I think the more critical issue at hand is, why do Brits say "maths"?
This thread lasting 40 pages is a direct result of a flawed public school system. How can you learn math from a system where 75% is supposed to be average and 80% of students get above the so called average grade.
Its now a crisis if someone reports that a school system is failing because 50% of students are below average.
Using math to disprove 0.999r = 1 is like having a web page where you prove that computers are impossible.
-Dr O
Its now a crisis if someone reports that a school system is failing because 50% of students are below average.
Using math to disprove 0.999r = 1 is like having a web page where you prove that computers are impossible.
-Dr O
I think that 0.9r ==1 is universally impossible.
Lets just say that you are accelarating at 0.9999r of lightspeed. You can never reach lightspeed since it would take an infinite amount of energy to do so. Hence 0.999r is never going to equal to 1.
Lets just say that you are accelarating at 0.9999r of lightspeed. You can never reach lightspeed since it would take an infinite amount of energy to do so. Hence 0.999r is never going to equal to 1.
QUOTE (krob+Mar 29 2007, 12:25 AM)
I think that 0.9r ==1 is universally impossible.
Lets just say that you are accelarating at 0.9999r of lightspeed. You can never reach lightspeed since it would take an infinite amount of energy to do so. Hence 0.999r is never going to equal to 1.
You realize that.. it takes an infinite amount of energy to reach light speed anyway, right?
Lets just say that you are accelarating at 0.9999r of lightspeed. You can never reach lightspeed since it would take an infinite amount of energy to do so. Hence 0.999r is never going to equal to 1.
You realize that.. it takes an infinite amount of energy to reach light speed anyway, right?
QUOTE (krob+Mar 29 2007, 12:25 PM)
Lets just say that you are accelarating at 0.9999r of lightspeed. You can never reach lightspeed since it would take an infinite amount of energy to do so. Hence 0.999r is never going to equal to 1.
That is a meaningless statement. The speed of light is a speed not acceleration, you don't accelerate at lightspeed or any fraction of it.
Nor does a physical representation like this have anything to do with a mathematical representation. Even if it did, lightspeed is not impossible. Doesn't light move at lightspeed?
That is a meaningless statement. The speed of light is a speed not acceleration, you don't accelerate at lightspeed or any fraction of it.
Nor does a physical representation like this have anything to do with a mathematical representation. Even if it did, lightspeed is not impossible. Doesn't light move at lightspeed?
Ah and so AlphaN returns with his "I am the all know'er, all see'er attitude".
And look at that. Another hypocritical post as well.
saying to Einstein "I think your theory of relativity will eventually be found flawed" is utterly different from saying to Euclid "I think your proof of infinitely many primes will be found flawed". The former is a justifiable statement, the latter is completely wrong.
If we define the rules there is one very simple truth that is absolutely undeniable. Human error. You claim math is entirely made up by a set of rules and figures created by man. True enough although the reason for the invention of math was to describe the physical world that is beside the point. The point is that it is something established by man and so WILL have human error. So to say that we define the rules then say it would be completely wrong to say "I think your proof (proof which is nothing more than rules we made up, kind of how the bible is proof that god exists) of infinitely many primes will be found flawed" is completely hypocritical.
Close minded? No more like narrow minded.
Also the rules of nature are not like the rules of math as you said. No see the rules of math are made up by us and are more than open to change when a better explanation comes along. The Laws of Nature are not made up by us. We can not change them when something we think could be better comes along. Now how we see and understand these laws may change/become more accurate but the laws themselves do not change.
As for graphing lines.. 1/3 is 1/3 is 1/3 and it doesn't matter if we are talking about a fraction, decimal representation, distance etc. Not to mention NOM that what you say is so called proof isn't. It's the same as saying 1/3 of 9 is 3 or 1/3 of 12 is 4. Where you take a line of finite length and divided it into three equal segments. That isn't even the argument. 9 times one third (1/3) equals three. 9 times 1 divided by three equals 3.
It means dividing into three equal segments. Where 9 * 1/3 = 3 (three equal segments of three) or 1 divided by 3 (1/3 of 1 where 1 is divided into three equal segments).
Tell me AlphaN what sounds more logical to you. That 1/3 exactly equals .3r where 1/3 represents a portion of a whole and is finite in size/value and .3r represents a number that has no end (not finite in size/value) and is a decimal with a repeating three that tends toward the finite value of 1/3. Where the repeating 3s are repeating BECAUSE of the remainder.
Or
1/3 equals .3_R1 where 1/3 represents a portion of a whole and is finite in size/value and .3_R1 is also finite in size/value and is easily shown using the foundation of mathematics (and associated rules) to equal 1/3 by using long division on paper. Simply divide 1 by three and you will get .3_R1. Only when you try to work out the remainder do you get more 3s but are still left with the remainder.
Remember that the more complex math was established based on the foundations of math. So if something can be shown using the simplest of math to be true than any more complex math used to try and show otherwise is one of two things.
Incorrect by the person showing
Incorrect when first established and left unquestioned/unanswered.
Also unless you come to terms that math was invented to be a tool and that is where it is belongs you will be nothing more than a kid playing a game of numbers.
Something that has been mentioned is that numbers (math itself) does not always represent a physical thing. That right there is the biggest mistake of all. That any math that has no physical equivalence simply isn't real. It only exists in our head by a set of rules that have been made up by us and manipulated to our liking. Regardless of it being physically correct or not. Well at least the foundations of math IS physical and is what I have used to actually prove.
Kinda like the argument that .9r = 1 because there is nothing between them on a number line. I guess you'll always believe that every number equals every other number.
Also the argument where you will travel 1 meter if you constantly travel half the distance remaining (which is also half the distance last traveled after the first distance is traveled). In the mean time I'll do what really needs to be done to walk that fool meter (travel the remaining distance which is equal to and not half of the last distance traveled after the first distance) and then keep on walking while saying goodbye.
Get real here, it's time to wake up and stop with the pretending and imagining.
I was wondering if anyone else was seeing that.
Anyway I've talked to Brits before so unless AlphaN uses the check spelling he certainly doesn't talk/type like what. I mean I have not even seen a 'wanker" in any of his posts yet.
Also with the level of maturity (or rather lack there of) where the insults are as bad as NOM's and Alpha's I highly question his so called claim to having any kind of degree, forget about going for his doctors. More like a teenager who still lives at home in the basement. He talks about logic a bit too much (Trekkie) despite his claims being completely illogical and contradicting. He's willing to claim that limits are something that are reached. Unless you're willing to say there are exceptions to the rules perhaps?
And look at that. Another hypocritical post as well.
QUOTE
. We define the rules,
QUOTE (->
| QUOTE |
| . We define the rules, |
saying to Einstein "I think your theory of relativity will eventually be found flawed" is utterly different from saying to Euclid "I think your proof of infinitely many primes will be found flawed". The former is a justifiable statement, the latter is completely wrong.
If we define the rules there is one very simple truth that is absolutely undeniable. Human error. You claim math is entirely made up by a set of rules and figures created by man. True enough although the reason for the invention of math was to describe the physical world that is beside the point. The point is that it is something established by man and so WILL have human error. So to say that we define the rules then say it would be completely wrong to say "I think your proof (proof which is nothing more than rules we made up, kind of how the bible is proof that god exists) of infinitely many primes will be found flawed" is completely hypocritical.
Close minded? No more like narrow minded.
Also the rules of nature are not like the rules of math as you said. No see the rules of math are made up by us and are more than open to change when a better explanation comes along. The Laws of Nature are not made up by us. We can not change them when something we think could be better comes along. Now how we see and understand these laws may change/become more accurate but the laws themselves do not change.
As for graphing lines.. 1/3 is 1/3 is 1/3 and it doesn't matter if we are talking about a fraction, decimal representation, distance etc. Not to mention NOM that what you say is so called proof isn't. It's the same as saying 1/3 of 9 is 3 or 1/3 of 12 is 4. Where you take a line of finite length and divided it into three equal segments. That isn't even the argument. 9 times one third (1/3) equals three. 9 times 1 divided by three equals 3.
It means dividing into three equal segments. Where 9 * 1/3 = 3 (three equal segments of three) or 1 divided by 3 (1/3 of 1 where 1 is divided into three equal segments).
Tell me AlphaN what sounds more logical to you. That 1/3 exactly equals .3r where 1/3 represents a portion of a whole and is finite in size/value and .3r represents a number that has no end (not finite in size/value) and is a decimal with a repeating three that tends toward the finite value of 1/3. Where the repeating 3s are repeating BECAUSE of the remainder.
Or
1/3 equals .3_R1 where 1/3 represents a portion of a whole and is finite in size/value and .3_R1 is also finite in size/value and is easily shown using the foundation of mathematics (and associated rules) to equal 1/3 by using long division on paper. Simply divide 1 by three and you will get .3_R1. Only when you try to work out the remainder do you get more 3s but are still left with the remainder.
Remember that the more complex math was established based on the foundations of math. So if something can be shown using the simplest of math to be true than any more complex math used to try and show otherwise is one of two things.
Incorrect by the person showing
Incorrect when first established and left unquestioned/unanswered.
Also unless you come to terms that math was invented to be a tool and that is where it is belongs you will be nothing more than a kid playing a game of numbers.
Something that has been mentioned is that numbers (math itself) does not always represent a physical thing. That right there is the biggest mistake of all. That any math that has no physical equivalence simply isn't real. It only exists in our head by a set of rules that have been made up by us and manipulated to our liking. Regardless of it being physically correct or not. Well at least the foundations of math IS physical and is what I have used to actually prove.
Kinda like the argument that .9r = 1 because there is nothing between them on a number line. I guess you'll always believe that every number equals every other number.
Also the argument where you will travel 1 meter if you constantly travel half the distance remaining (which is also half the distance last traveled after the first distance is traveled). In the mean time I'll do what really needs to be done to walk that fool meter (travel the remaining distance which is equal to and not half of the last distance traveled after the first distance) and then keep on walking while saying goodbye.
Get real here, it's time to wake up and stop with the pretending and imagining.
QUOTE
I think the more critical issue at hand is, why do Brits say "maths"?
I was wondering if anyone else was seeing that.
Anyway I've talked to Brits before so unless AlphaN uses the check spelling he certainly doesn't talk/type like what. I mean I have not even seen a 'wanker" in any of his posts yet.
Also with the level of maturity (or rather lack there of) where the insults are as bad as NOM's and Alpha's I highly question his so called claim to having any kind of degree, forget about going for his doctors. More like a teenager who still lives at home in the basement. He talks about logic a bit too much (Trekkie) despite his claims being completely illogical and contradicting. He's willing to claim that limits are something that are reached. Unless you're willing to say there are exceptions to the rules perhaps?
QUOTE (Precursor562+Mar 28 2007, 08:00 PM)
Also with the level of maturity (or rather lack there of) where the insults are as bad as NOM's and Alpha's I highly question his so called claim to having any kind of degree, forget about going for his doctors. More like a teenager who still lives at home in the basement. He talks about logic a bit too much (Trekkie) despite his claims being completely illogical and contradicting. He's willing to claim that limits are something that are reached. Unless you're willing to say there are exceptions to the rules perhaps?
Wow what a hypocritical post if I ever saw one!
What AN just posted but you seem to be too busy selectively posting your answers is that if you think there's a problem with the fundamental axioms of math then that's fine. What you've been posting in your past is that you don't seem to have as much a problem with the axoims as the concept of 0.9r = 1, which in of itself is a contradiction.
Wow what a hypocritical post if I ever saw one!
What AN just posted but you seem to be too busy selectively posting your answers is that if you think there's a problem with the fundamental axioms of math then that's fine. What you've been posting in your past is that you don't seem to have as much a problem with the axoims as the concept of 0.9r = 1, which in of itself is a contradiction.
PC: I'm afraid your line of argument is starting to sound a little homespun. Sort of "I'ze gets along just fine within' what I'ze already knows". Argument by "personal disbelief" is generally considered invalid; appealing to the "possibility of human error" without giving the slightest hint of if or where that error might be seems to be a non-starter, too.
I don't think you're going to get much sympathy for the position that mathematics is either unnecessarily complicated or somehow flawed simply because it doesn't sit well with you. You also come off poorly complaining that "despite so-and-so's pretty degree they learned it wrong in school", or "maybe they don't really have a degree after all". Exactly what are your credentials that you can make a credible charge that math is wrong? Oh, I forgot, anyone can point out the "possibility of human error". Is that all you've got, the hope that someday, someone will come up with something that'll prove your position --- is it your great "counter-proof" that because there is a slim possibility of that happening we should take your position seriously?
AlphaN has presented a fair summary of the difference between math and "observed knowledge". And I'm sure there are whole books on the subject, too. What he says is the way it is.
Here's the thing. If you stay with your "proofs by calculation" and your naive concept of infinity, your going to be stuck where you're at, well, forever. You need to add some new ideas to your knowledge base. For all your talk of being "open-minded", you haven't really walked the walk.
I don't think you're going to get much sympathy for the position that mathematics is either unnecessarily complicated or somehow flawed simply because it doesn't sit well with you. You also come off poorly complaining that "despite so-and-so's pretty degree they learned it wrong in school", or "maybe they don't really have a degree after all". Exactly what are your credentials that you can make a credible charge that math is wrong? Oh, I forgot, anyone can point out the "possibility of human error". Is that all you've got, the hope that someday, someone will come up with something that'll prove your position --- is it your great "counter-proof" that because there is a slim possibility of that happening we should take your position seriously?
AlphaN has presented a fair summary of the difference between math and "observed knowledge". And I'm sure there are whole books on the subject, too. What he says is the way it is.
Here's the thing. If you stay with your "proofs by calculation" and your naive concept of infinity, your going to be stuck where you're at, well, forever. You need to add some new ideas to your knowledge base. For all your talk of being "open-minded", you haven't really walked the walk.
QUOTE (AlphaNumeric+Mar 28 2007, 08:57 PM)
Notice there is no 'mainstream' and 'fringe' maths community, unlike physics. Maths is a case of "If it's proved, it's true. If it's disproved, it's false." You can disprove spurious maths claims with ease usually. 0.9r not being equal to 1 is just such a false statement which is easily disproved to any mathematicians and work continues elsewhere.
Want to argue about a physics theory? Fine. Want to argue about a pr oven maths result? It's a display of ignorance.
Great opus and clear thinking! Perfect!
But :
Wrong conclusion.
It is just more difficult in mathematics to find the flaws, as logic overwhelms reason. You need to let it go to able to see where the fundamental limitation of calculus and real numbers is. Maya...
Want to argue about a physics theory? Fine. Want to argue about a pr oven maths result? It's a display of ignorance.
Great opus and clear thinking! Perfect!
But :
Wrong conclusion.
It is just more difficult in mathematics to find the flaws, as logic overwhelms reason. You need to let it go to able to see where the fundamental limitation of calculus and real numbers is. Maya...
QUOTE (Precursor562+Mar 29 2007, 02:00 PM)
As for graphing lines.. 1/3 is 1/3 is 1/3 and it doesn't matter if we are talking about a fraction, decimal representation, distance etc. Not to mention NOM that what you say is so called proof isn't. It's the same as saying 1/3 of 9 is 3 or 1/3 of 12 is 4. Where you take a line of finite length and divided it into three equal segments. That isn't even the argument. 9 times one third (1/3) equals three. 9 times 1 divided by three equals 3.
