Let's get to the proofs.
1.
x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - x
10x - x = 9.9999... - 0.9999...
9x = 9.0000...
9x / 9 = 9.0000... / 9
x = 1.0000...
In fact this should be:
x = 0.9999...
10x = 9.999... (notice still only 4 nines unlike example above)
10x - x = 9.999... - x
10x - x = 9.999... - 0.9999...
9x = 8.999...1 (yes this is nine recurring with a one at the end of the number)
9x / 9 = 8.999...1 / 9
x = 0.9999...
2.
1/3 = 0.3333...
1/3 * 3 = 0.3333... * 3
1 = 0.9999....
In fact this should be:
1/3 = 0.3333... remainder 0.0...1 (yes this is 0 recurring with a one at the end)
1/3 x 3 = (0.3333... remainder 0.0...1) x 3
1 = 1
They try to get past the first one by saying that you can not stick a 1 at the end of a recurring sequence of nines. Nonsense.
They try to use the second one because we automatically assume that 1/3 = 0.3333... which it doesn't. No matter how far you work this sum out you always have to carry the one so there will always be a remainder of 0.0...1. You try working it out on paper without always having to carry the one.
Let's do it like this:
1-.9=.1
1-.99=.01
1-.999=.001
...
1-.999...999=.000...001
At no step of this recursion are the two values equal, so unless you can pick some magic number at infinity, they're never precisely equal
.000...001 != .000...000
A number ending in 1 never exactly equals the same number ending in zero, unless you're willing to weight the digits down to the point where you can close your eyes and ignore the difference. So, yes, for real world applications with successly smaller and smaller weighted digits it might be close enough but technically they aren't identical and can't be manipulated arbitrarily and see the same result apply.
If we could treat them indentically then we couldn't do this:
x=10^y
1-.9r = (1-.9)/x (where y is the number of 9 digits) = 0
But:
x*(1-.9)/x = x*0
1-.9=0
Is obviously incorrect.
Another way to see this is that it's a process continually approaching 1, but never reaching it. If it could actually reach 1 then no more 9s would be needed ... but that violates the repetition.
1-.9=.1
1-.99=.01
1-.999=.001
...
1-.999...999=.000...001
At no step of this recursion are the two values equal, so unless you can pick some magic number at infinity, they're never precisely equal
.000...001 != .000...000
A number ending in 1 never exactly equals the same number ending in zero, unless you're willing to weight the digits down to the point where you can close your eyes and ignore the difference. So, yes, for real world applications with successly smaller and smaller weighted digits it might be close enough but technically they aren't identical and can't be manipulated arbitrarily and see the same result apply.
If we could treat them indentically then we couldn't do this:
x=10^y
1-.9r = (1-.9)/x (where y is the number of 9 digits) = 0
But:
x*(1-.9)/x = x*0
1-.9=0
Is obviously incorrect.
Another way to see this is that it's a process continually approaching 1, but never reaching it. If it could actually reach 1 then no more 9s would be needed ... but that violates the repetition.
QUOTE (gonegahgah+)
1.
x = 0.9999... (here, you're assuming x=0.9r)
10x = 9.9999...
.....
And there it is. In the second line of the "proof" there's an assumption made that 9*0.9r=9 or that 0.9r=1, but that's cheating because that's what you're trying to prove and you can't use it yet until you prove it so the "proof" dies here.
If instead you're trying to do this via. multiplication then you should add a multiplicative error term and see if it remains the same:
So you should rewrite it as:
x=0.9999...+e
10x=10*(0.9999...+e)
10x=9.999...+10e
The error increased by 10 when you multiplied it and will continue to erase the validity of the result as you go along.
x = 0.9999... (here, you're assuming x=0.9r)
10x = 9.9999...
.....
And there it is. In the second line of the "proof" there's an assumption made that 9*0.9r=9 or that 0.9r=1, but that's cheating because that's what you're trying to prove and you can't use it yet until you prove it so the "proof" dies here.
If instead you're trying to do this via. multiplication then you should add a multiplicative error term and see if it remains the same:
So you should rewrite it as:
x=0.9999...+e
10x=10*(0.9999...+e)
10x=9.999...+10e
The error increased by 10 when you multiplied it and will continue to erase the validity of the result as you go along.
Steven, you are completely ignoring the fact 0.9r is the LIMIT of the sequence 0.9, 0.99, 0.999,.... While none of the elements a finite length along the sequence are equal to 1 their limit is.
How do I prove this? Well suppose they limit to something which isn't 1. Call is 1-X where X is some small, non infinitesimal number belonging to the Reals. It has to be non-infinitesimal because there's only one infinitesimal in the Reals and that's zero and that would make 1=0.9r if X=0.
As you point out, 1-0.999.....9 = (0.1)^n for n=(number of 9s in the decimal expansion). You'll also agree that (0.1)^(n) > (0.1)^(n+1), therefore the sequence 0.1, 0.01, 0.001, etc is always decreasing.
Now the question is "Can I find an n such that (0.1)^n < X for any non-zero X in the Reals?". The answer is a quite obvious yes. Give me any non-zero member of the Reals and I'll find you an n such that (0.1)^n < X.
But what does this mean? It means that 1-0.9r < X for ANY X>0. I don't think anyone claims that 0.9r > 1 and it's obviously not. That, by triality (ie a number is either positive, negative or zero), means that 0.9r=1.
Other proofs not so far mentioned include :
If 0.9r is not equal to 1, give me a number between them. If two numbers are not equal, then their average is betwee them and equal to neither of them, that being if A < B, then A < 0.5(A+B ) < B. So if by your claim 0.9r < 1, then what's the number (or numbers) between them?
If you say "There doesn't have to be a number between them" then you are ignoring the fact the Reals are a field under multiplication and addition and therefore for any A,B,C, C*(A+B ) is a Real number too. C=0.5, B=0.9r and A=1 means that 0.5(0.9r+1) must be in the Reals too. If it's not between 0.9r and 1, it must be equal to both of them, ie they are equal.
Remember that physics has nothing to do with this, before someone brings up "But what if space is quantised". We are not discussing anything which is contrained by physics or reality here but the algebraic structure of the field of Real numbers.
Anyone whose sat an introductory course in analysis or set theory will know of several ways to prove 0.9r=1 and in maths that's concrete (and you only need 1 proof anyway). It's not like physics where someone comes along later and disproves a theory, a correct proof in maths is true for all time. Someone proved 2+2=4 and noone will ever come along and from the same axioms prove 2+2=5, that's the nature of maths.
And if you don't accept my word, how about Professor Tim Gowers, Fields Medalist in Combinatorics and professor of maths at Cambridge uni who specifically states 0.9r=1 on that page I just linked to.
No doubt a bunch of muppets will now give their two cents worth with irrelevent comments about how they don't accept the proofs, it doesn't 'feel' right and Zephir will just have to post a pointless picture and mumble something about aether and how AWT explains all of this
/cue the idiots....
How do I prove this? Well suppose they limit to something which isn't 1. Call is 1-X where X is some small, non infinitesimal number belonging to the Reals. It has to be non-infinitesimal because there's only one infinitesimal in the Reals and that's zero and that would make 1=0.9r if X=0.
As you point out, 1-0.999.....9 = (0.1)^n for n=(number of 9s in the decimal expansion). You'll also agree that (0.1)^(n) > (0.1)^(n+1), therefore the sequence 0.1, 0.01, 0.001, etc is always decreasing.
Now the question is "Can I find an n such that (0.1)^n < X for any non-zero X in the Reals?". The answer is a quite obvious yes. Give me any non-zero member of the Reals and I'll find you an n such that (0.1)^n < X.
But what does this mean? It means that 1-0.9r < X for ANY X>0. I don't think anyone claims that 0.9r > 1 and it's obviously not. That, by triality (ie a number is either positive, negative or zero), means that 0.9r=1.
Other proofs not so far mentioned include :
If 0.9r is not equal to 1, give me a number between them. If two numbers are not equal, then their average is betwee them and equal to neither of them, that being if A < B, then A < 0.5(A+B ) < B. So if by your claim 0.9r < 1, then what's the number (or numbers) between them?
If you say "There doesn't have to be a number between them" then you are ignoring the fact the Reals are a field under multiplication and addition and therefore for any A,B,C, C*(A+B ) is a Real number too. C=0.5, B=0.9r and A=1 means that 0.5(0.9r+1) must be in the Reals too. If it's not between 0.9r and 1, it must be equal to both of them, ie they are equal.
Remember that physics has nothing to do with this, before someone brings up "But what if space is quantised". We are not discussing anything which is contrained by physics or reality here but the algebraic structure of the field of Real numbers.
Anyone whose sat an introductory course in analysis or set theory will know of several ways to prove 0.9r=1 and in maths that's concrete (and you only need 1 proof anyway). It's not like physics where someone comes along later and disproves a theory, a correct proof in maths is true for all time. Someone proved 2+2=4 and noone will ever come along and from the same axioms prove 2+2=5, that's the nature of maths.
And if you don't accept my word, how about Professor Tim Gowers, Fields Medalist in Combinatorics and professor of maths at Cambridge uni who specifically states 0.9r=1 on that page I just linked to.
No doubt a bunch of muppets will now give their two cents worth with irrelevent comments about how they don't accept the proofs, it doesn't 'feel' right and Zephir will just have to post a pointless picture and mumble something about aether and how AWT explains all of this
/cue the idiots....
LOL
Is it true the Fields medal is given out only once every 4 years?
Yep, but they give it to a number of people. Gowers got it late 90s I think. Weils just missed out on it and the guy who proved the Poincare Conjecture might have turned it down recently.
Alpha,
If we take two lines that are parrallel and one foot apart and mark them at exponentially increasing distances of 1 foot, 10 feet, 100 feet etc., we get the observed series of slopes 1,.1,.01,.001 ... etc.
Just because this value is forever approaching zero doesn't mean it truly reaches it. The lines are always parallel and never touch even though they seem to merge in the distance.
The existance of 0.9r relies upon an assumption of infinite precision. That assumption is self defeating and the result can be interpreted either way depending upon where you decide to make the value real (and at some point you need a finite proof, so you can prove it either way depending upon where you "clip" off the infinitesimal).
Likely this is the same over simplification that makes black holes appear as paradoxes or why people might imagine perfect circles can exist etc.
People can prove anything they want but those proofs rely on assumptions and not everyone will agree upon the underlying assumptions.
Yes, you can round an infinitesimal down to zero in some cases without any noticeable change in the result, but other times you can't and not all infinities are created equally (they just look a lot alike upon course inspection).
If we take two lines that are parrallel and one foot apart and mark them at exponentially increasing distances of 1 foot, 10 feet, 100 feet etc., we get the observed series of slopes 1,.1,.01,.001 ... etc.
Just because this value is forever approaching zero doesn't mean it truly reaches it. The lines are always parallel and never touch even though they seem to merge in the distance.
The existance of 0.9r relies upon an assumption of infinite precision. That assumption is self defeating and the result can be interpreted either way depending upon where you decide to make the value real (and at some point you need a finite proof, so you can prove it either way depending upon where you "clip" off the infinitesimal).
Likely this is the same over simplification that makes black holes appear as paradoxes or why people might imagine perfect circles can exist etc.
People can prove anything they want but those proofs rely on assumptions and not everyone will agree upon the underlying assumptions.
Yes, you can round an infinitesimal down to zero in some cases without any noticeable change in the result, but other times you can't and not all infinities are created equally (they just look a lot alike upon course inspection).
QUOTE (Alphanumeric+)
While none of the elements a finite length along the sequence are equal to 1 their limit is.
...
We are not discussing anything which is contrained by physics or reality here but the algebraic structure of the field of Real numbers.
I change my mind and have to actually disgree with even this as well because consider the following:
What's (1-.9r)/(1-(0.9r)^2)?
Now if you want to try to calculate the value in the limit, you actually have mutliple ways of doing it and getting different results (again, the problem is that people assume all infinities are equal)
(1-.9)/(1-.9^2)= .1/.19 ~= .526
(1-.99)/(1-.99^2)= .01/.0199 ~= .5025
...
So here we have (1-.9r)/(1-(0.9r)^2)=0.5
But if we instead mistakenly assume 0.9r is equal to 1, then we can get a range of results:
(1-.9r)/(1-1*.9r) = (1-0.9r)/(1-0.9r)
or
(1-0.9r)/(1-0.9r)=1/2!
