Here’s an old argument between myself and a friend:
Does .9999… (nine repeating forever) really equal 1? I’ve had people swear it does, and try to prove it w/ equations, but the equations just boiled down to saying “.999 =1 is true because 1 = .999”. Can anyone offer any decent proof either way?
This has been discussed before. .9999 is shorthand for “The limit as n goes to infinity of .910^(-1)+.910^(-2)+…10^(-n)” which is indeed one.
Rounding. Odds are you learned about it in first grade. No, .999… does not equal one. It equals .999… . Oh, you said “decent proof”. Well, in that case, I can’t answer your question. But I’m guessing that it equals .999… . Anyway, if it really did equal one, then 1.999… would really equal two. So we wouldn’t even have decimal points or places.
Well damn. I didn’t even see Ryan’s post. Okay, if you want to get all technical. 
Pay no attention to my previous post.
This is often misunderstood because people forget the infinite repeating part.
Here goes the equations. Remember that a number with a line over it represents a repeating pattern.
Let:
_
N = 0.9
then
_
10N = 9.9
Since there's an infinite number of nines after the decimal:
_ _ _
10N - N = 9.9 - 0.9 = 9.0
10N - N = 9.0
9N = 9
N = 1
In case you’re wondering if this technique is valid, here it is used to determine the fractional representation of another repeating decimal:
_
N = 0.06
_
10N = 0.66
_
100N = 6.66
_ _
100N - 10N = 6.66 - 0.66
_
90N = 6.0 = 6
6 2 1
N = --- = --- = ---
90 30 15
N = 1/15
Well, how do you distinguish between two points (in this case one and .999…)? Well, if there’s a point between them, then they’re separate points, otherwise they’re the same point, right?
Okay, no equations, just the simple fact that there is not a single point between .999… and 1. Not a one.
As for the equations:
10 * .999… = 9.999…
9.999… - .999… = 9
Now, 10 * .999… can be written as 10x and as 9.999… so
10x = 9.999…
and .999… would be x
so 10x - x = 9x
or 9.999… - .999… = 9 * .999…
HOWEVER we noted that 9.999… - .999… = 9 so if 9x = 9, what does x equal? (the last step is left to the reader)
Oh, and slvr’s right, 1.999… does indeed equal 2. However, she’s wrong in that this does not invalidate the decimal system. 1.888… is still distinct from every number other than 1.888… Just think of it in terms of “is there a point in between A and B?” What about 1.888… and 1.9? 1.89
Nice and simple, this is not a contradiction of the rest of mathematics, but an inevitable result.
Just out of curiosity, where is the thread that discussed this? I’ve searched but I apparently can’t seem to hit on the right keyword or something… BTW, 0.999… is the same as 1. The algebraic proof shows that well enough. However, calculus works a bit differently. In calculus you can meaningfully talk of two numbers with an infinitely small difference between them. Imagine this: You take Zeno’s paradox and express it in a program that calculates it infinitely. That is, the program adds 1/2 to 1/4 to 1/8 to … without end. In theory, the program can be said to count in a fashion that takes it infinitely close to 1 but never lets it reach 1. In fact, the program does reach 1 because the system can only hold so many decimal places before it begins to round. This concept finds practical use in relativistic physics, where the acceleration of a particle can be plotted along a curve that acts the same way that Zeno’s curve works, replacing 1 with c, the speed of light in a vacuum. In theory and in practice the particle’s speed can get very close to c but can never quite reach it. Interesting, in an odd sort of way.
Well, shit, didn’t even notice that by the time I got around to answering, it’d already been answered better.
The proof that I always like to give goes like this:
1/3 = .333333…
(1/3)*3 = 1
.33333… * 3 = .999999999999
Therefore, .9999 = 1
It usually shuts them up.
mouthbreather
Thank you. It’s like a trip down Memory Lane. There were at least three such treads at that time. Here is another for your education/amusement.
Another way to put it, I guess is, if .9999 doesn’t equal 1, then Zeno’s paradoxes are true. Which throws the concept of Calculus, and algebra for that matter, out the window. One of them was, a man standing in a room cannot touch the other wall, because in order to do so he would have to walk half the distance, then half again, and half again, etc. You have to say the limit of the sequence a[n]/10^n is the full distance of the room (1) as n approaches infinity. Of course nowadays, one can get nitpicky and say he never actually touches the wall because the atoms of his finger never actually contact the atoms of the wall (unless a nuclear bomb drops on him and splatters him all over the wall…).