It means dividing into three equal segments. Where 9 * 1/3 = 3 (three equal segments of three) or 1 divided by 3 (1/3 of 1 where 1 is divided into three equal segments).
Yep, you've missed the point yet again.
How long is each of the 1/3 sections of the 1cm line?
It is a line, not a process, so your process argument doesn't hold.
Where is your mythical remainder?
It means dividing into three equal segments. Where 9 * 1/3 = 3 (three equal segments of three) or 1 divided by 3 (1/3 of 1 where 1 is divided into three equal segments).
Yep, you've missed the point yet again.
How long is each of the 1/3 sections of the 1cm line?
It is a line, not a process, so your process argument doesn't hold.
Where is your mythical remainder?
QUOTE (Precursor562+Mar 29 2007, 02:00 AM)
True enough although the reason for the invention of math was to describe the physical world that is beside the point.
Numbers weren't invented to describe the universe, they were invented to keep track of things.. like money.
Then, later, people found all sorts of uses for them. The amazing thing about mathematics is that emergent behavior and structure in the universe takes its queue from math. If you look at the fundamental scale of things (that we know at least), all you have is wooshy washy probability clouds. Yet the emergent behavior is such that we get whole things, like apples or fingers, that illustrate perfectly concepts in our mind: numbers.
But inherently, the universe does not operate on numbers, nor does it care about them.
Math is just a game in our heads. Most mathematicians couldn't care less about potential applications, or how much their rules "mimic" nature. There are hundreds and hundreds of different number systems and systems too weird to be called number systems. They are studied for their own sake.
Which one describes reality perfectly? I'm guessing none of them. The universe will turn out to be weirder than any math we've thought of yet. But the beauty of mathematics is that it all ends up describing some aspect of our world, somewhere on some level. That, to me, is amazing.
You can establish a number system where .9r doesn't mean 1. But in the predominantly used number system, aka the "real" numbers, they are one in the same. Accepted proofs of very, very complicated things have been wrong on rare occasions, but this one is solid.
Numbers weren't invented to describe the universe, they were invented to keep track of things.. like money.
Then, later, people found all sorts of uses for them. The amazing thing about mathematics is that emergent behavior and structure in the universe takes its queue from math. If you look at the fundamental scale of things (that we know at least), all you have is wooshy washy probability clouds. Yet the emergent behavior is such that we get whole things, like apples or fingers, that illustrate perfectly concepts in our mind: numbers.
But inherently, the universe does not operate on numbers, nor does it care about them.
Math is just a game in our heads. Most mathematicians couldn't care less about potential applications, or how much their rules "mimic" nature. There are hundreds and hundreds of different number systems and systems too weird to be called number systems. They are studied for their own sake.
Which one describes reality perfectly? I'm guessing none of them. The universe will turn out to be weirder than any math we've thought of yet. But the beauty of mathematics is that it all ends up describing some aspect of our world, somewhere on some level. That, to me, is amazing.
You can establish a number system where .9r doesn't mean 1. But in the predominantly used number system, aka the "real" numbers, they are one in the same. Accepted proofs of very, very complicated things have been wrong on rare occasions, but this one is solid.
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
Another long post...
You seem to have missed my earlier question, after responding some vehemently to my initial comment on the matter:
You seem to have missed my earlier question, after responding some vehemently to my initial comment on the matter:
QUOTE (Euler+)
What would you attribute it to then? You seem fairly adamant that all those trained in elementary mathematics, all the world leaders in the area, all those in history who have made astounding contributions to the subject, have all got it wrong.
Whereas you, someone with little-to-no mathematical knowledge (which is fairly apparent btw), has got it right?
If you were a betting man, who would you say is having a misunderstanding? Regardless of what you currently believe about the fact that 1=0.9r, do you think it likely that you'd have more of an understanding about ANY mathematical result, than the experts?
Whereas you, someone with little-to-no mathematical knowledge (which is fairly apparent btw), has got it right?
If you were a betting man, who would you say is having a misunderstanding? Regardless of what you currently believe about the fact that 1=0.9r, do you think it likely that you'd have more of an understanding about ANY mathematical result, than the experts?
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
Ah and so AlphaN returns with his "I am the all know'er, all see'er attitude".
Nope, just that I know something with a bit of confidence which happens to be relevent to the discussion and that something can be found in a great many books and other resources available to everyone.
I find it funny you say that to me, when I'm agreeing with the view of many people while you think you know better than ALL the great minds in maths and physics.
So think about it, which of us is really trying to come across as knowing more than they actually do? Me, pointing out books, giving proofs, quoting geniuses. Or you, who thinks that he knows 'the truth' which people like Einstein, Feynman, Newton etc were all blind to, despite you knowing nothing of this area of maths?
Nope, just that I know something with a bit of confidence which happens to be relevent to the discussion and that something can be found in a great many books and other resources available to everyone.
I find it funny you say that to me, when I'm agreeing with the view of many people while you think you know better than ALL the great minds in maths and physics.
So think about it, which of us is really trying to come across as knowing more than they actually do? Me, pointing out books, giving proofs, quoting geniuses. Or you, who thinks that he knows 'the truth' which people like Einstein, Feynman, Newton etc were all blind to, despite you knowing nothing of this area of maths?
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
Tell me AlphaN what sounds more logical to you. That 1/3 exactly equals .3r where 1/3 represents a portion of a whole and is finite in size/value and .3r represents a number that has no end (not finite in size/value) and is a decimal with a repeating three that tends toward the finite value of 1/3. Where the repeating 3s are repeating BECAUSE of the remainder.
Or
1/3 equals .3_R1 where 1/3 represents a portion of a whole and is finite in size/value and .3_R1 is also finite in size/value and is easily shown using the foundation of mathematics (and associated rules) to equal 1/3 by using long division on paper. Simply divide 1 by three and you will get .3_R1. Only when you try to work out the remainder do you get more 3s but are still left with the remainder.
Or
1/3 equals .3_R1 where 1/3 represents a portion of a whole and is finite in size/value and .3_R1 is also finite in size/value and is easily shown using the foundation of mathematics (and associated rules) to equal 1/3 by using long division on paper. Simply divide 1 by three and you will get .3_R1. Only when you try to work out the remainder do you get more 3s but are still left with the remainder.
The former, not the latter. There is no remainder in the limit because the remainder only remains after finitely many steps. In the limit it's nothing.
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
Get real here, it's time to wake up and stop with the pretending and imagining.
Same to you. You aren't exactly fooling anyone here.
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
Anyway I've talked to Brits before so unless AlphaN uses the check spelling he certainly doesn't talk/type like what.
I say 'maths' instead of 'math'. Are you now saying that not only am I lying about what I do but also my nationality? To what end?
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
Also with the level of maturity (or rather lack there of) where the insults are as bad as NOM's and Alpha's I highly question his so called claim to having any kind of degree, forget about going for his doctors. More like a teenager who still lives at home in the basement.
Because everyone who gets a degree automatically becomes a polite person who never swears or calls an idiot an idiot? Yes, degrees somehow infer such a level of maturity 
I knew a few people during my undergrad times who I considered extremely childish and annoying and others who were extremely polite. People who have gone through university still come in all shapes and sizes and attitudes. But rather than realising that blatently obvious fact, you attempt to use my level of maturity and patience with fools as a way of gauging my knowledge. Even if I pepper my posts with insults towards people who quite clearly deserve the 'titles' I give them, the things like proofs and descriptions of the differences between maths and physics are still valid.
I knew a few people during my undergrad times who I considered extremely childish and annoying and others who were extremely polite. People who have gone through university still come in all shapes and sizes and attitudes. But rather than realising that blatently obvious fact, you attempt to use my level of maturity and patience with fools as a way of gauging my knowledge. Even if I pepper my posts with insults towards people who quite clearly deserve the 'titles' I give them, the things like proofs and descriptions of the differences between maths and physics are still valid.
QUOTE (Precursor562+Mar 29 2007, 03:00 AM)
He talks about logic a bit too much (Trekkie) despite his claims being completely illogical and contradicting. He's willing to claim that limits are something that are reached.
Given the structure of maths and logic is somewhat relevent (particularly when someone erroneously brought up Godel), explaining why maths doesn't default to physical reality as some kind of 'absolute truth' of itself is important. The fact you don't understand and you don't bother to learn about limits is not a failing of mine but yours.
QUOTE (N O M+Mar 29 2007, 06:29 AM)
Yep, you've missed the point yet again.
How long is each of the 1/3 sections of the 1cm line?
It is a line, not a process, so your process argument doesn't hold.
Where is your mythical remainder?
I'm no math whiz but I see some silly stuff here.
First... .9999... does not equal 1!
If you can measure or envision a difference...then there is a difference!
I may reach a point where I will "tolerate" an error, but they are still not equal.
I know a lady that works on injector pumps.
The operative word is "tolerances". How wrong are they allowed to be before it causes trouble. The tolerances they work with are extremely low. They actually have no devices that are accurate enough to make a proper fit.
They work on matching bores with pistons by feel and they last 100s of thousands of miles.
I see the question about how long are each of the 3 equal sections of 1cm and reply
"1/3 of a centimeter".
Had we been evolved into 12 fingered individuals and used 12 as our basis, the 1/3 argument would be clearer.
1/3 of 12 = 4.
Bruce
How long is each of the 1/3 sections of the 1cm line?
It is a line, not a process, so your process argument doesn't hold.
Where is your mythical remainder?
I'm no math whiz but I see some silly stuff here.
First... .9999... does not equal 1!
If you can measure or envision a difference...then there is a difference!
I may reach a point where I will "tolerate" an error, but they are still not equal.
I know a lady that works on injector pumps.
The operative word is "tolerances". How wrong are they allowed to be before it causes trouble. The tolerances they work with are extremely low. They actually have no devices that are accurate enough to make a proper fit.
They work on matching bores with pistons by feel and they last 100s of thousands of miles.
I see the question about how long are each of the 3 equal sections of 1cm and reply
"1/3 of a centimeter".
Had we been evolved into 12 fingered individuals and used 12 as our basis, the 1/3 argument would be clearer.
1/3 of 12 = 4.
Bruce
QUOTE (egnorant+Mar 29 2007, 02:10 PM)
I'm no math whiz but I see some silly stuff here.
First... .9999... does not equal 1!
If you can measure or envision a difference...then there is a difference!
... but if you were a math wiz, then you'd know .99... == 1.
The answer lies, strangely enough, in your second sentance: at "infinity" you can't measure a difference.
Moral: You can't carry "common sense experience" out to infinity.
First... .9999... does not equal 1!
If you can measure or envision a difference...then there is a difference!
... but if you were a math wiz, then you'd know .99... == 1.
The answer lies, strangely enough, in your second sentance: at "infinity" you can't measure a difference.
Moral: You can't carry "common sense experience" out to infinity.
can you even do infinity - infinity?
I guess it would be 0 but would it take an infinite amount of posts to firgue it out?
I guess it would be 0 but would it take an infinite amount of posts to firgue it out?
QUOTE (TenGig+Mar 29 2007, 03:40 PM)
can you even do infinity - infinity?
No, it's undefined. However, just as 0/0 or infinity/infinity are undefined, if you're given a function which tends to that, you can extract more information and a possible result if you're careful and know how to go about it.
For instance, consider f(x) = g(x) + h(x) where g(x) = x, h(x) = 5-x. In the limits of x->infinity you have g->infinity, h->infinity, but obviously f(x) = 5 always.
This is akin to the problem Precursor and/or StevenA has earlier. They considered f(x) = (a-x)/(a^2 - x^2) and complained that f(a) = 0/0 yet also equalled other things. They failed to understand that precisely because of that problem if you're sloppy, we have the notion of limits and results like L'Hopitals rule which allow for such things to be computed with consistency.
Otherwise you just prove 1+1=4 and that you didn't pay attention in maths class.
No, it's undefined. However, just as 0/0 or infinity/infinity are undefined, if you're given a function which tends to that, you can extract more information and a possible result if you're careful and know how to go about it.
For instance, consider f(x) = g(x) + h(x) where g(x) = x, h(x) = 5-x. In the limits of x->infinity you have g->infinity, h->infinity, but obviously f(x) = 5 always.
This is akin to the problem Precursor and/or StevenA has earlier. They considered f(x) = (a-x)/(a^2 - x^2) and complained that f(a) = 0/0 yet also equalled other things. They failed to understand that precisely because of that problem if you're sloppy, we have the notion of limits and results like L'Hopitals rule which allow for such things to be computed with consistency.
Otherwise you just prove 1+1=4 and that you didn't pay attention in maths class.
^ Just noticed I dropped a minus sign on the h->-infinity. While it's obvious it's a typo I'm sure someone would jump on it in the hopes of it somehow backing up their claims that 0.9r isn't 1.
INDEED 1=.9r
it follows from simple law of geometric progression.
0.9r= 0.9+0.09+0.009+0.0009+...............
=0.9[1+(1/10)+(1/100)+...........]
=0.9*[1/(1-1/10)]
=0.9*10/9
=1
the point is there is no number between 0.9r and 1 and so they are same. the debate about limits and all that are apllicable for problems like is 0.3r=0.34,for they
are not same,but not for1=0.9r.
IF I AM WRONG PLEASE POINT IT OUT
it follows from simple law of geometric progression.
0.9r= 0.9+0.09+0.009+0.0009+...............
=0.9[1+(1/10)+(1/100)+...........]
=0.9*[1/(1-1/10)]
=0.9*10/9
=1
the point is there is no number between 0.9r and 1 and so they are same. the debate about limits and all that are apllicable for problems like is 0.3r=0.34,for they
are not same,but not for1=0.9r.
IF I AM WRONG PLEASE POINT IT OUT
QUOTE (egnorant+Mar 30 2007, 02:10 AM)
I see the question about how long are each of the 3 equal sections of 1cm and reply
"1/3 of a centimeter".
But state this in decimal. You can't, it is threes all the way to infinity.
Add them up and you get 1cm
"1/3 of a centimeter".
But state this in decimal. You can't, it is threes all the way to infinity.
- No end
- No remainder
- No tolerance
- No Error
Add them up and you get 1cm
- No remainder
- No tolerance
- No Error
How many whole numbers are there between 1 and 2?
Whats that? None?! Why then 1 and 2 must be the same.
Therefore, because there are no whole numbers between 1 and 2...
1 = 2
Just because the nature of 0.9R implies that it is the closest possible decimal number to 1, does not immediately imply that it must be equal to 1.
It's not even a matter of accuracy versus approximation. It's a matter of pure identity.
Whats that? None?! Why then 1 and 2 must be the same.
Therefore, because there are no whole numbers between 1 and 2...
1 = 2
Just because the nature of 0.9R implies that it is the closest possible decimal number to 1, does not immediately imply that it must be equal to 1.
It's not even a matter of accuracy versus approximation. It's a matter of pure identity.
An interesting, but ultimately flawed, argument SSU. You're on the right track, though.