But wait, that can't be because:
(1-.9)/(1-.9)= .1/.1 = 1
(1-.99)/(1-.99)= .01/.01 = 1
(1-.999)/(1-.999)= .001/.001 = 1
...
So (1-.9r)/(1-.9r)=1 or 1/2?!
The problem remains that the underlying process of recursion that the r represents is removed when you assign them all the same representation as a single letter and the differences can easily go unnoticed.
You can get the recursive function to do all sorts of twists and turns along the way before it reaches infinity (in the complex plane you have it draw a smiley face before circling forever!) or even make it ballon up to some value and appear to be approaching infinity when you'd assume it should end up as some finite value or produce an irrational sequence of jumps along the way etc.. It all depends upon the manner in which the recursion is applied which isn't specified by the letter r alone.
Now I have to admit that it may not be possible to do all this by remaining entirely within the realm of real numbers but that still doesn't negate the fact that you can't calculate the limit of a value unless you have the process being used to generate that limit precisely defined.
...
We are not discussing anything which is contrained by physics or reality here but the algebraic structure of the field of Real numbers.
I change my mind and have to actually disgree with even this as well because consider the following:
What's (1-.9r)/(1-(0.9r)^2)?
Now if you want to try to calculate the value in the limit, you actually have mutliple ways of doing it and getting different results (again, the problem is that people assume all infinities are equal)
(1-.9)/(1-.9^2)= .1/.19 ~= .526
(1-.99)/(1-.99^2)= .01/.0199 ~= .5025
...
So here we have (1-.9r)/(1-(0.9r)^2)=0.5
But if we instead mistakenly assume 0.9r is equal to 1, then we can get a range of results:
(1-.9r)/(1-1*.9r) = (1-0.9r)/(1-0.9r)
or
(1-0.9r)/(1-0.9r)=1/2!
But wait, that can't be because:
(1-.9)/(1-.9)= .1/.1 = 1
(1-.99)/(1-.99)= .01/.01 = 1
(1-.999)/(1-.999)= .001/.001 = 1
...
So (1-.9r)/(1-.9r)=1 or 1/2?!
The problem remains that the underlying process of recursion that the r represents is removed when you assign them all the same representation as a single letter and the differences can easily go unnoticed.
You can get the recursive function to do all sorts of twists and turns along the way before it reaches infinity (in the complex plane you have it draw a smiley face before circling forever!) or even make it ballon up to some value and appear to be approaching infinity when you'd assume it should end up as some finite value or produce an irrational sequence of jumps along the way etc.. It all depends upon the manner in which the recursion is applied which isn't specified by the letter r alone.
Now I have to admit that it may not be possible to do all this by remaining entirely within the realm of real numbers but that still doesn't negate the fact that you can't calculate the limit of a value unless you have the process being used to generate that limit precisely defined.
QUOTE (Alphanumeric+)
Steven, you are completely ignoring the fact 0.9r is the LIMIT of the sequence 0.9, 0.99, 0.999,.... While none of the elements a finite length along the sequence are equal to 1 their limit is.
...
There doesn't have to be a number between them" then you are ignoring the fact the Reals are a field under multiplication and addition and therefore for any A,B,C, C*(A+B ) is a Real number too. C=0.5, B=0.9r and A=1 means that 0.5(0.9r+1) must be in the Reals too. If it's not between 0.9r and 1, it must be equal to both of them, ie they are equal.
Ok, and I just treated 0.9r as if it lay within an infinite field of numbers, and used the value in the limit and even assumed 0.9r was equal to 1 and still got an indeterminant answer after doing all this. Now make it harder and try to define a set of rules by which these infinite sequences can be deterministically compressed in a real expression and always act like they approach the same limit. I doubt you can pack all that into the letter r.
...
There doesn't have to be a number between them" then you are ignoring the fact the Reals are a field under multiplication and addition and therefore for any A,B,C, C*(A+B ) is a Real number too. C=0.5, B=0.9r and A=1 means that 0.5(0.9r+1) must be in the Reals too. If it's not between 0.9r and 1, it must be equal to both of them, ie they are equal.
Ok, and I just treated 0.9r as if it lay within an infinite field of numbers, and used the value in the limit and even assumed 0.9r was equal to 1 and still got an indeterminant answer after doing all this. Now make it harder and try to define a set of rules by which these infinite sequences can be deterministically compressed in a real expression and always act like they approach the same limit. I doubt you can pack all that into the letter r.
Simple proof
1/9=0.1111...
2/9=0.2222...
....
7/9=0.7777...
8/9=0.8888...
9/9=0.9999...
So
9/9=1=0.9999
Youu need more than your caculator.
PS
I agree /set 'idiots' = 92 '%'/
-Dr O
1/9=0.1111...
2/9=0.2222...
....
7/9=0.7777...
8/9=0.8888...
9/9=0.9999...
So
9/9=1=0.9999
Youu need more than your caculator.
PS
I agree /set 'idiots' = 92 '%'/
-Dr O
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
Just because this value is forever approaching zero doesn't mean it truly reaches it. The lines are always parallel and never touch even though they seem to merge in the distance.
You assume physical requirements there. There's no 'limit' here on time or a process. Does it take 'mathematical time' to add together 2 and 2? Does it need 1 second for me to do that? Me, yes, it takes me a short time to do mental arithmetic, but that's because I'm a physical entity, not a mathematical system. 2+2=4, it's not a matter of "Wait, I have to do it", it's 4.
Lets do another thing like that. You mention e^x. What's that? It's an infinite series
e^x = 1 + x + (1/2)x^2 + (1/3!)x^3 + ...
Yet does the fact I cannot physically add together infinitely many terms mean that e^x doesn't exist? Bollocks it does.
log(1 +x ) = x + ... (another infinite series)
Can I therefore not say with absolute certainty that e^(log x) = x because both log and exp involve infinite series? Of course I can say it with certainty! I am talking about mathematical entities and systems not some monkey with a calculator doing it by hand!
You assume physical requirements there. There's no 'limit' here on time or a process. Does it take 'mathematical time' to add together 2 and 2? Does it need 1 second for me to do that? Me, yes, it takes me a short time to do mental arithmetic, but that's because I'm a physical entity, not a mathematical system. 2+2=4, it's not a matter of "Wait, I have to do it", it's 4.
Lets do another thing like that. You mention e^x. What's that? It's an infinite series
e^x = 1 + x + (1/2)x^2 + (1/3!)x^3 + ...
Yet does the fact I cannot physically add together infinitely many terms mean that e^x doesn't exist? Bollocks it does.
log(1 +x ) = x + ... (another infinite series)
Can I therefore not say with absolute certainty that e^(log x) = x because both log and exp involve infinite series? Of course I can say it with certainty! I am talking about mathematical entities and systems not some monkey with a calculator doing it by hand!
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
The existance of 0.9r relies upon an assumption of infinite precision. That assumption is self defeating and the result can be interpreted either way depending upon where you decide to make the value real (and at some point you need a finite proof, so you can prove it either way depending upon where you "clip" off the infinitesimal).
The existence of 0.9r requires a concept known as 'completeness'. I suggest you look it up along with related things like convergences, compactness and Cauchy sequences. You'll find that ALL the requirements and framework to completely justify and uniquely define 0.9r exist.
It's not self defeating, it's an extremely common notion in plenty of mathematical systems. Infact, it's in plenty of physical systems. You do NOT require infinitesimals to perfectly define a Real number, by definition! If you required an infinitesimal (other than 0) to define a Real number, X, X would not be Real but something like HyperReal.
I REALLY suggest you get your hands on a book of mathematical analysis (such as the truely excellent Burkhill - Mathematical Analysis) because it would explain rigorously the utter validity of what I'm saying within mathematics.
For instance, there's a theorem which says "If an infinite sequence is monotonic and bounded, then it converges". In other words, if you have a sequence A, B, C such that A>B>C> .... or A<B<C<.... then the limit of this sequence is a set number. This seperates such sequences from oscillating ones like 0, 1, 0, 1 etc which don't converge despite being bounded (but aren't monotonic).
Is the sequence 0.1, 0.01, 0.001,. ... monotonic? Yes. Is it bounded below? Yes, by -1, -2, -3 or any set negative number. Therefore it converges. And as I demonstrated in my last post, it converges to 0, because it doesn't converge to a negative number and it doesn't converge to any positive number, so it must converge to 0. Therefore 1-0.9r = 0. Alternatively you can consider the sequence 0.9, 0.99, 0.999, ... which converges to 1.
It's not self defeating, it's an extremely common notion in plenty of mathematical systems. Infact, it's in plenty of physical systems. You do NOT require infinitesimals to perfectly define a Real number, by definition! If you required an infinitesimal (other than 0) to define a Real number, X, X would not be Real but something like HyperReal.
I REALLY suggest you get your hands on a book of mathematical analysis (such as the truely excellent Burkhill - Mathematical Analysis) because it would explain rigorously the utter validity of what I'm saying within mathematics.
For instance, there's a theorem which says "If an infinite sequence is monotonic and bounded, then it converges". In other words, if you have a sequence A, B, C such that A>B>C> .... or A<B<C<.... then the limit of this sequence is a set number. This seperates such sequences from oscillating ones like 0, 1, 0, 1 etc which don't converge despite being bounded (but aren't monotonic).
Is the sequence 0.1, 0.01, 0.001,. ... monotonic? Yes. Is it bounded below? Yes, by -1, -2, -3 or any set negative number. Therefore it converges. And as I demonstrated in my last post, it converges to 0, because it doesn't converge to a negative number and it doesn't converge to any positive number, so it must converge to 0. Therefore 1-0.9r = 0. Alternatively you can consider the sequence 0.9, 0.99, 0.999, ... which converges to 1.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
Likely this is the same over simplification that makes black holes appear as paradoxes or why people might imagine perfect circles can exist etc.
Black holes are (supposedly) physical objects. Their existence relies on our assumption that our current understanding of the universe is valid, such as the strong energy condition. In maths WE DEFINE OUR RULES, our axioms, the rules of the game and so we are CERTAIN that our base conditions are right because we define them to be and we see what structure we develop from those base rules.
No offence but you're coming at this like someone whose done very little maths, thinks of maths as nothing more than a tool of physics and hence bound by the same rules as physics and also as someone who doesn't understand how the logic of maths works.
No offence but you're coming at this like someone whose done very little maths, thinks of maths as nothing more than a tool of physics and hence bound by the same rules as physics and also as someone who doesn't understand how the logic of maths works.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
People can prove anything they want but those proofs rely on assumptions and not everyone will agree upon the underlying assumptions.
I suggest you look up the axioms of maths and have a go rewriting them then. ALL the maths you've ever seen comes from the same set of axioms. You're welcome to try to rewrite them but it'll be ahell of a job and I don't think many people would want to try it from the ground up.
It took two of the greatest logicistic mathematicians of the early 20th century a decade to write out 3 volumes of maths from axioms and they didn't even get to geometry. It took 360 pages to get from the axioms to defining the number 1. The axioms are that basic!
It took two of the greatest logicistic mathematicians of the early 20th century a decade to write out 3 volumes of maths from axioms and they didn't even get to geometry. It took 360 pages to get from the axioms to defining the number 1. The axioms are that basic!
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
Yes, you can round an infinitesimal down to zero in some cases without any noticeable change in the result, but other times you can't and not all infinities are created equally (they just look a lot alike upon course inspection).
The sequence 0.9, 0.99, 0.999, .... is a sequence of countably many terms. There are extensions within HyperReals which have uncountably many terms in the sequence but the end result is the same. 0.9r* = 1*, where * signifies the HyperReal version of the number.
And again, no offence, but if you're struggling with this level of analysis, I don't think you're really up to talking about various types of cardinality in infinite sets unless you just skipped all the foundations which lead to that result.
And besides, if you accept such notions of cardinality along with basic arithmetic, calculus, geometry, set theory and everything else then you must accept 0.9r=1 because they ALL come from the same axioms.
And again, no offence, but if you're struggling with this level of analysis, I don't think you're really up to talking about various types of cardinality in infinite sets unless you just skipped all the foundations which lead to that result.
And besides, if you accept such notions of cardinality along with basic arithmetic, calculus, geometry, set theory and everything else then you must accept 0.9r=1 because they ALL come from the same axioms.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
What's (1-.9r)/(1-(0.9r)^2)?
It's undefined because you've asked me "What's 0/0", which is undefined.
Now you can define sequences which limit to that, but to simply ask what the number is is undefined.
