douchebagatronCustom member titleJoin Date: 2003-12-20Member: 24581Members, Constellation, Reinforced - Shadow
i have an idea that just might prove this..
ok take the integral of the function y=1 from 0 to .99_ it has to be EXACT. if it is exact and the answer is 1, then .99 is 1. if it is anything other than 1, then .99 is NOT 1. the only problem of this would be how to take the integral from 0 to .99_
someone needs to call a calculus teacher, i would but i work during the day and thats the only time my calc teacher is available.
This confounds me... why would anyone think one number was any other number? Do you think four is two? Do you count things by repeatedly saying "one, one, one, one" because "one" can equal any number?! It doesn't take a rocket scientist to see that an infinite number (.99 repeating) can never be a finite number (1).
<!--QuoteBegin-coris+Jun 21 2004, 06:25 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (coris @ Jun 21 2004, 06:25 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Cereal_KillR: 1/3 = 0.33_ so it will be as close as you get one third. BUT 0.33_ + 0.33_ + 0.33_ = 0.99_
I still want to see someone giving me a mathematical PROOF that 0.99_ is EXACTLY 1, because no-one has proveded a calculation that is correct.
0.99_ in infinity is as close to 1 you can get, but it will never be exactly one if you do not add 0.00_1 to it.
You just have to realise that this is the way it is, I dont care about your opinions, can you give me some actual proof that strengthens your theory? <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> well every single proof I've told you came from one of my math teachers so yeah I believe they should be convincing.
There's NOTHING like "as close as you get" to 1/3. You're either on it, or you're not. I think I'm going to go with what I've been taught ever since I started fractions and that 0.3333 repeating (forever until infinity) is precisely 1/3
[WHO]ThemYou can call me DaveJoin Date: 2002-12-11Member: 10593Members, Constellation
Have you ever asked yourself <i>why</i> 1/3 is commonly written as 0.<u>3333</u> ?
0.3 isn't quite there, let's go another digit 0.33 isn't quite there, let's go another digit 0.333 isn't quite there, let's go another digit 0.3333 isn't quite there, let's go another digit 0.33333 isn't quite there, let's go another digit 0.333333 isn't quite there, let's go another digit 0.3333333 isn't quite there, let's go another digit ad infinitum, no matter how many discrete points you pick, it never adds up, saying that we're going out to infinity never changes the fact that as far out as you go, it <b>never</b> adds up.
Even the paper that ziggy posted confirmed that there *was* a difference, but that it was too insignificant for them to care.
And for everyone that is taking the counterpoint of how you can't stick a 1 at the end of an infinite number of 0's.... Hogwash. Just because we happen to have notation to represent a repeating decimal doesn't mean that you can actually write that one down either.
<!--QuoteBegin-[WHO]Them+Jun 21 2004, 10:36 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> ([WHO]Them @ Jun 21 2004, 10:36 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> And for everyone that is taking the counterpoint of how you can't stick a 1 at the end of an infinite number of 0's.... Hogwash. Just because we happen to have notation to represent a repeating decimal doesn't mean that you can actually write that one down either. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> Wait... you know how to do it? You can reach the end of an infinite number, and place a "1" there? Unless I'm missing your point.
[WHO]ThemYou can call me DaveJoin Date: 2002-12-11Member: 10593Members, Constellation
edited June 2004
<!--QuoteBegin-MedHead+Jun 21 2004, 08:39 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (MedHead @ Jun 21 2004, 08:39 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Wait... you know how to do it? You can reach the end of an infinite number, and place a "1" there? Unless I'm missing your point. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> As easily as I can go to the end of an already infinitely long string of 9's and put down another 9.
douchebagatronCustom member titleJoin Date: 2003-12-20Member: 24581Members, Constellation, Reinforced - Shadow
<!--QuoteBegin-[WHO]Them+Jun 21 2004, 09:47 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> ([WHO]Them @ Jun 21 2004, 09:47 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> <!--QuoteBegin-MedHead+Jun 21 2004, 08:39 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (MedHead @ Jun 21 2004, 08:39 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Wait... you know how to do it? You can reach the end of an infinite number, and place a "1" there? Unless I'm missing your point. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd--> As easily as I can go to the end of an already infinitely long string of 9's and put down another 9. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> you got served.