In the reals, no two numbers are "next" to each other but are separated by infinitely more additional reals (by definition). In the case of .9r and 1, it can be shown that there in fact cannot be any additional numbers between .9r and 1 (care to try to name one?). If this is the case, then .9r and 1 cannot be distinct since if they were distinct there would have to be additional reals separating them.
Therefore, the only logical conclusion is that they are one and the same number (differing only in representation, not value).
So with a little extra machinery, your conclusion is valid if you replace the "1=2" with ".9r = 1".
In the reals, no two numbers are "next" to each other but are separated by infinitely more additional reals (by definition). In the case of .9r and 1, it can be shown that there in fact cannot be any additional numbers between .9r and 1 (care to try to name one?). If this is the case, then .9r and 1 cannot be distinct since if they were distinct there would have to be additional reals separating them.
Therefore, the only logical conclusion is that they are one and the same number (differing only in representation, not value).
So with a little extra machinery, your conclusion is valid if you replace the "1=2" with ".9r = 1".
Name me a whole number that exists between 1 and 2.
It's pure semantics and argumentative clowning around meant to prove that 2 + 2 = 5.
Seriously. There's a whole line of thought that starts with accepting this kind of denial of identity and it's called 'doublethink'.
I am 1 individual.
0.9R is not my equal.
It's pure semantics and argumentative clowning around meant to prove that 2 + 2 = 5.
Seriously. There's a whole line of thought that starts with accepting this kind of denial of identity and it's called 'doublethink'.
I am 1 individual.
0.9R is not my equal.
Note that I edited my previous post while you were making yours ...
Your argument for the whole numbers is perfectly fine ... but we aren't talking about the whole numbers but instead the reals. As I said, with the reals, no two lie "next" to one another. With the wholes, there are no interlopers between nextdoor neighbors. So you can't extend your argument about the wholes into the reals. Sorry.
But your comments about "identify" are on the mark; strangely, it is exactly your logic that is used to demonstrate .9r = 1.
Your argument for the whole numbers is perfectly fine ... but we aren't talking about the whole numbers but instead the reals. As I said, with the reals, no two lie "next" to one another. With the wholes, there are no interlopers between nextdoor neighbors. So you can't extend your argument about the wholes into the reals. Sorry.
But your comments about "identify" are on the mark; strangely, it is exactly your logic that is used to demonstrate .9r = 1.
What NCN said.
The real line forms a continuum, whereas the whole numbers do not. If points in a continuum, such as the real line, had nearest neighbors, you'd run into all sorts of trouble. For example, you could enumerate the real numbers, which is impossible.
Follow your line of thought. If .9r is the number right before 1, what is the number right before .9r? And the number right before that? I would love to see a list, say of the first 20 numbers immediately to the left of 1.
The real line forms a continuum, whereas the whole numbers do not. If points in a continuum, such as the real line, had nearest neighbors, you'd run into all sorts of trouble. For example, you could enumerate the real numbers, which is impossible.
Follow your line of thought. If .9r is the number right before 1, what is the number right before .9r? And the number right before that? I would love to see a list, say of the first 20 numbers immediately to the left of 1.
QUOTE (N O M+Mar 29 2007, 08:35 PM)
QUOTE
(egnorant @ Mar 30 2007, 02:10 AM)
But state this in decimal. You can't, it is threes all the way to infinity.
Add them up and you get 1cm
Alas, I am but a simple man. I see the limitations of our counting system and its foibles. Infinity is a slippery beast.
I understand that it is tolerated that 3 X .33r =1 because of the imperfection of our method of counting.
3 X .33r= .99r is the basic truth.
3 X .33r = 1 because we agree to ignore the error and our counting method has no more accurate way to portray it.
To demand complete accuracy on one side and allow inaccuracy on the other is
a little disturbing.
.99r either exists as .99r and come up short of 1......or .99r does not exist because it is agreed that it is 1.
Are we going for absolute accuracy or are we agreeing to forgo the inaccuracies?
Bruce
No pointing out any spelling mistake. With the equation f(x) = g(x) + h(x) where g(x) = x and h(x) = 5 - x then f(x) = x + 5 - x
Giving x any finite value will give f(x) = 5. If we give x the limit of infinity x->infinity simply says that there is no limit to how large or how small the finite value of x can be. So this doesn't show anything regarding .9r = 1/.9r != 1.
No pointing out any spelling mistake. With the equation f(x) = g(x) + h(x) where g(x) = x and h(x) = 5 - x then f(x) = x + 5 - x
Giving x any finite value will give f(x) = 5. If we give x the limit of infinity x->infinity simply says that there is no limit to how large or how small the finite value of x can be. So this doesn't show anything regarding .9r = 1/.9r != 1.
INDEED 1=.9r
it follows from simple law of geometric progression.
0.9r= 0.9+0.09+0.009+0.0009+...............
=0.9[1+(1/10)+(1/100)+...........]
=0.9*[1/(1-1/10)]
=0.9*10/9
Congrats you just showed that .1r = .9r with the assumption that .1r = 1/9. 1/9 != .1r
it equals .1_R1. .1r has the limit 1/9 where it value is forever approaching the value 1/9.
You can see threes for as long as you try to work out the remainder. Without the remainder you won't get repeated threes.
There is just one thing here that doesn't work.
There is just one thing here that doesn't work.
If you are still "always" doing something, you haven't yet reached infinity,
You can never reach infinity only approach it. Approaching it is exactly what .9r does and to get closer to something there has to be a process. Physically the process for a person to get closer to something requires that person to travel. So traveling is the process.
Mathematically the process is working out a remainder that yields another remainder. For the case of a single number repeating the remainder you get is the same over and over again. So for 1/3 you get .3 with a remainder of 1.
For pi where it is a different number each time you get a different remainder each time but a remainder none the less.
The remainder is important here and does hold value. Now the longer you try to work the remainder out the less of an error you will get by dropping this info but an error you still get.
Now you got it right it's just too bad it goes against what you had said earlier. Saying "you haven't yet reached infinity" implies that at some point you can. But then you say ""infinity is that place beyond always". It has no single value, or for that matter any value at all".
Now you got it right it's just too bad it goes against what you had said earlier. Saying "you haven't yet reached infinity" implies that at some point you can. But then you say ""infinity is that place beyond always". It has no single value, or for that matter any value at all".
So far our area is 1/2 + 1/4 + 1/4 = 1. Continue this process, ad infinitum. Clearly, we are geometrically generating the series 1/2 + 1/4 + 1/8 + ... .
You might say, well OK, but no matter how many times you do the division, there's always going to remain a little nick left. But we are going to do it an infinite number of times ... and I have said infinity is "beyond always", so the "nick" is going to disappear!
In order to reach one you must add the remaining amount which in this case is the last amount added. Eg. 1/2 + 1/2 = 1, 1/2 + 1/4 + 1/4 = 1, 1/2 + 1/4 + 1/8 + 1/8 = 1
If you don't do this at some point then you will always fall short of reaching 1. The difference becomes smaller and smaller and no it doesn't disappear. It's an endless equation that equals a number that has no value because the value is always changing. However we can represent the answer. Let's call it x. So 1 - x = y where y has a value depended on the value of both 1 and x. Now 1 is solid but x is changing since x is the sum of an equation that doesn't end. So y will not have a set value either.
.9r = y where x = 1/10^n n->infinity and 1/10^n becomes an endless equation where n->infinity (n can not equal infinity). This equation is 1 / (10 * 10 * 10 * 10.....) = x
1 - x = y
1 - 1 / (10 * 10 * 10 * 10.....) = .9r
y = .9r
1 - x = y
1 = x + y
1 != y where y = .9r
.9r != 1
Mathematics suffers from the "middle world" syndrome, just like physics.
It's "conceptual microscope" is limited.
1/3 (of a metre) =.3r
.3r*3=.9r
.9r !=1 metre
Intersecting lines are no better.
Take a close look at the thickness of the lines.
Now take a closer look.
Still closer...
Tired of looking?
Need something now?
OK. .9r=1
Welcome to "middle world mathematics".
It's terribly convenient, but not entirely accurate.
True.
True.
Now let's move on to our normal rules where we can continue on "forever". Consider that somehow the "infinite'th" term is no longer divisible because "infinity isn't a real number".
Here is the mess up. Yes you can't divide by infinith term because you can't divide infinity but you can't equal infinith either because you can't equal infinity. It's always beyond as you said.
So with 1/2 + 1/4 + 1/8 + 1/16.... you are adding finite fractions where each finite fraction is a half of the previous and can be added, there is just no end to how many is being added.
And so when you do just that when you say "you are adding an infinite number of fractions where each fraction is half of the previous" you are not actually adding an infinite number of fractions. You are adding "n" number of fractions where n has the limit n->infinity meaning there is no limit to the number of fractions you can add.
In order to reach 1 you must add a fraction equal to the last fraction added. Since you are adding fractions that are only half of the last fraction added and you are doing this an unlimited number of times (there is no limit to how many times you can) you get infinitely closer to equaling 1 but you will never actually equal it. Therefore the endless equation 1/2 + 1/4 + 1/8 + 1/16..... has the limit 1.
QUOTE (->
| QUOTE |
| (egnorant @ Mar 30 2007, 02:10 AM) I see the question about how long are each of the 3 equal sections of 1cm and reply "1/3 of a centimeter". |
But state this in decimal. You can't, it is threes all the way to infinity.
- No end
- No remainder
- No tolerance
- No Error
Add them up and you get 1cm
- No remainder
- No tolerance
- No Error
Alas, I am but a simple man. I see the limitations of our counting system and its foibles. Infinity is a slippery beast.
I understand that it is tolerated that 3 X .33r =1 because of the imperfection of our method of counting.
3 X .33r= .99r is the basic truth.
3 X .33r = 1 because we agree to ignore the error and our counting method has no more accurate way to portray it.
To demand complete accuracy on one side and allow inaccuracy on the other is
a little disturbing.
.99r either exists as .99r and come up short of 1......or .99r does not exist because it is agreed that it is 1.
Are we going for absolute accuracy or are we agreeing to forgo the inaccuracies?
Bruce
Or, even more generally...
If there exist two real numbers, x and y, that are nearest neighbors, then what value is (x + y) / 2?
What is (x - y)? You can call it .0r1, but then what is (x - y) / 2? (x - y) / 4?
If there exist two real numbers, x and y, that are nearest neighbors, then what value is (x + y) / 2?
What is (x - y)? You can call it .0r1, but then what is (x - y) / 2? (x - y) / 4?
Welcome to the land of the Paradox.
To your left you'll see a number of mathematicians inventing contrived arguments designed to contradict themselves and force those who don't immediately understand the queer language of 'mathspeak' to suspend disbelief concerning their rigorously contradictory claims of omnisense.
You'll find them selling their wares next to priests, lawyers, holocaust deniers, animal rights activists and Pro-Lifers who're willing to kill to make their point.
If you'll look to your left... you'll quickly go mad.
My advice?
Keep on truckin' straight and don't look back until you get to Kentucky.
To your left you'll see a number of mathematicians inventing contrived arguments designed to contradict themselves and force those who don't immediately understand the queer language of 'mathspeak' to suspend disbelief concerning their rigorously contradictory claims of omnisense.
You'll find them selling their wares next to priests, lawyers, holocaust deniers, animal rights activists and Pro-Lifers who're willing to kill to make their point.
If you'll look to your left... you'll quickly go mad.
My advice?
Keep on truckin' straight and don't look back until you get to Kentucky.
You always have to add that next nine. 
MITCH RAEMSCH -- LIGHT FELL --
MITCH RAEMSCH -- LIGHT FELL --
QUOTE (egnorant+Mar 30 2007, 03:28 PM)
3 X .33r = 1 because we agree to ignore the error and our counting method has no more accurate way to portray it.
No error at all.
By definition: 0.3r = 1/3
that is what 0.3r is, in decimal
so 3 * 0.3r is 1
the fact that is also equals 0.9r is because 0.9r equals 1, no other reason.
again, no error, no tollerance, no remainder
No error at all.
By definition: 0.3r = 1/3
that is what 0.3r is, in decimal
so 3 * 0.3r is 1
the fact that is also equals 0.9r is because 0.9r equals 1, no other reason.
again, no error, no tollerance, no remainder
QUOTE (Solid State Universe+Mar 30 2007, 04:02 AM)
Welcome to the land of the Paradox.
To your left you'll see a number of mathematicians inventing contrived arguments designed to contradict themselves and force those who don't immediately understand the queer language of 'mathspeak' to suspend disbelief concerning their rigorously contradictory claims of omnisense.
You'll find them selling their wares next to priests, lawyers, holocaust deniers, animal rights activists and Pro-Lifers who're willing to kill to make their point.
If you'll look to your left... you'll quickly go mad.
My advice?
Keep on truckin' straight and don't look back until you get to Kentucky.
Did you seriously just compare me to a holocaust denier?
I mean this with offense, go f yourself.
To your left you'll see a number of mathematicians inventing contrived arguments designed to contradict themselves and force those who don't immediately understand the queer language of 'mathspeak' to suspend disbelief concerning their rigorously contradictory claims of omnisense.
You'll find them selling their wares next to priests, lawyers, holocaust deniers, animal rights activists and Pro-Lifers who're willing to kill to make their point.
If you'll look to your left... you'll quickly go mad.
My advice?
Keep on truckin' straight and don't look back until you get to Kentucky.
Did you seriously just compare me to a holocaust denier?
I mean this with offense, go f yourself.
QUOTE (Solid State Universe+Mar 30 2007, 01:05 AM)
How many whole numbers are there between 1 and 2?
Whats that? None?! Why then 1 and 2 must be the same.
Therefore, because there are no whole numbers between 1 and 2...
1 = 2
Just because the nature of 0.9R implies that it is the closest possible decimal number to 1, does not immediately imply that it must be equal to 1.
It's not even a matter of accuracy versus approximation. It's a matter of pure identity.
The point is that given 2 distinct real numbers, there are an infinite number of real numbers between them. You can easily prove this:
Given 2 distinct real numbers numbers, call x the smaller of the 2 and y the larger. So x < y.
Then x + x < x + y
so 2x < x+y
and x < (x+y)/2 (their average, since the real numbers are closed on division by any number other than 0, this is a real number)
Also, x + y < 2*y
so (x+y)/2 < y
Therefore x < (x+y)/2 < y, so between any 2 distinct real numbers x and y, there is at least 1 real number. We have found their average, but there are others obviously. In fact...
Since x and (x+y)/2 are distinct numbers, there is at least 1 number between them, and so on (and similarly between (x+y)/2 and y), so we have found 3 numbers between x and y, and since this keeps going, there are an infinite amount of numbers between any 2 distinct real numbers.
To show how this is relevant to the discussion:
Between every 2 distinct real numbers, there is an infinite amount of real numbers.
Therefore, it is not possible to have 2 real numbers where there is a finite (but non-zero) amount of real numbers between them (since if the 2 numbers are unique there are infinitely many between them, so having a non-zero finite amount between 2 numbers is a contradiction) and if you have 2 real numbers with 0 real numbers between them, then they are the same number.
------------
Note that this proof relies on the fact that (x+y)/2 is always a real number. This does not work for the integers, so the same results cannot be extended to them.
Whats that? None?! Why then 1 and 2 must be the same.
Therefore, because there are no whole numbers between 1 and 2...
