I think it's one of those weird math things. They're not actually equal, but the numbers work that way. Wikipedia uses this as an example:
1/9= 0.111...
9 times 1/9 = 9 times 0.111...
9 times 1/9 = 1.
Therefore, 1 = 0.999...
They lost me. Forgot me. Made you from parts of me. If you're the One, my father's son, what am I supposed to be?@DDR, midget, Wulf: No, the difference is actually zero. .999... == 1. Exactly.
The 1/3 = .333... thing is a good proof of this. Another is:
- 1 - .99 = .01
- 1 - .999 = .001
- And so on
- Therefore, 1 - .999... = .000... (that is, an infinite string of zeros)
- An infinite string of zeros is equal to zero.
- Therefore since the difference between .999... and 1 is 0, .999... = 1
@Black Humor- Doesn't make sense. They're functionally the same because there's no end to infinity, but if that end did come, there'd be a last nine and a one.
They lost me. Forgot me. Made you from parts of me. If you're the One, my father's son, what am I supposed to be?Oh great it's this thing again.
Sure the difference at a large enough chain of digits is insignificant, but .999 repeating is not equal to 1. (I mean seriously, Blizzard Entertainment did this exact thing as an April Fools Joke about 7 years ago.)
@OP: It's a fairly common debate that crops up everywhere. Don't expect any kind of general agreement on which side is right.
On the bright side, at least your friend isn't arguing that division by zero should be possible. I don't think we need the message of "everymathaxiomever.exe has stopped working" popping up on this wiki.
edited 9th Sep '11 4:20:48 PM by TotemicHero
Expergiscēre cras, medior quam hodie. (Awaken tomorrow, better than today.)No, mathematicians are pretty unified on this one.
If anybody is not convinced, there are far too many proofs right on Wikipedia
. Note the deeper implications about real numbers.
edited 9th Sep '11 4:58:23 PM by Tzetze
[1] This facsimile operated in part by synAC.It's exactly one, not nearly, not almost but exactly. The "thirds" thing is the easiest to understand proof, I think, and issues regarding it are to do with limitations in numerical representation.
It's easy to understand that 3 x 1/3 = 1. That is, three thirds are one whole. But representing that decimally is the issue, because numbers do not allow that. Thinking about it, they're actually pretty limited in that regard. Thus, the best we have is 0.3333 repeating.
When 0.333 is taken as representative of a third, three times that number has to equal three thirds, or one whole. So, 0.999 is three thirds, or, one.
That said, 0.999 does not necessarily have to equal 1, because it can also be the closest number to 1 in its own right. That is, 0.333 need not be a decimal representation of 1/3, it exists as a number on its own anyway. As does 0.666, as does 0.999 or any other fraction.
Conclusion: If 0.333 is simply a decimalisation of the fraction "one third" then 0.999 as a decimalisation of "three thirds" does indeed equal exactly one. But when 0.999 is simply a number in itself, then it means no more than that. Everything depends on the context.
My name is Addy. Please call me that instead of my username.![]()
Yep. Cross-multiplication is all you need to think of this. (this would be easier if I could use multiple lines for the equations effectively)
3/3 = 1/1. Cross multiply, and you get 1 = 3 x 1/3. Divide both sides of the equation by three and decimalise the result on the right hand side and you get 1/3 = 0.333 repeating, and from that you simply multiply both sides of the equation by three to give you 3/3 = 0.999 repeating.
Thus, 1 = 0.999 repeating.
Note that it is fallacious to assume there can ever be a "last 9". When we're talking about infinity, the notion of a final number simply goes against logic - by definition, infinity does not have an end.
edited 9th Sep '11 6:01:49 PM by CaissasDeathAngel
My name is Addy. Please call me that instead of my username.No, mathematicians are pretty unified on this one.
QFT. To a mathematician, this is like arguing whether 1 + 1 = 2* In the integers
Blind Final Fantasy 6 Let's PlayOH this again...
Every fraction has a decimal equivalent, and by equivalent, they're equal. In decimal system, certain numbers have more than one decimal forms.
1/1 = 1.0000... = 0.999...
1/2 = 0.5000... = 0.499...
1/3 = 0.333...
edited 9th Sep '11 7:07:19 PM by abstractematics
Now using Trivialis handle.

Kexruct and I have been arguing for hours about this. (Read title.) Does any one have evidence they would like to present on this topic?
DEM PIGEONS