<!--QuoteBegin-[WHO]Them+Jun 21 2004, 10:47 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> ([WHO]Them @ Jun 21 2004, 10:47 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> As easily as I can go to the end of an already infinitely long string of 9's and put down another 9. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> You're not. There is no end to the number. You're thinking that there is an end where you may place another number. There is not. That is the point of infinite.
[WHO]ThemYou can call me DaveJoin Date: 2002-12-11Member: 10593Members, Constellation
<!--QuoteBegin-MedHead+Jun 21 2004, 09:56 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (MedHead @ Jun 21 2004, 09:56 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> You're not. There is no end to the number. You're thinking that there is an end where you may place another number. There is not. That is the point of infinite. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> It's nice that you completely missed the point of my newest post. I've shown you a notation system that allows such a number to be written down.
Writing only represents ideas. And the idea of an infinitely small nonzero number cannot be denied.
I don't see what that proves, though. I could have easily said .9 ∞ 1, and said I proved I could write down an infinite number with a 1 at the end of it. It doesn't mean it's possible. What's the point in making a number system of impossible numbers?
[WHO]ThemYou can call me DaveJoin Date: 2002-12-11Member: 10593Members, Constellation
* [WHO]Them smashes his head on his keyboard several times.
You're COMPLETELY missing my point. You've failed to even hit the state that the point lives in.
Your side is that we can't write the number down, and I'm telling you that writing is an expression of an idea, and you're telling me that we can't write it down, and I'm telling you that some ideas can't be written down, and you're telling me we can't write it down.
WRITING IT DOWN HAS NOTHING TO DO WITH WHETHER IT EXISTS.
For example, without using shorthand notation, write down your infinitely repeating 9. I want to see it written down..... I know you can't write it down. Yet somehow you cling to the idea that one number that can't possibly be written exists, while another that can't be written doesn't, and only doesn't exist because it can't be written.
Go read up on limits.... lim x->infinity 1/x
You can't deny that the limit above gets closer to zero the further you go, but never reaches zero, proving the existance of infinitely small numbers.
<!--QuoteBegin-[WHO]Them+Jun 21 2004, 10:36 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> ([WHO]Them @ Jun 21 2004, 10:36 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> And for everyone that is taking the counterpoint of how you can't stick a 1 at the end of an infinite number of 0's.... Hogwash. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> That's what my comments have been about. You said that it was possible to place a number at the end of an infinite number. It's not possible, which is why I commented on it. I wasn't commenting on writing down small numbers, nor was I commenting on the existance of infinitely small numbers. Never was. I have been trying to figure out how you can place a 1 at the end of an infinite amount of numbers. My "side" has never been that you or anyone else can't write a number down. That's silly. You can write whaever number you want down on a piece of paper. Doesn't mean it exists. Because, in this case, a .9 repeating with a 1 at the end can't exist. Which was my point.
[WHO]ThemYou can call me DaveJoin Date: 2002-12-11Member: 10593Members, Constellation
I'm not arguing the existance of 0.<u>9</u>1 I'm arguing the existance of 0.<u>0</u>1
start with 0.1, place a 0 after the decimal place, now we have 0.01 place a 0 after the decimal place, now we have 0.001 place a 0 after the decimal place, now we have 0.0001 place a 0 after the decimal place, now we have 0.00001 place a 0 after the decimal place, now we have 0.000001 lather, rinse, repeat......
I just "stuck" a 1 at the end of a string of infinitely long string of zero's by adding the zeros last.
Both sides are correct but are taking different perspectives on the problem. WHO is taking the idea of .9_ as a concept and therefore can only equal the concept itself. Therefore .9_ would only = .9_. The other sides define the idea as a limit and therefore .9_ = 1 would be correct.
Where one see's a defined picture another see's an imaginary concept. Both are correct in how they can manipulate each thing even though they are equal to two different things.
Realize you guys are looking at this differently and that both sides are correct.
Edit: God united mandkind like I united the people in this thread! I should be given a cult of some kind. Plus I needed to fix some grammOOAEOOR
<!--QuoteBegin-lagger+Jun 22 2004, 03:38 PM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> (lagger @ Jun 22 2004, 03:38 PM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Plus I needed to fix some grammOOAEOOR <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> You need to fix a little more. It's "sees", not "see's".
Did you know grammooaeoor is spelled wrong too? /end sarcasm
Cough* math thread Cough* (By cough I mean that grammar is by far very unimportant in this discussion)
I did say some, ie: not all.
