Originally posted by obvekSorry, I was having a laugh, because the point of the story is that he can buy beer indefinitely because of the tangled loop in the exchange rate. I'm not sure I quite understand what you said at first. Could you explain again for my benefit?
so am i right, wrong, or in-between? I'm not talking just about pesos, of course, but about all types of money, like the franc, which is no longer used, or the euro, which has replaced the franc and many other currencies, too.
Originally posted by obvekI don't see specifically why it has to be money, but in fact the Argentinian unit of currency (I forgot what it's called) was worth $0.999... for quite a while IIRC.
What I said was, if the pesos or some other form of money equaled another form of money, could it possibly equal .9999999..., and would this be another way to answer the question originally asked in this forum?
Originally posted by TheMaster37here what i think is the absolute final proof...
a) Proof given in post is true, hence 0.999... = 1
b) As result you are a nut.
,99... can be rewritten as:
[9/(10^1) + 9/(10^2) + 9/(10^3) + ... + 9/(10^n-1) + 9/10^n]
where n APPROACHES infinity.
The LIMIT of .999... as n approaches infinity is 1.
HOWEVER!!! n never can actually reach infinity because a number n+1 can always be found... so while .999 becomes infinitesimally close to 1, it never can actually equal 1.
Originally posted by Gambitzoidactually i was wrong...
here what i think is the absolute final proof...
,99... can be rewritten as:
[9/(10^1) + 9/(10^2) + 9/(10^3) + ... + 9/(10^n-1) + 9/10^n]
where n APPROACHES infinity.
The LIMIT of .999... as n approaches infinity is 1.
HOWEVER!!! n never can actually reach infinity because a number n+1 can always be found... so while .999 becomes infinitesimally close to 1, it never can actually equal 1.
.99... repeating =
the limit as n approaches infiniity of
[9/(10^1) + 9/(10^2) + 9/(10^3) + ... + 9/(10^n-1) + 9/10^n]
that limit equals 1 which is equal to .999...repeating
Originally posted by Gambitzoid*bows*
actually i was wrong...
.99... repeating =
the limit as n approaches infiniity of
[9/(10^1) + 9/(10^2) + 9/(10^3) + ... + 9/(10^n-1) + 9/10^n]
that limit equals 1 which is equal to .999...repeating
There you are, you understand 🙂
The proofs in this thread are somewhat more cumbersome, as some people do not like basic proof involving limits.
You could have saved yourself alot of typing by writing 0.999... as 1- 10^(-n) with n tending to infinity 😛
Well, after reading the original post, and a few there after it. I decided to explain my reasonings here.
Note:I did not read many posts(8 pages was a bit to tedious)
This whole problem derives from the assumption that 1/3 =0.333333...So on and so on. But the problem is that .33333... is not equal to 1/3. It is merely as close as possible. There is not finite number that can represent 1/3.
The problem is that attempting to multiply the answer to 1/3 by 3 will give you .99999999...ect. ect. Many represent this number as 3/3, which equals one. As before, this number in NOT equal to 1, but is the closest number possible to being one.
I have my own logic that I work through, I just merely attempted to explain the workings of my head to you. If it didn't make sense, I'm sorry, but I can't really put that into clearer words.
You're close 🙂
From what i've heard of it, Non-standard analasis allows for infinitely small numbers to exist. In that NS-analasis you are completely right.
However, in the standard analasis those kind of numbers do not exist. You said it yourself; 0,999... is the closest number to 1. The difference is smaller then anything positive you can write down, hence the difference must be 0.
The point where you make a 'mistake' in standard analasis it that 1/3 is exactly 0,333... The reason for that is the dots. They say that the row of 3's is infinite. No matter how many you write down, the actual number has more 3's. An approximation for 1/3 is 0,33 or even 0,33333333, however 0,333... is the exact value.
If you have problems with what i said about the difference being 0, look back to one of my earlier posts, i gave a nice proof about it 🙂