[split] 0.999... equals 1

[CliveStaples]

But it's mathematical chicanery. The mind always wants to make numbers real. Here we go: 5.13 x 10^61. The number of the universe; well, age in Planck time. But the difference between finite and infinite is not a number, it is a perspective. One cannot write 9/10^(n) on to every n quark in the universe to add to to one, just ain't done; so whether or not it really sums to one is a metaphysical concern.

If you want to deal only with finite quantities, then just use some real analysis.

Take x = .999...

If we agree that x is indeed a real number, then by trichotomy either x > 1, x < 1, or x = 1. Clearly x > 1 is false; thus x < 1 or x = 1.

Now, suppose x < 1. Then |1 - x| must be positive; call it c. But for any positive c you give, I can show that c must be smaller. Thus c cannot be positive. Thus x < 1 is false.

Hence x = 1.

[Categories+Sheaves]

I'mma drop some links

See: Wikipeida or Ed Nelson's introduction to nonstandard analysis for systems of analysis where this sort of thing need not happen.
There systems do treat infinitesimals as genuine, nonzero quantities. Here we have that .999... ~= 1 i.e. they are infinitely close, but we do not insist that .999... = 1.

Now, I'm a big fan of the standard real numbers, and I strongly prefer using them to the other systems out there. But the other systems are still out there. We do get this equivalence of .999... and 1 once we accept all that business with cauchy sequences and epsilon-delta. But definitely not before then.

[CliveStaples]

I'mma drop some links

See: Wikipeida or [url=http://www.math.princeton.edu/~nelson/books/1.pdf]Ed Nelson's introduction to nonstandard analysis[/i] for systems of analysis where this sort of thing need not happen.
There systems do treat infinitesimals as genuine, nonzero quantities. Here we have that .999... ~= 1 i.e. they are infinitely close, but we do not insist that .999... = 1.

Now, I'm a big fan of the standard real numbers, and I strongly prefer using them to the other systems out there. But the other systems are still out there. We do get this equivalence of .999... and 1 once we accept all that business with cauchy sequences and epsilon-delta. But definitely not before then.

Well, of course I assume we're talking about the standard presentation of the real numbers, which are an Archimedean field.

(But I suspect that even in the hyperreals, .999... = 1.)

[Categories+Sheaves]

(But I suspect that even in the hyperreals, .999... = 1.)

Mmk. Goldblatt's Lectures on the Hyperreals came in the mail this week...

Without getting into any business with ultrafilters, it turns out that there are too many elements in *R that I would have wanted .999... to be.
The issue is that we don't know whether .999... is referring to the equivalence class of [.9, .99, .999, ...] or [.99, .9999, .999999, ...] etc. (yes, these are two distinct elements of *R, and the first is less than the second).

So if '.999...' is a well-defined symbol for a single thing, we'd have to treat it as an element of R and not *R - R...

So: Touché. If our writing '.999...' is to stand any chance of being intelligible, .999... = 1

[Tiberius]

Another win for mathematics. :-)

[john_gabriel]

All the proofs in favour of the equality 0.999... = 1 are debunked in the following article:

<snip!>

Allow me to debunk the most common fallacious proofs:

1. 1/3 = 0.333....
Well, a little known fact is that 1/3 is NOT equal to 0.333...
See Pages 33-36 of article.

2. There is no number between 0.999... and 1.
True. This is due to the fact that 0.999... is not a well-defined number.
Article explains more.

3. x = 1(0.999...)
10x= 10 (0.999...)
9x = 9(0.999...)
x = 0.999...

Wha?! Yes. If you don't do anything stupid, like try to multiply a quasi-number object by 10, you can predict the output of the algorithm exactly.
Arithmetic is designed to work with well-defined mathematical objects called the rational numbers.

For more on this, see pages 12-18.

To learn much more, read the entire article. Do visit my New Calculus site for the first rigorous formulation of calculus in history!

[Aractus]

Firstly welcome to the forums.

Secondly, you're full of shit.

[Tiberius]

I quickly skimmed over the first few pages of your article. I stopped when I came across a sentence blaming an apparent confusion in set theory on the fact that the father of set theory was a "bipolar Jew".

[Dee Dee Ramone]

Bi = 2... I don't see the problem :-)

[Aractus]

I quickly skimmed over the first few pages of your article. I stopped when I came across a sentence blaming an apparent confusion in set theory on the fact that the father of set theory was a "bipolar Jew".

ROFL! I stopped when john_gabriel couldn't even present an argument without referencing this mythical material in a PDF (in other words, I didn't bother with the PDF).