That looks like a well thought out proof. I'll think about it while I sleep and see if I can come up with any challenges.
What you people need isn't proof. You've been provided with ample proof. What you need is some way in which to reconcile the mathematical truth with your own intuitions.
Or we need to fix math so that it describes reality.
Only if you assume (as you do) that .999... doesn't equal 1.
Right now my thinking is that saying .999... = 1 in some way contradicts the notion of infinity, and in effect putting .999... equal to 1 defines an infinite sequence as terminating. Because if the sequence truly is infinite the 9s will go on forever and, as Zeno said, the rabbit will never reach the wall.
So I'm going to go to sleep.
You are not grasping the notion of infinity. This is the problem with all the objections in the thread.
0.999... is not just a really long chain of digits. It has absolutely nothing to do with a long chain of digits, in fact. 0.999999999999999999999999 is no closer to 0.999... than 0.9, or 12, or 516.3. Infinite 9's is not just more nines; it is an entirely different realm of nines.
if the sequence truly is infinite the 9s will go on forever and, as Zeno said, the rabbit will never reach the wall.
This is the key problem, right here. You are conceiving of 0.999... as something that almost reaches one but never gets there, as if it is a process. It is not a process. There is no "never" involved here. By virtue of infinitely repeating, it becomes equal to one. The rabbit reaches the wall because it comes infinitely close. There is no difference between "there" and "infinitely close to there." Zeno's paradox traps you in a corrupted thought experiment, as if infinity is a process that can be resolved into finite steps. That is NOT how it should be conceived.
Why don't you understand infinity? It isn't a terribly difficult concept.
Everything ends. Nothing is infinite.
So to posit that there exists something infinite and to base a system on that notion of infinity doesn't make any sense.
Show me an infinite thing other than some formula which posits the existence of some infinite thing which is itself based upon self-maintained definitions and axioms and assumptions that exist only insofar as they are assumed to be true.
I know the thread is moving fast, but seriously, did you look at my triangles two pages ago?
Everything is finite? Are you going to tell me that you've actually had a moment in your life when you've been able to measure a distance of exactly 1 meter?
This is the key problem, right here. You are conceiving of 0.999... as something that almost reaches one but never gets there, as if it is a process. It is not a process. There is no "never" involved here. By virtue of infinitely repeating, it becomes equal to one. The rabbit reaches the wall because it comes infinitely close. There is no difference between "there" and "infinitely close to there." Zeno's paradox traps you in a corrupted thought experiment, as if infinity is a process that can be resolved into finite steps. That is NOT how it should be conceived.
But that whole notion of infinity only exists within the mathematical system which supports it. To remove infinity from finite sets may be sensible within mathematics in order to allow infinity to function as it must for the purposes of mathematics, but when viewed from any other perspective it borders on lunacy.
Your description makes it slightly more sensible to me. Your "by virtue of infinitely repeating" gets me away from the Zeno and the question of "how" to an understanding of some inherent nature to infinite sequences.
But it still doesn't make much sense outside of the mathematical system which requires it.
Why don't you understand infinity? It isn't a terribly difficult concept.
Everything ends. Nothing is infinite.
So to posit that there exists something infinite and to base a system on that notion of infinity doesn't make any sense.
Show me an infinite thing other than some formula which posits the existence of some infinite thing which is itself based upon self-maintained definitions and axioms and assumptions that exist only insofar as they are assumed to be true.
Edit: I love Drez.
Edit edit: Basically, I view the mathematical infinity argument as basically the same thing as "The Bible must be true because it says it is." Because nothing other than itself supports it.
Hey you know what's great though
What's great is that math is actually really useful and can be applied to the real world, and a lot of really important things happen when you mess around with the concept of infinity
It doesn't matter whether or not you can see something infinite; mathematics uses the concept of infinity not in some fit of masturbatory self-affirmation, but because it is demonstrably useful and leads to greater understanding of other things.