1 = 2
Just because the nature of 0.9R implies that it is the closest possible decimal number to 1, does not immediately imply that it must be equal to 1.
It's not even a matter of accuracy versus approximation. It's a matter of pure identity.
The point is that given 2 distinct real numbers, there are an infinite number of real numbers between them. You can easily prove this:
Given 2 distinct real numbers numbers, call x the smaller of the 2 and y the larger. So x < y.
Then x + x < x + y
so 2x < x+y
and x < (x+y)/2 (their average, since the real numbers are closed on division by any number other than 0, this is a real number)
Also, x + y < 2*y
so (x+y)/2 < y
Therefore x < (x+y)/2 < y, so between any 2 distinct real numbers x and y, there is at least 1 real number. We have found their average, but there are others obviously. In fact...
Since x and (x+y)/2 are distinct numbers, there is at least 1 number between them, and so on (and similarly between (x+y)/2 and y), so we have found 3 numbers between x and y, and since this keeps going, there are an infinite amount of numbers between any 2 distinct real numbers.
To show how this is relevant to the discussion:
Between every 2 distinct real numbers, there is an infinite amount of real numbers.
Therefore, it is not possible to have 2 real numbers where there is a finite (but non-zero) amount of real numbers between them (since if the 2 numbers are unique there are infinitely many between them, so having a non-zero finite amount between 2 numbers is a contradiction) and if you have 2 real numbers with 0 real numbers between them, then they are the same number.
------------
Note that this proof relies on the fact that (x+y)/2 is always a real number. This does not work for the integers, so the same results cannot be extended to them.
If A and B are two distinct real numbers, A != B, and C is a real, then f(A,B,N) = if (C>=0, (A + B(C+1))/(C+2), (A(C-1) + B )/(C-2) ) is a real number between A and B. In fact this is a 1-to-1 mapping between all the reals and the (open) interval ( A,B ).
f(e, pi, (6 - e - pi)/(pi - 3)) = 3, exactly.
f(0, 1, 198) = 0.995, exactly.
f(1, 0, -198) = 0.995, also.
Can you prove that f(x,y,z) = f(y,x,-z) ?
Can you prove that 0 < f(0,1,z) < 1 ?
What is the supremum of { f(0,1,n) } where n is in Z? What about when n is in R?
f(e, pi, (6 - e - pi)/(pi - 3)) = 3, exactly.
f(0, 1, 198) = 0.995, exactly.
f(1, 0, -198) = 0.995, also.
Can you prove that f(x,y,z) = f(y,x,-z) ?
Can you prove that 0 < f(0,1,z) < 1 ?
What is the supremum of { f(0,1,n) } where n is in Z? What about when n is in R?
QUOTE
For instance, consider f(x) = g(x) + h(x) where g(x) = x, h(x) = 5-x. In the limits of x->infinity you have g->infinity, h->infinity, but obviously f(x) = 5 always.
No pointing out any spelling mistake. With the equation f(x) = g(x) + h(x) where g(x) = x and h(x) = 5 - x then f(x) = x + 5 - x
Giving x any finite value will give f(x) = 5. If we give x the limit of infinity x->infinity simply says that there is no limit to how large or how small the finite value of x can be. So this doesn't show anything regarding .9r = 1/.9r != 1.
QUOTE (->
| QUOTE |
| For instance, consider f(x) = g(x) + h(x) where g(x) = x, h(x) = 5-x. In the limits of x->infinity you have g->infinity, h->infinity, but obviously f(x) = 5 always. |
No pointing out any spelling mistake. With the equation f(x) = g(x) + h(x) where g(x) = x and h(x) = 5 - x then f(x) = x + 5 - x
Giving x any finite value will give f(x) = 5. If we give x the limit of infinity x->infinity simply says that there is no limit to how large or how small the finite value of x can be. So this doesn't show anything regarding .9r = 1/.9r != 1.
INDEED 1=.9r
it follows from simple law of geometric progression.
0.9r= 0.9+0.09+0.009+0.0009+...............
=0.9[1+(1/10)+(1/100)+...........]
=0.9*[1/(1-1/10)]
=0.9*10/9
Congrats you just showed that .1r = .9r with the assumption that .1r = 1/9. 1/9 != .1r
it equals .1_R1. .1r has the limit 1/9 where it value is forever approaching the value 1/9.
QUOTE
But state this in decimal. You can't, it is threes all the way to infinity.
* No end
* No remainder
* No tolerance
* No Error
* No end
* No remainder
* No tolerance
* No Error
You can see threes for as long as you try to work out the remainder. Without the remainder you won't get repeated threes.
QUOTE (Nick+Mar 30 2007, 04:04 AM)
You always have to add that next nine.
This is typically the flawed concept of infinity people have that makes it impossible for them to see what is going on at "the limit". If you are still "always" doing something, you haven't yet reached inifinity, so you can't say anything about what's going on "at infinity".
As I said before, "infinity is that place beyond always". It has no single "value", or for that matter any value at all. I could go into a long, involved justification of why this is a good definition of infinity, but those who refuse to believe are just going to refuse to believe and so it's a waste of time. Suffice it to say, your understanding has stopped short of infinity.
Since we are in a "land beyond always", you can't start pontificating about division, remainders, the next digit ... anything.
But maybe I'll blow a few bytes anyway in what will probably be a futile gesture.
Base-10 isn't the only notation system that exhibits ".9r = 1" (as it were). All bases do. For example, in base-6 the number ".5r = 1", in hex ".Fr = 1", etc. But perhaps most illuminating of all, in base-2 ".1r = 1".
Why is that example so enlightening? Because .1r base-2 is the series 1/2 + 1/4 + ... whose sum can be shown to be exactly 1. Now I suppose there are a few of you who are going to complain the demonstration of that sum contains "limits" or some "sleight of hand", so let's ignore algebra and use geometry.
Take a 1x1 square (whose area is clearly exactly 1 and start successively dividing it in half. The first division results in two equal pieces, dividing one of those halves again in half results in two quarter pieces along with the remaining half piece. So far our area is 1/2 + 1/4 + 1/4 = 1. Continue this process, ad infinitum. Clearly, we are geometrically generating the series 1/2 + 1/4 + 1/8 + ... .
You might say, well OK, but no matter how many times you do the division, there's always going to remain a little nick left. But we are going to do it an infinite number of times ... and I have said infinity is "beyond always", so the "nick" is going to disappear!
How can one justify this? The fact is, the square does have an area of 1 by simple geometry. A little piece of it hasn't fallen off simply because we chose to draw lines in it. At infinity, mathematical infinity, the dividing process doesn't "end" so much as it no longer has meaning because what's left isn't divisible ... not because it's 0, but because it's "something that can't be divided". So the fact that the square does have a precise area of 1 by definition justifies the concept of infinity must be somewhere beyond the process of successive division. There is no other choice.
By the way, this is also one case of where the so-called "limit" is the exact answer ... the "limit is reached", not just "approached".
Sure, there are those that are going to still argue "but the dividing never ends, there's always something left over" ... and they are right, the dividing never ends. Nevertheless, "never" is not the mathematician's infinity. If you can still perform operations, then you are just at a stop along the way to infinity.
Again, if you think hard about the fact the square has a precise area of exactly 1, you are forced into the idea that infinity lies beyond always.
This is typically the flawed concept of infinity people have that makes it impossible for them to see what is going on at "the limit". If you are still "always" doing something, you haven't yet reached inifinity, so you can't say anything about what's going on "at infinity".
As I said before, "infinity is that place beyond always". It has no single "value", or for that matter any value at all. I could go into a long, involved justification of why this is a good definition of infinity, but those who refuse to believe are just going to refuse to believe and so it's a waste of time. Suffice it to say, your understanding has stopped short of infinity.
Since we are in a "land beyond always", you can't start pontificating about division, remainders, the next digit ... anything.
But maybe I'll blow a few bytes anyway in what will probably be a futile gesture.
Base-10 isn't the only notation system that exhibits ".9r = 1" (as it were). All bases do. For example, in base-6 the number ".5r = 1", in hex ".Fr = 1", etc. But perhaps most illuminating of all, in base-2 ".1r = 1".
Why is that example so enlightening? Because .1r base-2 is the series 1/2 + 1/4 + ... whose sum can be shown to be exactly 1. Now I suppose there are a few of you who are going to complain the demonstration of that sum contains "limits" or some "sleight of hand", so let's ignore algebra and use geometry.
Take a 1x1 square (whose area is clearly exactly 1 and start successively dividing it in half. The first division results in two equal pieces, dividing one of those halves again in half results in two quarter pieces along with the remaining half piece. So far our area is 1/2 + 1/4 + 1/4 = 1. Continue this process, ad infinitum. Clearly, we are geometrically generating the series 1/2 + 1/4 + 1/8 + ... .
You might say, well OK, but no matter how many times you do the division, there's always going to remain a little nick left. But we are going to do it an infinite number of times ... and I have said infinity is "beyond always", so the "nick" is going to disappear!
How can one justify this? The fact is, the square does have an area of 1 by simple geometry. A little piece of it hasn't fallen off simply because we chose to draw lines in it. At infinity, mathematical infinity, the dividing process doesn't "end" so much as it no longer has meaning because what's left isn't divisible ... not because it's 0, but because it's "something that can't be divided". So the fact that the square does have a precise area of 1 by definition justifies the concept of infinity must be somewhere beyond the process of successive division. There is no other choice.
By the way, this is also one case of where the so-called "limit" is the exact answer ... the "limit is reached", not just "approached".
Sure, there are those that are going to still argue "but the dividing never ends, there's always something left over" ... and they are right, the dividing never ends. Nevertheless, "never" is not the mathematician's infinity. If you can still perform operations, then you are just at a stop along the way to infinity.
Again, if you think hard about the fact the square has a precise area of exactly 1, you are forced into the idea that infinity lies beyond always.
QUOTE (N O M+Mar 30 2007, 04:23 AM)
No error at all.
By definition: 0.3r = 1/3
that is what 0.3r is, in decimal
so 3 * 0.3r is 1
the fact that is also equals 0.9r is because 0.9r equals 1, no other reason.
again, no error, no tollerance, no remainder
I still see the disclaimer "by definition".
Is this because it is as close as this method of counting will allow?
Kind of a square peg, round hole thing here.
.99r exists to show an amount of less than 1.
No error...by acclimation.
No tolerance...that is measurable.
No remainder...that our language can define.
If I ship a package that costs a buck for under a pound and 2 bucks for a pound and over, how much should I pay for a package that weighs .99r pounds?
Bruce
By definition: 0.3r = 1/3
that is what 0.3r is, in decimal
so 3 * 0.3r is 1
the fact that is also equals 0.9r is because 0.9r equals 1, no other reason.
again, no error, no tollerance, no remainder
I still see the disclaimer "by definition".
Is this because it is as close as this method of counting will allow?
Kind of a square peg, round hole thing here.
.99r exists to show an amount of less than 1.
No error...by acclimation.
No tolerance...that is measurable.
No remainder...that our language can define.
If I ship a package that costs a buck for under a pound and 2 bucks for a pound and over, how much should I pay for a package that weighs .99r pounds?
Bruce
QUOTE (egnorant+Mar 29 2007, 02:10 PM)
I'm no math whiz but I see some silly stuff here.
First... .9999... does not equal 1!
If you can measure or envision a difference...then there is a difference!
I may reach a point where I will "tolerate" an error, but they are still not equal.
Bruce,
First, one non-math whiz (technically, anyway...) to another (I am far more the philosopher...) I happen to agree with you that .9999.... and 1 are not the exact same thing. Only "virtually" so in a kind of Princess Bride "almost dead" kind of way, BUT... the notion that they are equal is not something I view as "silly" whatsoever. It's a question of framing.
Are you familiar with the notion of parallel lines that converge somewhere at infinity? In a sense, near as I can figure, that is the "logic" behind .9999... = 1, although, for the record, I would modify the parallel line concept to say that the convergence happens somewhere "beyond" infinity.
It's all a question on some level of where one is looking at the number in a kind of "elephant of many parts" manner. Similarly, does Man = star dust? Go back in time far enough and the answer is yes... kind of. All the seeds that became Man were there.
Also not unrelated... does 4/4= 1? Ala your comments regarding "imagining a difference" I would apply the same logic: 4/4 = 4 pieces together AS IF 1, which equals an indivisible unity and so I would have to say they are only "virtually" identitcal, not actually identical.
I wonder what Liebniz would have to say on this with respect to his "Identity of indiscernibles" (i.e. Two things are identical if and only if they share the same properties.). By the logic presented here, can any two things really be viewed as "identical"? If not, the slope becomes pretty slippery pretty fast...
First... .9999... does not equal 1!
If you can measure or envision a difference...then there is a difference!
I may reach a point where I will "tolerate" an error, but they are still not equal.
Bruce,
First, one non-math whiz (technically, anyway...) to another (I am far more the philosopher...) I happen to agree with you that .9999.... and 1 are not the exact same thing. Only "virtually" so in a kind of Princess Bride "almost dead" kind of way, BUT... the notion that they are equal is not something I view as "silly" whatsoever. It's a question of framing.
Are you familiar with the notion of parallel lines that converge somewhere at infinity? In a sense, near as I can figure, that is the "logic" behind .9999... = 1, although, for the record, I would modify the parallel line concept to say that the convergence happens somewhere "beyond" infinity.
It's all a question on some level of where one is looking at the number in a kind of "elephant of many parts" manner. Similarly, does Man = star dust? Go back in time far enough and the answer is yes... kind of. All the seeds that became Man were there.
Also not unrelated... does 4/4= 1? Ala your comments regarding "imagining a difference" I would apply the same logic: 4/4 = 4 pieces together AS IF 1, which equals an indivisible unity and so I would have to say they are only "virtually" identitcal, not actually identical.
I wonder what Liebniz would have to say on this with respect to his "Identity of indiscernibles" (i.e. Two things are identical if and only if they share the same properties.). By the logic presented here, can any two things really be viewed as "identical"? If not, the slope becomes pretty slippery pretty fast...
QUOTE (ez ezz+Mar 30 2007, 04:50 AM)
Did you seriously just compare me to a holocaust denier?
I mean this with offense, go f yourself.
Are you a lawyer or a priest?
Or perhaps an Pro-Lifer willing to kill to save a baby?
I mean this with offense, go f yourself.
Are you a lawyer or a priest?
Or perhaps an Pro-Lifer willing to kill to save a baby?
QUOTE
This is typically the flawed concept of infinity people have that makes it impossible for them to see what is going on at "the limit". If you are still "always" doing something, you haven't yet reached infinity, so you can't say anything about what's going on "at infinity".
There is just one thing here that doesn't work.
QUOTE (->
| QUOTE |
| This is typically the flawed concept of infinity people have that makes it impossible for them to see what is going on at "the limit". If you are still "always" doing something, you haven't yet reached infinity, so you can't say anything about what's going on "at infinity". |
There is just one thing here that doesn't work.
If you are still "always" doing something, you haven't yet reached infinity,
You can never reach infinity only approach it. Approaching it is exactly what .9r does and to get closer to something there has to be a process. Physically the process for a person to get closer to something requires that person to travel. So traveling is the process.
