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[Jreengus]

The problem with infinity is that so many operations are undefined.
Also, people use it without regards as to where to came from!

what spontaneously caused "a 1/infinity" in your system?

I mean, seriously!
Also, there were about 10 examples of 0.999...9=1 on wikipedia,
as well as 3 examples of 0.999 not equal to 1.
So thats an awfully poor way to resolve this dispute!

If you stop using infinity the problem might loose this "infinite difficulty"
: P

The thing is in order to fully understand this you have to have a good working understanding or limits which is actually quite hard because people have trouble comprehending that there is no smallest value of Z you can't put any set value on it without it becoming 1 or having a smaller number possible.

As I said earlier mathematicians just say it's arbitrarily close to 1 and let it go at that.

Although according to my lecturer when he was at uni they used to play a game saying things like I can get to within  0.000001 of the value and then someone else would get within 0.0000001 etc.

Spoiler (click to show/hide)

That story makes me eternally grateful for the internet.

[PTTG??]

How exactly can you have more zeros that the unending value implied by the ellipses? 0.000...1 has no lower value other than 0 (And negative numbers, of course).

And let's just be clear on something here... 0.999... = 1. That's pretty much proven, right? It makes sense, after all. three thirds is one AND 0.999..., after all.

Let me explain where I got infinity here...

Look, X>Y, X-Y=Z is our equation. Z must be positive... and can't be zero.
So basically we're asking what's the smallest possible positive number.
That's kind of the way I approached this. I figured that the best way to reach that would be 0.000...1. Then I figured that 1/infinity would also be pretty low too. I assumed that they where similar numbers- though I admit that I can't prove that.

[stummel]

http://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit

you all should take a look at this. maybe then you get, that there won't be a neighboured number within the real numbers, as it works for all numbers above the rational ones (those included of course).

[Jreengus]

First up you can't have infinity as a number, just like you cant have 1/0 as a number it's more an idea than a number.

The problem with saying 0.000...1 is really that the 0.000... never ends, you never get a 1 because the zeros never end.

You could try coming up with a proof for your idea to show I'm wrong but that would be at a level of mathematics of ungodly complication.

[eerr]

How exactly can you have more zeros that the unending value implied by the ellipses? 0.000...1 has no lower value other than 0 (And negative numbers, of course).

And let's just be clear on something here... 0.999... = 1. That's pretty much proven, right? It makes sense, after all. three thirds is one AND 0.999..., after all.

Let me explain where I got infinity here...

Look, X>Y, X-Y=Z is our equation. Z must be positive... and can't be zero.
So basically we're asking what's the smallest possible positive number.
That's kind of the way I approached this. I figured that the best way to reach that would be 0.000...1. Then I figured that 1/infinity would also be pretty low too. I assumed that they where similar numbers- though I admit that I can't prove that.

unfortunately, 0.000...1 isn't a real number.
because for any value of Y,
(Y+X)/2 <Y must remain true.

of course, you end up doing division on a number that is already the smallest it could ever be!

1/infinity /2 < 1/infinity

This impossible by the very definition you've given!

[bjlong]

Easiest proof:

.999...+.000...1=1
.000...1=1-.999...

But .999=1

.000...1=0

This is really bad notation.

[Jreengus]

of course, you end up doing division on a number that is already the smallest it could ever be!

1/infinity /2 < 1/infinity

This impossible by the very definition you've given!

Or you know you just proved infinity doesn't act like a real number.

Seriously guys stop trying to use infinity like a number.

[Neonivek]

Well there are three types in infinites I know

Infinity: Unlimitedly high
Infantesimal: Unlimitedly low
Infinitum: Unlimitedly repeating

[Jreengus]

And as far as I know none can be used as a number.

Mathematically using infinty like you would use say 3 or 2 is like playing chess with someone but using dreadnoughts as rooks.

[stummel]

i never should have taken a look into this thread. aarrgh.

[G-Flex]

Performing operations on any sort of infinity does get very, very tricky and often undefined. That's where you get things like indeterminate forms; something that evaluates to 1^(inf) doesn't even necessarily give you the same answer all the time, nor does it always give you an answer of 1, surprisingly.

[qwertyuiopas]

As to the .999...9=1, use .999...99998, as the "proof" fails in that case.

For a computer, there IS a definite answer, but it varies with the method of storage.

An int, 1 is the smallest diffrence, a float is probably 10^-99(or maybe -64?) or so, a double can get much smaller.

It would take a custom number representation, that takes up all of the RAM and virtual memory in the world to get an infinith of the way to 0.000...0001.

[eerr]

calculus has infinity, but I have not found it used for anything but:

finding area under the curve, with the express definition that you do not actually touch infinity.

limit of a as a goes to positive infinity, of the area  of the curve f(x).


area under the curve can also produce infinite area.


but you know what? those answers answers aren't good for anything!

Now as to the concept of limit, it uses terminology to get as close as possible.

Also, X=2 is  a line, and lines have no area. Therefore area of the curve between 0<x<2 is actually valid, and furthermore gives the same result as
0 =< x =< 2


Limits are specifically only valid when talking about area under the curve though.
They discount the possiblity of actually reaching said line.

[Virex]

Well, we can summarise the problem to 3 equations:
X < Y
X - Y = Z
Z(min) = lim(Y\/X) Z         (We let Y approach to X from above it, and look at what happens)

Now this yields Z = 0, which is inconsistent with the first two equations. This means that the system is inconsystent and thus there is no mathamatical solution.

[eerr]

Ah but you used limits improperly.
You change the meaning fundamentally when you remove the word "limit" from that equation.


The limit of Y>X as Y approaches X from the right= X
Therefore Y>X
for Y>X, Y approaches X from the right


The limit of Z>Y as Z approaches Y from the right = Y
Therefore Z>Y
When Z>Y as Z approaches Y from the right


I think...