Omc your grammar corrections are insulting my extreme reflection of god and therefore must be punished by death!
edit: I have gone off course and this thread is being derailed by spamming (By spamming I mean one post) grammer corrections. In reaction I will stop posting here in hope that this 10 page monster will be stopped in its tracks.
edit: Wait! By re-posting about this I have actually gone against what I previously stated and therefore I must restate that I will stop posting after this post assuming I dont find another reason to post here that is relavent to the main topic of this thread. Except I still feel I have the right to post somewhat off topic if it also has some pertanence to the discussion.
Seriously though, posts like this and the one above should rarely ever happen. Actually I'm proabably no better for constructing this then anyone else who has made irrelavent posts, but I feel it drives the point home.
If 1- 0.<u>99</u>= 1.0000infinite amount of '0's' with a 1 at the end i.e. 0.<u>00</u>1
but as you can never reach the end of infinite then it's called 0.
Does this mean that 0.<u>99</u>8 is the same as 0.<u>99</u> as you'll never reach the 8? And of course 0.<u>99</u> seems to equal 1 so... does that make 0.<u>99</u>8 = 1
0.<u>99</u>1=1 as well?
1 is a big number
And I like this thread, nice to have something to think about even if, for all practical purposes, it's pretty useless to me.
I think I might try and melt my co-workers brains later <!--emo&:D--><img src='http://www.unknownworlds.com/forums/html//emoticons/biggrin.gif' border='0' style='vertical-align:middle' alt='biggrin.gif' /><!--endemo-->
<!--QuoteBegin-[WHO]Them+Jun 22 2004, 07:36 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> ([WHO]Them @ Jun 22 2004, 07:36 AM)</td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> Go read up on limits.... lim x->infinity 1/x
You can't deny that the limit above gets closer to zero the further you go, but never reaches zero, proving the existance of infinitely small numbers. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> well limits....
f(x)=x²
lim f(x) = 0 x -> 0
As long as x gets closer to 0, f(x) gets closer as well. Yet as long as x != 0 then f(x) != lim f(x)
Same thing here. As long as x gets closer to infinity. if X is the numbers of 9's, then as long as x gets closer to infinity it gets closer to 1. BUT when it <i>does</i> hit infinity, it does hit 1.
The problem would just be actually getting to infinity, but that's not what we're discussing right now.
Try asking your teacher or any trustable math source (anything more trustable than a gaminf forum) I'd be surprised your math teacher will disagree with 0.999_ = 1
I've posted this in the 64=65 thread, because we began discussing there 1=0.(9) <!--emo&:p--><img src='http://www.unknownworlds.com/forums/html//emoticons/tounge.gif' border='0' style='vertical-align:middle' alt='tounge.gif' /><!--endemo-->
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin--> I can even give you two proofs for this.
First is: Lets take the following theorem: Between any two different numbers A and B there is always a number C. (I use "number" here to simplify because i don't know the english words.. racional numbers? not sure). Anyway, let's take A and B as A=1 and B=0.(9). So what's the number C between then? What's the number between 1 and 0.(9)? There is none. So A and B cannot be different numbers.
The second proof is: Let's write 0.(9) = sum from k=1 to k=+infinity of 9/10^k. This is a geometric progression. If you search the web for "geometric progression" you'll find websites that tell you how to calculate them. You'll see that equals 1. <!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
BTW, this also answers the problem someone raised about "some ideas you can't write down". This is true, but this particular idea you CAN write down. 0.(9) = lim with x->+infinity of sum from k=1 to k=x of 9/10^k. Calculate this expression and you'll see it equals 1. Not almost 1. Exactly 1.
I vote we stop this discussion with this: it's very difficult to reach the number .(9). It is, however, possible to reach .(3), which is essentially... wait, scratch that, EXACTLY 1/3. I don't see how any of this should apply to standard math. I also don't think any professors would appreciate you "rounding" any .(9)'s up. So in conclusion, I vote "WHO CARES."
clear and simple: what most people don't realize is that if you're using decimal notation to express a fraction, your reults are automaticaly rounded to the number of digits of your least precise number.
as in, when i add 1.0 and 2.0, my answer is 3.0, not 3.00 because my original numbers were only precise to 1/10th
by using the number 0.99_ you have skewed your answer, as it has been rounded to 1/infinityeth
now, using that: 0.66_ rounded to 1/infinityeth would actualy be 0.66...67
as, 0.33_ * 3 = 0.33_ + 0.33_ + 0.33_ and by the law of additive association 0.33_ + 0.33_ + 0.33_ = 0.66_ + 0.33_ o.66_ actually being 0.66 ..67: 0.33_ + 0.66...67 == 1
clear and simple: what most people don't realize is that if you're using decimal notation to express a fraction, your reults are automaticaly rounded to the number of digits of your least precise number.