But no, you're right, let's just toss out higher math because we can't give examples based on apples or count it on our fingers.
Why don't you understand infinity? It isn't a terribly difficult concept.
Everything ends. Nothing is infinite.
So to posit that there exists something infinite and to base a system on that notion of infinity doesn't make any sense.
Show me an infinite thing other than some formula which posits the existence of some infinite thing which is itself based upon self-maintained definitions and axioms and assumptions that exist only insofar as they are assumed to be true.
Edit: I love Drez.
Edit edit: Basically, I view the mathematical infinity argument as basically the same thing as "The Bible must be true because it says it is." Because nothing other than itself supports it.
Hey you know what's great though
What's great is that math is actually really useful and can be applied to the real world, and a lot of really important things happen when you mess around with the concept of infinity
It doesn't matter whether or not you can see something infinite; mathematics uses the concept of infinity not in some fit of masturbatory self-affirmation, but because it is demonstrably useful and leads to greater understanding of other things.
But no, you're right, let's just toss out higher math because we can't give examples based on apples or count it on our fingers.
Well, in a way what he's saying is true. The idea of, for example, the instantaneous velocity of a car does not represent reality in that, in an instant, the car is stationary.
This is the key problem, right here. You are conceiving of 0.999... as something that almost reaches one but never gets there, as if it is a process. It is not a process. There is no "never" involved here. By virtue of infinitely repeating, it becomes equal to one. The rabbit reaches the wall because it comes infinitely close. There is no difference between "there" and "infinitely close to there." Zeno's paradox traps you in a corrupted thought experiment, as if infinity is a process that can be resolved into finite steps. That is NOT how it should be conceived.
But that whole notion of infinity only exists within the mathematical system which supports it. To remove infinity from finite sets may be sensible within mathematics in order to allow infinity to function as it must for the purposes of mathematics, but when viewed from any other perspective it borders on lunacy.
Your description makes it slightly more sensible to me. Your "by virtue of infinitely repeating" gets me away from the Zeno and the question of "how" to an understanding of some inherent nature to infinite sequences.
But it still doesn't make much sense outside of the mathematical system which requires it.
And the word "tree" doesn't make sense outside the linguistic system which makes use of it.
It's really important to see math in the same way as a language, and not as an attempt to objectively describe and categorize reality.
There is no other perspective from which to interpret the word "tree," or any word, except the language from which it is taken; there is no other perspective from which to interpret the concept of infinity. Math is as semiotically arbitrary as any language, and it is just as useful. Saying that "infinity" does not inherently connect to reality is like saying that a word does not inherently connect to reality. It's missing the point, really; they're not supposed to.
Oooookiedokie. What's the difference between 0.222... and 3?
In what base system?
Man, you can read it for yourself on the damn wiki.
Decimal system (base 10) 1 = 1 Ternary system (base 3)
Decimal system (base 10) 1/3 = .1 Ternary system (base 3)
I am not understand the notion of why a Base 3 system would have to represent a whole number like 1 as a decimal.
For the same reason you'd do it in the decimal system?
No, you represent 1 as 1 in the decimal system last I checked? I mean, you could represent it as 1.0 if you really wanted to, but you could do the same in the ternary system.
Look, a base 3 system just means you count 1,2,10,11,12,20,21,22,100
as opposed to 1,2,3,4,5,6,7,8,9
Man, you can read it for yourself on the damn wiki.
Decimal system (base 10) 1 = 1 Ternary system (base 3)
Decimal system (base 10) 1/3 = .1 Ternary system (base 3)
I am not understand the notion of why a Base 3 system would have to represent a whole number like 1 as a decimal.
Oh dear god.
Look, ternary has a finite representation of 1/3. However it doesn't have one of 1/2. 1/2 in ternary is .111... . It thus follows that 2/2, or 1, is .222... .
Oooookiedokie. What's the difference between 0.222... and 3?
In what base system?
Man, you can read it for yourself on the damn wiki.