Now you can define sequences which limit to that, but to simply ask what the number is is undefined.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
(1-.9)/(1-.9^2)= .1/.19 ~= .526
You are playing with numbers and actually missing the fact you're simply demonstrating a well known factorisation.
What is (1-x)/(1-x^2) ? I can factorise the bottom to give me (1-x)/(1-x)(1+x) = 1/(1+x)
Therefore if you put in x->1, you get (1-x)/(1-x^2) -> 1/2
So infact you've proved nothing about your own claim, simply that you shouldn't directly evaluate (1-x)/(1-x^2) at x=1 but should consider a limiting process x->1. Well done, that's high school algebra sorted for you.
You mistakenly thought that (1-x)/(1-x^2) = 1 if x=1 since the top and bottom end up being the same value. WRONG. You have ignored a basic result which I remember teaching to 1st year geographers last term, never mind 1st year mathematicians! If f(x) = g(x)/h(x) and f(a) = 0/0, then you must FIRST factorise g(x) and h(x) and cancel the common factors of (x-a) (which must exist by the fundamental theorem of algebra) and THEN evaluate f(a), if you can.
If you don't, you can prove some otherwise seemingly off things. Lets consider f(x) = (2-x)/(4-x^2). By your initial logic f(2) = 1, but if I put in x->2 I get 1/4. Oh no!! Somehow I've destroyed algebra as we know it!!!
No, I've just been sloppy and made a mistake a 1st year should know better of.
What is (1-x)/(1-x^2) ? I can factorise the bottom to give me (1-x)/(1-x)(1+x) = 1/(1+x)
Therefore if you put in x->1, you get (1-x)/(1-x^2) -> 1/2
So infact you've proved nothing about your own claim, simply that you shouldn't directly evaluate (1-x)/(1-x^2) at x=1 but should consider a limiting process x->1. Well done, that's high school algebra sorted for you.
You mistakenly thought that (1-x)/(1-x^2) = 1 if x=1 since the top and bottom end up being the same value. WRONG. You have ignored a basic result which I remember teaching to 1st year geographers last term, never mind 1st year mathematicians! If f(x) = g(x)/h(x) and f(a) = 0/0, then you must FIRST factorise g(x) and h(x) and cancel the common factors of (x-a) (which must exist by the fundamental theorem of algebra) and THEN evaluate f(a), if you can.
If you don't, you can prove some otherwise seemingly off things. Lets consider f(x) = (2-x)/(4-x^2). By your initial logic f(2) = 1, but if I put in x->2 I get 1/4. Oh no!! Somehow I've destroyed algebra as we know it!!!
No, I've just been sloppy and made a mistake a 1st year should know better of.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
You can get the recursive function to do all sorts of twists and turns along the way before it reaches infinity (in the complex plane you have it draw a smiley face before circling forever!) or even make it ballon up to some value and appear to be approaching infinity when you'd assume it should end up as some finite value or produce an irrational sequence of jumps along the way etc.. It all depends upon the manner in which the recursion is applied which isn't specified by the letter r alone.
Unless you care to rephrase that, I'm going to have to say that's bollocks. The sequence 0.9, 0.99, 0.999, .... is well behaved (monotonic and bounded!), has nothing to do with the complex plane and it sounds distinctly like you're trying to BS your way past me.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
Now I have to admit that it may not be possible to do all this by remaining entirely within the realm of real numbers but that still doesn't negate the fact that you can't calculate the limit of a value unless you have the process being used to generate that limit precisely defined.
It can and is done entirely within the Real numbers and the sequence is well defined, it's properties understood and by various results in basic analysis it's known to convert to exactly 1.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
Ok, and I just treated 0.9r as if it lay within an infinite field of numbers, and used the value in the limit and even assumed 0.9r was equal to 1 and still got an indeterminant answer after doing all this.
I don't follow what you're saying because you're just being too vague. I notice you failed to give me a number between 0.9r and 1.
Recursiveness has a limit? Based on a finite derived from infinity?
I'll give you a 1, but only as "representing" something, that you can't "describe" any greater than .9r.
QUOTE (StevenA+Mar 6 2007, 08:04 PM)
I doubt you can pack all that into the letter r.
Probably because what you said was nonsense.
If you're going to talk about all that stuff, at least learn some mathematical analysis first. Learn what 'complete' means, what 'compact' means. Why a monotonic, bounded function always converges. I mean, this isn't like physics where someone can slip up or misinterpret data, it's logic and once a maths proof is done, it's true forever, that's the essence of proof.
I just get the feeling you're trying to bluff your way past me by throwing nonsense my way but all the time you didn't counter any of my proofs, pointing out the flaw in the logic. Burkhill's book would explain this nicely. Infact....
/checked book on shelf
No, it's not specifically mentioned in there, but things like a geometric convergent series within infinitely many terms are, which 0.9r is an example of.
And just to let you know, I did maths as an undergrad and this stuff is 1st year work so short of decending into gibberish (which wouldn't help your case anyway), you're going to have to do better than that
If you're going to talk about all that stuff, at least learn some mathematical analysis first. Learn what 'complete' means, what 'compact' means. Why a monotonic, bounded function always converges. I mean, this isn't like physics where someone can slip up or misinterpret data, it's logic and once a maths proof is done, it's true forever, that's the essence of proof.
I just get the feeling you're trying to bluff your way past me by throwing nonsense my way but all the time you didn't counter any of my proofs, pointing out the flaw in the logic. Burkhill's book would explain this nicely. Infact....
/checked book on shelf
No, it's not specifically mentioned in there, but things like a geometric convergent series within infinitely many terms are, which 0.9r is an example of.
And just to let you know, I did maths as an undergrad and this stuff is 1st year work so short of decending into gibberish (which wouldn't help your case anyway), you're going to have to do better than that
QUOTE
Steven, you are completely ignoring the fact 0.9r is the LIMIT of the sequence 0.9, 0.99, 0.999,.... While none of the elements a finite length along the sequence are equal to 1 their limit is.
Recursiveness has a limit? Based on a finite derived from infinity?
I'll give you a 1, but only as "representing" something, that you can't "describe" any greater than .9r.
QUOTE (Dr. Obvious+)
1/9=0.1111...
But why?
If you perform each step of the long division you'll find that there never is a point where 0.1111... is exactly equal to 1/9.
If someone made the claim that 1/9!=0.1111... then the proof you presented would instead say that 1!=0.9999...
Is faith enough to make something true? That may very well end up being the case but then someone could assume 0.1111... is not equal to 1/9 and be able to prove it as much as someone else claiming they are equal, so the only way to resolve this paradox is to not treat them identically or things become confusing.
The best way to do this is to show in what cases .9r=1 and in what cases it doesn't equal 1. Then you're covered on both sides and can see much more accurately when and where the division between 0 and infinitesimals lies.
But why?
If you perform each step of the long division you'll find that there never is a point where 0.1111... is exactly equal to 1/9.
If someone made the claim that 1/9!=0.1111... then the proof you presented would instead say that 1!=0.9999...
Is faith enough to make something true? That may very well end up being the case but then someone could assume 0.1111... is not equal to 1/9 and be able to prove it as much as someone else claiming they are equal, so the only way to resolve this paradox is to not treat them identically or things become confusing.
The best way to do this is to show in what cases .9r=1 and in what cases it doesn't equal 1. Then you're covered on both sides and can see much more accurately when and where the division between 0 and infinitesimals lies.
QUOTE (Eric England+Mar 6 2007, 09:45 PM)
I'll give you a 1, but only as "representing" something, that you can't "describe" any greater than .9r.
That's known as 'word salad' Eric. Rather than using logic to refute what I've said you throw out pseudophilisophical crap.
That's known as 'word salad' Eric. Rather than using logic to refute what I've said you throw out pseudophilisophical crap.
QUOTE (StevenA+Mar 6 2007, 09:48 PM)
Is faith enough to make something true?
No, rock solid logic is. You seem to be taking your faith over logic about the logical results of statements.
QUOTE (StevenA+Mar 6 2007, 09:48 PM)
The best way to do this is to show in what cases .9r=1 and in what cases it doesn't equal 1.
Sorry, do you actually think what you just said is valid?! 
That's like saying "Lets find when 2+2=4 and when is doesn't". Why would a number's value change? Does the value of 2 ever change? Are there cases when 2 is actually 3? But then what's 2? 1? 7? -3?
Numbers don't change, 0.9r=1 and once you've proved it (and I have, several times in several ways), it's an unchanging fact. Or do you need a course in the basic workings of maths?
That's like saying "Lets find when 2+2=4 and when is doesn't". Why would a number's value change? Does the value of 2 ever change? Are there cases when 2 is actually 3? But then what's 2? 1? 7? -3?
Numbers don't change, 0.9r=1 and once you've proved it (and I have, several times in several ways), it's an unchanging fact. Or do you need a course in the basic workings of maths?
QUOTE (StevenA+Mar 6 2007, 09:48 PM)
Then you're covered on both sides and can see much more accurately when and where the division between 0 and infinitesimals lies.
Might I suggest learning what the definitions and properties of fields containing infinitesimals and their relation to Reals or extensions of the Reals before making such comments because it seems very much like you're just stumbling in the dark, trying to hide behind pseudo-sounding nonsense the fact you can't logically prove your claim so if you just go round and round in circles with vague comments then maybe you'll somehow come out right?
The problem is, all the people who claim 0.9r isn't 1 haven't done maths. They have little to no idea the machinary which underpins the result and so when I say something like that theorem about bounded monotonic sequences, it means nothing to them. But rather than think "Hey, perhaps the guy with the maths degree and the guy who lectures this stuff at Cambridge know a little bit about this area and might be right?", it's back to the crank logic where you can't possibly be wrong and to hell with the people who know about this stuff. Only this time, it's not a matter of debate like physics can be, it's a cold hard, irrefutable (but people try!) fact that the axioms of maths undeniably result in the proof that 0.9r=1.
To claim otherwise is just to show you haven't actually got the knowledge to understand how the result is proved. It also means you don't accept 1+1=2, 4/2 = 2 and 1+0 = 1, along with ALL other maths, because they are all interconnected with one another.
But hey, don't let a little thing like knowledge and logic get in your way, most cranks seem to manage well without either.
The problem is, all the people who claim 0.9r isn't 1 haven't done maths. They have little to no idea the machinary which underpins the result and so when I say something like that theorem about bounded monotonic sequences, it means nothing to them. But rather than think "Hey, perhaps the guy with the maths degree and the guy who lectures this stuff at Cambridge know a little bit about this area and might be right?", it's back to the crank logic where you can't possibly be wrong and to hell with the people who know about this stuff. Only this time, it's not a matter of debate like physics can be, it's a cold hard, irrefutable (but people try!) fact that the axioms of maths undeniably result in the proof that 0.9r=1.
To claim otherwise is just to show you haven't actually got the knowledge to understand how the result is proved. It also means you don't accept 1+1=2, 4/2 = 2 and 1+0 = 1, along with ALL other maths, because they are all interconnected with one another.
But hey, don't let a little thing like knowledge and logic get in your way, most cranks seem to manage well without either.
QUOTE (Alphanumeric+)
You are playing with numbers and actually missing the fact you're simply demonstrating a well known factorisation.
What is (1-x)/(1-x^2) ? I can factorise the bottom to give me (1-x)/(1-x)(1+x) = 1/(1+x)
Therefore if you put in x->1, you get (1-x)/(1-x^2) -> 1/2
So infact you've proved nothing about your own claim, simply that you shouldn't directly evaluate (1-x)/(1-x^2) at x=1 but should consider a limiting process x->1. Well done, that's high school algebra sorted for you.
I DID use the limit of x->1 and got 1/2 as you said. That was my point! If 0.9r can only be equal to one, then I should be able to do this simplification:
(1-x)/(1-x^2) = (1-x)/(1-1*x) = (1-x)/(1-x)
but that gives a different result in the limit. So analyzing a repeating value in the limit isn't reliable and simplifying it to a finite expression removes the ratios of infinity underlying it.
You're repeating what I said and then ignoring the paradox of it. You can't say it's only ok to evaluate the limit of an expression approaching zero in some cases and not others. The reason why this is no paradox to me is because I know why it works in some cases and gets certain results versus when it gets a different result.
0.9r=1 is based upon an assumption that's not always true and again, not all infinities are created equal.
What's x/x?
What's x^2/x or x/x^2?
