2=1

[IAmACalculator]

Numbers are merely abstract ideas that express quantity. You can never have .9999 repeating of something, so the number is meaningless. Therefore, we must use the closest possible meaningful approximation, assuming we want to actually use this number for something (if we don't, what's the point?), which would be one. Also, π is irrational, and so, doesn't exist. Since we cannot find the whole thing, we cannot prove that any given answer is correct, unless given in meaningless terms of π. Therefore, all irrational/repeating numbers do not exist.</gobbledegook>

[chipmaster]

[Weregoose]

My current position is this:

0.999…≠1

The limit as n→∞ of 1-1/10ⁿ=1

Thoughts? I'm sure there are a few from those who have taken calculus, because I haven't yet.

[EDIT]

Three posts all at once. Yikes. :roll:

[IAmACalculator]

[DarkerLine]

[WikiGuru]

[DarkerLine]

I don't think you quite understand what I did there. I cross-multiplied. The flaw is in a different line.

[WikiGuru]

I see...
2^nd line, right :biggrin: ?

[chipmaster]

[IAmACalculator]

[JoeImp]

I can't believe I've been slowly refreshing the first page for a few hours, waiting for a new response, and never noticed it went onto a second page.

[c_plus_plus]

OK. I must aploligise for my horible proof. But, to add on to this, even the math teachers in my school are divided over the subject about if .9 repeating does or doesn't equal 1.

my opinion is that it does.

[JoeImp]

What I think, anyways. Makes sense.

[chipmaster]

I can't believe I forgot about inifinite series. Well, I don't think that, that will settle it. The thing about the infinite series is that it approaches that number as the number of terms of the finite series grows. You could do sum(seq(9/(10^X),X,1,999999999 and if the calculator had enough precision (and was able to make a list with 999999999 elements, and sum them in a quick enough time) it would not find the answer to be 1. You would just get closer and closer as the elements of series increased, but there would always be a difference (yes, an infinitely small difference--wait, now I think they are the same).

I think this issue is all about how you look at it. I, myself, have flip-flopped back and forth from one side to the other. Even as I type up this post I feel I have switched my position again (midway through). It's an interesting thing to think about, but I don't think it really matters in the scheme of things. .999999^- is close enough to 1 or is 1 (whatever your side of the matter is) that we can just call it one. You may not agree that it is or isn't one, but you have to agree that it is so infinitesimally different such that the difference between the two (if it exists) is of no use to anything in the real world (I mean that cutting a board to .99999^-in is no different than 1 in, I realize mathmatics may be affected).

[Brazucs]

[IAmACalculator]

[c_plus_plus]

Thats what I said!

Edit: i meen the last paragraph.

Edit2: I have found more proof:

The comon way to change a repeating decimal into a fraction is to devide the repeating decimals by that many 9's.

for example
__
.21 is equal to 7/33

so:
if you put 9/9 you get 1 which is somewhat returning to my original idea.

[chipmaster]

When you look at it from the point of view of limits you can have a certain degree of confidence. The whole concept of limits, is that as a number n approaches a number x, the function f will become closer and closer to a variable a. With limits we accept that when n = x, f(n) = a, but this can never be proven in most limits because we can't divide by zero. We can see a clear pattern and observe that it should be that.

Interestingly, with this logic, we can prove that x/0 = ∞ (where x=/= 0). Let's say that x = 1. 1/.01 = 100, 1/.001 = 1000, 1/.0001 = 10000... and so on. So as n approaches zero in the function f(x) = x/n, f(x) approaches ∞. Therefore, with limits, we conclude that the limit as n→∞ of x/n is = ∞.

Now we can make this assumption, even though we can never witness a number being divided by zero. In the same sense, we can't ever see an irrational number, but we see that as x→∞ in the formula f(x)=9/(10ⁿ), f(x) approaches 1. Can't we conclude that .999^- = 1 with this logic? I say so. I now am firmly convinced that .999^- = 1, and I'd like to see anybody prove me otherwise.

Edit: Here's a good read!

[c_plus_plus]

I just saw that site also the interesting thing is a friend told my einstien devided by 0 and got finite results in his proces of calculating E=MC^2

[chipmaster]

He found a finite number big enough that when multiplied by 0 it resulted in a non-zero number? Astounding!...and almost unbelievable