Decimal system (base 10) 1 = 1 Ternary system (base 3)
Decimal system (base 10) 1/3 = .1 Ternary system (base 3)
I am not understand the notion of why a Base 3 system would have to represent a whole number like 1 as a decimal.
For the same reason you'd do it in the decimal system?
No, you represent 1 as 1 in the decimal system last I checked? I mean, you could represent it as 1.0 if you really wanted to, but you could do the same in the ternary system.
Look, a base 3 system just means you count 1,2,10,11,12,20,21,22,100
as opposed to 1,2,3,4,5,6,7,8,9
We've been talking about 0.999... = 1.0 in decimal for pages. 0.222... = 1.0 in ternary is no different. You've done nothing but waste our time.
EDIT: This is just conjecture on my part, actually. I've never thought about decimals in bases other than 10! I'll read the wiki.
So Marty was right, then. 0.222... = 1.0.
Sorry, I am tired and misreading shit. I misinterpreted the ... as a dramatic pause. Not as repeating, durr durr durr.
Yeah, .2 repeating would be 1 in a base 3 system.
The whole point of converting to a base 3 system is it is another, different way to illustrate that .9 repeating = 1, and it's just a failure of the base 10 system. Just like .2 repeating = 1 in base 3 is a failure of its system.
Edit: Yeah, yeah, apologizing again for misreading the ... I personally have never seen that notation outside of this thread to mean repeating, so, I kinda breeze over it. Hopefully I help explained the point of switching to other bases, as you can always switch to another base where the repeating wont happen. Once again, just another way to illustrate that .9 repeating = 1.
EDIT: This is just conjecture on my part, actually. I've never thought about decimals in bases other than 10! I'll read the wiki.
So Marty was right, then. 0.222... = 1.0.
Sorry, I am tired and misreading shit. I misinterpreted the ... as a dramatic pause. Not as repeating, durr durr durr.
Yeah, .2 repeating would be 1 in a base 3 system.
The whole point of converting to a base 3 system is it is another, different way to illustrate that .9 repeating = 1, and it's just a failure of the base 10 system. Just like .2 repeating = 1 in base 3 is a failure of its system.
Ha! A "dramatic pause"! It's okay, I think 2:30 AM is the wrong time to be debating math so I'll be going to bed too.
Drez on
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SmasherStarting to get dizzyRegistered Userregular
Why don't you understand infinity? It isn't a terribly difficult concept.
Everything ends. Nothing is infinite.
So to posit that there exists something infinite and to base a system on that notion of infinity doesn't make any sense.
Show me an infinite thing other than some formula which posits the existence of some infinite thing which is itself based upon self-maintained definitions and axioms and assumptions that exist only insofar as they are assumed to be true.
Edit: I love Drez.
Edit edit: Basically, I view the mathematical infinity argument as basically the same thing as "The Bible must be true because it says it is." Because nothing other than itself supports it.
I don't understand why _J_ and many others are getting so hung up on whether this is in "reality" or not. Math in general is a tool that has shown an amazing proclivity for being able to describe how the world works in a concise and useful form. If I'm designing a circuit and want to know if it is stable or not, what do I do? I use a mathematical model and look at its behavior as, guess what, it goes towards infinity. Now, I know that my circuit won't last for an infinite amount of time, yet it is still a good test to see if my circuit will blow up or not. Here the concept of infinity is a useful tool, since modeling the exact behavior at a particular time is sometimes computationally infeasible.
We use the natural numbers so "naturally" (hence the name) that, at least I find it hard to imagine any world that wouldn't develop a system by which to manipulate them. From this system, it flows naturally that a concept of "infinity" arises, because there is no limit to the natural numbers. Why? Because natural numbers start with the concept of 0, 1, and this crazy concept of addition.
chrono_traveller on
The trouble with having an open mind, of course, is that people will insist on coming along and trying to put things in it. ~ Terry Pratchett
EDIT: This is just conjecture on my part, actually. I've never thought about decimals in bases other than 10! I'll read the wiki.