In all these cases x can't be equal to zero, but if you know what's generating this then you can attempt to calculate a limit instead, but that's only possible by knowing what process is generating the limit!. Otherwise zeroes cannot be treated as infinitesimals, because you end up creating an artificial structure where it didn't exist before or on the other hand you take a structure and attempt to replace it with a symbolic and then detach it from an ability to verify whether it's true or not.
In the case of the equation 1-0.9r=0, you're doing the exact same thing but it's arithmetic so the error falls off by a factor of 10^-y (where y is the resolution of the error), but anything more sensitive than that (and many functions are) will cause it to explode into an infinite error instead of an infinitesimal so you have to know what's causing the repetition to occur in order to determine if the relationship can be evaluated. It's not about simple symbolics - there are processes behind these that the symbolics are intended to represent and separating the two means bogus results.
What is (1-x)/(1-x^2) ? I can factorise the bottom to give me (1-x)/(1-x)(1+x) = 1/(1+x)
Therefore if you put in x->1, you get (1-x)/(1-x^2) -> 1/2
So infact you've proved nothing about your own claim, simply that you shouldn't directly evaluate (1-x)/(1-x^2) at x=1 but should consider a limiting process x->1. Well done, that's high school algebra sorted for you.
I DID use the limit of x->1 and got 1/2 as you said. That was my point! If 0.9r can only be equal to one, then I should be able to do this simplification:
(1-x)/(1-x^2) = (1-x)/(1-1*x) = (1-x)/(1-x)
but that gives a different result in the limit. So analyzing a repeating value in the limit isn't reliable and simplifying it to a finite expression removes the ratios of infinity underlying it.
You're repeating what I said and then ignoring the paradox of it. You can't say it's only ok to evaluate the limit of an expression approaching zero in some cases and not others. The reason why this is no paradox to me is because I know why it works in some cases and gets certain results versus when it gets a different result.
0.9r=1 is based upon an assumption that's not always true and again, not all infinities are created equal.
What's x/x?
What's x^2/x or x/x^2?
In all these cases x can't be equal to zero, but if you know what's generating this then you can attempt to calculate a limit instead, but that's only possible by knowing what process is generating the limit!. Otherwise zeroes cannot be treated as infinitesimals, because you end up creating an artificial structure where it didn't exist before or on the other hand you take a structure and attempt to replace it with a symbolic and then detach it from an ability to verify whether it's true or not.
In the case of the equation 1-0.9r=0, you're doing the exact same thing but it's arithmetic so the error falls off by a factor of 10^-y (where y is the resolution of the error), but anything more sensitive than that (and many functions are) will cause it to explode into an infinite error instead of an infinitesimal so you have to know what's causing the repetition to occur in order to determine if the relationship can be evaluated. It's not about simple symbolics - there are processes behind these that the symbolics are intended to represent and separating the two means bogus results.
Edit.
Nevermind lol. I should have seen that sooner. 1 - .9 = .1, 1 - .99 = .01 etc. so the n doesn't represent the number of 1's after the decimal it represents the number of zeros between the decimal and the one.
Nevermind lol. I should have seen that sooner. 1 - .9 = .1, 1 - .99 = .01 etc. so the n doesn't represent the number of 1's after the decimal it represents the number of zeros between the decimal and the one.
QUOTE (StevenA+Mar 6 2007, 10:26 PM)
I DID use the limit of x->1 and got 1/2 as you said. That was my point! If 0.9r can only be equal to one, then I should be able to do this simplification:
(1-x)/(1-x^2) = (1-x)/(1-1*x) = (1-x)/(1-x)
but that gives a different result in the limit.
That's not valid algebra. That's like saying :
1-x^2 = 0. I know x=1 is a solution so I can say 1-1*x = 0, therefore 1-x=0, so x=1 is the only solution.
You can't just sub back in SOME of the variable terms with ONE possible value. It's complete nonsense and hence not suprising you get nonsense as a result. As you'll notice from my example, by doing that you completely missed x=-1 is a valid solution to.
(1-x)/(1-x^2) = (1-x)/(1-1*x) = (1-x)/(1-x)
but that gives a different result in the limit.
That's not valid algebra. That's like saying :
1-x^2 = 0. I know x=1 is a solution so I can say 1-1*x = 0, therefore 1-x=0, so x=1 is the only solution.
You can't just sub back in SOME of the variable terms with ONE possible value. It's complete nonsense and hence not suprising you get nonsense as a result. As you'll notice from my example, by doing that you completely missed x=-1 is a valid solution to.
QUOTE (StevenA+Mar 6 2007, 10:26 PM)
You're repeating what I said and then ignoring the paradox of it. You can't say it's only ok to evaluate the limit of an expression approaching zero in some cases and not others. The reason why this is no paradox to me is because I know why it works in some cases and gets certain results versus when it gets a different result.
I'm not ignoring the paradox of anything and you don't know why it works that way because it doesn't.
QUOTE (StevenA+Mar 6 2007, 10:26 PM)
0.9r=1 is based upon an assumption that's not always true and again, not all infinities are created equal.
No it isn't! It doesn't even require you to know anything about cardinalities of infinite sets! 0.9r=1 was proved hundreds of years before Cantor did his work on infinite cardinality.
You're just making unfounded claims about somethign you haven't bothered to learn about.
You're just making unfounded claims about somethign you haven't bothered to learn about.
QUOTE (StevenA+Mar 6 2007, 10:26 PM)
In the case of the equation 1-0.9r=0, you're doing the exact same thing but it's arithmetic so the error falls off by a factor of 10^-y (where y is the resolution of the error), but anything more sensitive than that (and many functions are) will cause it to explode into an infinite error instead of an infinitesimal so you have to know what's causing the repetition to occur in order to determine if the relationship can be evaluated. It's not about simple symbolics - there are processes behind these that the symbolics are intended to represent and separating the two means bogus results.
You talk about the processes behind these things but it's clear you've never bothered to learn such processes.
You are arguing about this kind of maths from the point of view of someone whose done high school maths. High school maths is devoid of any analysis (at least it is in the UK) and the notion of rigorous proof is hence lost on plenty of people. You therefore try to argue by armwaving. That doesn't work.
Stop arm waving and rambling (which boils down to 'Because I don't like it') and instead try to disprove using only maths and only logic the proofs people have given. There's plenty of results in maths which people 'don't like' because they seem counter intuitive, but that's because logic and intuition are not always the same thing, particularly in abstract systems.
Consider the sequence : (0.9, 0.99, 0.999, .....)
If that doesn't limit to 1, what does it limit to? Be precise, no arm waving, use mathematical statements.
You are arguing about this kind of maths from the point of view of someone whose done high school maths. High school maths is devoid of any analysis (at least it is in the UK) and the notion of rigorous proof is hence lost on plenty of people. You therefore try to argue by armwaving. That doesn't work.
Stop arm waving and rambling (which boils down to 'Because I don't like it') and instead try to disprove using only maths and only logic the proofs people have given. There's plenty of results in maths which people 'don't like' because they seem counter intuitive, but that's because logic and intuition are not always the same thing, particularly in abstract systems.
Consider the sequence : (0.9, 0.99, 0.999, .....)
If that doesn't limit to 1, what does it limit to? Be precise, no arm waving, use mathematical statements.
QUOTE (Precursor562+Mar 6 2007, 10:26 PM)
(0.1)^(n) where n=5
0.11111
(0.1)^(n+1) where n=5
0.111111
0.11111
(0.1)^(n+1) where n=5
0.111111
You misunderstand the notation. (0.1)^n = 0.1*0.1*...*0.1 n times, so 0.1^n = 0.000....01 with n-1 zeros.
(0.1)^(n+1) = 0.1*(0.1)^n < 0.1^n
(0.1)^(n+1) = 0.1*(0.1)^n < 0.1^n
QUOTE (Precursor562+Mar 6 2007, 10:26 PM)
(0.1)^(n) > (0.1)^(n-1)
That's actually exactly what I said, just both sides divided by 0.1.
QUOTE (Alphanumeric+)
I suggest you look up the axioms of maths and have a go rewriting them then. ALL the maths you've ever seen comes from the same set of axioms. You're welcome to try to rewrite them but it'll be ahell of a job and I don't think many people would want to try it from the ground up.
It took two of the greatest logicistic mathematicians of the early 20th century a decade to write out 3 volumes of maths from axioms and they didn't even get to geometry. It took 360 pages to get from the axioms to defining the number 1. The axioms are that basic!
If it takes 360 pages to "prove" (and using Occam's Razor I'll assume that's just what it took to obfiscate things to the point of incomprehension) that parallel lines can't exist except when superimposed on each other then we need a new system of mathematics.
In some ways I could agree that physically separate objects couldn't be seen to exist in reality unless they interacted (which allows possibilities for them violating the idea of being parallel), but this could end up causing quite a stir in other areas of math.
Let me ask you this: If two separate points are not coincident, but as you move away they appear to become vanishingly close together on the horizon do they actually intersect?
Normally we'd just say if the distance d!=0 then they don't intersect, but you seem to be telling me that if they're far enough away then they DO intersect because the subjective separation between them appears to get smaller and smaller until they merge together as a single object and you're willing to just say they're identical.
But then how do you resolve the conflict with the guy who's still next to them and never saw them getting any closer together?
I agree that everything is relative and if it "looks close enough", then it is, but only for that specific result.
My opinion is that you should rely on the close up opinion as to whether they truly can be treated identically ... appearances can be deceiving. Though understanding why each sees it differently is even better.
Now you've said that this is just for abstract math, but even math is based upon logic and as far as I can tell all of the "proofs" here make the assumption that if something gets small enough to ignore we can just swap it with a 0 ... sort of like a mental back hole that sucks everything near it into becoming an object that no longer possesses any distinctive atrributes ... this lack of distinction makes the use of infinitesimal a tricky proposition as sometimes they will be ignored and other times not and depending upon when and how, you get a different answer - sometimes separate points are actually not and visa versa and the funky world of superposition begins to emerge where "logic" begins to resemble wavelike fluids without reliable solutions, instead of delicate and precisely defined quantum objects in which no room for error is tolerated.
If everything gets crammed into 0 then how are we suppose to sort out the infinitesimals?
Instead, .9r~=1
And then all the problems are acknowledged (there are valid reasons to indicate when a relationship is an identity or an approximate). Just like you can't integrate without ignoring the possible existance of a non-zero offset or you can't always toss out the imaginary solutions for a root etc. So identities often have specific conditions under which they're true and this should be made clear.
For example, a specific precision to computations could be specified when using r and then the form of the deviation resulting from it included as part of the solution. Many mathematical techniques use this method of applying boundary analysis and placing limits on the errors.
For example we could rewrite the above more precisely as:
.9r=1-10^n (where n is number of digits extension)
Then
.9r=1-10^n=1-e
and e=10^-n
This can be seen as a complex inward spiralling rotation sampled at integer values of n ... but that depends upon what numbering base is being used and this isn't explicitly specified by a real numbering system.
So the base of the repeating representation isn't specified by the real numbers either and could result in irrational values!
Also how can infinity always be a power of integer bases! If so, what about infinity + 1?
There are entirely too many conflicts that arise if these paradoxes are ignored.
It took two of the greatest logicistic mathematicians of the early 20th century a decade to write out 3 volumes of maths from axioms and they didn't even get to geometry. It took 360 pages to get from the axioms to defining the number 1. The axioms are that basic!
If it takes 360 pages to "prove" (and using Occam's Razor I'll assume that's just what it took to obfiscate things to the point of incomprehension) that parallel lines can't exist except when superimposed on each other then we need a new system of mathematics.
In some ways I could agree that physically separate objects couldn't be seen to exist in reality unless they interacted (which allows possibilities for them violating the idea of being parallel), but this could end up causing quite a stir in other areas of math.
Let me ask you this: If two separate points are not coincident, but as you move away they appear to become vanishingly close together on the horizon do they actually intersect?
Normally we'd just say if the distance d!=0 then they don't intersect, but you seem to be telling me that if they're far enough away then they DO intersect because the subjective separation between them appears to get smaller and smaller until they merge together as a single object and you're willing to just say they're identical.
But then how do you resolve the conflict with the guy who's still next to them and never saw them getting any closer together?
I agree that everything is relative and if it "looks close enough", then it is, but only for that specific result.
My opinion is that you should rely on the close up opinion as to whether they truly can be treated identically ... appearances can be deceiving. Though understanding why each sees it differently is even better.