Mathematically the process is working out a remainder that yields another remainder. For the case of a single number repeating the remainder you get is the same over and over again. So for 1/3 you get .3 with a remainder of 1.
For pi where it is a different number each time you get a different remainder each time but a remainder none the less.
The remainder is important here and does hold value. Now the longer you try to work the remainder out the less of an error you will get by dropping this info but an error you still get.
QUOTE
As I said before, "infinity is that place beyond always". It has no single "value", or for that matter any value at all.
Now you got it right it's just too bad it goes against what you had said earlier. Saying "you haven't yet reached infinity" implies that at some point you can. But then you say ""infinity is that place beyond always". It has no single value, or for that matter any value at all".
QUOTE (->
| QUOTE |
| As I said before, "infinity is that place beyond always". It has no single "value", or for that matter any value at all. |
Now you got it right it's just too bad it goes against what you had said earlier. Saying "you haven't yet reached infinity" implies that at some point you can. But then you say ""infinity is that place beyond always". It has no single value, or for that matter any value at all".
So far our area is 1/2 + 1/4 + 1/4 = 1. Continue this process, ad infinitum. Clearly, we are geometrically generating the series 1/2 + 1/4 + 1/8 + ... .
You might say, well OK, but no matter how many times you do the division, there's always going to remain a little nick left. But we are going to do it an infinite number of times ... and I have said infinity is "beyond always", so the "nick" is going to disappear!
In order to reach one you must add the remaining amount which in this case is the last amount added. Eg. 1/2 + 1/2 = 1, 1/2 + 1/4 + 1/4 = 1, 1/2 + 1/4 + 1/8 + 1/8 = 1
If you don't do this at some point then you will always fall short of reaching 1. The difference becomes smaller and smaller and no it doesn't disappear. It's an endless equation that equals a number that has no value because the value is always changing. However we can represent the answer. Let's call it x. So 1 - x = y where y has a value depended on the value of both 1 and x. Now 1 is solid but x is changing since x is the sum of an equation that doesn't end. So y will not have a set value either.
.9r = y where x = 1/10^n n->infinity and 1/10^n becomes an endless equation where n->infinity (n can not equal infinity). This equation is 1 / (10 * 10 * 10 * 10.....) = x
1 - x = y
1 - 1 / (10 * 10 * 10 * 10.....) = .9r
y = .9r
1 - x = y
1 = x + y
1 != y where y = .9r
.9r != 1
Forget it, PC, you're stuck short of infinity. You need to be looking for an "ah ha" moment to get it. Grinding it out with your current inventory of concepts isn't doing it ... you need to add some machinery, which I have no doubt you are capable of doing. Don't feel bad, I'm a few concepts short of Einstein myself.
In particular consider the end of the series 1/2+1/4+1/8+1/8=1. What if the "rules" of arithmetic did not permit dividing 1/8 further? The series has stopped with the sum being 1. Now let's move on to our normal rules where we can continue on "forever". Consider that somehow the "infinite'th" term is no longer divisible because "infinity isn't a real number". So then the last two terms contribute nothing, since they can't be divided therefore they aren't real numbers and won't work with addition. The sum is still "1" at this point, as it always is before we start another halving procedure.
I know this is pushing it a bit, well, quite a bit, but I think that's the picture out there at infinity.
And since each term of 1/2 + 1/4 + ... can be represented by the binary string ".111 ..." we therefore must conclude that ".1r base-2 == 1".
The same is true for any base.
In particular consider the end of the series 1/2+1/4+1/8+1/8=1. What if the "rules" of arithmetic did not permit dividing 1/8 further? The series has stopped with the sum being 1. Now let's move on to our normal rules where we can continue on "forever". Consider that somehow the "infinite'th" term is no longer divisible because "infinity isn't a real number". So then the last two terms contribute nothing, since they can't be divided therefore they aren't real numbers and won't work with addition. The sum is still "1" at this point, as it always is before we start another halving procedure.
I know this is pushing it a bit, well, quite a bit, but I think that's the picture out there at infinity.
And since each term of 1/2 + 1/4 + ... can be represented by the binary string ".111 ..." we therefore must conclude that ".1r base-2 == 1".
The same is true for any base.
QUOTE (Precursor562+)
Forget it, PC, you're stuck short of infinity. You need to be looking for an "ah ha" moment to get it.
That "ah ha" moment never occurs because the expression can never equal 1 without terminating.
Watch this:
lim(x->infinity) (1-1/x)^x=1/e
Whereas 1^x=1
(Yes, you can substitute 10^x instead of x if you want)
1/e!=1
Go back to basics and look at the definition of what a limit is in calculus. The limit of a function isn't reached.
Also,
.9r<.9r^.5<1
So there are an infinite number of values that can still fit between .9r and 1. The real numbers are infinitely greater in quantity. For example, there are an infinite number of integers, but the real numbers are infinitely greater than that. There are infinitely more rationals than integers, but the real numbers are still infinitely more than those and there are infinitely more irrationals than rations etc. etc. etc. The real numbers include all these and more (not all infinities are created equally).
That "ah ha" moment never occurs because the expression can never equal 1 without terminating.
Watch this:
lim(x->infinity) (1-1/x)^x=1/e
Whereas 1^x=1
(Yes, you can substitute 10^x instead of x if you want)
1/e!=1
Go back to basics and look at the definition of what a limit is in calculus. The limit of a function isn't reached.
Also,
.9r<.9r^.5<1
So there are an infinite number of values that can still fit between .9r and 1. The real numbers are infinitely greater in quantity. For example, there are an infinite number of integers, but the real numbers are infinitely greater than that. There are infinitely more rationals than integers, but the real numbers are still infinitely more than those and there are infinitely more irrationals than rations etc. etc. etc. The real numbers include all these and more (not all infinities are created equally).
Mathematics suffers from the "middle world" syndrome, just like physics.
It's "conceptual microscope" is limited.
1/3 (of a metre) =.3r
.3r*3=.9r
.9r !=1 metre
Intersecting lines are no better.
Take a close look at the thickness of the lines.
Now take a closer look.
Still closer...
Tired of looking?
Need something now?
OK. .9r=1
Welcome to "middle world mathematics".
It's terribly convenient, but not entirely accurate.
StevenA, Eric: It's not that what you know is wrong, it's that you don't know enough yet.
I'm heading off now to be a play ski instructor for the next few days; that should give you adequete time to actually research the subject of limits, etc., rather than simply trot out what seems to be guesswork on your parts. I'll think you'll find after some study that math isn't quite so "arbitrary", "narrow minded", "lacking in vision", or whatever phrase you are currently using to coverup your lack of knowledge.
You need to focus on the concept that mathematical infinity is somewhat more sophisticated than just being "a really, really big number".
I'm heading off now to be a play ski instructor for the next few days; that should give you adequete time to actually research the subject of limits, etc., rather than simply trot out what seems to be guesswork on your parts. I'll think you'll find after some study that math isn't quite so "arbitrary", "narrow minded", "lacking in vision", or whatever phrase you are currently using to coverup your lack of knowledge.
You need to focus on the concept that mathematical infinity is somewhat more sophisticated than just being "a really, really big number".
QUOTE
In particular consider the end of the series 1/2+1/4+1/8+1/8=1. What if the "rules" of arithmetic did not permit dividing 1/8 further? The series has stopped with the sum being 1.
True.
QUOTE (->
| QUOTE |
| In particular consider the end of the series 1/2+1/4+1/8+1/8=1. What if the "rules" of arithmetic did not permit dividing 1/8 further? The series has stopped with the sum being 1. |
True.
Now let's move on to our normal rules where we can continue on "forever". Consider that somehow the "infinite'th" term is no longer divisible because "infinity isn't a real number".
Here is the mess up. Yes you can't divide by infinith term because you can't divide infinity but you can't equal infinith either because you can't equal infinity. It's always beyond as you said.
So with 1/2 + 1/4 + 1/8 + 1/16.... you are adding finite fractions where each finite fraction is a half of the previous and can be added, there is just no end to how many is being added.
And so when you do just that when you say "you are adding an infinite number of fractions where each fraction is half of the previous" you are not actually adding an infinite number of fractions. You are adding "n" number of fractions where n has the limit n->infinity meaning there is no limit to the number of fractions you can add.
In order to reach 1 you must add a fraction equal to the last fraction added. Since you are adding fractions that are only half of the last fraction added and you are doing this an unlimited number of times (there is no limit to how many times you can) you get infinitely closer to equaling 1 but you will never actually equal it. Therefore the endless equation 1/2 + 1/4 + 1/8 + 1/16..... has the limit 1.
QUOTE (NoCleverName+Mar 30 2007, 11:07 AM)
StevenA, Eric: It's not that what you know is wrong, it's that you don't know enough yet.
I'm heading off now to be a play ski instructor for the next few days; that should give you adequete time to actually research the subject of limits, etc., rather than simply trot out what seems to be guesswork on your parts. I'll think you'll find after some study that math isn't quite so "arbitrary", "narrow minded", "lacking in vision", or whatever phrase you are currently using to coverup your lack of knowledge.
You need to focus on the concept that mathematical infinity is somewhat more sophisticated than just being "a really, really big number".
NCN,
"Middle World" is not my phrase. It belongs to Richard Dawkins. The problem it represents is quite valid.
And please, don't quote me incorrectly.
Don't pick words, attribute them to me, and then insult me by saying "or whatever you said".
I said what I said.
"Convenience."
I never said, "mathematical infinity is just a really, really big number".
So obviously, any reference to my "lack of knowledge", will have to be taken in the context, that you didn't really read what I said.
How convenient.
I'm heading off now to be a play ski instructor for the next few days; that should give you adequete time to actually research the subject of limits, etc., rather than simply trot out what seems to be guesswork on your parts. I'll think you'll find after some study that math isn't quite so "arbitrary", "narrow minded", "lacking in vision", or whatever phrase you are currently using to coverup your lack of knowledge.
You need to focus on the concept that mathematical infinity is somewhat more sophisticated than just being "a really, really big number".
NCN,
"Middle World" is not my phrase. It belongs to Richard Dawkins. The problem it represents is quite valid.
And please, don't quote me incorrectly.
Don't pick words, attribute them to me, and then insult me by saying "or whatever you said".
I said what I said.
"Convenience."
I never said, "mathematical infinity is just a really, really big number".
So obviously, any reference to my "lack of knowledge", will have to be taken in the context, that you didn't really read what I said.
How convenient.
QUOTE (Guest_StevenA+Mar 30 2007, 05:41 PM)
Go back to basics and look at the definition of what a limit is in calculus.
I find it funny you say that but then say :
QUOTE (Guest_StevenA+Mar 30 2007, 05:41 PM)
Watch this:
lim(x->infinity) (1-1/x)^x=1/e
Whereas 1^x=1
(Yes, you can substitute 10^x instead of x if you want)
1/e!=1
lim(x->infinity) (1-1/x)^x=1/e
Whereas 1^x=1
(Yes, you can substitute 10^x instead of x if you want)
1/e!=1
You say "Go back to the definition" but then ignore what 0.9r^infinity would be defined as and write down something else. You attempt to make it seem like you're doing something rigorous by paying lip service to definitions but you don't actually use them.
Could this thread be the next "plane of a conveyor belt" immortal thread?
I'm guess that 0.9r must equal 1, because there could never be enough paper to write 9's to infinity. Therefore I would be forced to round up
I'm guess that 0.9r must equal 1, because there could never be enough paper to write 9's to infinity. Therefore I would be forced to round up
Definition: The limit of f(x) as x approaches a is L if and only if, given e > 0, there exists d > 0 such that 0 < |x - a| < d implies that |f(x) - L| < e.
0,9r is a function of f(n) n such that lim n-> infinity 0,9( n times) = ?
so n has to approach infinity in such a way that 0</n-infinity/<d for every d exists epsilon such that implies /0,9(n)-1/ < epsilon.
what does it mean 0</n-infinity/ < every d?
What is the value of: n-infinity?
If n is integer, what is n-infinity? I do not understand....
0,9r is a function of f(n) n such that lim n-> infinity 0,9( n times) = ?
so n has to approach infinity in such a way that 0</n-infinity/<d for every d exists epsilon such that implies /0,9(n)-1/ < epsilon.
what does it mean 0</n-infinity/ < every d?
What is the value of: n-infinity?
If n is integer, what is n-infinity? I do not understand....
It's kinda silly this has gone on so long.
How about this:
If 0.9R = 1
Then R = (+0.1)
This isn't the conventional use of the notation. But screw conventional notation. That's what got us into this discussion in the first place. It conventional notation wants to insist it's bloody correct to ignore identity, then we say "Fine. Have it your way. But I'll just choose my own definition of what that notation means."
A contradiction in the formalism is no reason to have a lack of respect for the concept of Infinity.
Some of the greatest minds in history have had issues with the way we mathematically add, subtract, or just plain ignore infinite answers.
Does anyone on here claim to be smarter than Feynman, Dirac or Einstein?
No?
Then maybe it's time we just accepted the contradiction as one inherently built into the Universe, just like pi or sqrt(2) and stop arguing over it.
How about this:
If 0.9R = 1
Then R = (+0.1)
This isn't the conventional use of the notation. But screw conventional notation. That's what got us into this discussion in the first place. It conventional notation wants to insist it's bloody correct to ignore identity, then we say "Fine. Have it your way. But I'll just choose my own definition of what that notation means."
A contradiction in the formalism is no reason to have a lack of respect for the concept of Infinity.
Some of the greatest minds in history have had issues with the way we mathematically add, subtract, or just plain ignore infinite answers.
Does anyone on here claim to be smarter than Feynman, Dirac or Einstein?
No?
Then maybe it's time we just accepted the contradiction as one inherently built into the Universe, just like pi or sqrt(2) and stop arguing over it.
Math is an international language. From when math was first used (created to describe and measure the physical world around us) the ground based rules were established.
1 + 1 = 2 always and nothing else. Additional rules have been established based on these rules and have allowed math to go beyond the physical. Now if these newer rules are indeed true then there should be physical properties where these rules can be used. Even if we have not discovered these physical properties yet. If we find that there is no physical value then the math and associated rules are imaginary. If the imaginary conflicts with the physical (real) than it is the imaginary that is wrong.
Infinity is real physically however the rules of math become as clear as mud when dealing with infinities. You can find work by people in the field to support your views however to sit and hear or read what they have to say, you will soon realize that they are speculating. Of course they will try and "prove" (more likely show) their case but such showmanship is based on the assumption that what they are saying is true to begin with. This is what falsifies their proof as proof. All it is that they are doing is saying what they already said (in whatever language) again in the mathematical language.
Now if they can show what they are saying to be true using the math and associated rules that have been physically shown to be correct then and only then are they proving it to be true. With no previous assumptions required.
Often the biggest mistake is saying that something can equal infinity. Nothing can equal infinity because it does not have value. Infinity can't even equal infinity. It then becomes a true mathematical limit. A variable can have the limit of infinity but not equal infinity. This simply means that there is no limit to how big or small the value of the variable can be. This can be shown physically when describing a volume of space.
(1*10^n) * (1*10^n) * (1*10^n) = (1 * 10^n)^3 n->infinity
(1*10^n)^3 = 1*10^3n n->infinity
1*10^3n becomes the end of the line until we give n a finite value. n can not equal infinity. It can be any number and there is no limit to how big or small that number can be and so n->infinity shows us this.