as in, when i add 1.0 and 2.0, my answer is 3.0, not 3.00 because my original numbers were only precise to 1/10th
by using the number 0.99_ you have skewed your answer, as it has been rounded to 1/infinityeth
now, using that: 0.66_ rounded to 1/infinityeth would actualy be 0.66...67
as, 0.33_ * 3 = 0.33_ + 0.33_ + 0.33_ and by the law of additive association 0.33_ + 0.33_ + 0.33_ = 0.66_ + 0.33_ o.66_ actually being 0.66 ..67: 0.33_ + 0.66...67 == 1
if you want to be purely mathmatical, you would state that: the limit of 0.33_ * 3 is equal to 1 <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd--> that's used in physics. not in mathematics.
Anyways. If this was true, I could not imagine the existance of asymptopes or anything of that regard. Hell, parabolas could never exist because it would eventually reach 1 and stop. (If an asymptope were to be at 1.)
Comments
ok take the integral of the function y=1 from 0 to .99_ it has to be EXACT. if it is exact and the answer is 1, then .99 is 1. if it is anything other than 1, then .99 is NOT 1. the only problem of this would be how to take the integral from 0 to .99_
someone needs to call a calculus teacher, i would but i work during the day and thats the only time my calc teacher is available.
1/3 = 0.33_ so it will be as close as you get one third. BUT 0.33_ + 0.33_ + 0.33_ = 0.99_
I still want to see someone giving me a mathematical PROOF that 0.99_ is EXACTLY 1, because no-one has proveded a calculation that is correct.
0.99_ in infinity is as close to 1 you can get, but it will never be exactly one if you do not add 0.00_1 to it.
You just have to realise that this is the way it is, I dont care about your opinions, can you give me some actual proof that strengthens your theory? <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
well every single proof I've told you came from one of my math teachers so yeah I believe they should be convincing.
There's NOTHING like "as close as you get" to 1/3. You're either on it, or you're not. I think I'm going to go with what I've been taught ever since I started fractions and that 0.3333 repeating (forever until infinity) is precisely 1/3
0.3 isn't quite there, let's go another digit
0.33 isn't quite there, let's go another digit
0.333 isn't quite there, let's go another digit
0.3333 isn't quite there, let's go another digit
0.33333 isn't quite there, let's go another digit
0.333333 isn't quite there, let's go another digit
0.3333333 isn't quite there, let's go another digit
ad infinitum, no matter how many discrete points you pick, it never adds up, saying that we're going out to infinity never changes the fact that as far out as you go, it <b>never</b> adds up.
Even the paper that ziggy posted confirmed that there *was* a difference, but that it was too insignificant for them to care.
And for everyone that is taking the counterpoint of how you can't stick a 1 at the end of an infinite number of 0's.... Hogwash. Just because we happen to have notation to represent a repeating decimal doesn't mean that you can actually write that one down either.
Wait... you know how to do it? You can reach the end of an infinite number, and place a "1" there? Unless I'm missing your point.
As easily as I can go to the end of an already infinitely long string of 9's and put down another 9.
As easily as I can go to the end of an already infinitely long string of 9's and put down another 9. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
you got served.
Consider what the symbol 1 means.
Consider what 0.1 means.
The inception of standard number notation was completely arbitrary, all it needed to do was convey an idea of an amount.
So I propose to you a new notation, used exclusively for amounts smaller than one.
This new notation is always smaller than one, so there is absolutely no need for a decimal place, so throw that out.
And just for the hell of it, I'm going to write the numbers backwards.
I'm going to borrow the infinite repition notation (specifically underlines for this forum) and write the number I'm thinking of.
Here are a few examples of my new numbering notation....
Common System: 0.6
My System: 6
Common System: 0.15348
My System: 84351
Common System: 0.52
My System: 25
Common System: 0.<u>9</u>
My System: <u>9</u>
Common System: 1 - 0.<u>9</u>
My System: 1<u>0</u>
Look how easy it is to express, simply because the notation suddenly supports it.
The lesson? Just because it's not easy to write in the common system doesn't mean it doesn't exist.
