i absolutely hate this thread. and the sad thing is, i started the first one :(
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i absolutely hate this thread. and the sad thing is, i started the first one :(
Indeed!Quote:
Originally Posted by lefteyewilly
And so far i have only seen proof that three thirds of one equals one.
And that our math system cant properly work with a number that has an infinite number of digits after the decimal.
random25: I agree that maths might not be perfect but let's not go on and blame it again and again from time to time. Rather looking at the downside of it, I rather look at what remarkable and astounding position maths has got us in the current world. I know I sound geeky/angry/lecturer type, but hey, maths is one of the reason why we are posting on this 'forum'.
i have to jump in and give my opinion about this subject since i am a math nerd. like a few of you here, haha.
to me 0.999...!=1
it comes down to basic calculus and limits. the limit of 1/x where x>0 gets infinitely smaller and closer to 0 BUT NEVER reaches 0. so i am thinking that somehow 0.999...+p=1, p=lm 1/x?
would i be wrong in saying that? since 0.000...1 does not exist, and 0.999... does not equal 1, so there must be a describable gap between 0.999... and 1. the gap is an infinitely small number, hence the limit of 1/x.
now i know that i havent had calculus in a few years, so my exact explanation of the limit of 1/x is probably wrong, but you get the point. that gap is an infinitely small number.
just my 2 cents :)
I do enjoy such in depth debates on such theoretical problems like this.
I will never believe that 0.999... = 1,
because i know that we are all way to primitive to truly grasp what these numbers actually represent.
Any one that says that they do completely understand what these numbers actually represent, and imply that they can do math with them is wrong, and too primitive to understand that they are too primitive to truly grasp what these numbers actually represent.
And that is my 0.999... cents.
I find that kind of funny because that talks about gps and goes on to explain about how there are 4 problems with using special relativity or general relativity. Lorentz is the theory that I forgot about and is the one that works with the GPS.Quote:
Originally Posted by ihoss.com
There is nothing in between 1 and 0.999... If you can say that then you are thinking in terms of time. It's like saying that there's an integer between 1 and 0. I think that until we die (unless you are an atheist or have a religion the goes against this or we get a better math system) and know most if not all the secrets of the universe then this will always be argued. By our math system which is an abstract idea not a tangible object you can deduce that 0.999... = 1 or you can argue that it does not. This is an abstract idea. You can never be sure that an abstract idea is a 100% correct. Even for the laws of gravity. We know that there is gravity and we know how to project it good enough but it's still a thought. Until we can attach to each theory or idea that it's the absolute truth, we can't state it as a fact. This is not one of the ages past where you would take any common knowledge at face value. It's too comfortable to do that. Take nothing for granted.
Hello! I already declared Yasunobu the winner of this. :D
Do we need to make two versions of math to make everyone feel good the same way schools sometimes play with grading systems to avoid giving Fs? Ok. We'll have the one version that all serious scholars of math subscribe to and then the friendly, gentle one where 1 only equals 1 because 0.999... looks different and that is just way too confusing!
Do some research. You can disagree and even find an occasional nut in a forum that will agree. But, virtually every person qualified to confirm this point agrees that 1 = 0.999...
Look here:
http://en.wikipedia.org/wiki/0.999...
http://www.relativesanity.com/archive/192
http://www.purplemath.com/modules/howcan1.htm
and my favorite, a lesson for little kids:
http://www.mathsisfun.com/9recurring.html
Perhaps the more interesting part of this debate, why people have such a hard time accepting this fact, is touched on in the purplemath link (just before proving the fact from several perspectives):
Had to add this part as well in response to this "leapfrog" theory of a number "getting closer". Numbers don't move.Quote:
There are many different proofs of the fact that 0.9999... does indeed equal 1. So why does this question keep coming up?
Students don't generally argue with 0.3333... being equal to 1/3, but then, one-third is a fraction. Maybe it's just that it "feels" "wrong" that something as nice and neat and well-behaved as the number "1" could also be written in such a messy form as "0.9999..." Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.