Now you've said that this is just for abstract math, but even math is based upon logic and as far as I can tell all of the "proofs" here make the assumption that if something gets small enough to ignore we can just swap it with a 0 ... sort of like a mental back hole that sucks everything near it into becoming an object that no longer possesses any distinctive atrributes ... this lack of distinction makes the use of infinitesimal a tricky proposition as sometimes they will be ignored and other times not and depending upon when and how, you get a different answer - sometimes separate points are actually not and visa versa and the funky world of superposition begins to emerge where "logic" begins to resemble wavelike fluids without reliable solutions, instead of delicate and precisely defined quantum objects in which no room for error is tolerated.
If everything gets crammed into 0 then how are we suppose to sort out the infinitesimals?
Instead, .9r~=1
And then all the problems are acknowledged (there are valid reasons to indicate when a relationship is an identity or an approximate). Just like you can't integrate without ignoring the possible existance of a non-zero offset or you can't always toss out the imaginary solutions for a root etc. So identities often have specific conditions under which they're true and this should be made clear.
For example, a specific precision to computations could be specified when using r and then the form of the deviation resulting from it included as part of the solution. Many mathematical techniques use this method of applying boundary analysis and placing limits on the errors.
For example we could rewrite the above more precisely as:
.9r=1-10^n (where n is number of digits extension)
Then
.9r=1-10^n=1-e
and e=10^-n
This can be seen as a complex inward spiralling rotation sampled at integer values of n ... but that depends upon what numbering base is being used and this isn't explicitly specified by a real numbering system.
So the base of the repeating representation isn't specified by the real numbers either and could result in irrational values!
Also how can infinity always be a power of integer bases! If so, what about infinity + 1?
There are entirely too many conflicts that arise if these paradoxes are ignored.
The 9's go on forever. That is not ONE.
QUOTE (Eric England+Mar 6 2007, 01:45 PM)
Recursiveness has a limit? Based on a finite derived from infinity?
I'll give you a 1, but only as "representing" something, that you can't "describe" any greater than .9r.
Here's my post again AN.
You skipped the first part which directly related to a statement you made.
The second part you laugh off as "word salad".
Take a "crouton". 1 crouton, amongst 8 croutons, amongst 1 salad, in 1 bowl, amongst whatever.
Now try to "describe" what I just "represented". Tell you what, I'll make it easy for you. Just the crouton and it's outermost surface, all the way down to it's infinitesimal constituent parts and their respective surfaces.
No cheating now. No... "it's made of 1 this and however many of that, etc".
If you can do that, I'll eat my word salad.
I'll give you a 1, but only as "representing" something, that you can't "describe" any greater than .9r.
Here's my post again AN.
You skipped the first part which directly related to a statement you made.
The second part you laugh off as "word salad".
Take a "crouton". 1 crouton, amongst 8 croutons, amongst 1 salad, in 1 bowl, amongst whatever.
Now try to "describe" what I just "represented". Tell you what, I'll make it easy for you. Just the crouton and it's outermost surface, all the way down to it's infinitesimal constituent parts and their respective surfaces.
No cheating now. No... "it's made of 1 this and however many of that, etc".
If you can do that, I'll eat my word salad.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
If it takes 360 pages to "prove" (and using Occam's Razor I'll assume that's just what it took to obfiscate things to the point of incomprehension) that parallel lines can't exist except when superimposed on each other then we need a new system of mathematics.
No, it took 360 pages because the authors (Russel and Whitehead) started from extremely basic and fundamental statements of logic and took NOTHING else for granted if it wasn't proven previously in the book.
The length wasn't to make it uncomprehensible to others, they were doing it as much for themselves as other people. It was because the maths that you and I are familiar with on a working level is a long long long way down the line from the founding statements of mathematics.
Remember, they couldn't (and shouldn't!) use physical examples of the truthfulness of things, they had to do it with formal logic from the axioms, that was the purpose of the work.
The fact you seem oblivious to the importance of such methods of proof illustrates why your claims here are somewhat flawed.
No, it took 360 pages because the authors (Russel and Whitehead) started from extremely basic and fundamental statements of logic and took NOTHING else for granted if it wasn't proven previously in the book.
The length wasn't to make it uncomprehensible to others, they were doing it as much for themselves as other people. It was because the maths that you and I are familiar with on a working level is a long long long way down the line from the founding statements of mathematics.
Remember, they couldn't (and shouldn't!) use physical examples of the truthfulness of things, they had to do it with formal logic from the axioms, that was the purpose of the work.
The fact you seem oblivious to the importance of such methods of proof illustrates why your claims here are somewhat flawed.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
In some ways I could agree that physically separate objects couldn't be seen to exist in reality unless they interacted (which allows possibilities for them violating the idea of being parallel), but this could end up causing quite a stir in other areas of math.
A stir in maths? What the heck are you talking about? That paragraph is nonsense. Physically interacting? They're numbers! 1 and 2 don't interact like two electrons bouncing off one another, they are elements in a set which have two binary operations defined on them, addition and multiplication. They can be derived from more basic statements like the Peano axioms or even more basic ones again, such as those used by Russel and Whitehead.
It won't cause a stir in maths because what you said was mumbo jumbo, not maths. Parallel? How is 1 parallel to 2 or whatever it is you're talking about. Points aren't parallel to one another, extended objects are.
It won't cause a stir in maths because what you said was mumbo jumbo, not maths. Parallel? How is 1 parallel to 2 or whatever it is you're talking about. Points aren't parallel to one another, extended objects are.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
Let me ask you this: If two separate points are not coincident, but as you move away they appear to become vanishingly close together on the horizon do they actually intersect?
That's using physical considerations. Your eyes ability to resolve two close objects is related to the size of your pupil and the frequency of light you see in (hence why blue is better to resolve things than red). That has NOTHING to do with the actual different (or not) of 0.9r and 1.
Even in physical situations, your analogy is flawedd. In an eclipse, the Moon and Sun seem to line up. Are they touching or is it simply our point of view? Does our point of view have anything to do with actual physical distance between the Moon and Sun? Of course not!
Even in physical situations, your analogy is flawedd. In an eclipse, the Moon and Sun seem to line up. Are they touching or is it simply our point of view? Does our point of view have anything to do with actual physical distance between the Moon and Sun? Of course not!
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
Normally we'd just say if the distance d!=0 then they don't intersect, but you seem to be telling me that if they're far enough away then they DO intersect because the subjective separation between them appears to get smaller and smaller until they merge together as a single object and you're willing to just say they're identical.
Nothing to do with subjectivity, distance, 'willingness', they ARE the same by the proofs I've given. It's not a case of "They are 'roughly' the same", they are the same. Maths deals in a lot of absolutes. Does 100000000000000 = 100000000000001 ? No, irrelevent of how you try to swing it. In physics, if you are only interested in an order of magnitude assessment, then you can say 100000000000000 ~ 100000000000001 and in your calculations call them equal for ease of algebra, but physical approximations is NOT mathematical truth.
0.9r=1, exactly, not 'about', exactly.
0.9r=1, exactly, not 'about', exactly.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
But then how do you resolve the conflict with the guy who's still next to them and never saw them getting any closer together?
For about the 6th time, physical considerations are irrelevent here. We're talking about limits of mathematical entities.
If you insist on such folly, then this is how it works :
Your 'man' says "They are seperated by X". I can then show that he's wrong and their true seperation is less than X. So he might try "They are seperated by X/10", but I can show it's smaller than that! Whatever he says, it's smaller than that.
Your problem is you think if it as a long but finite sequence, where there's always a small but non-zero difference. WRONG.
For all the cranks telling me to 'free my mind' and other Matrix Neo pseudo-quotes, I seem to manage to grasp non-physical abstract systems a damn sight better than them!
If you insist on such folly, then this is how it works :
Your 'man' says "They are seperated by X". I can then show that he's wrong and their true seperation is less than X. So he might try "They are seperated by X/10", but I can show it's smaller than that! Whatever he says, it's smaller than that.
Your problem is you think if it as a long but finite sequence, where there's always a small but non-zero difference. WRONG.
For all the cranks telling me to 'free my mind' and other Matrix Neo pseudo-quotes, I seem to manage to grasp non-physical abstract systems a damn sight better than them!
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
I agree that everything is relative and if it "looks close enough", then it is, but only for that specific result.
It doesn't 'look close', they are the same.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
My opinion is that you should rely on the close up opinion as to whether they truly can be treated identically ... appearances can be deceiving. Though understanding why each sees it differently is even better..
There's no 'appearances', it's not a physical system! If you prove it algebraicly, you prove it!
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
Now you've said that this is just for abstract math, but even math is based upon logic and as far as I can tell all of the "proofs" here make the assumption that if something gets small enough to ignore we can just swap it with a 0
No, the proofs don't make that assumption but because you're attempting to argue about maths you've never bothered to learn you don't see the logic in it, not least because you've not looked at how all these things are derived and cemented into logical truth.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
... sort of like a mental back hole that sucks everything near it into becoming an object that no longer possesses any distinctive atrributes
No, that isn't true and there are plenty of things in maths where the differences between two objects can be extremely small but of vital importance.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
this lack of distinction makes the use of infinitesimal a tricky proposition as sometimes they will be ignored and other times not
No it doesn't. The Real numbers are complete, which gives a precise definition and validity to the notion of convergence sequences. They don't require infinitesimals for that.
The HyperReal numbers are the Real numbers but with a vast extension by adding infinitesimals into the set upon which the binary operators are defined. In HyperReals there is an analogue notion of 0.9r* (* represents hyperreals) and wether it equals 1*.
There is a theorem in analysis (a theorem is a result which has been proven true) that any statement in standard analysis (ie on the Reals of Complex numbers) is the same in it's version of non-standard analysis (ie on hyperreals). Therefore, even in hyperreals, which have infinitesimals in it, 0.9r* = 1*.
This isn't a 'blind spot' or 'black hole' for maths, it's pretty much the simplest, basic bunch of results you learn when you start doing maths at uni because everything else develops on from here.
The HyperReal numbers are the Real numbers but with a vast extension by adding infinitesimals into the set upon which the binary operators are defined. In HyperReals there is an analogue notion of 0.9r* (* represents hyperreals) and wether it equals 1*.
There is a theorem in analysis (a theorem is a result which has been proven true) that any statement in standard analysis (ie on the Reals of Complex numbers) is the same in it's version of non-standard analysis (ie on hyperreals). Therefore, even in hyperreals, which have infinitesimals in it, 0.9r* = 1*.
This isn't a 'blind spot' or 'black hole' for maths, it's pretty much the simplest, basic bunch of results you learn when you start doing maths at uni because everything else develops on from here.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
sometimes separate points are actually not and visa versa and the funky world of superposition begins to emerge where "logic" begins to resemble wavelike fluids without reliable solutions, instead of delicate and precisely defined quantum objects in which no room for error is tolerated.
Irrelevent nonsense.
QUOTE (StevenA+Mar 7 2007, 12:05 AM)
If everything gets crammed into 0 then how are we suppose to sort out the infinitesimals?
I'll type this slowly so you get it :
There are no infinitesimals other than 0 in the Real numbers
It's not a case of we're ignoring them or calling them zero or we can't handle them, the set of entities which makes up the Reals doesn't contain them. It's like being given a normal bunch of dice and saying "But I can't throw a 7 with a die! Something is wrong!". No, that's what dice are, 6 sided shapes, just as the Reals are a set which doesn't contain infinitesimals other than 0.
The set which is like the Reals but does have infinitesimals is the hyperreals and 0.9r* = 1*, so it's true there too. No 'black hole', no 'quantum wave-like properties' (whatever the hell you were going for there), no problem.
You said in the "Philosophy of a Theory of Everything" thread when I corrected you to "give you a bit more credit". I would have given you enough credit before this thread to be the kind of person to actually READ about something before spouting crap about it because you don't know about it, as people like Zephir and SolidStateUniverse do. Unfortunately it would seem to have the same failing of plenty of cranks here. Don't know, don't care to know, don't want to know, don't listen to the people who do know.
Maths results aren't a matter of opinion or open to the possibility of being proven false, like physics. Once proven, they are proven for all time. As sure as 1+1=2, 0.9r=1.
There are no infinitesimals other than 0 in the Real numbers
It's not a case of we're ignoring them or calling them zero or we can't handle them, the set of entities which makes up the Reals doesn't contain them. It's like being given a normal bunch of dice and saying "But I can't throw a 7 with a die! Something is wrong!". No, that's what dice are, 6 sided shapes, just as the Reals are a set which doesn't contain infinitesimals other than 0.