You're not. There is no end to the number. You're thinking that there is an end where you may place another number. There is not. That is the point of infinite.
Infinity is a funny thing.
Get it? INFINITY? GET IT?! HAHAHH!H!H!h11h111111111
It's nice that you completely missed the point of my newest post. I've shown you a notation system that allows such a number to be written down.
Writing only represents ideas. And the idea of an infinitely small nonzero number cannot be denied.
You're COMPLETELY missing my point. You've failed to even hit the state that the point lives in.
Your side is that we can't write the number down, and I'm telling you that writing is an expression of an idea, and you're telling me that we can't write it down, and I'm telling you that some ideas can't be written down, and you're telling me we can't write it down.
WRITING IT DOWN HAS NOTHING TO DO WITH WHETHER IT EXISTS.
For example, without using shorthand notation, write down your infinitely repeating 9. I want to see it written down..... I know you can't write it down. Yet somehow you cling to the idea that one number that can't possibly be written exists, while another that can't be written doesn't, and only doesn't exist because it can't be written.
Go read up on limits....
lim x->infinity 1/x
You can't deny that the limit above gets closer to zero the further you go, but never reaches zero, proving the existance of infinitely small numbers.
That's what my comments have been about. You said that it was possible to place a number at the end of an infinite number. It's not possible, which is why I commented on it. I wasn't commenting on writing down small numbers, nor was I commenting on the existance of infinitely small numbers. Never was. I have been trying to figure out how you can place a 1 at the end of an infinite amount of numbers. My "side" has never been that you or anyone else can't write a number down. That's silly. You can write whaever number you want down on a piece of paper. Doesn't mean it exists. Because, in this case, a .9 repeating with a 1 at the end can't exist. Which was my point.
I'm arguing the existance of 0.<u>0</u>1
start with 0.1,
place a 0 after the decimal place, now we have 0.01
place a 0 after the decimal place, now we have 0.001
place a 0 after the decimal place, now we have 0.0001
place a 0 after the decimal place, now we have 0.00001
place a 0 after the decimal place, now we have 0.000001
lather, rinse, repeat......
I just "stuck" a 1 at the end of a string of infinitely long string of zero's by adding the zeros last.
Both sides are correct but are taking different perspectives on the problem. WHO is taking the idea of .9_ as a concept and therefore can only equal the concept itself. Therefore .9_ would only = .9_. The other sides define the idea as a limit and therefore .9_ = 1 would be correct.
Where one see's a defined picture another see's an imaginary concept. Both are correct in how they can manipulate each thing even though they are equal to two different things.
Realize you guys are looking at this differently and that both sides are correct.
Edit: God united mandkind like I united the people in this thread! I should be given a cult of some kind. Plus I needed to fix some grammOOAEOOR
You need to fix a little more. It's "sees", not "see's".
Cough* math thread Cough* (By cough I mean that grammar is by far very unimportant in this discussion)
I did say some, ie: not all.
Omc your grammar corrections are insulting my extreme reflection of god and therefore must be punished by death!
edit: I have gone off course and this thread is being derailed by spamming (By spamming I mean one post) grammer corrections. In reaction I will stop posting here in hope that this 10 page monster will be stopped in its tracks.
edit: Wait! By re-posting about this I have actually gone against what I previously stated and therefore I must restate that I will stop posting after this post assuming I dont find another reason to post here that is relavent to the main topic of this thread. Except I still feel I have the right to post somewhat off topic if it also has some pertanence to the discussion.
Seriously though, posts like this and the one above should rarely ever happen. Actually I'm proabably no better for constructing this then anyone else who has made irrelavent posts, but I feel it drives the point home.
If 1- 0.<u>99</u>= 1.0000infinite amount of '0's' with a 1 at the end i.e. 0.<u>00</u>1
but as you can never reach the end of infinite then it's called 0.
Does this mean that 0.<u>99</u>8 is the same as 0.<u>99</u> as you'll never reach the 8? And of course 0.<u>99</u> seems to equal 1 so... does that make 0.<u>99</u>8 = 1
0.<u>99</u>1=1 as well?
1 is a big number
And I like this thread, nice to have something to think about even if, for all practical purposes, it's pretty useless to me.