(1*10^-n) * (1*10^-n) * ( 1*10^-n) = (1*10^-n)^3 = 1*10^-3n n->infinity
So there is no limit to how big or small a volume of space can be. The universe is physical example to how there is no limit to how big (vast) space can be.
As for .1r, .3r, .9r etc. the r means repeat not infinite. The number of 1s, 3s, or 9s after the decimal can not equal infinity but it can have the limit of infinity which means there is no limit to how many you can have. As for repeating ( r ), something repeating is the direct result of a process that can not end and so the answer (result) is ever changing in value. This makes the answer not a set value but rather an equivalent process. Only when the process ends does it become a set (finite) value.
Adding the remaining amount ends the process giving a finite value of 1.
Adding the remaining amount ends the process giving a finite value of 1.
If you don't do this at some point then you will always fall short of reaching 1. The difference becomes smaller and smaller and no it doesn't disappear. It's an endless equation that equals a number that has no value because the value is always changing. However we can represent the answer. Let's call it x. So 1 - x = y where y has a value depended on the value of both 1 and x. Now 1 is solid but x is changing since x is the sum of an equation that doesn't end and so y will not have a set value either.
.9r = y where x = 1/10^n n->infinity and 1/10^n becomes an endless equation where n->infinity (n can not equal infinity). This equation is 1 / (10 * 10 * 10 * 10.....) = x
1 - x = y
1 - 1 / (10 * 10 * 10 * 10.....) = .9r
y = .9r
1 - x = y
1 = x + y
1 != y where y = .9r
.9r != 1
Where .9r is the result of an endless equation .9 + .09 + .009.....
1 subtract an endless equation (endless process) will equal an endless equation (endless process).
The other big mistake is saying that two things equal each other because there is no gap between them. For numbers on a number line the number is a dimensionless point and so no gap is needed. This is where point value individuality is critical. In the case of .9r it represents a point on the number line that is getting closer to the point of 1. There is no limit to how close it can get because each point is dimensionless. So we can say that the point .9r can get infinitely close to the point 1 because you will always be able to fit an infinite number of dimensionless points between two dimensionless points. This means there is no limit to the number of dimensionless points we can put between two dimensionless points. So since the point .9r can get infinitely closer to the point 1 and is doing so, it is moving toward but will never reach the point of 1. Hence .9r has the limit of 1.
So start with the finite point of .9 and the finite point of 1
As the point of .9 moves closer to the point of 1 on the number line its value changes.
.9 -> .91 -> .92 -> .93 ....... .99 -> .991 -> .992 -> .993 -> ...... .999
.999 -> .9991 -> .9992 ->...... .99999
So it started with the value of .9 which is 1/10 (.1) away from the point 1. It then increased in 'closeness' by increments of 1/100 (.01) until reaching .99 then it slowed and began increasing in 'closeness' by increments of 1/1000 (.001). It became an endless process where the point .9 moved toward the point of 1 (increasing in value accordingly) but as it gets closer it reduces the speed at which the gap between the two points decrease. So with .9r since there is an ever increasing number of nines and the difference between .9r and 1 is an ever decreasing amount so the difference becomes 1/10^n. The value of 'n' depends on the number of nines and since we have an infinite number of nines (which means there is no limit nor set amount to the number of nines since the number of nines can't equal infinity (nothing can)) then the same is applied to the variable 'n' and so 1 - .9r = 1/10^n n->infinity.
Now if the point .9 had no limit to how far it can move it no longer becomes .9r with the limit of 1 but instead it becomes X where X->infinity. There becomes no limit to how large the value of X can be when there is no limit to how far the point .9 can go (changing in value accordingly). The only limit is we started with the point .9 and it is only going one way so X !< .9.
1 + 1 = 2 always and nothing else. Additional rules have been established based on these rules and have allowed math to go beyond the physical. Now if these newer rules are indeed true then there should be physical properties where these rules can be used. Even if we have not discovered these physical properties yet. If we find that there is no physical value then the math and associated rules are imaginary. If the imaginary conflicts with the physical (real) than it is the imaginary that is wrong.
Infinity is real physically however the rules of math become as clear as mud when dealing with infinities. You can find work by people in the field to support your views however to sit and hear or read what they have to say, you will soon realize that they are speculating. Of course they will try and "prove" (more likely show) their case but such showmanship is based on the assumption that what they are saying is true to begin with. This is what falsifies their proof as proof. All it is that they are doing is saying what they already said (in whatever language) again in the mathematical language.
Now if they can show what they are saying to be true using the math and associated rules that have been physically shown to be correct then and only then are they proving it to be true. With no previous assumptions required.
Often the biggest mistake is saying that something can equal infinity. Nothing can equal infinity because it does not have value. Infinity can't even equal infinity. It then becomes a true mathematical limit. A variable can have the limit of infinity but not equal infinity. This simply means that there is no limit to how big or small the value of the variable can be. This can be shown physically when describing a volume of space.
(1*10^n) * (1*10^n) * (1*10^n) = (1 * 10^n)^3 n->infinity
(1*10^n)^3 = 1*10^3n n->infinity
1*10^3n becomes the end of the line until we give n a finite value. n can not equal infinity. It can be any number and there is no limit to how big or small that number can be and so n->infinity shows us this.
(1*10^-n) * (1*10^-n) * ( 1*10^-n) = (1*10^-n)^3 = 1*10^-3n n->infinity
So there is no limit to how big or small a volume of space can be. The universe is physical example to how there is no limit to how big (vast) space can be.
As for .1r, .3r, .9r etc. the r means repeat not infinite. The number of 1s, 3s, or 9s after the decimal can not equal infinity but it can have the limit of infinity which means there is no limit to how many you can have. As for repeating ( r ), something repeating is the direct result of a process that can not end and so the answer (result) is ever changing in value. This makes the answer not a set value but rather an equivalent process. Only when the process ends does it become a set (finite) value.
QUOTE
In order to reach one you must add the remaining amount which in this case is the last amount added. Eg. 1/2 + 1/2 = 1, 1/2 + 1/4 + 1/4 = 1, 1/2 + 1/4 + 1/8 + 1/8 = 1
Adding the remaining amount ends the process giving a finite value of 1.
QUOTE (->
| QUOTE |
| In order to reach one you must add the remaining amount which in this case is the last amount added. Eg. 1/2 + 1/2 = 1, 1/2 + 1/4 + 1/4 = 1, 1/2 + 1/4 + 1/8 + 1/8 = 1 |
Adding the remaining amount ends the process giving a finite value of 1.
If you don't do this at some point then you will always fall short of reaching 1. The difference becomes smaller and smaller and no it doesn't disappear. It's an endless equation that equals a number that has no value because the value is always changing. However we can represent the answer. Let's call it x. So 1 - x = y where y has a value depended on the value of both 1 and x. Now 1 is solid but x is changing since x is the sum of an equation that doesn't end and so y will not have a set value either.
.9r = y where x = 1/10^n n->infinity and 1/10^n becomes an endless equation where n->infinity (n can not equal infinity). This equation is 1 / (10 * 10 * 10 * 10.....) = x
1 - x = y
1 - 1 / (10 * 10 * 10 * 10.....) = .9r
y = .9r
1 - x = y
1 = x + y
1 != y where y = .9r
.9r != 1
Where .9r is the result of an endless equation .9 + .09 + .009.....
1 subtract an endless equation (endless process) will equal an endless equation (endless process).
The other big mistake is saying that two things equal each other because there is no gap between them. For numbers on a number line the number is a dimensionless point and so no gap is needed. This is where point value individuality is critical. In the case of .9r it represents a point on the number line that is getting closer to the point of 1. There is no limit to how close it can get because each point is dimensionless. So we can say that the point .9r can get infinitely close to the point 1 because you will always be able to fit an infinite number of dimensionless points between two dimensionless points. This means there is no limit to the number of dimensionless points we can put between two dimensionless points. So since the point .9r can get infinitely closer to the point 1 and is doing so, it is moving toward but will never reach the point of 1. Hence .9r has the limit of 1.
So start with the finite point of .9 and the finite point of 1
As the point of .9 moves closer to the point of 1 on the number line its value changes.
.9 -> .91 -> .92 -> .93 ....... .99 -> .991 -> .992 -> .993 -> ...... .999
.999 -> .9991 -> .9992 ->...... .99999
So it started with the value of .9 which is 1/10 (.1) away from the point 1. It then increased in 'closeness' by increments of 1/100 (.01) until reaching .99 then it slowed and began increasing in 'closeness' by increments of 1/1000 (.001). It became an endless process where the point .9 moved toward the point of 1 (increasing in value accordingly) but as it gets closer it reduces the speed at which the gap between the two points decrease. So with .9r since there is an ever increasing number of nines and the difference between .9r and 1 is an ever decreasing amount so the difference becomes 1/10^n. The value of 'n' depends on the number of nines and since we have an infinite number of nines (which means there is no limit nor set amount to the number of nines since the number of nines can't equal infinity (nothing can)) then the same is applied to the variable 'n' and so 1 - .9r = 1/10^n n->infinity.
Now if the point .9 had no limit to how far it can move it no longer becomes .9r with the limit of 1 but instead it becomes X where X->infinity. There becomes no limit to how large the value of X can be when there is no limit to how far the point .9 can go (changing in value accordingly). The only limit is we started with the point .9 and it is only going one way so X !< .9.
QUOTE (Precursor562+Mar 31 2007, 11:39 PM)
1 + 1 = 2 always and nothing else. Additional rules have been established based on these rules and have allowed math to go beyond the physical.
No, the axioms of maths were formulated to derive what had already been done but from much more basic statements, allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
As such, maths is not bound by physics.
No, the axioms of maths were formulated to derive what had already been done but from much more basic statements, allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
As such, maths is not bound by physics.
QUOTE (Precursor562+Mar 31 2007, 11:39 PM)
If the imaginary conflicts with the physical (real) than it is the imaginary that is wrong.
Nonsense. You're saying "If you can imagine somethign which isn't physics, you're imagination is wrong". How can someone's imagination, if it's logically consistent, be 'wrong' ? If you can imagine it, you can imagine it. If you can create a self contained logic system, then it's self contained nature means it doesn't have to be linked to anything else.
The validity of English as a language is not determined by the validity of French. If noone spoke French, it wouldn't mean noone could speak English. Similarly, maths can talk about plenty of logical constructs which are not talked about by nature. To bound your horizons only by what physically exists is enormously restricting. I thought you cranks were always telling us mainstream people to 'look outside the box' ? Now you're saying we should never leave a box!
Not to mention that argument boils down to "In my opinion, if it's not physical it's not valid". Thankfully some of us have larger horizons than that.
The validity of English as a language is not determined by the validity of French. If noone spoke French, it wouldn't mean noone could speak English. Similarly, maths can talk about plenty of logical constructs which are not talked about by nature. To bound your horizons only by what physically exists is enormously restricting. I thought you cranks were always telling us mainstream people to 'look outside the box' ? Now you're saying we should never leave a box!
Not to mention that argument boils down to "In my opinion, if it's not physical it's not valid". Thankfully some of us have larger horizons than that.
QUOTE (Precursor562+Mar 31 2007, 11:39 PM)
but such showmanship is based on the assumption that what they are saying is true to begin with
Plenty of maths is aside from reality in every way so there is no 'truth' to default to.
What you're claiming is akin to going to an American Football player "You're not allowed to throw the ball forwards! It's true!". His reply would be "But the rules say I can, that's what the game is defined as". Your argument "No!!! It's not!" doesn't cut it. Someone defined the rules of American Football and you're allowed to throw forwards. In maths, the base rules are defined and they give 0.9r=1. They also give 'physical' systems like 1+1=2 etc but they also move beyond that to much more subtle and complex things. Heck, even the straight line is a purely mathematical construct. Should we now deny the existence of straight lines only because they exist in our imagination? Of course not.
So you can walk through walls can you? How is it you can fly without the use of wings, or a device of some sort (eg. plane, glider, suspension cables)? I'd really like to know.
Walking through walls and being able to fly (by sheer will alone) are example of things we can imagine, even dream (and do within those dreams) but are physically impossible. Hence why when the imaginary conflict with the real/physical (reality) it is the imaginary that is wrong/impossible and the physical that is correct/possible.
Now you can imagine yourself flying a plane or swimming (dream as well) and such things you can do physically even if you don't know how. This is an example of imaginary not conflicting with real and so both are true. Only until you try and learn (find the physical equivalent) will you truly know if you can or not.
For instance not everyone can whistle using two fingers. No matter how many times shown or how long they practice they just can't. They can imagine doing it but can't physically. Now before being shown how or practicing (trying to learn) they can imagine doing it and there is no confliction because it is something that can be done. They then try and learn and find out it is impossible for them (not just anyone can do it) suddenly there is a confliction between the imaginary and the physical. Which is wrong? Well congrats AlphaN you claim the imaginary being wrong to be nonsense. Which means the physical (reality) is wrong.
So you can walk through walls can you? How is it you can fly without the use of wings, or a device of some sort (eg. plane, glider, suspension cables)? I'd really like to know.
Walking through walls and being able to fly (by sheer will alone) are example of things we can imagine, even dream (and do within those dreams) but are physically impossible. Hence why when the imaginary conflict with the real/physical (reality) it is the imaginary that is wrong/impossible and the physical that is correct/possible.
Now you can imagine yourself flying a plane or swimming (dream as well) and such things you can do physically even if you don't know how. This is an example of imaginary not conflicting with real and so both are true. Only until you try and learn (find the physical equivalent) will you truly know if you can or not.
For instance not everyone can whistle using two fingers. No matter how many times shown or how long they practice they just can't. They can imagine doing it but can't physically. Now before being shown how or practicing (trying to learn) they can imagine doing it and there is no confliction because it is something that can be done. They then try and learn and find out it is impossible for them (not just anyone can do it) suddenly there is a confliction between the imaginary and the physical. Which is wrong? Well congrats AlphaN you claim the imaginary being wrong to be nonsense. Which means the physical (reality) is wrong.
No, the axioms of maths were formulated to derive what had already been done but from much more basic statements, allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
Axiom. I just love your choice of words. Wiki defines an Axiom as a starting assumption. That such is not "demonstrable by formal proofs". Hmm how interesting. All this time you were trying to "prove" something using assumptions (something I had pointed out before but I just felt it was necessary to make a clearer point) with no formal proof.
1 is defined and can be proven using formal proofs. Shown as a single entity or object. 1 + 1 = 2 where 1 whole apple + 1 whole apple = 2 whole apples.
Math is a formal system and so to say "axiom" is to say "rule". Within math you have rules based on rules based on rules and so on but sooner or later you will get the foundation of math and the associated rules that haven been physically proven. Unfortunately an axiom of math is a rule not based upon previously proven rules.
So first you start with...
No, the axioms of maths were formulated to derive what had already been done but from much more basic statements, allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
Axiom. I just love your choice of words. Wiki defines an Axiom as a starting assumption. That such is not "demonstrable by formal proofs". Hmm how interesting. All this time you were trying to "prove" something using assumptions (something I had pointed out before but I just felt it was necessary to make a clearer point) with no formal proof.
