Phage, you seem very knowledgeable and sure of yourself, are you a mathematician?
OK Phage, the best way to show you that 0.999… can be evaluated to 1 is to use a real life situation:
one of Xeno’s paradoxes was the arrow, I’ll assume you know the basics of it. Now imagine the arrow travels 9/10, 99/100, 999/1000…, etc.of length l (where l is defined as the length between the bow and the target), by your logic the arrow will never reach there as you say that 9/10+9/100/+9/1000… cannot be peformed.
(to infinity, that is)
IANAM, but if Phage wants a logical proof, perhaps a geometric example would help. (I trust that the experts will keep me honest here.)
Instead of a number line, take an equilateral triangle of area = 1. Draw a line from each vertex which bisects the opposing side. You should now have six smaller triangles. Now, take the pairs of adjacent triangles along the sides of the original triangle so that you have three triangles of equal area.
Compute the area of each of the three triangles. The result should be 0.333… for each triangle. What is the sum of the three triangles? 0.333… + 0.333… + 0.333 = 0.999…
But since the original triangle had an area of exactly one, if 0.999… is not equal to one, then where did the missing infinitesimal area go? Can your logic explain that, Phage?
Phage says:
Skogcat asks:
Unless Phage is just yanking our chain – a very real possibility – then the answer has to be “no”. Open any text on calculus and you’ll find plenty on defining and calculating limits.
I don’t see any point in prolonging this. To a mathematician Phage’s last post is nonsensical. I don’t mean that as an offense, it’s just the case. To Phage, the mathematical posts are nonsensical. I can’t see how it would be possible to bring either harmony or a conversion (forgive me), and I can’t see a third result.
By the way, Phage – Orbifold did not define a real number as having that property – he said that was a property of real numbers, a very different statement. And he did not say that 0.999, is not a real number. He says that it is, and is equal to 1.000…
Phage wrote:
As has been already stated, one of the rules of math says that between any two distinct real numbers, there are an infinite number of real numbers.
You claim that 0.999… and 1.0 are two distinct real numbers, not equal to each other.
Since you haven’t yet proven that there can be two distinct real numbers without an infinite number of reals between them, your statement that 0.999… is the closest real number to 1.0 is ludicrous.
Orbifold and kabbes are not saying that 0.999… is not a real number, they are saying it is not different from 1.0, or that the two numbers are not distinct.
Only the representation of the two are different. They are the same number, represented in two different ways. Much like 1.0 and 1.0000 are the same number. Or 1/3 and 0.333… Or 3/9 and 0.4[sub]12[/sub].
This is nothing but a rejection of the standard notation. There’s no reason to write out the entire decimal to do math with it if you know what all of the digits are. Nobody is going to sneak a 7 into 0.999… while you’re trying to multiply it by ten, which is why, since multiplying by ten is simply a decimal shift to the right, we know that 0.999… is equal to 9.999… without needing an inifite amount of time. Or are you claiming that I actually need to do ten additions to find the result of 498764202376.0 times 10, and simply shifting the decimal point is invalid?!?
I like Phage’s idea that we arbitrarily choose to say that .9… is equal to 1 even though it isn’t, just because that makes it easier for us. If I could’ve pulled anything like that, integration theory would’ve been so much easier…
A number does not have to be calculated to exist. It may take infinite time to count to it, but existing, it equals 1.
Luckily, there are faster ways to examine, manipulate, and define numbers than adding. Calculus is handy that way as it examines areas.
It’s time for Phage to state his entire math education so that we can know where he received his unique understanding of what a limit is.
That will help us determine whether he should sue his school for malpractice or whether the school should be suing him for defamation. 
Don’t mistake me for some dumbass that dumps other people’s proofs with a blind eye. I don’t deny that mathematics can prove .99… = 1. But I do get an itch when I hear it, that’s all.
Originally posted by Donut: you could at least take one of the definitions provided that is specifically mentioned as being used in mathematics.