The set which is like the Reals but does have infinitesimals is the hyperreals and 0.9r* = 1*, so it's true there too. No 'black hole', no 'quantum wave-like properties' (whatever the hell you were going for there), no problem.
You said in the "Philosophy of a Theory of Everything" thread when I corrected you to "give you a bit more credit". I would have given you enough credit before this thread to be the kind of person to actually READ about something before spouting crap about it because you don't know about it, as people like Zephir and SolidStateUniverse do. Unfortunately it would seem to have the same failing of plenty of cranks here. Don't know, don't care to know, don't want to know, don't listen to the people who do know.
Maths results aren't a matter of opinion or open to the possibility of being proven false, like physics. Once proven, they are proven for all time. As sure as 1+1=2, 0.9r=1.
QUOTE (Nick+Mar 7 2007, 12:05 AM)
The 9's go on forever. That is not ONE.
You suffer from the misconception that a number is defined by it's unique decimal expansion. That's not true. Some numbers can have 2 decimal expansions. Any number which has a terminating decimal expansion can also have another expansion in terms of 9s.
For example 17/100 = 0.17 = 0.170000000.... I can replace those 0s with 9s in the same way I did with 1 and 0.9r, so 0.17000.... = 0.1699999....
Completely equal, no error, no rounding, no 'perceptions', no doubt.
For example 17/100 = 0.17 = 0.170000000.... I can replace those 0s with 9s in the same way I did with 1 and 0.9r, so 0.17000.... = 0.1699999....
Completely equal, no error, no rounding, no 'perceptions', no doubt.
QUOTE (Eric England+Mar 7 2007, 12:20 AM)
You skipped the first part which directly related to a statement you made.
Because it's not really worth responding to. Recursive methods can have finite or infinitely many steps. Depends what you're doing.
Because it's not really worth responding to. Recursive methods can have finite or infinitely many steps. Depends what you're doing.
QUOTE (Eric England+Mar 7 2007, 12:20 AM)
If you can do that, I'll eat my word salad.
You mean if I can respond to a completely irrelevent nonsensical ramblings of someone who for all his claims about understanding logic doesn't get maths, the language of logic, to a level even a high school student should have?
Your question about croutons has nothing to do with 0.9r=1 and you and I have had such a discussion before. You seem to think your pointless and irrelevent nonsense somehow counts as a logical argument.
If you can prove mathematically that 0.9r =/= 1, I'll eat my hat. However, I'm willing to bet you'll either not try at all or you'll reply with something irrelevent and completely unmathematical.
Your question about croutons has nothing to do with 0.9r=1 and you and I have had such a discussion before. You seem to think your pointless and irrelevent nonsense somehow counts as a logical argument.
If you can prove mathematically that 0.9r =/= 1, I'll eat my hat. However, I'm willing to bet you'll either not try at all or you'll reply with something irrelevent and completely unmathematical.
QUOTE (Alphanumeric+)
There's no 'appearances', it's not a physical system! If you prove it algebraicly, you prove it!
Ok, how about repeating to the left instead of the right?
9!=10
99!=100
999!=1,000
9999!=10,000
...
etc.
Ok then let's do a better proof that 1!=.9r
If we assume an integer n which represents the number of digits for which the 9 repeats, and we assume this value to be infinite, then "infinity" for this expression must be of the form 10^n.
If n is an integer, then the value is real. If n is not an integer, then the value can be complex and not real, and hence violate the supposition that this problem lies within the realm of real numbers. (Now there remains a question of what fractional values of n could exist that would be able to maintain this relationship, but I'm not going to bother with that as that would need be addressed by the person providing the proof of the identity and not the inequality).
Now n is assumed to be infinite, so
n="infinity"
and "infinity" is equal to an integer.
Yet,
.9r = (10^n-1)/10^n (where n is an integer)
And we have a new construct, 10^"infinity", which is also assumed to approach "infinity".
but this places even tighter restrictions on "infinity" and requires it to be both an integer and a power of ten.
So we have to prove that "infinity"^10="infinity" is an identity.
But if we rewrite the .9r into base 3, we get the equation:
.2r = 1
.2r = (3^n-1)/3^n
And in this case "infinity" must be an integer power of 3.
But "infinity" cannot both simultaneously be an integer power of 3 and an integer power of 10 because they are both relatively prime.
Hence "infinity" cannot meet this restriction, unless it's 1, but even then it violates the equation "infinity"^10="infinity" etc. (unless we're working in binary numbers).
Long live boolean logic!
Of course you can try to use fractional powers but then you leave the imagined comfort in the world of the reals.
The problem stems from trying to treat infinity as if it was a single number that can morph at will to become whatever value you want it to be, but of course as soon as you begin creating different flavors of infinity to mix and match in order to proof something specifically is true, it no longer applies in general to other things.
.9r!=1
By the way, I appreciate you playing devil's advocate in this, Alpha. It's no fun if someone doesn't take the losing side in a discussion. So, seriously, I thank you for having the balls to do it.
Ok, how about repeating to the left instead of the right?
9!=10
99!=100
999!=1,000
9999!=10,000
...
etc.
Ok then let's do a better proof that 1!=.9r
If we assume an integer n which represents the number of digits for which the 9 repeats, and we assume this value to be infinite, then "infinity" for this expression must be of the form 10^n.
If n is an integer, then the value is real. If n is not an integer, then the value can be complex and not real, and hence violate the supposition that this problem lies within the realm of real numbers. (Now there remains a question of what fractional values of n could exist that would be able to maintain this relationship, but I'm not going to bother with that as that would need be addressed by the person providing the proof of the identity and not the inequality).
Now n is assumed to be infinite, so
n="infinity"
and "infinity" is equal to an integer.
Yet,
.9r = (10^n-1)/10^n (where n is an integer)
And we have a new construct, 10^"infinity", which is also assumed to approach "infinity".
but this places even tighter restrictions on "infinity" and requires it to be both an integer and a power of ten.
So we have to prove that "infinity"^10="infinity" is an identity.
But if we rewrite the .9r into base 3, we get the equation:
.2r = 1
.2r = (3^n-1)/3^n
And in this case "infinity" must be an integer power of 3.
But "infinity" cannot both simultaneously be an integer power of 3 and an integer power of 10 because they are both relatively prime.
Hence "infinity" cannot meet this restriction, unless it's 1, but even then it violates the equation "infinity"^10="infinity" etc. (unless we're working in binary numbers).
Long live boolean logic!
Of course you can try to use fractional powers but then you leave the imagined comfort in the world of the reals.
The problem stems from trying to treat infinity as if it was a single number that can morph at will to become whatever value you want it to be, but of course as soon as you begin creating different flavors of infinity to mix and match in order to proof something specifically is true, it no longer applies in general to other things.
.9r!=1
By the way, I appreciate you playing devil's advocate in this, Alpha. It's no fun if someone doesn't take the losing side in a discussion.
So, seriously, I thank you for having the balls to do it.
If 0.9r != 1 in base 10, then it is consistent to argue that 0.1r != 1 in base two.
In base 2:
0.1 = 1/2 = 1-1/2 = 1 - 1/2 = 1 - (1/2)^1
0.11 = 1/2 + 1/4 = 3/4 = 1 - 1/4 = 1 - (1/2)^2
0.111 = 1/2 + 1/4 + 1/8 = 7/8 = 1 - 1/8 = 1 - (1/2)^3
and
0.111 ... {n} ... 111 = 1 - (1/2)^n
So saying that 0.1r != 1 in base 2 is saying Zeno was right, that You can't get anywhere without having travelled halfway first and that logically there is no way to get from here to there.
http://www.mathacademy.com/pr/prime/articles/zeno_tort/
http://www.mathpages.com/rr/s3-07/3-07.htm
Since, experimentally, my message did get from me to you, Zeno must be wrong about the nature of physical reality and that 0.1r = 1 in base two, and therefore 0.9r = 1 in physical reality.
Sidebar:
With p-adic numbers, the infinite integer K = ...99999 = -1.
Proof: 10 K = K - 9 -> 9 K = -9 -> K = -1
In base 2:
0.1 = 1/2 = 1-1/2 = 1 - 1/2 = 1 - (1/2)^1
0.11 = 1/2 + 1/4 = 3/4 = 1 - 1/4 = 1 - (1/2)^2
0.111 = 1/2 + 1/4 + 1/8 = 7/8 = 1 - 1/8 = 1 - (1/2)^3
and
0.111 ... {n} ... 111 = 1 - (1/2)^n
So saying that 0.1r != 1 in base 2 is saying Zeno was right, that You can't get anywhere without having travelled halfway first and that logically there is no way to get from here to there.
http://www.mathacademy.com/pr/prime/articles/zeno_tort/
http://www.mathpages.com/rr/s3-07/3-07.htm
Since, experimentally, my message did get from me to you, Zeno must be wrong about the nature of physical reality and that 0.1r = 1 in base two, and therefore 0.9r = 1 in physical reality.
Sidebar:
With p-adic numbers, the infinite integer K = ...99999 = -1.
Proof: 10 K = K - 9 -> 9 K = -9 -> K = -1
A challenge to Alphanumeric: Go find the differences between "philosophical" and "mathematical" logic. They do exist.
This discussion is silly. .9R as a representation can be set equal to the immediate point to the left of 1.0R on a real number line. When this assumption - and statement - is made - they are not equal.
End of discussion.
This discussion is silly. .9R as a representation can be set equal to the immediate point to the left of 1.0R on a real number line. When this assumption - and statement - is made - they are not equal.
End of discussion.
So what's the distance between 0.9r and 1 ?
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
Ok, how about repeating to the left instead of the right?
9!=10
99!=100
999!=1,000
9999!=10,000
...
etc.
Ok then let's do a better proof that 1!=.9r
That wasn't a proof that 1 != 0.9r. The sequence 9, 99, 999, 9999, ... is NOT the same type as 0.9, 0.99, 0.999, ..... they are fundamentally different.
9, 99, 999, 9999, .... is unbounded, while 0.9, 0.99, 0.999, ..... is bounded. Hence, the theorem I mentioned about monotonic and bounded doesn't apply to 9, 99, 999, ....
9!=10
99!=100
999!=1,000
9999!=10,000
...
etc.
Ok then let's do a better proof that 1!=.9r
That wasn't a proof that 1 != 0.9r. The sequence 9, 99, 999, 9999, ... is NOT the same type as 0.9, 0.99, 0.999, ..... they are fundamentally different.
9, 99, 999, 9999, .... is unbounded, while 0.9, 0.99, 0.999, ..... is bounded. Hence, the theorem I mentioned about monotonic and bounded doesn't apply to 9, 99, 999, ....
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
If we assume an integer n which represents the number of digits for which the 9 repeats, and we assume this value to be infinite, then "infinity" for this expression must be of the form 10^n.
Firstly, that's an assumption you don't justify, you just claim. Secondly, you make the mistake of treating infinity like a number. You can't just evaluate things at infinite, you take limits.
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
If n is an integer, then the value is real. If n is not an integer, then the value can be complex and not real, and hence violate the supposition that this problem lies within the realm of real numbers.
n is the number of terms in a sequence, it is therefore ALWAYS integer. Unless you want to try and claim that you can have half a term in a sequence?
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
(Now there remains a question of what fractional values of n could exist that would be able to maintain this relationship, but I'm not going to bother with that as that would need be addressed by the person providing the proof of the identity and not the inequality).
It doesn't need to be addressed because it's irrelevent.
Also, you speak of the fact the onus is on me rather than you. Bollocks it is. I've given numerous proofs and you'll find even more on the Wikipedia page for this and so the onus is on you to disprove it. So far you've failed and you'll continue to fail because it can't be disproved without altering the axioms of maths and then you'd be changing the question.
Also, you speak of the fact the onus is on me rather than you. Bollocks it is. I've given numerous proofs and you'll find even more on the Wikipedia page for this and so the onus is on you to disprove it. So far you've failed and you'll continue to fail because it can't be disproved without altering the axioms of maths and then you'd be changing the question.
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
And we have a new construct, 10^"infinity", which is also assumed to approach "infinity".
but this places even tighter restrictions on "infinity" and requires it to be both an integer and a powe
but this places even tighter restrictions on "infinity" and requires it to be both an integer and a powe
No, you think it does because you've done a series of flawed steps.