1 is defined and can be proven using formal proofs. Shown as a single entity or object. 1 + 1 = 2 where 1 whole apple + 1 whole apple = 2 whole apples.
Math is a formal system and so to say "axiom" is to say "rule". Within math you have rules based on rules based on rules and so on but sooner or later you will get the foundation of math and the associated rules that haven been physically proven. Unfortunately an axiom of math is a rule not based upon previously proven rules.
So first you start with...
No
Then continue to say....
Which is much the same of what I had said in the quote and so you contradicted yourself within the very same sentence. First by disagreeing with me then by agreeing. No surprise coming from you.
Of course this is based upon my assumption that you incorrectly used the word axiom as being a rule of math based on previous rules of math.
Which is much the same of what I had said in the quote and so you contradicted yourself within the very same sentence. First by disagreeing with me then by agreeing. No surprise coming from you.
Of course this is based upon my assumption that you incorrectly used the word axiom as being a rule of math based on previous rules of math.
allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
Then you completely miss what I said (showing a complete lack of comprehension) and talk about the irrelevant. I wasn't talking about physical intuition, I was talking about the physical world.
As for the axioms of math. Such are started assumption. Considered rules that are not based upon any predecessor. Such are not and can not be physically shown and so are imaginary (existing within our own minds). If there is a case where an axiom of math conflicts with a physically proven rule then we must change the axiom.
Eg. we have an axiom that ultimately gives us 1+1 = 3 (much like the one that gives us SqRt2 = 1) but a fundamental and physically proven rule of math is 1 + 1 = 2 and that 2 != 3. Do we say 1 + 1 = 2 is wrong? No we say 1 + 1 = 3 is wrong and rule out the axiom that got us that as being false.
However congrats on you in saying that 1 + 1 = 2 would be wrong and showing once again how conceited you are. Can't teach God anything after all.
There have been arguments made that .9 != 1 and .99 != 1 and .999 != 1 and so on but as soon as you have an infinite number of 9s it does equal 1. These arguments supporting that .9r = 1 were claiming that you could have an infinite number of nines as in x number of nines x=infinity. Such arguments are flat out wrong because nothing can equal infinity. Infinity is another way of saying without limit.
So what does that mean? That .9r = 1 when there is no limit to the number of nines we can have after the decimal place? Which would be better represented as .9n and not .9r since r stands for repeat (which means ever increase, the equivalent to an endless cycle/process) and n can represent the quantity of nines present.
So .9n n->infinity means there is no limit to the number of nines we can have after the decimal. We can have any finite amount (there is no limit hence limit is infinite) and any finite amount will give us a value less then 1. .9n n->infinity can never gives us a value equal to 1.
.9r is the result of an endless process and thus becomes an equivalent endless process shown a different way. This process does not end and will not have a value until it does. If it does and no matter when it does you will get a value less then 1.
There have been arguments made that .9 != 1 and .99 != 1 and .999 != 1 and so on but as soon as you have an infinite number of 9s it does equal 1. These arguments supporting that .9r = 1 were claiming that you could have an infinite number of nines as in x number of nines x=infinity. Such arguments are flat out wrong because nothing can equal infinity. Infinity is another way of saying without limit.
So what does that mean? That .9r = 1 when there is no limit to the number of nines we can have after the decimal place? Which would be better represented as .9n and not .9r since r stands for repeat (which means ever increase, the equivalent to an endless cycle/process) and n can represent the quantity of nines present.
So .9n n->infinity means there is no limit to the number of nines we can have after the decimal. We can have any finite amount (there is no limit hence limit is infinite) and any finite amount will give us a value less then 1. .9n n->infinity can never gives us a value equal to 1.
.9r is the result of an endless process and thus becomes an equivalent endless process shown a different way. This process does not end and will not have a value until it does. If it does and no matter when it does you will get a value less then 1.
But you don't seem to be interested in the rigorous approach or even learning the philosophy of maths, just inventing your own version of it and then complaining your version doesn't seem to square right with you. Wow, that's a shock rolleyes.gif
First of the rules of math were made up by people. I would not be out of place to do some such thing myself (it would be an axiom). However this is NOT the case where I am using very simple math with physical evidence of being correct. Such math and associated rules have been established and physically proven correct long ago by experts. It "squares right" with me just fine when it is you that it doesn't square right with. However I wouldn't put it past you to be able to show more ignorance than that.
What you're claiming is akin to going to an American Football player "You're not allowed to throw the ball forwards! It's true!". His reply would be "But the rules say I can, that's what the game is defined as". Your argument "No!!! It's not!" doesn't cut it. Someone defined the rules of American Football and you're allowed to throw forwards. In maths, the base rules are defined and they give 0.9r=1. They also give 'physical' systems like 1+1=2 etc but they also move beyond that to much more subtle and complex things. Heck, even the straight line is a purely mathematical construct. Should we now deny the existence of straight lines only because they exist in our imagination? Of course not.
QUOTE (Precursor562+Mar 31 2007, 11:39 PM)
Often the biggest mistake is saying that something can equal infinity.
0.9r=1 has nothing like that to say. It's people like StevenA who start pulling out "To the power of infinity" to try and back up his erroneous claim. Any mathematician worth his salt will only use infinite in calculus as a notion of limits.
But you don't seem to be interested in the rigorous approach or even learning the philosophy of maths, just inventing your own version of it and then complaining your version doesn't seem to square right with you. Wow, that's a shock
But you don't seem to be interested in the rigorous approach or even learning the philosophy of maths, just inventing your own version of it and then complaining your version doesn't seem to square right with you. Wow, that's a shock
YOU WILL BE TOO BUSY ADDING THAT NEXT NINE TO EVER REACH 1.
MITCH RAEMSCH -- LIGHT FALL --
MITCH RAEMSCH -- LIGHT FALL --
QUOTE
Nonsense. You're saying "If you can imagine somethign which isn't physics, you're imagination is wrong". How can someone's imagination, if it's logically consistent, be 'wrong' ?
So you can walk through walls can you? How is it you can fly without the use of wings, or a device of some sort (eg. plane, glider, suspension cables)? I'd really like to know.Walking through walls and being able to fly (by sheer will alone) are example of things we can imagine, even dream (and do within those dreams) but are physically impossible. Hence why when the imaginary conflict with the real/physical (reality) it is the imaginary that is wrong/impossible and the physical that is correct/possible.
Now you can imagine yourself flying a plane or swimming (dream as well) and such things you can do physically even if you don't know how. This is an example of imaginary not conflicting with real and so both are true. Only until you try and learn (find the physical equivalent) will you truly know if you can or not.
For instance not everyone can whistle using two fingers. No matter how many times shown or how long they practice they just can't. They can imagine doing it but can't physically. Now before being shown how or practicing (trying to learn) they can imagine doing it and there is no confliction because it is something that can be done. They then try and learn and find out it is impossible for them (not just anyone can do it) suddenly there is a confliction between the imaginary and the physical. Which is wrong? Well congrats AlphaN you claim the imaginary being wrong to be nonsense. Which means the physical (reality) is wrong.
QUOTE (->
| QUOTE |
| Nonsense. You're saying "If you can imagine somethign which isn't physics, you're imagination is wrong". How can someone's imagination, if it's logically consistent, be 'wrong' ? |
So you can walk through walls can you? How is it you can fly without the use of wings, or a device of some sort (eg. plane, glider, suspension cables)? I'd really like to know.Walking through walls and being able to fly (by sheer will alone) are example of things we can imagine, even dream (and do within those dreams) but are physically impossible. Hence why when the imaginary conflict with the real/physical (reality) it is the imaginary that is wrong/impossible and the physical that is correct/possible.
Now you can imagine yourself flying a plane or swimming (dream as well) and such things you can do physically even if you don't know how. This is an example of imaginary not conflicting with real and so both are true. Only until you try and learn (find the physical equivalent) will you truly know if you can or not.
For instance not everyone can whistle using two fingers. No matter how many times shown or how long they practice they just can't. They can imagine doing it but can't physically. Now before being shown how or practicing (trying to learn) they can imagine doing it and there is no confliction because it is something that can be done. They then try and learn and find out it is impossible for them (not just anyone can do it) suddenly there is a confliction between the imaginary and the physical. Which is wrong? Well congrats AlphaN you claim the imaginary being wrong to be nonsense. Which means the physical (reality) is wrong.
QUOTE
QUOTE (Precursor562 @ Mar 31 2007, 11:39 PM)
1 + 1 = 2 always and nothing else. Additional rules have been established based on these rules and have allowed math to go beyond the physical.
1 + 1 = 2 always and nothing else. Additional rules have been established based on these rules and have allowed math to go beyond the physical.
No, the axioms of maths were formulated to derive what had already been done but from much more basic statements, allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
Axiom. I just love your choice of words. Wiki defines an Axiom as a starting assumption. That such is not "demonstrable by formal proofs". Hmm how interesting. All this time you were trying to "prove" something using assumptions (something I had pointed out before but I just felt it was necessary to make a clearer point) with no formal proof.
1 is defined and can be proven using formal proofs. Shown as a single entity or object. 1 + 1 = 2 where 1 whole apple + 1 whole apple = 2 whole apples.
Math is a formal system and so to say "axiom" is to say "rule". Within math you have rules based on rules based on rules and so on but sooner or later you will get the foundation of math and the associated rules that haven been physically proven. Unfortunately an axiom of math is a rule not based upon previously proven rules.
So first you start with...
QUOTE (->
| QUOTE |
| QUOTE (Precursor562 @ Mar 31 2007, 11:39 PM) 1 + 1 = 2 always and nothing else. Additional rules have been established based on these rules and have allowed math to go beyond the physical. |
No, the axioms of maths were formulated to derive what had already been done but from much more basic statements, allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
Axiom. I just love your choice of words. Wiki defines an Axiom as a starting assumption. That such is not "demonstrable by formal proofs". Hmm how interesting. All this time you were trying to "prove" something using assumptions (something I had pointed out before but I just felt it was necessary to make a clearer point) with no formal proof.
1 is defined and can be proven using formal proofs. Shown as a single entity or object. 1 + 1 = 2 where 1 whole apple + 1 whole apple = 2 whole apples.
Math is a formal system and so to say "axiom" is to say "rule". Within math you have rules based on rules based on rules and so on but sooner or later you will get the foundation of math and the associated rules that haven been physically proven. Unfortunately an axiom of math is a rule not based upon previously proven rules.
So first you start with...
No
Then continue to say....
QUOTE
the axioms of maths were formulated to derive what had already been done but from much more basic statements,
Which is much the same of what I had said in the quote and so you contradicted yourself within the very same sentence. First by disagreeing with me then by agreeing. No surprise coming from you
.Of course this is based upon my assumption that you incorrectly used the word axiom as being a rule of math based on previous rules of math.
QUOTE (->
| QUOTE |
| the axioms of maths were formulated to derive what had already been done but from much more basic statements, |
Which is much the same of what I had said in the quote and so you contradicted yourself within the very same sentence. First by disagreeing with me then by agreeing. No surprise coming from you
.Of course this is based upon my assumption that you incorrectly used the word axiom as being a rule of math based on previous rules of math.
allowing for the decoupling of physical intuition from logic, since intuition of often wrong.
Then you completely miss what I said (showing a complete lack of comprehension) and talk about the irrelevant. I wasn't talking about physical intuition, I was talking about the physical world.
As for the axioms of math. Such are started assumption. Considered rules that are not based upon any predecessor. Such are not and can not be physically shown and so are imaginary (existing within our own minds). If there is a case where an axiom of math conflicts with a physically proven rule then we must change the axiom.
Eg. we have an axiom that ultimately gives us 1+1 = 3 (much like the one that gives us SqRt2 = 1) but a fundamental and physically proven rule of math is 1 + 1 = 2 and that 2 != 3. Do we say 1 + 1 = 2 is wrong? No we say 1 + 1 = 3 is wrong and rule out the axiom that got us that as being false.
However congrats on you in saying that 1 + 1 = 2 would be wrong and showing once again how conceited you are. Can't teach God anything after all.
QUOTE
0.9r=1 has nothing like that to say.
There have been arguments made that .9 != 1 and .99 != 1 and .999 != 1 and so on but as soon as you have an infinite number of 9s it does equal 1. These arguments supporting that .9r = 1 were claiming that you could have an infinite number of nines as in x number of nines x=infinity. Such arguments are flat out wrong because nothing can equal infinity. Infinity is another way of saying without limit.
So what does that mean? That .9r = 1 when there is no limit to the number of nines we can have after the decimal place? Which would be better represented as .9n and not .9r since r stands for repeat (which means ever increase, the equivalent to an endless cycle/process) and n can represent the quantity of nines present.
So .9n n->infinity means there is no limit to the number of nines we can have after the decimal. We can have any finite amount (there is no limit hence limit is infinite) and any finite amount will give us a value less then 1. .9n n->infinity can never gives us a value equal to 1.
.9r is the result of an endless process and thus becomes an equivalent endless process shown a different way. This process does not end and will not have a value until it does. If it does and no matter when it does you will get a value less then 1.
QUOTE (->
| QUOTE |
| 0.9r=1 has nothing like that to say. |
There have been arguments made that .9 != 1 and .99 != 1 and .999 != 1 and so on but as soon as you have an infinite number of 9s it does equal 1. These arguments supporting that .9r = 1 were claiming that you could have an infinite number of nines as in x number of nines x=infinity. Such arguments are flat out wrong because nothing can equal infinity. Infinity is another way of saying without limit.
So what does that mean? That .9r = 1 when there is no limit to the number of nines we can have after the decimal place? Which would be better represented as .9n and not .9r since r stands for repeat (which means ever increase, the equivalent to an endless cycle/process) and n can represent the quantity of nines present.
So .9n n->infinity means there is no limit to the number of nines we can have after the decimal. We can have any finite amount (there is no limit hence limit is infinite) and any finite amount will give us a value less then 1. .9n n->infinity can never gives us a value equal to 1.
.9r is the result of an endless process and thus becomes an equivalent endless process shown a different way. This process does not end and will not have a value until it does. If it does and no matter when it does you will get a value less then 1.
But you don't seem to be interested in the rigorous approach or even learning the philosophy of maths, just inventing your own version of it and then complaining your version doesn't seem to square right with you. Wow, that's a shock rolleyes.gif
First of the rules of math were made up by people. I would not be out of place to do some such thing myself (it would be an axiom). However this is NOT the case where I am using very simple math with physical evidence of being correct. Such math and associated rules have been established and physically proven correct long ago by experts. It "squares right" with me just fine when it is you that it doesn't square right with. However I wouldn't put it past you to be able to show more ignorance than that.
QUOTE (Precursor562+Apr 1 2007, 01:49 AM)
Well congrats AlphaN you claim the imaginary being wrong to be nonsense. Which means the physical (reality) is wrong.
No, I claim that if you're trying to build a logical construction, the only requirement is that it's not inconsistent. It doesn't require a physical meaning and it doesn't have to refer to physical reality to check it's consistency.
I am not claiming you can walk through walls if you can imagine it. I am saying that if a construct exists only within our minds, it's not automatically invalidated as a construct.
No, I claim that if you're trying to build a logical construction, the only requirement is that it's not inconsistent. It doesn't require a physical meaning and it doesn't have to refer to physical reality to check it's consistency.