So Marty was right, then. 0.222... = 1.0.
Sorry, I am tired and misreading shit. I misinterpreted the ... as a dramatic pause. Not as repeating, durr durr durr.
Yeah, .2 repeating would be 1 in a base 3 system.
The whole point of converting to a base 3 system is it is another, different way to illustrate that .9 repeating = 1, and it's just a failure of the base 10 system. Just like .2 repeating = 1 in base 3 is a failure of its system.
.
Oh.
Well, in any case, I don't really think it makes sense to describe a problem that affects all bases equally as a failure of any base.
For that matter, I don't see rational numbers having multiple representations as a problem at all. Well, aside from the obvious.
I was seriously pulling my hair out over why people was saying .22 = 1, stupid ... notation.
Evil, it's really pretty simply. Imagine a base 3 system as a odometer that turns the next digit over by 1 every time it would reach 3 instead of 10.
Adrien: Yeah, I mean. I'm not saying that base 10 is a failure, or something. But base 10 does a poor job representing 1/3, is all. Every base has a hard time representing some numbers (hell, base 3 is awful at representing any fraction other then 1/3). And yeah, .9 repeating being 1 really isn't a big deal, at the end of the day. It's just another way to tackle the issue, is all.
Inquisitor on
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SmasherStarting to get dizzyRegistered Userregular
edited October 2007
At the risk of repeating what other people have said in different words, your problem _J_ is that you're insisting that mathematics is not equivalent to reality (which is true), yet insisting that a mathematical construct (.999...) must have a physical equivalent.
It doesn't matter that you can't physically "have" an infinite number of 9s (whatever that really means), because mathematically you can. And the rules of infinity are governed by math, not physics, so trying to argue about infinity using the latter is pointless.
I was seriously pulling my hair out over why people was saying .22 = 1, stupid ... notation.
Evil, it's really pretty simply. Imagine a base 3 system as a odometer that turns the next digit over by 1 every time it would reach 3 instead of 10.
Adrien: Yeah, I mean. I'm not saying that base 10 is a failure, or something. But base 10 does a poor job representing 1/3, is all. Every base has a hard time representing some numbers (hell, base 3 is awful at representing any fraction other then 1/3). And yeah, .9 repeating being 1 really isn't a big deal, at the end of the day. It's just another way to tackle the issue, is all.
I seriously thought we were about to do the whole thread over again in base 3
I was seriously pulling my hair out over why people was saying .22 = 1, stupid ... notation.
Evil, it's really pretty simply. Imagine a base 3 system as a odometer that turns the next digit over by 1 every time it would reach 3 instead of 10.
Adrien: Yeah, I mean. I'm not saying that base 10 is a failure, or something. But base 10 does a poor job representing 1/3, is all. Every base has a hard time representing some numbers (hell, base 3 is awful at representing any fraction other then 1/3). And yeah, .9 repeating being 1 really isn't a big deal, at the end of the day. It's just another way to tackle the issue, is all.
I seriously thought we were about to do the whole thread over again in base 3
Bwahahah, the thought of that amuses me to no end. Because some people are already confused enough, it would be even more confusing for most if we were working in a different base.
Person A: .2... equals 1!!
Prson B: But that is like .8... away from 1!!! WTFBBQ!~
I was seriously pulling my hair out over why people was saying .22 = 1, stupid ... notation.
Evil, it's really pretty simply. Imagine a base 3 system as a odometer that turns the next digit over by 1 every time it would reach 3 instead of 10.
Adrien: Yeah, I mean. I'm not saying that base 10 is a failure, or something. But base 10 does a poor job representing 1/3, is all. Every base has a hard time representing some numbers (hell, base 3 is awful at representing any fraction other then 1/3). And yeah, .9 repeating being 1 really isn't a big deal, at the end of the day. It's just another way to tackle the issue, is all.