Why are you trying to do mathematical analysis when you've obviously never studied it? You're trying to use logic to prove 0.9r =! 1, all the while trying to use the machinary of maths which PROVES 0.9r=1. You're being inconsistent.
Why are you trying to do mathematical analysis when you've obviously never studied it? You're trying to use logic to prove 0.9r =! 1, all the while trying to use the machinary of maths which PROVES 0.9r=1. You're being inconsistent.
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
The problem stems from trying to treat infinity as if it was a single number
The problem stems from the fact you've treated infinity as a number at all.
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
By the way, I appreciate you playing devil's advocate in this, Alpha. It's no fun if someone doesn't take the losing side in a discussion.
I find it funny you think you've proved it false.
I'm not playing devils advocate, I'm playing the guy with a maths degree, who was lectured this material by the aformentioned Fields Medalist and who agrees with EVERY maths textbook, every maths graduate and everyone whose actually learnt maths rather than trying to pull it from their backside on the fly.
I'm not playing devils advocate, I'm playing the guy with a maths degree, who was lectured this material by the aformentioned Fields Medalist and who agrees with EVERY maths textbook, every maths graduate and everyone whose actually learnt maths rather than trying to pull it from their backside on the fly.
QUOTE (StevenA+Mar 7 2007, 01:16 AM)
So, seriously, I thank you for having the balls to do it.
I used to think you were someone who did a bit of reading and just liked 'out there' theories. Now I realise you're just as stupid a crank as the rest of them but you manage a slightly more polished presentation of your crap than most others.
Make no mistake Steven, you haven't 'shaken my belief' in the fact 0.9r=1. All you've done is display you lack of understanding of maths, you're stupidity at thinking you can bluf your way past me, your apathy for not bothering to look up other people's work on it, and your arrogance to think you know better than the last 4 centurys of the maths community.
Make no mistake Steven, you haven't 'shaken my belief' in the fact 0.9r=1. All you've done is display you lack of understanding of maths, you're stupidity at thinking you can bluf your way past me, your apathy for not bothering to look up other people's work on it, and your arrogance to think you know better than the last 4 centurys of the maths community.
QUOTE (Aerohead+Mar 7 2007, 01:16 AM)
Go find the differences between "philosophical" and "mathematical" logic. They do exist.
And? Last time I checked this was a maths discussion about a maths equation and wether, within maths, 0.9r and 1 are mathematically equal.
I've proven it several times now.
I've proven it several times now.
QUOTE (Aerohead+Mar 7 2007, 01:16 AM)
This discussion is silly.
You're right, it is a silly discussion. A bunch of people who have never actually done analysis attempting, and failing, to grasp and then refute a proven result even when presented with various proofs.
QUOTE (Aerohead+Mar 7 2007, 01:16 AM)
9R as a representation can be set equal to the immediate point to the left of 1.0R on a real number line. When this assumption - and statement - is made - they are not equal.
There is no 'immediate point to the left of 1". The Reals are 'dense', in that if you give me A and B in the Reals and they aren't equal, their average 0.5(A+B ) is between them. ALWAYS. Hence, if 0.9r is a different point from 1, then (0.9r + 1)/2 is between them. But then 0.9r can't be the point immediately to the left of 1. There is either a non-zero difference between them which is a value within the Reals or they are equal. Since you can easily disprove the former (and I have, twice!), it must be the latter. They are equal.
But I'm sure the fact you're completely ignorant of things like 'denseness', 'completeness', 'compactnessn' and the general properties of fields within mathematics will not make you stop and think that perhaps, just perhaps you might be wrong and the guy who studied this stuff and has given numerous proofs might, just might, be right. No, because you're supremely confident in your own ignorance for some reason.
It baffles me how cranks can call me arrogant and close minded and then display the exact same characteristics only 100 times worse.
But I'm sure the fact you're completely ignorant of things like 'denseness', 'completeness', 'compactnessn' and the general properties of fields within mathematics will not make you stop and think that perhaps, just perhaps you might be wrong and the guy who studied this stuff and has given numerous proofs might, just might, be right. No, because you're supremely confident in your own ignorance for some reason.
It baffles me how cranks can call me arrogant and close minded and then display the exact same characteristics only 100 times worse.
Zero points in between. An infinitely small distance. Ad nauseum, ad infinitum, ad absurdum.
QUOTE (rpenner+Mar 7 2007, 12:21 AM)
If 0.9r != 1 in base 10, then it is consistent to argue that 0.1r != 1 in base two.
In base 2:
0.1 = 1/2 = 1-1/2 = 1 - 1/2 = 1 - (1/2)^1
0.11 = 1/2 + 1/4 = 3/4 = 1 - 1/4 = 1 - (1/2)^2
0.111 = 1/2 + 1/4 + 1/8 = 7/8 = 1 - 1/8 = 1 - (1/2)^3
and
0.111 ... {n} ... 111 = 1 - (1/2)^n
So saying that 0.1r != 1 in base 2 is saying Zeno was right, that You can't get anywhere without having travelled halfway first and that logically there is no way to get from here to there.
http://www.mathacademy.com/pr/prime/articles/zeno_tort/
http://www.mathpages.com/rr/s3-07/3-07.htm
Again, the differences between how infinitesimals are used enters the picture and creates what appears to be a paradox only if the processes which generate them are ignored. Not all infinite(simals) are created equally.
So while the number of subdivisions becomes infinite, the time taken to cross them becomes proportionally infinitesimal and it's only be keeping track of their relative values that the paradox can be avoided.
If we create an artificial clock that ticks progressively faster and faster, it doesn't impede the runner and the two equations become separate, but this would not be true if at some point we considered 1/2^-n to be precisely 0 as we lose the ability to distinguish between these infinitesimals because in that case we're creating an artificial perspective and then rounding the distortion created by it's new approximation and assuming that subjective distortion represents the original equations!
1/2^-n!=0
In base 2:
0.1 = 1/2 = 1-1/2 = 1 - 1/2 = 1 - (1/2)^1
0.11 = 1/2 + 1/4 = 3/4 = 1 - 1/4 = 1 - (1/2)^2
0.111 = 1/2 + 1/4 + 1/8 = 7/8 = 1 - 1/8 = 1 - (1/2)^3
and
0.111 ... {n} ... 111 = 1 - (1/2)^n
So saying that 0.1r != 1 in base 2 is saying Zeno was right, that You can't get anywhere without having travelled halfway first and that logically there is no way to get from here to there.
http://www.mathacademy.com/pr/prime/articles/zeno_tort/
http://www.mathpages.com/rr/s3-07/3-07.htm
Again, the differences between how infinitesimals are used enters the picture and creates what appears to be a paradox only if the processes which generate them are ignored. Not all infinite(simals) are created equally.
So while the number of subdivisions becomes infinite, the time taken to cross them becomes proportionally infinitesimal and it's only be keeping track of their relative values that the paradox can be avoided.
If we create an artificial clock that ticks progressively faster and faster, it doesn't impede the runner and the two equations become separate, but this would not be true if at some point we considered 1/2^-n to be precisely 0 as we lose the ability to distinguish between these infinitesimals because in that case we're creating an artificial perspective and then rounding the distortion created by it's new approximation and assuming that subjective distortion represents the original equations!
1/2^-n!=0
QUOTE (rpenner+)
Since, experimentally, my message did get from me to you, Zeno must be wrong about the nature of physical reality and that 0.1r = 1 in base two, and therefore 0.9r = 1 in physical reality.
Oops, I believe you made a mistake here. Since experimentally we are able to communicate, 1!=0.9r or Zeno would be right that the velocity can precisely stop and become 0. In reality the infinitesimals always remain significant and trying to claim they ever become precisely equal to a finite value causes the paradoxes.
Oops, I believe you made a mistake here. Since experimentally we are able to communicate, 1!=0.9r or Zeno would be right that the velocity can precisely stop and become 0. In reality the infinitesimals always remain significant and trying to claim they ever become precisely equal to a finite value causes the paradoxes.
QUOTE (rpenner+)
Sidebar:
With p-adic numbers, the infinite integer K = ...99999 = -1.
Proof: 10 K = K - 9 -> 9 K = -9 -> K = -1
Thank you. Yes, in binary two compliment format 111111.... represents -1 and not 0, though you can create an artificial superposition of two results by rounding the infinite string to zero but that would make the computations inaccurate and allow for multiple states to be merged into an inreversible result.
So (10^n-1) mod (10^n)=-1 (for finite n)
I still see no identity for 0 or 1 in any of the infinite representations.
For example, if I created a number system that mapped things into this binary state where 111111... was transformed into 00000...
The there's no way to represent 111111... and if I tried to count down, it would keep folding the values "back up" (depending upon the decoding) to 0 and erase all information. (Kind of like the Pentium floating point bug that was overlooked for quite a while). You need precise definitions if you want precise answers.
With p-adic numbers, the infinite integer K = ...99999 = -1.
Proof: 10 K = K - 9 -> 9 K = -9 -> K = -1
Thank you. Yes, in binary two compliment format 111111.... represents -1 and not 0, though you can create an artificial superposition of two results by rounding the infinite string to zero but that would make the computations inaccurate and allow for multiple states to be merged into an inreversible result.
So (10^n-1) mod (10^n)=-1 (for finite n)
I still see no identity for 0 or 1 in any of the infinite representations.
For example, if I created a number system that mapped things into this binary state where 111111... was transformed into 00000...
The there's no way to represent 111111... and if I tried to count down, it would keep folding the values "back up" (depending upon the decoding) to 0 and erase all information. (Kind of like the Pentium floating point bug that was overlooked for quite a while). You need precise definitions if you want precise answers.
QUOTE (AlphaNumeric+Mar 6 2007, 04:50 PM)
Your question about croutons has nothing to do with 0.9r=1 and you and I have had such a discussion before.
So, you can't describe the surface(s) or the extents other than to call them names and represent them with a number. Oh well.
So what if we've had this discussion before.
Finite recursiveness is a load of oxymoronic horseshit. I'd don't care who says it's not.
Gowers (as one example), says he doesn't have a definition for infinity, yet he says 1 divided by 0 = Infinity.
If you don't have a definition for one side of the equals sign, don't try to tell anybody you know what equals it.
Unless of course, they're as gullible as the teacher is.
How dare anyone with a brain, think they can treat infinity so casually and stick it inside finite, in any way, shape, or form?
And no, I'm not some dummy that simply looks at .9=1 and says it doesn't make sense. There's a lot more to it than that, and if you stick around your chums too long, you'll just become one of them. Oh, you already are. What was I thinking?
Theory hasn't made a major advance in 50 years. No, M-theory is just a rehash of infinitesimal.
Whoopie! Tight enough and small enough, to give me an "oxymoronic wedgie".
So, you can't describe the surface(s) or the extents other than to call them names and represent them with a number. Oh well.
So what if we've had this discussion before.
Finite recursiveness is a load of oxymoronic horseshit. I'd don't care who says it's not.
Gowers (as one example), says he doesn't have a definition for infinity, yet he says 1 divided by 0 = Infinity.
If you don't have a definition for one side of the equals sign, don't try to tell anybody you know what equals it.
Unless of course, they're as gullible as the teacher is.
How dare anyone with a brain, think they can treat infinity so casually and stick it inside finite, in any way, shape, or form?
And no, I'm not some dummy that simply looks at .9=1 and says it doesn't make sense. There's a lot more to it than that, and if you stick around your chums too long, you'll just become one of them. Oh, you already are. What was I thinking?
Theory hasn't made a major advance in 50 years. No, M-theory is just a rehash of infinitesimal.
Whoopie! Tight enough and small enough, to give me an "oxymoronic wedgie".
Dr Obvious
1 / 9 = 0.1... r 0.0...1
2 / 9 = 0.2... r 0.0...2
3 / 9 = 0.3... r 0.0...3
4 / 9 = 0.4... r 0.0...4
5 / 9 = 0.5... r 0.0...5
6 / 9 = 0.6... r 0.0...6
7 / 9 = 0.7... r 0.0...7
8 / 9 = 0.8... r 0.0...8
9 / 9 = 1
Let's work out 7 / 9 for example:
1. 9 ) 7 (can't do)
2. 9 ) 7.0 = .7 r .7 (as .7 x 9 = 6.3)
3. 9 ) 7.00 = .77 r .07 (as .77 x 9 = 6.93)
4. 9 ) 7.000 = .777 r .007 (as .777 x 9 = 6.993)
5. 9 ) 7.0000 = .7777 r .0007 (as .7777 x 9 = 6.9993)
6. 9 ) 7.00000 = .77777 r .00007 (as .77777 x 9 = 6.99993)
All the above answers (1)-(6) are correct.
ie 9 / 7 = 0.7 r 0.7 = 0.77 r 0.07 = 0.777 r 0.007 etc.