I am not claiming you can walk through walls if you can imagine it. I am saying that if a construct exists only within our minds, it's not automatically invalidated as a construct.
QUOTE (Precursor562+Apr 1 2007, 01:49 AM)
Axiom. I just love your choice of words. Wiki defines an Axiom as a starting assumption. That such is not "demonstrable by formal proofs". Hmm how interesting. All this time you were trying to "prove" something using assumptions (something I had pointed out before but I just felt it was necessary to make a clearer point) with no formal proof.
As I've said several times, 0.9r and 1 are defined from the axioms of maths. As such saying "I don't agree with your axioms and therefore 0.9r isn't 1 in my maths" doesn't change the fact in the maths everyone talks about, 0.9r=1.
It's like saying "The french word for 'good day' isn't bon jour, because in German it's 'Gutten tag'." You attempt to argue with someone's system because your supposed system is different.
When someone says "0.9r" or "1" they refer to the entities defined from the current axioms of maths. Those entities are equal, as follows from the axioms. If you say "I disagree, I prefer these axioms...." then you no longer talk about the same 0.9r and the same 1, just as if you allow infinitesimals you no longer talk about 0.9r and 1, because those are Real numbers, you talk about their Hyperreal counterparts.
The question "Does 0.9r=1" is short hand for "Given the current axioms of maths, do the entitites 0.9r and 1 which follow from those axioms display equality under the mathematical equivalence relation 'equals' defined from those axioms?". The answer is an undeniable 'Yes'.
What you're trying to say is that there's some underlying 'true' 0.9r and 1 which aren't equal. Firstly, there's no underlying truth, just the constructs we create. Secondly, even if there are other contructs where entities like 0.9r and 1 exist (and there are), that has no effect on the 0.9r and 1 typically refered to by people. As such, trying to say "I disagree with your axioms" or "What about hyperreals?" has no relevence to the discussion.
It's like saying "The french word for 'good day' isn't bon jour, because in German it's 'Gutten tag'." You attempt to argue with someone's system because your supposed system is different.
When someone says "0.9r" or "1" they refer to the entities defined from the current axioms of maths. Those entities are equal, as follows from the axioms. If you say "I disagree, I prefer these axioms...." then you no longer talk about the same 0.9r and the same 1, just as if you allow infinitesimals you no longer talk about 0.9r and 1, because those are Real numbers, you talk about their Hyperreal counterparts.
The question "Does 0.9r=1" is short hand for "Given the current axioms of maths, do the entitites 0.9r and 1 which follow from those axioms display equality under the mathematical equivalence relation 'equals' defined from those axioms?". The answer is an undeniable 'Yes'.
What you're trying to say is that there's some underlying 'true' 0.9r and 1 which aren't equal. Firstly, there's no underlying truth, just the constructs we create. Secondly, even if there are other contructs where entities like 0.9r and 1 exist (and there are), that has no effect on the 0.9r and 1 typically refered to by people. As such, trying to say "I disagree with your axioms" or "What about hyperreals?" has no relevence to the discussion.
QUOTE (Precursor562+Apr 1 2007, 01:49 AM)
Eg. we have an axiom that ultimately gives us 1+1 = 3 (much like the one that gives us SqRt2 = 1) but a fundamental and physically proven rule of math is 1 + 1 = 2 and that 2 != 3. Do we say 1 + 1 = 2 is wrong? No we say 1 + 1 = 3 is wrong and rule out the axiom that got us that as being false.
On the surface, there's nothing wrong with 1+1=3. That equation on it's own is fine, at least in formal structure without referring to physical reality. However, you'd be able to use such an equation to prove that 0<0, yet < has the property that x<x is completely false. As such, you'd have undeniable evidence of a flaw in your workings. That is what would make you reject the equation.
Simply saying "It's not valid in reality" doesn't cut it. After all, in reality, 3 lots of 5 is 5 lots of 3, a*b = b*a. Maths are awash with things where ab-ba is NOT zero. If things like a*b+b*a=0. That seems nonsense in physical terms of our everyday life.
Simply saying "It's not valid in reality" doesn't cut it. After all, in reality, 3 lots of 5 is 5 lots of 3, a*b = b*a. Maths are awash with things where ab-ba is NOT zero. If things like a*b+b*a=0. That seems nonsense in physical terms of our everyday life.
QUOTE (Precursor562+Apr 1 2007, 01:49 AM)
Such arguments are flat out wrong because nothing can equal infinity. Infinity is another way of saying without limit..
That amounts to "I don't like it". It generate a completely consistent mathematical structure which is all that matters to maths. You think of 'limit to infinity' as some kind of physical process you need to do by hand (like Nick seems to think too) but you don't. Can you imagine the limit of the process? Yes, you don't have to imagine doing infinitely many steps. Things like 'proof by induction' are taught to school kids, ways of proving infiniteluy many results are true in only a few lines using logic rather than brute force. Only those who miss the point of such questions try to do something by brute force, particularly when there's an elegant, airtight, short method which does all the work for you. That's what mathematicians often look for, a different method of attack which reduces an otherwise insurmountable problem to almost effortless. You don't seem to appreciate that and continue to say "But you can't ever do all the steps!". You don't have to, the algebra does it for you.
Once again you missed what I said. You're right, an axiom doesn't require a physical meaning and it doesn't refer to physical reality. It just needs to be consistent. However when an axiom is proven true then it is no longer an axiom but a proven rule of math. How do you prove it to be true? You prove it physically. If you can physically show it to be true than it is true. If you physically prove it to be false then it is false. If you can't physically prove it at all then it remains an axiom.
If an axiom conflicts with a physically proven rule then it is the axiom that is wrong.
I have been showing that .9r != 1 NOT with axioms but with physically proven rules of math. Physically proven rules that have been around for centuries.
Once again you missed what I said. You're right, an axiom doesn't require a physical meaning and it doesn't refer to physical reality. It just needs to be consistent. However when an axiom is proven true then it is no longer an axiom but a proven rule of math. How do you prove it to be true? You prove it physically. If you can physically show it to be true than it is true. If you physically prove it to be false then it is false. If you can't physically prove it at all then it remains an axiom.
If an axiom conflicts with a physically proven rule then it is the axiom that is wrong.
I have been showing that .9r != 1 NOT with axioms but with physically proven rules of math. Physically proven rules that have been around for centuries.
I am not claiming you can walk through walls if you can imagine it. I am saying that if a construct exists only within our minds, it's not automatically invalidated as a construct.
I didn't say that either. I simply said that if the construct that exists only within our mind conflicts with physical constructs then it is the one that is within our minds that is wrong and not the physical.
First off thank you for admitting that .9r = 1 is unproven and is an assumption.
Second, I don't have axioms. None. The math I used and associated rules are within the very foundation of math. Such rules have been physically shown and proven true centuries ago.
If I ever come across .9r will I ever change it to 1? Absolutely but only because the difference is infinitely small and so rounding up will leave a more than acceptable error for any practical means. However that is not the point of this thread. The point of this is are they exactly the same number? Where the answer to that is no.
Just to start one has a fixed value (1) and the other (.9r) doesn't have a fixed value. Since r stands for repeat you have a repeating number of nines which is the same as saying an ever increasing number of nines. So the value is also ever increasing. Ever increasing towards the value of 1.
1 = fixed value
.9r = no fixed value
fixed value != no fixed value
1 != .9r
.9r has the limit 1.
First off thank you for admitting that .9r = 1 is unproven and is an assumption.
Second, I don't have axioms. None. The math I used and associated rules are within the very foundation of math. Such rules have been physically shown and proven true centuries ago.
If I ever come across .9r will I ever change it to 1? Absolutely but only because the difference is infinitely small and so rounding up will leave a more than acceptable error for any practical means. However that is not the point of this thread. The point of this is are they exactly the same number? Where the answer to that is no.
Just to start one has a fixed value (1) and the other (.9r) doesn't have a fixed value. Since r stands for repeat you have a repeating number of nines which is the same as saying an ever increasing number of nines. So the value is also ever increasing. Ever increasing towards the value of 1.
1 = fixed value
.9r = no fixed value
fixed value != no fixed value
1 != .9r
.9r has the limit 1.
That amounts to "I don't like it". It generate a completely consistent mathematical structure which is all that matters to maths.
So you believe infinity is a value that some variable can equal....
So you think that you will be able to reach the end of a road that has no end....
That says a lot about your common sense and logic.
Nah I changed my mind. It's not that you fail to comprehend but rather you either don't read what I post (despite replying to it) or you hear only what you want to. I have on more than one case defined infinite as a limitless limit. A true mathematical limit where nothing can equal it and only approach it. So no I don't think limit to infinity as some kind of physical process but rather the equation that has no end is a physical process that can be done by hand if need be.
1/2 + 1/4 + 1/8 + 1/16......
is an endless equation (process) where the next fraction added is half of the previous after the first. There is no limit to the number of times a fraction can be added and so the limit is infinity.
1 - (1/10 * 10 * 10 * 10 * 10......)
is an endless equation (process) where there is no limit to the number of times 10 can be multiplied by itself. The number of times therefore has the limit of infinity. This can be better shown as 1-(1/10^n) n->infinity.
1 - (1/10^n) = .9r
n->infinity
1/10^n n->infinity shows n having a value that is constantly increasing with no end to how large the value can become. 1 has a fixed value that is neither increasing nor decreasing. 1 - (1/10^n) = Y; n->infinity. Y has a value that depends on the value of 1 and 1/10^n. Since the value of 1/10^n is constantly getting smaller as the value of n is getting constantly bigger the value of Y will not have a fixed value but rather a value that is forever getting larger but has the limit of 1.
QUOTE (Precursor562+Apr 1 2007, 01:49 AM)
Can't teach God anything after all.
Funny, coming from the guy whose never done any mathematical analysis but knows better than the last 4 centuries of all the people who have
QUOTE
It doesn't require a physical meaning and it doesn't have to refer to physical reality to check it's consistency.
Once again you missed what I said. You're right, an axiom doesn't require a physical meaning and it doesn't refer to physical reality. It just needs to be consistent. However when an axiom is proven true then it is no longer an axiom but a proven rule of math. How do you prove it to be true? You prove it physically. If you can physically show it to be true than it is true. If you physically prove it to be false then it is false. If you can't physically prove it at all then it remains an axiom.
If an axiom conflicts with a physically proven rule then it is the axiom that is wrong.
I have been showing that .9r != 1 NOT with axioms but with physically proven rules of math. Physically proven rules that have been around for centuries.
QUOTE (->
| QUOTE |
| It doesn't require a physical meaning and it doesn't have to refer to physical reality to check it's consistency. |
Once again you missed what I said. You're right, an axiom doesn't require a physical meaning and it doesn't refer to physical reality. It just needs to be consistent. However when an axiom is proven true then it is no longer an axiom but a proven rule of math. How do you prove it to be true? You prove it physically. If you can physically show it to be true than it is true. If you physically prove it to be false then it is false. If you can't physically prove it at all then it remains an axiom.
If an axiom conflicts with a physically proven rule then it is the axiom that is wrong.
I have been showing that .9r != 1 NOT with axioms but with physically proven rules of math. Physically proven rules that have been around for centuries.
I am not claiming you can walk through walls if you can imagine it. I am saying that if a construct exists only within our minds, it's not automatically invalidated as a construct.
I didn't say that either. I simply said that if the construct that exists only within our mind conflicts with physical constructs then it is the one that is within our minds that is wrong and not the physical.
QUOTE
As I've said several times, 0.9r and 1 are defined from the axioms of maths. As such saying "I don't agree with your axioms and therefore 0.9r isn't 1 in my maths" doesn't change the fact in the maths everyone talks about, 0.9r=1.
First off thank you for admitting that .9r = 1 is unproven and is an assumption.
Second, I don't have axioms. None. The math I used and associated rules are within the very foundation of math. Such rules have been physically shown and proven true centuries ago.
If I ever come across .9r will I ever change it to 1? Absolutely but only because the difference is infinitely small and so rounding up will leave a more than acceptable error for any practical means. However that is not the point of this thread. The point of this is are they exactly the same number? Where the answer to that is no.
Just to start one has a fixed value (1) and the other (.9r) doesn't have a fixed value. Since r stands for repeat you have a repeating number of nines which is the same as saying an ever increasing number of nines. So the value is also ever increasing. Ever increasing towards the value of 1.
1 = fixed value
.9r = no fixed value
fixed value != no fixed value
1 != .9r
.9r has the limit 1.
QUOTE (->
| QUOTE |
| As I've said several times, 0.9r and 1 are defined from the axioms of maths. As such saying "I don't agree with your axioms and therefore 0.9r isn't 1 in my maths" doesn't change the fact in the maths everyone talks about, 0.9r=1. |
First off thank you for admitting that .9r = 1 is unproven and is an assumption.
Second, I don't have axioms. None. The math I used and associated rules are within the very foundation of math. Such rules have been physically shown and proven true centuries ago.
If I ever come across .9r will I ever change it to 1? Absolutely but only because the difference is infinitely small and so rounding up will leave a more than acceptable error for any practical means. However that is not the point of this thread. The point of this is are they exactly the same number? Where the answer to that is no.
Just to start one has a fixed value (1) and the other (.9r) doesn't have a fixed value. Since r stands for repeat you have a repeating number of nines which is the same as saying an ever increasing number of nines. So the value is also ever increasing. Ever increasing towards the value of 1.
1 = fixed value
.9r = no fixed value
fixed value != no fixed value
1 != .9r
.9r has the limit 1.
That amounts to "I don't like it". It generate a completely consistent mathematical structure which is all that matters to maths.
So you believe infinity is a value that some variable can equal....
So you think that you will be able to reach the end of a road that has no end....
That says a lot about your common sense and logic.
QUOTE
You think of 'limit to infinity' as some kind of physical process you need to do by hand
Nah I changed my mind. It's not that you fail to comprehend but rather you either don't read what I post (despite replying to it) or you hear only what you want to. I have on more than one case defined infinite as a limitless limit. A true mathematical limit where nothing can equal it and only approach it. So no I don't think limit to infinity as some kind of physical process but rather the equation that has no end is a physical process that can be done by hand if need be.
1/2 + 1/4 + 1/8 + 1/16......
is an endless equation (process) where the next fraction added is half of the previous after the first. There is no limit to the number of times a fraction can be added and so the limit is infinity.
1 - (1/10 * 10 * 10 * 10 * 10......)
is an endless equation (process) where there is no limit to the number of times 10 can be multiplied by itself. The number of times therefore has the limit of infinity. This can be better shown as 1-(1/10^n) n->infinity.
1 - (1/10^n) = .9r
n->infinity
1/10^n n->infinity shows n having a value that is constantly increasing with no end to how large the value can become. 1 has a fixed value that is neither increasing nor decreasing. 1 - (1/10^n) = Y; n->infinity. Y has a value that depends on the value of 1 and 1/10^n. Since the value of 1/10^n is constantly getting smaller as the value of n is getting constantly bigger the value of Y will not have a fixed value but rather a value that is forever getting larger but has the limit of 1.
QUOTE (->
| QUOTE |
| You think of 'limit to infinity' as some kind of phys |