To argue that point just use the second paradox as a arguement, of course we know Archilles actually passes the tortoise, because his arguement was that in order to actually pass the tortoise he would have to first be at the same point he is - then overtake him. In other words, in order for you to actually say he passes him you have to say he actually gets to that point, in essence you say that .999… converges to 1, THEN you can get numbers greater than 1 again, and at precisely that point, 1, he is right beside him, and then runs past him.
This is more of a “logic”-type and popular arguement settler for this question.
Did you try “.999”?
Actually, Ryan, I was looking for an obscenely old thread I remember from before I registered. I haven’t been able to find that exact thread, but I’ve found ones that are essentially clones of it. BTW, links to threads created in the UBB system don’t work. Ferexample, follow this link: http://boards.straightdope.com/ubb/Forum3/HTML/007496-2.html You won’t get to the thread (not by just pointing and clicking, anyway) but you’ll instead be routed to http://boards.straightdope.com/sdmb/ which is the front page to all the several SDMB boards. I’m almost certain this has to do with the ‘/ubb/’ in the addy, which was dropped when we switched from UBB to vB. It’s kind of annoying to those with so little to do we read old threads.
All arguments about .999… = 1 in my mind boil down to what do you believe 1/infinity to be. If you say 0 then they are equal. If you say no, its a really really really… Small number, then they are NOT equal. Some people don’t believe in infinity, some people don’t believe in 1/infinity (other than saying it is zero, which is just saying it doesn’t exist).
People use informal proofs like:
1/3 = .333…
And since 1/3 x 3 = 1
Then 3 x .333… = .999… = 1
The assumption here is that 1/3 in fact exactly = .333…
But if you do the math for your self you can see after every (3) in the .333… There is always a remainder. What we are supposed to believe is some how after infinitely many 3s this remainder vanishes. And why should that be? I would say it’s still there, you could say its remainder is in fact 1/(3 * infinity).
So lets look at the informal proof again:
1/3 = .333… + 1/(3 * infinty)
3 * (.333… + 1/(3 * infinity)) = .999… + 1/infinity = 1
This is not the same as saying .999… = 1 !
All algebraic proofs use this trick to make you forget about the 1/infinity.
.99 * 10 is not 9.99 but rather 9.90
Yet people will say matter of factly .999… * 10 = 9.999…(9) that extra 9/infinity just gets convenirntly thrown in there. Why not = 9.999…(0)? Okay so that number is hard to think about and so is 1/infinity. But the proof is none the less… A cheat.
There are various calculus based proofs that are supposedly more rigorous, but built into the foundation of calculus is the concept of limits which allow us to say 1/infinity is arbitrarily close to zero, so lets just call it zero. For practical purposes, this makes sense. Lets face it if .999… Is not equal to 1, it’s damn close. We all agree. But that’s different than saying it IS EXACTLY 1.
Let me ask you this. Is there a number bigger than infinity? How many rational numbers are? Infinity? Yes. How many real numbers are there? Infinity, yes, but also provably more infinite than the number of rationals. How can a number be bigger than infinity? I don’t know but clearly there can be. So when people try to prove by contadiction that 1 - .999… Can’t exist because it would be the smallest number > zero, yet certainly that number divided by 2 would be a contradiction. Its like arguing infinity can’t exist because than what’s infinity + 1, or infinity * 2.
I have never seen a proof for .999… = 1 that at its basis doesn’t assume 1/infinity = 0. And we all know 1/infinity is undefined, so at best this question is undefined.
Every time I begin to think I am the smartest person in the world, some udiot comes along to prove me even smarter. ;-p
Just more proof that zombies are bad at math.
I believe that .999… equals 1 and that 1/infinity is not equal to zero. What now?
Please don’t laugh at me, but can you have a repeating decimal when measuring time?
edit: If I say that I got there in 10.33… seconds and Jamie got there in 10.34 seconds, did we arrive at the same time?
No, it’s not the same time. 10.33… is equal to 10 and 1/3, which is a different number than 10.34.