0.99999~ = 1 ?

It was intended as satire?

I guess it's commendable, Rrhain, in some sort of twisted way, but you are so wrong in your beliefs here.All evidence is on my side, and NONE on yours.You have failed already.You of all people should realise that you CANNOT teach a pig to sing...Edited by cavediver, : No reason given.

Funnily enough in a way there is nothing wrong with this. There are mathematical systems with 1 > 0.999..., such as a cute system I found out about in Nottingham called Hackenstrings. It's just not true in the real numbers because of how they are defined.
The problem with mathematical systems which do have 1 > 0.999..., is that you can't create calculus with them. Yeah, infinity is definitely wierd. Oh don't worry, I know you're not arguing. I'm glad you found the explanation that worked for you in asymptotes, for me it was a formal definition of the reals that made me understand it. At the time I didn't really understand what real number were mathematically, I just had some intuition about them being rationals and irrationals put together. It's funny how it's often just a case of the right picture for the right person.Edited by Son Goku, : Clarity.

Dr Adequate has already outlined my logic, so rather I will take the only problem you seem to have: First of all, in later posts than this one, you appear to be getting hung up on the word real. The real in real number means nothing more than the complex in complex number. In the Renaissance it was chosen for a reason, to contrast with the recently discovered square roots of negative numbers. However it has no meaning now.0.999... is a real number because you can prove that the sum it represents converges to a finite value. That's all there is to it.
Take the following facts:
1. 0.999... > 0.9
2. 0.999.. < 1.1
3. As every additional 9 is added, 0.999.. only grows larger.So 0.999... as you add every 9 is trapped between 0.9 and 1.1, it can only move in between them.
Also it doesn't move up and down in between them, it only keeps growing. This growth combined with the fact that there is a upper limit to its growth (1.1), is enough to show that the series settles down to a finite value.Basically if something keeps growing it can either grow out to infinity or have its growth get slower and slower, until it settles down on a specific number. Those are the only two possibilities. The fact that 0.999.. < 1.1 shows that it cannot grow to infinity. So we're left with the other alterntive.0.999.. is indeed a real number.Edited by Son Goku, : More explanation

I believe I see your question. Let's take three decimals of accuracy:
1 + 0.999 = 1.999
1.999/2 = 0.9995Which differs from 0.999 by 0.0005.So if I use n decimals of accuracy, they will differ by 5 x 10^-(n+1). In the limit as n tends to infinity this converges to 0 and so:
(1 + 0.999...)/2 = 0.999...You question is basically related to the definition of arithmetic on infinite series.Edited by Son Goku, : Little more information.

RAZD responds to me:

quote:

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I have

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Incorrect.Are you denying that division of 1.999... by 2 results in a pair-wise identity in decimal place with 0.999...? For that is what was given as justification (Message 4): Are you saying that this statement isn't true? That 1.999.../2 <> 0.999...?You don't just get to whine about your doubt. You need to provide your evidence as to why the equality isn't true or why it is that 0.999... <> 0.999.... Is there something specific that is bothering you? If so, what is it?Be specific.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Rrhain's post just made me want to clear up the long division component of my proof. I've given a technical demonstration in post #140, but it isn't need.1 + 0.999... = 1.999... Just to be clear, since there are an infinite number of decimals, you should never stop and compare things at a finite point in the long division. This is because stopping at a finite point would give you a result which is the division of a number with a finite number of 9s. You proceed until the number ends (which of course it doesn't).The first step of the division produces a 0
The second step 0.9
The third 0.99and so on....You generate more and more 9s and you never stop and hence you have 0.999999.... as your answer.

Hi lyx2no2, The proof was intended to show that 1 ≡ 0.999~ by assuming that it wasn't, and then showing that this results in a contradiction. In the process it uses another version of 0.999~ and the problem is that if one is not 1 then the other isn't either and it remains half way between. One can't use the conclusion as part of the proof eh?A much simpler process is take 0.999~, multiply it by two (=1.999~), where the 9's are exactly aligned from the decimal, and subtract the original (≡ 1), QEDEnjoy

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I don't understand this.You went logically along until the very end. I don't (logically) see any reason why you put a "≠" in instead of a "=" at the end. What would make you do so?Is it just because "0.9999|" isn't written using the same symbol as "1"?The minor proof you used here logically shows that we have two different symbols "0.9999|" and "1" and that they are both exactly equal to each other.Unless you also disagree that 1/3 = 0.3333|?
But... if you disagree with that, why would you use it within your own proof? That doesn't make any logical sense either.What if we don't use the symbol "0.9999|"? What if I replace that with the symbol "#"?Then your proof reads (without the final step):1 = 3/3 = 1/3 + 2/3 = 0.3333| + 0.6666| = #So, with equal signs all the way through... wouldn't you logically agree that the symbol "#" = the symbol "1"?Where is your disconnect?I am strongly starting to think that you do not agree that 1/3 = 0.3333|
If that's so, you should not use such symbology as valid within your own explanations... it will only add confusion. It's very illogical as well.

Yes, and neither are assumed to be one. However, they are reasoned to be the same as each other, by virtue of the continued long division. In the same way that you reason that your .999~ multiplied by 2 and with 1 subtracted is also the same, depsite the fact that it would not be true for a terminating decimal .9999.....9

IIRC, pi and e are not irrational, they are transcendental. The reals are all of the rational, irrational, and transcendental.

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If a nation expects to be ignorant and free, in a state of civilization, it expects what never was and never will be. --Thomas Jefferson

I could have, you know, looked it up. Yea transcendentals are contained in the irrationals. My bad.

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If a nation expects to be ignorant and free, in a state of civilization, it expects what never was and never will be. --Thomas Jefferson

They are both What they are not is algebraic.

Ah, now I see the problem. cavediver has already explained, but I'll make a go as well. That's not quite what is happening, although maybe I explained it poorly. Rather I take 0.999.... and 1 and divide them by two to find the number half way between them.
By the axioms of the real numbers, if they are distinct this must result in a number different from them both.
However when you perform the division you get 0.999...
So the number half between 0.999... and 1 is 0.999..., which is impossible unless they are the same by the axioms.A proof of the fact follows from a bit of algebra. If I have two numbers A and B. The number halfway between them is:
(A + B)/2If this equals one of them, say B, then:
(A+B)/2 = B
(A + B) = 2B
A = BIn my case we have A = 1 and B = 0.999... and indeed:
(1 + 0.999...)/2 = 0.999...So,
1 = 0.999...At no point did I assume that either version (I'm still not sure what this versions of 0.999... thing is about, maybe I'm missing something) was equal to 1. Or if I did, could you point it out?

He was c&p'ing a line from my demonstration that if we say that 0.9999| ≠ 1 then we embroil ourselves in a paradox.