I think I might try and melt my co-workers brains later <!--emo&:D--><img src='http://www.unknownworlds.com/forums/html//emoticons/biggrin.gif' border='0' style='vertical-align:middle' alt='biggrin.gif' /><!--endemo-->
lim x->infinity 1/x
You can't deny that the limit above gets closer to zero the further you go, but never reaches zero, proving the existance of infinitely small numbers. <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
well limits....
f(x)=x²
lim f(x) = 0
x -> 0
As long as x gets closer to 0, f(x) gets closer as well. Yet as long as x != 0 then f(x) != lim f(x)
Same thing here. As long as x gets closer to infinity. if X is the numbers of 9's, then as long as x gets closer to infinity it gets closer to 1. BUT when it <i>does</i> hit infinity, it does hit 1.
The problem would just be actually getting to infinity, but that's not what we're discussing right now.
Try asking your teacher or any trustable math source (anything more trustable than a gaminf forum) I'd be surprised your math teacher will disagree with 0.999_ = 1
<!--QuoteBegin--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td><b>QUOTE</b> </td></tr><tr><td id='QUOTE'><!--QuoteEBegin-->
I can even give you two proofs for this.
First is:
Lets take the following theorem: Between any two different numbers A and B there is always a number C. (I use "number" here to simplify because i don't know the english words.. racional numbers? not sure). Anyway, let's take A and B as A=1 and B=0.(9). So what's the number C between then? What's the number between 1 and 0.(9)? There is none. So A and B cannot be different numbers.
The second proof is:
Let's write 0.(9) = sum from k=1 to k=+infinity of 9/10^k. This is a geometric progression. If you search the web for "geometric progression" you'll find websites that tell you how to calculate them. You'll see that equals 1.
<!--QuoteEnd--></td></tr></table><div class='postcolor'><!--QuoteEEnd-->
BTW, this also answers the problem someone raised about "some ideas you can't write down". This is true, but this particular idea you CAN write down. 0.(9) = lim with x->+infinity of sum from k=1 to k=x of 9/10^k. Calculate this expression and you'll see it equals 1. Not almost 1. Exactly 1.
clear and simple:
what most people don't realize is that if you're using decimal notation to express a fraction, your reults are automaticaly rounded to the number of digits of your least precise number.
as in, when i add 1.0 and 2.0, my answer is 3.0, not 3.00 because my original numbers were only precise to 1/10th
by using the number 0.99_ you have skewed your answer, as it has been rounded to 1/infinityeth
now, using that: 0.66_ rounded to 1/infinityeth would actualy be 0.66...67
as, 0.33_ * 3 = 0.33_ + 0.33_ + 0.33_ and by the law of additive association
0.33_ + 0.33_ + 0.33_ = 0.66_ + 0.33_
o.66_ actually being 0.66 ..67: 0.33_ + 0.66...67 == 1
<a href='http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm' target='_blank'>http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm</a>
if you want to be purely mathmatical, you would state that:
the limit of 0.33_ * 3 is equal to 1
clear and simple:
what most people don't realize is that if you're using decimal notation to express a fraction, your reults are automaticaly rounded to the number of digits of your least precise number.
as in, when i add 1.0 and 2.0, my answer is 3.0, not 3.00 because my original numbers were only precise to 1/10th
by using the number 0.99_ you have skewed your answer, as it has been rounded to 1/infinityeth
now, using that: 0.66_ rounded to 1/infinityeth would actualy be 0.66...67
as, 0.33_ * 3 = 0.33_ + 0.33_ + 0.33_ and by the law of additive association
0.33_ + 0.33_ + 0.33_ = 0.66_ + 0.33_
o.66_ actually being 0.66 ..67: 0.33_ + 0.66...67 == 1
<a href='http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm' target='_blank'>http://www.physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm</a>
if you want to be purely mathmatical, you would state that:
the limit of 0.33_ * 3 is equal to 1 <!--QuoteEnd--> </td></tr></table><div class='postcolor'> <!--QuoteEEnd-->
that's used in physics. not in mathematics.
<img src='http://nasa.perbang.dk/billeder/bGk5a5RQ_stor.jpg' border='0' alt='user posted image' />
Xx8=64
Xx13=65
64=65
Lets do some simple algebra then shall we?
Since you claim 64 = 65, replace those two numbers in the equations, and put X in place of the first number.
Xx8=65
Xx13=64
Now continue by moving one of the numbers on the left over, to leave the X.
X=65/8
X=64/13
X=8.125
X=4.92 some big number repeating.
These values are close, but not equal to the original numbers.
GG screwing with math.