I seriously thought we were about to do the whole thread over again in base 3
If we do it in base 2 it'd be really easy to make some computers do the arguing for us, while we go ride bikes or something
Posts
You are not grasping the notion of infinity. This is the problem with all the objections in the thread.
0.999... is not just a really long chain of digits. It has absolutely nothing to do with a long chain of digits, in fact. 0.999999999999999999999999 is no closer to 0.999... than 0.9, or 12, or 516.3. Infinite 9's is not just more nines; it is an entirely different realm of nines.
This is the key problem, right here. You are conceiving of 0.999... as something that almost reaches one but never gets there, as if it is a process. It is not a process. There is no "never" involved here. By virtue of infinitely repeating, it becomes equal to one. The rabbit reaches the wall because it comes infinitely close. There is no difference between "there" and "infinitely close to there." Zeno's paradox traps you in a corrupted thought experiment, as if infinity is a process that can be resolved into finite steps. That is NOT how it should be conceived.
Convert to any base system (which you can do in math) that can handle 1/3 easier, such as a base 3 system, and the issue isn't even an issue anymore.
Yes I did.
In base 3, you have 1=0.222... so it's not really any better.
Incorrect.
Uh, in base 3 1 =1 man.
And how am I incorrect? Maybe I am remembering my math wrong Adrien, but I'd love to be proved wrong.
http://en.wikipedia.org/wiki/Ternary_numeral_system btw
:googly-eyes:
But that whole notion of infinity only exists within the mathematical system which supports it. To remove infinity from finite sets may be sensible within mathematics in order to allow infinity to function as it must for the purposes of mathematics, but when viewed from any other perspective it borders on lunacy.
Your description makes it slightly more sensible to me. Your "by virtue of infinitely repeating" gets me away from the Zeno and the question of "how" to an understanding of some inherent nature to infinite sequences.
But it still doesn't make much sense outside of the mathematical system which requires it.
Hey you know what's great though
What's great is that math is actually really useful and can be applied to the real world, and a lot of really important things happen when you mess around with the concept of infinity
It doesn't matter whether or not you can see something infinite; mathematics uses the concept of infinity not in some fit of masturbatory self-affirmation, but because it is demonstrably useful and leads to greater understanding of other things.
But no, you're right, let's just toss out higher math because we can't give examples based on apples or count it on our fingers.
...
Oooookiedokie. What's the difference between 0.222... and 3?
2?
edit: Nevermind, I was taking Inquisitor's post as fact.
A perfect circle would be. Or sphere, if you prefer.
Well, in a way what he's saying is true. The idea of, for example, the instantaneous velocity of a car does not represent reality in that, in an instant, the car is stationary.
It's still a very useful concept, though.
In what base system?
Man, you can read it for yourself on the damn wiki.
Decimal system (base 10) 1 = 1 Ternary system (base 3)
Decimal system (base 10) 1/3 = .1 Ternary system (base 3)
I am not understand the notion of why a Base 3 system would have to represent a whole number like 1 as a decimal.
For the same reason you'd do it in the decimal system? The concept of 0.999... = 1.0 exists in decimal. How does moving to Ternary solve anything?
And the word "tree" doesn't make sense outside the linguistic system which makes use of it.
It's really important to see math in the same way as a language, and not as an attempt to objectively describe and categorize reality.
There is no other perspective from which to interpret the word "tree," or any word, except the language from which it is taken; there is no other perspective from which to interpret the concept of infinity. Math is as semiotically arbitrary as any language, and it is just as useful. Saying that "infinity" does not inherently connect to reality is like saying that a word does not inherently connect to reality. It's missing the point, really; they're not supposed to.
0.1, 0.2, 1.0, 1.1, 1.2, 2.0, 2.1, 2.2, 10.0
EDIT: This is just conjecture on my part, actually. I've never thought about decimals in bases other than 10! I'll read the wiki.
No, you represent 1 as 1 in the decimal system last I checked? I mean, you could represent it as 1.0 if you really wanted to, but you could do the same in the ternary system.