So, is the definition of “infinite” that I linked NOT the same as the one being referenced in the OP? I really am asking. If they are different definitions, please post the proper definition.
No. An infinite sum is the value approached by a finite sum as you add more and more numbers. When an infinite sum exists, you can approach it with arbitrary precision by adding enough terms.
One more time:
If 1/3 = 0.3333333…
then 1/3 + 1/3 + 1/3 = 0.3333… + 0.33333… + 0.33333…
then 1 = 0.999999…
Ah, but Mr. Phage insists 1/3 != 0.333333…
Fair enough.
But I’ve shown that dividing 1 by 3 gives you a result of 0.333… and that by simple long division, the value of the nth decimal place is 3.
So for 1/3 != 0.3333…, there must be a decimal place that does not have the value of 3.
Where is it?
(Note that Phage has been ducking my points time and again. He cannot find anything wrong with my proofs, so refuses to address the issue.)
What’s confusing the issue is the fact that 0.99999… is not infinite. You can clearly show it on a number line (even if you don’t belive it’s one). What it does have is an infinite number of decimal places. That does not make it an infinite number, since any number has an infinite number of decimal places.
I’ll assume Phage won’t answer this, either.
criminalcatalog: consider that an infinity can be bounded by the finite. There are infinities all around you. It is you who cannot grasp them.
Right. Here’s an example that I think is helpful: how many numbers are between 1 and 2?
I can accept that .9999… = 1 if I can accept that an asymptotic function will ultimately equal its asymptote as the function extends to infinity.
I have a problem with that.
I have read the whole thread and I have to say that no matter how far, for how long, for how many decimal places you calculate, 0.9999… will never = 1.
Quod Erat Demonstrandum (As it was to have been shown)
Stopping by to say I’ve been reading with both growing alarm and amusement at what should have been a very simple question to answer.
I’ll add my two bits. I’m a computer scientist by career. I’ve done my fair share of math. I’ll toss my considered opinion in the 0.999…=1 camp.
Mostly because I have no problem believing that mathematicians can perform ‘infinite’ operations in finite time provided the problems are well formed.
Also because I’ve performed those calculations myself.
And I’m still alive.
And I’m not still working on those problems.
And I’m confident that by application of logic and mathematics that my answer is correct, and this seems to be agreed upon by mathematicians much much brighter than I.
Calculus itself would be entirely useless without the ability to understand and perform limiting operations to get an exact answer. Ways of working out exact limits of arithmetic, geometric, and exponential series have been around a long time in pure math and in applied math (physics, chemistry for example).
The assumptions, definitions, and behaviours are well known, and they can perform useful work and prove useful and correct things in a consistent and rigorous manner.
I can’t contribute more proof than the true mathematicians of the board.
I still await Phage’s proof that 0.999…!=1, or that 0.999… is a distinct real number from 1. Because everyone on the other side has provided theirs.
Phage has more than willingly provided a definition of what he thinks 0.999… is (next smallest number below, but not equal to 1), but has provided nothing to show that this is true, or that such a number can exist. Counterarguments, on the other hand, have been provided.
My mind just boggles at this thread.
My best of luck, actually, to Phage for willing to try to provide extraordinary proof for such an extraordinary claim… 
Nope, sorry. Q. E. D. (among others) has already shown that 0.999… = 1.
I, of course, having posted all that, put forth the caveat that you are willing to accept various mathematical axioms on which the various necessary number lines are built, and on which higher level math such as series, limits, and calculus are well defined.
Feel free to put forth new axioms to invent math or logic that follows different rules. After all, that’s how non-Euclidean geometry came about, and its perfectly good.
You’re wrong, but you’re wrong in an interesting way, so I’ll go ahead and attempt to enlighten you.
The statement “After any finite number of decimal places, .9… is not equal to 1” is correct. It’s also not what we’re talking about. .9… has an infinite number of decimal places, and is in fact equal to 1, by the definition of a decimal representation. That’s what we’re talking about.
And knock off the QED. Asserting something with no proof doesn’t constitute showing a damn thing.