One of the major sticking points seems to be notational, so let me get that out of the way first: When I say "0.9999...", I don't mean "0.9" or "0.99" or "0.9999" or "0.999 followed by some large but finite (limited) number of 9's". The ellipsis (the "dot, dot, dot" after the last 9) means "goes on forever in like manner". In other words, "0.9999..." never ends. There will always be another "9" to tack onto the end of 0.9999.... So don't object to 0.9999... = 1 on the basis of "however far you go out, you still won't be equal to 1", because there is no "however far" to "go out" to; you can always go further.
"But", some say, "there will always be a difference between 0.9999... and 1." Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between 0.999...9 and 1. That is, if you do the subtraction, 1 – 0.999...9 will not equal zero. But the point of the "dot, dot, dot" is that there is no end; 0.9999... is inifinte. There is no "last" digit. So the "there's always a difference" argument betrays a lack of understanding of the infinite. (That's not a "criticism", per se; infinity is a messy topic.)
Quote:
This "getting arbitrarily close" process refers to something called "limits". You'll learn about limits later, probably in calculus. And, according to limit theory, "getting arbitrarily close" means that they're equal: 0.999... does indeed equal 1.
As I was saying earlier, this has already been debated and decided. The decision was 0.9999.... == 1 returns true. FlashLacky basically stated a lot of what I was trying to explain.
Maybe it's best put like this:
1.000... - 0.999... = 0.000...
1.000... - 0.999... = an infinitely small number. If it best suits your purposes to pretend that an infinitely small number is equivelant to 0, go ahead. But by defenition of inifinity itself, an infinitely small number, while however mathematically intangible, will always remain SOMETHING above 0.
1.000... - 0.999... != 0.000... / 0.333... != 1/3 / 0.999... != 1
:D
The tribe has spoken. Go back to your business.
Seriously though, this could be disputed forever, due to varying opinions and theories about the mechanics of inifity. Where you see 0.999... expanding infinitely into a whole number, I see an infinitely expanding difference. That difference isn't just made irrelevent because we can't process something without an end.
It doesn't matter that experts agree one way or another. It's unprovable.
I think the problem is that infinity really does'nt exist in the real world, it's just a human made construct, an idea, a philosophical theorem. Because of that, if you start depending on it in your math, you can only fail.Quote:
Originally Posted by plang
Fredi
It does exist in the real world. It just doesn't exist in the world that we can mentally encapsulate.
Fine, tell me what the edge of outer space looks like when you get there. :)Quote:
Originally Posted by Fredi
Black, no light, no radiation, nothing. But you never get there, it's impossible. Only light from the big bang can reach the edge (or be the edge).
@plang: where?
Fredi
You're talking as if going the speed of light is imposable. Tell me what experiment shows this.
If you have mass, reaching the speed of light is impossible by definition. Just ask the CERN.
Fredi
It exists in lots of ways. Take a look around the room you're in. There's an infinite number of points in space right in front you.
It just falls appart when we apply mathematics to it because equations need finite numbers for our heads not to hurt. When you use infinity in Maths...well, like you said; it doesn't really fit properly. It's so abstract that it becomes open to interpretation and cannot be proven either way. It falls to personal belief and perception.
A point in space is just geometry, it's not something that actually exist, it's just a construct to apply our formulas.Quote:
Originally Posted by plang
Geometry, math != reality
Fredi
Yes. We could debate subjective concepts of infinite all day. The real questions to ask are what definition of infinite is most useful to us and which is the most well founded? That definition has, in fact, been worked out at least to the point where the vast majority of people who must deal with the concept on a serious level agree on ONE well thought out interperetation. See Axiom of Choice.Quote:
Originally Posted by plang
So, you can choose to see infinite differently than everyone else, including all of the people whose ideas mathematics and most sciences are based around. Certainly, that is your perogative. Has this tribe yet mastered fire?
Anyway, there is no such thing as "infinitely small". "Small" is a relative term. What do you mean by small? Small to me might mean the size of a pin-head. But, to an ant, a pin-head might mean medium. 1-.2345 results in a value that is no more deserving of the word "small" than it is "medium", "large", "purple" or "blue".
Again, I think the best way to demonstrate that 0.999...= 1 is to simply sit down and start doing the math on:
1.000... - 0.999...
Try it out! Keep carrying over that one about thirty times. Pretty soon you will realize that '...' means the one never happens. There is no limit exception for infinite. Infinite zeros means all zeros. No eventual 1.