interesting (from a Cornell Mathematics supplement):
If someone accepts/assumes the Archimedean Axiom (which can't be proven or disproven per se), and still refuses to accept the 1 = 0.9999... "proofs" (quotes borrowed from the Cornell link) given here a few hundred times over, then they deserve the scorn heaped on them by the folks in this thread.
If you aren't willing to take the Archimedean Axiom as a given (it doesn't seem that self-evident to me), then you have every right to assert that 1 != 0.9999... since there appears to be some legitimate schools of mathematics that operate just fine where this is the case, published theories that certainly haven't been discredited in this thread.
Hmm... the debate has been going on for at least two thousand years eh? Judging by this thread we may manage two thousand more.
P.S. RossGr - your proof in your sig appears to implicitly assume the Archimedean Axiom as well.
This is not so clear cut as that.
What does my proof say?
It proves that 1 and .999... coexist in every interval which surrounds 1. This statement is true for ALL integers. I make no claim on anything being zero. I have simply proven that we cannot find an interval on the real line which contians only one of the numbers.
Where is zero used here?
This proof coupled with the common definition of equality of real numbers (Which is proven) gives the equality of the numbers.
You will have to prove to me that the Archimedian princple is involved with this.
It says:
"As can be proven using the Axioms and theorems of Real Analysis the sum of a convergent geometric series is found by..."
The Archimedean Axiom falls into the category defined by your bolded words (Link Presented here as Lemma 1.1).
I only said "appears", as I don't offhand recall the process of that "can be proven" part, and whether it relies in any way upon the Archimedean Axiom/Lemma specifically.
The most interesting counter examples in this thread have often reconstructed the reals in some way or another. The reals you build your proof upon are constructed upon axioms, axioms that are convenient to take as given factors, but which are replaced with different sets of axioms for certain fields of advanced mathematics.
To say that the debate over the nature of the reals is completely over is a very iconclast view, and doesn't appear to hold with several of the links in this thread, specifically suggested by the quote given in this link, originally thrown into the debate by luvly.
Unless the reals have been locked down since that quote (one of the authors was a professor of mine at University - so it is a modern era reference), the reals are very much still up for debate (at least at the level of the axioms). String theory is one of our generation's mathematical objectives, and the reconstructed Reals in Knuth's article make certain "insoluble" string theory problems solvable. Another reexamination for our generation, not invalid in any sense other than the questioning of some of the more assailable axioms used by the version of the reals that is currently in favor.
I've brought up much of this before in this thread (other than challenging your 3rd proof, which is clearly quite sound given your assumptions). I know we're beating a dead horse, and I'm accomplishing little more than goading you on. Not that I'm the first to do so.
If you care to reconstruct the sum of a convergent geometric series used in your proof without the use of the Archimedean Axiom/Lemma, that would be quite impressive. I don't doubt your mathematical talent.
Perhaps you need to read beyond the sum of a series proof, there is a second, more fundamental proof also on that link. That is the one I am referring to. We at least should be speaking of the same proof.
Can you make such a strong claim about about 0.1^N for "ANY N>0" without falling back on the standard construction of the Reals?
That second proof breaks down under non-standard reals at the point where you claim:
"This is a statement that shows that for ANY N>0 0.99? + .1^N > 1"
That is simply not true for all N with Knuth's reals, or for that matter any set of reals constructed to permit the existence of infinitessimal numbers. The fact that you observed a pattern for a few example integers is immaterial. With Knuth's reals, for N being infinity, .1^N being * (dark number, or the now allowed "infinitessimal", which in your reals does not exist), 0.99... + * = 1 exactly and is not greater than one. To assume that 1^N for all N > 0 is not infinitessimal at infinity, but rather always enough to push 0.99... past 1 is just the Archimedean Axiom reworded.
For N being infinity, the combined equation (allowing infinitessimal numbers with Knuth's construction of the reals) is:
1-.1^N = .999.... < 1+.1^N
Which does not allow the possibility of 0.999.... = 1
OokiiNeko put it well when he said:
"Bottom line is no one knows for sure what happens at infinity, not even you." Perhaps the smartest poll voters in this thread are ironically the ones who answered "Huh?"
That second proof breaks down under non-standard reals at the point where you claim:
"This is a statement that shows that for ANY N>0 0.99? + .1^N > 1"
That is simply not true for all N with Knuth's reals, or for that matter any set of reals constructed to permit the existence of infinitessimal numbers. The fact that you observed a pattern for a few example integers is immaterial. With Knuth's reals, for N being infinity, .1^N being * (dark number, or the now allowed "infinitessimal", which in your reals does not exist), 0.99... + * = 1 exactly and is not greater than one. To assume that 1^N for all N > 0 is not infinitessimal at infinity, but rather always enough to push 0.99... past 1 is just the Archimedean Axiom reworded.
For N being infinity, the combined equation (allowing infinitessimal numbers with Knuth's construction of the reals) is:
1-.1^N = .999.... < 1+.1^N
Which does not allow the possibility of 0.999.... = 1
OokiiNeko put it well when he said:
"Bottom line is no one knows for sure what happens at infinity, not even you." Perhaps the smartest poll voters in this thread are ironically the ones who answered "Huh?"
MadRat
Lifer
- Oct 14, 1999
- 11,999
- 307
- 126
I want to see .999... represented on a number line and then the numberline broken down to intervals where we can compare .999... to any other fraction, like .5 or even .9 for that matter. No matter how many slices are taken out between 1 and whichever fraction is chosen, we will never arrive at .999... along said interval to be able to make the direct comparison of values between .999... and 1.
PowerEngineer
Diamond Member
- Oct 22, 2001
- 3,606
- 786
- 136
Why do you keep dragging the non standard analysis into this. I have made it clear my proof and my discussion apply to standard Real Analysis. The whole non standard discussion is a red herring. If you which to work with that system then I will have to take your word for the results, I personally have not studied it and know nothing about it.
Now since I am dealing the standard approach, I must disagree with your comments about infinity. Infinity is defined as an extension to the real number system, its behavior is defined, so yes we do know how infinity behaves.
What do you propose will happen to the integers for large N that is different from what we are familiar with? Will 10^-N some how behave differently when N passes some magic threshold? I would really like to see some proof of this proposed odd behavior.
BTW I will point out once again my 2nd proof does not even mention infinity or zero. It is simply a construction of open intervals surrounding 1.
Now since I am dealing the standard approach, I must disagree with your comments about infinity. Infinity is defined as an extension to the real number system, its behavior is defined, so yes we do know how infinity behaves.
What do you propose will happen to the integers for large N that is different from what we are familiar with? Will 10^-N some how behave differently when N passes some magic threshold? I would really like to see some proof of this proposed odd behavior.
BTW I will point out once again my 2nd proof does not even mention infinity or zero. It is simply a construction of open intervals surrounding 1.
Consider it done.
Take a piece of paper, draw a number line showing the interval between 0 and 1, break it into subintervals corresponding to .1 .2 etc Then number each of the boxes 0 -9
Should look like this
_____________________________________________
...|.0.|.1.|.2.|.3.|.4.|.5.|.6.|.7.|.8.|.9.|
...0...1...2....3...4...5...6....7...8...9...1
The bottom row off number represent the most significant digit .1 .2 etc the number in the box represents the next digit of all number in that box. For example in the first number box the digit 0 appears all numbers in that box start 0.0 that would be all numbers x such that 0 < x < .1.
Now, go to the 9th box, repeat this procedure. when that is complete go to the 9th box repeat the procedure. Repeeat....
to say that .999.... does not exist is to say that this process has some forced ending point. Getting tired of writing does not count.
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