Questions in the title.
0.999… + 0.999… = 2.
if .999…= 1 does .999…+.999…= 2 or 1.999…8?
If in fact we look at the fact that .99999… never ends, there will be no rounding. And since we are getting microscopically closer and closer to one, there is no way that you will get to one. No matter how many 9’s you add, you’ll never get all the way to 1. Just really close.
However, you can look at it as if you are moving away from one, however you’ll never go very far. You never really move away from it because .9999 is never ending and any fraction of a step will be far too far. (support … http://mathforum.org/library/drmath/view/55746.html)
Thus, to answer your question 
, it will equal 2. You cannot round because .9999… is never ending and any no matter how far you go you’re over the edge.
- The doper formally known as Vixen_Llama
If 0.999 … = 1, then, by definition 0.999… + 0.999… must equal 2.
i.e. If 0.999 … = A, then 0.999… + 0.999 … = A + A
Please look here, especially at the post with all the linked threads.
1.999…8 is not meaningful. That would mean that you put on an endless string of nines, and then after then end of that string, an eight. But we just said that the string of nines was endless.
Now, you could say that 0.999… + 0.999… = 1.999…, but that’s just another way of writing 2.
You’re kidding, right? 0.999… is exactly equal to 1. It is not a matter of “closer and closer, but not quite.”
Arithmetic shows us that:
x = 0.999...
10x = 9.999...
9.999... = 10x
-.999... = - x
-------- ---
9 = 9x
9x = 9
x = 9/9
x = 1
Not “close to” 1. Exactly 1.
Oh yeah!
1.000…
- 0.999…
= ?
Math can’t handle infinity, so we pretend.
1 - 0.999… = 0.
It must be zero. If it isn’t zero, what do you claim the answer is?
1/3 = 0.333…
2/3 = 0.666…
3/3 = 0.999…
3/3 = 1
Therefore, 1 = 0.999…
I claim the answer is the assumption that infinity is indeed infinite. That’s just a convenience math uses because there’s no other way to handle it.
I also claim that .999… = 1 because we can’t prove otherwise.
The difference between .999… and 1 is… “The Big Collapse”.
This’ll all work out when Hawking finds the unified thingy.
:eek:
Damn, I’m a masochist.
.9… = sum( 9/10[sup]i[/sup], 1 < i) = 1. There’s no convenience about it. Check Rudin’s Principles of Mathematical Analysis, 3e if you doubt me.
There’s no assumption. “Infinity” is defined as “infinite”. C’mon.
Another way to think of it: If I say that “Glorpth = 3”, and then perform the equation “Glorpth + 3 = 6”, can you criticize it by stating “You just assume that Glorpth is equal to 3!” Well, no, it’s not an assumption… it’s a given.
Of course, this is ignoring the fact that Glorpth is actually the General of the alien space forces coming to enslave our puny race, and is not, in fact, equal to 3. But hey, it’s a hypothetic. What’re you gonna do?
Where did you get your 0.99999, in the first place?
Suppose it was something like this.
How do you express 1 divided by 5 as a decimal number?
Easy, it is 0.2 you just do the long division.
Ah, but what about 1 divided by 3?
So, you try it, and after about six hundred iterations of long division, it dawns on you that this sucker is not only never going to come out even, it is always going to keep repeating the value 3 for every position to the left of the decimal point.
Hmmmmm, well, I will just draw a line over the third three after the decimal point (or in our example, follow it with four commas), indicating that it will repeat forever.
So, What is .333, plus .333, plus .333, equal to?
One.
But how can you say that, since it works out to .999,?
Well, I defined it to equal 1 divided by three. That is how I can say it. It is my decimal convention for the rational number one third. And one third, plus one third, plus one third is equal to one. So, it turns out that my decimal notation shorthand has not changed the fact that three thirds is one. Accepting that .999, is equal to one is just deciding to follow my own choice of notation in decimal numbers.
Tris
Sorry, math can and does handle many infinities with astonishing grace.
If it makes you happier you can say it equals 0.000000…
which of course equals 0.
Sadist and masochist sitting together. Masochist begs, “Beat me! Oh, please, beat me.” Sadist folds his arms and with a wicked grin responds, “No.”
[Twin Peaks]
IT IS HAPPENING AGAIN.
IT IS HAPPENING AGAIN.
[/Twin Peaks]
Seriously, folks: 1 == 0.999… == 1.000…
There is no discussion. There is no loophole. There is no philosophical handwaving.
This is mathematics, for crying out loud, not politics. Rules work in mathematics.
Okay, okay. I’m just messing with all you math geeks. I was convinced that
1=.999, a couple of years ago, by another (very good) thread on these very same boards.
But what’s wrong with saying that 1/3 can’t be expressed as a decimal?
I can eat 1/3 of a (pecan) pie. I can’t eat 33.3333, of a pie, even though it’s exactly the same thing.
BTW; “The Big Collapse” is the opposite of the “Big Bang”, when physics don’t count no more. All our rules die.
1/3 can’t be expressed exactly in a decimal expansion. It can be expressed as the limit of a sequence of decimal expansions.
Incidentally, the “big collapse” isn’t really the opposite of the big bang. For one, there’s a unique initial singularity in the big bang, while the collapse (if it happens at all) may well split into multiple final singularities.
Doesn’t look like there’s going to be a big crunch in our universe’s future. All evidence points to the contrary: the eventual “heat-death” of maximum entropy, long after the last star has burned out and the last black hole has evaporated, one particle at a time.
I was bummed out when I heard about this discovery. It appears that the universe will have active stars for as much of its lifespan as the earth has had humans in its history up to now. An almost negligably small fraction. 