Look, a base 3 system just means you count 1,2,10,11,12,20,21,22,100
as opposed to 1,2,3,4,5,6,7,8,9
So Marty was right, then. 0.222... = 1.0.
Oh dear god.
Look, ternary has a finite representation of 1/3. However it doesn't have one of 1/2. 1/2 in ternary is .111... . It thus follows that 2/2, or 1, is .222... .
Hence, ternary has the same problem.
We've been talking about 0.999... = 1.0 in decimal for pages. 0.222... = 1.0 in ternary is no different. You've done nothing but waste our time.
Thanks.
BOOM
good times!
.222... = 1 is base 3's equivalent of .999... = 1 in base 10. Any base is going to have that issue.
Sorry, I am tired and misreading shit. I misinterpreted the ... as a dramatic pause. Not as repeating, durr durr durr.
Yeah, .2 repeating would be 1 in a base 3 system.
The whole point of converting to a base 3 system is it is another, different way to illustrate that .9 repeating = 1, and it's just a failure of the base 10 system. Just like .2 repeating = 1 in base 3 is a failure of its system.
Edit: Yeah, yeah, apologizing again for misreading the ... I personally have never seen that notation outside of this thread to mean repeating, so, I kinda breeze over it. Hopefully I help explained the point of switching to other bases, as you can always switch to another base where the repeating wont happen. Once again, just another way to illustrate that .9 repeating = 1.
That's because all your base are belong to us.
Ha! A "dramatic pause"! It's okay, I think 2:30 AM is the wrong time to be debating math so I'll be going to bed too.
Well played
I don't understand why _J_ and many others are getting so hung up on whether this is in "reality" or not. Math in general is a tool that has shown an amazing proclivity for being able to describe how the world works in a concise and useful form. If I'm designing a circuit and want to know if it is stable or not, what do I do? I use a mathematical model and look at its behavior as, guess what, it goes towards infinity. Now, I know that my circuit won't last for an infinite amount of time, yet it is still a good test to see if my circuit will blow up or not. Here the concept of infinity is a useful tool, since modeling the exact behavior at a particular time is sometimes computationally infeasible.
We use the natural numbers so "naturally" (hence the name) that, at least I find it hard to imagine any world that wouldn't develop a system by which to manipulate them. From this system, it flows naturally that a concept of "infinity" arises, because there is no limit to the natural numbers. Why? Because natural numbers start with the concept of 0, 1, and this crazy concept of addition.
The wiki isn't very helpful
We should talk about semiotics instead
Oh.
Well, in any case, I don't really think it makes sense to describe a problem that affects all bases equally as a failure of any base.
For that matter, I don't see rational numbers having multiple representations as a problem at all. Well, aside from the obvious.
Evil, it's really pretty simply. Imagine a base 3 system as a odometer that turns the next digit over by 1 every time it would reach 3 instead of 10.
Adrien: Yeah, I mean. I'm not saying that base 10 is a failure, or something. But base 10 does a poor job representing 1/3, is all. Every base has a hard time representing some numbers (hell, base 3 is awful at representing any fraction other then 1/3). And yeah, .9 repeating being 1 really isn't a big deal, at the end of the day. It's just another way to tackle the issue, is all.
It doesn't matter that you can't physically "have" an infinite number of 9s (whatever that really means), because mathematically you can. And the rules of infinity are governed by math, not physics, so trying to argue about infinity using the latter is pointless.
Yeah that clicked about thirty seconds before I read your post
In fact in a fit of comprehension I ran through 1 to 17 or so in ternary on a notepad file
Awesome.
I seriously thought we were about to do the whole thread over again in base 3
Bwahahah, the thought of that amuses me to no end. Because some people are already confused enough, it would be even more confusing for most if we were working in a different base.
Person A: .2... equals 1!!
Prson B: But that is like .8... away from 1!!! WTFBBQ!~
If we do it in base 2 it'd be really easy to make some computers do the arguing for us, while we go ride bikes or something