I can keep going but I will never be able to get rid of the remainder at any time.
All the proofs that exist to show 1 = 0.9... - though they look clever - are basically all derivations of exactly the same thing. It is a blatent attempt to bamboozle initiates and the less prepared into accepting the false belief.
We are told to simply forget about the remainder.
We are told that 0.0...1 is nonsense and to simply just accept that. Once (and only when) you accept that then it is possible to push this false belief without challenge.
It does worry me.
Why is it so vital that 0.9... = 1? Why is it so vigourously defended?
What is at stake?
1 / 9 = 0.1... r 0.0...1
2 / 9 = 0.2... r 0.0...2
3 / 9 = 0.3... r 0.0...3
4 / 9 = 0.4... r 0.0...4
5 / 9 = 0.5... r 0.0...5
6 / 9 = 0.6... r 0.0...6
7 / 9 = 0.7... r 0.0...7
8 / 9 = 0.8... r 0.0...8
9 / 9 = 1
Let's work out 7 / 9 for example:
1. 9 ) 7 (can't do)
2. 9 ) 7.0 = .7 r .7 (as .7 x 9 = 6.3)
3. 9 ) 7.00 = .77 r .07 (as .77 x 9 = 6.93)
4. 9 ) 7.000 = .777 r .007 (as .777 x 9 = 6.993)
5. 9 ) 7.0000 = .7777 r .0007 (as .7777 x 9 = 6.9993)
6. 9 ) 7.00000 = .77777 r .00007 (as .77777 x 9 = 6.99993)
All the above answers (1)-(6) are correct.
ie 9 / 7 = 0.7 r 0.7 = 0.77 r 0.07 = 0.777 r 0.007 etc.
I can keep going but I will never be able to get rid of the remainder at any time.
All the proofs that exist to show 1 = 0.9... - though they look clever - are basically all derivations of exactly the same thing. It is a blatent attempt to bamboozle initiates and the less prepared into accepting the false belief.
We are told to simply forget about the remainder.
We are told that 0.0...1 is nonsense and to simply just accept that. Once (and only when) you accept that then it is possible to push this false belief without challenge.
It does worry me.
Why is it so vital that 0.9... = 1? Why is it so vigourously defended?
What is at stake?
This is far easier to understand when you realize that written numbers are just symbols for values that exist outside the writing system. In the "decimal" symbolic system, the value 1/3 cannot be expressed as a finite symbol (it's an infinitely long writing). Similarly, the 0.9r is an alternative way of saying "1" in the same decimal writing system. It's just a quirk of the symbolic writing system, not mathematics. The values are still the values ... sort of like objects remain objects, even if you don't have words to describe them.
QUOTE (Eric England+Mar 7 2007, 02:13 AM)
So what if we've had this discussion before.
Because you seem to have the same nonsensical rants each time.
Because you seem to have the same nonsensical rants each time.
QUOTE (Eric England+Mar 7 2007, 02:13 AM)
So, you can't describe the surface(s) or the extents other than to call them names and represent them with a number. Oh well.
That's how people describe things, with words which produce concepts in peoples minds.
Oh course if you want to debate philosophy rather than maths, I'm sure you can start another thread.
Oh course if you want to debate philosophy rather than maths, I'm sure you can start another thread.
QUOTE (Eric England+Mar 7 2007, 02:13 AM)
Finite recursiveness is a load of oxymoronic horseshit. I'd don't care who says it's not.
You don't care about much beyond your own opinions it would seem.
start another thread.
start another thread.
QUOTE (Eric England+Mar 7 2007, 02:13 AM)
Theory hasn't made a major advance in 50 years. No, M-theory is just a rehash of infinitesimal.
Whoopie! Tight enough and small enough, to give me an "oxymoronic wedgie".
Whoopie! Tight enough and small enough, to give me an "oxymoronic wedgie".
And there's the nonsensical rant... You obviously know nothing of M theory or infinitesimals in maths.
QUOTE (gonegahgah+Mar 7 2007, 03:22 AM)
1 / 9 = 0.1... r 0.0...1
Invalid notation. You can't have r 0.00... 01, it's meaningless. Want to know specifically why? You'll have to put in the effort to learn maths.
QUOTE (gonegahgah+Mar 7 2007, 03:22 AM)
All the proofs that exist to show 1 = 0.9... - though they look clever - are basically all derivations of exactly the same thing. It is a blatent attempt to bamboozle initiates and the less prepared into accepting the false belief.
No, some of them approach the result in various ways.
Also, it's not an attempt to bamboozle people with strange concepts, it's using standard maths results and notation to be precise about logical steps. Those terms and notation I've used are taught to 1st year university maths students. Hundreds of thousands of people as year learn it.
This is the classic problem. You don't put the effort into learning the language of maths and then you complain you don't understand the answer!? It's like me going to Japan and complaining everyone is speaking Japanese! You wanted a maths answer, you got one. The fact you don't understand it doesn't make it wrong, it means you'll have to put in some effort if you want to get to grips with it fully.
It's like just SolidStateUniverse and various quantum field theory results. He doesn't accept the layman explaination but is completely unwilling to learn enough to understand the proper explainations.
Also, it's not an attempt to bamboozle people with strange concepts, it's using standard maths results and notation to be precise about logical steps. Those terms and notation I've used are taught to 1st year university maths students. Hundreds of thousands of people as year learn it.
This is the classic problem. You don't put the effort into learning the language of maths and then you complain you don't understand the answer!? It's like me going to Japan and complaining everyone is speaking Japanese! You wanted a maths answer, you got one. The fact you don't understand it doesn't make it wrong, it means you'll have to put in some effort if you want to get to grips with it fully.
It's like just SolidStateUniverse and various quantum field theory results. He doesn't accept the layman explaination but is completely unwilling to learn enough to understand the proper explainations.
QUOTE (gonegahgah+Mar 7 2007, 03:22 AM)
It does worry me.
Why is it so vital that 0.9... = 1? Why is it so vigourously defended?
What is at stake?
Why is it so vital that 0.9... = 1? Why is it so vigourously defended?
What is at stake?
The basic understanding and premise of maths and logic.
Most people seem perfectly happy to accept complex algebra because it's so foreign to them, they don't get any of it. But when it comes to something they feel they should be able to grasp without effort like decimal expansions, they get cheesed if they don't understand immediately and then threads like this occur.
Maths is about taking statements and seeing where they lead via irrefutable logic. If 1+1=2 and 2+2=4, then (1+1)+(2+2) = 2+4. Given the first two statements, you get the third. If 3<4 and 4<6 then 3<6. See?
To start saying "No, 0.9r isn't 1!!!" first is to be completely ignorant of SIMPLE mathematics (not bamboozling if you take the time to TRY and learn it!) and secondly to basically say to everty mathematician "You don't know anything about the work you do and I know it all better than you without even trying!". I'm not denying sometimes someone new to maths comes up with something never before seen, but this isn't a case of proving a new result, it's a case of showing the PROOF of it and then someone whose never done maths properly saying "I don't care about your proof, your logic or the entire ethos of maths, I refuse to admit I'm wrong, despite me knowing nothing of this".
To me, being a mathematician, the evident truth of 0.9r=1 is clear. I learnt to understand the methods and workings which surround and develop this area of maths so many years ago that it's second nature to me. For someone to say "But it's not!!" initially tells me "Oh, you've not seen a proof them" but after that, when you continue to deny it, it not only tells me you didn't know the proof, it says to me you don't understand pretty simple logic. For all the ravings of the cranks on these forums telling me I don't think logically or have an open mind, they are UTTERLY unwilling to open their mind to counter intuitive results or strange results.
Look at Eric. In this thread he tells me I'm gullible and a sheep because I can open my mind to non-intuitive thinking Yet when someone else does it it's a good thing, or when I don't talk about counter intuitive results, but more maindaine ones, I'm just parroting books.
Basically threads like this just serve to highlight the mind warping hypocrisy of cranks and their stupidity and ignorance. 0.9r=1 exeplifies the "I know better than everyone, despite knowing nothing of this subject" attitude of cranks on this forum and many others. Somehow you all think pseudologic and ignorance make up for proper logic, reasoning and knowledge. It might in Crazyland but not in maths.
Most people seem perfectly happy to accept complex algebra because it's so foreign to them, they don't get any of it. But when it comes to something they feel they should be able to grasp without effort like decimal expansions, they get cheesed if they don't understand immediately and then threads like this occur.
Maths is about taking statements and seeing where they lead via irrefutable logic. If 1+1=2 and 2+2=4, then (1+1)+(2+2) = 2+4. Given the first two statements, you get the third. If 3<4 and 4<6 then 3<6. See?
To start saying "No, 0.9r isn't 1!!!" first is to be completely ignorant of SIMPLE mathematics (not bamboozling if you take the time to TRY and learn it!) and secondly to basically say to everty mathematician "You don't know anything about the work you do and I know it all better than you without even trying!". I'm not denying sometimes someone new to maths comes up with something never before seen, but this isn't a case of proving a new result, it's a case of showing the PROOF of it and then someone whose never done maths properly saying "I don't care about your proof, your logic or the entire ethos of maths, I refuse to admit I'm wrong, despite me knowing nothing of this".
To me, being a mathematician, the evident truth of 0.9r=1 is clear. I learnt to understand the methods and workings which surround and develop this area of maths so many years ago that it's second nature to me. For someone to say "But it's not!!" initially tells me "Oh, you've not seen a proof them" but after that, when you continue to deny it, it not only tells me you didn't know the proof, it says to me you don't understand pretty simple logic. For all the ravings of the cranks on these forums telling me I don't think logically or have an open mind, they are UTTERLY unwilling to open their mind to counter intuitive results or strange results.
Look at Eric. In this thread he tells me I'm gullible and a sheep because I can open my mind to non-intuitive thinking
Yet when someone else does it it's a good thing, or when I don't talk about counter intuitive results, but more maindaine ones, I'm just parroting books. Basically threads like this just serve to highlight the mind warping hypocrisy of cranks and their stupidity and ignorance. 0.9r=1 exeplifies the "I know better than everyone, despite knowing nothing of this subject" attitude of cranks on this forum and many others. Somehow you all think pseudologic and ignorance make up for proper logic, reasoning and knowledge. It might in Crazyland but not in maths.
Hi Alphanumeric
My mathematics is just fine. Only senior high school but Very High Achievements in Math I and Math II when it was called that here.
I understand each of the proofs including the ones with integrals.
I'm talking with you rationally. You are the one who feels the need to belittle and downplay without knowing any facts about what the other person understands.
Otherwise you would appreciate that I have shown my counter arguments with new ways of talking about it and not just the same old counter arguments that people initially think of.
My explanations are always to help people understand and not to confound them.
I don't walk around with a chip on my shoulder. I can't help but feel that you do have a chip on your shoulder.
My mathematics is just fine. Only senior high school but Very High Achievements in Math I and Math II when it was called that here.
I understand each of the proofs including the ones with integrals.
I'm talking with you rationally. You are the one who feels the need to belittle and downplay without knowing any facts about what the other person understands.
Otherwise you would appreciate that I have shown my counter arguments with new ways of talking about it and not just the same old counter arguments that people initially think of.
My explanations are always to help people understand and not to confound them.
I don't walk around with a chip on my shoulder. I can't help but feel that you do have a chip on your shoulder.
wow this thread is taking right off.
Yeah I caught myself right after I pressed the post button.
My edit date is one minute before your post date.
QUOTE
You misunderstand the notation. (0.1)^n = 0.1*0.1*...*0.1 n times, so 0.1^n = 0.000....01 with n-1 zeros.
Yeah I caught myself right after I pressed the post button.
My edit date is one minute before your post date.
QUOTE ( gonegahgah+)
It does worry me.
Why is it so vital that 0.9... = 1? Why is it so vigourously defended?
What is at stake?
Because they would go gahgah, if they let go of it.
They would be faced with an infinity they wouldn't know what to do with and their finite measurements would always have to come with an * (*approximation).
Mathematics and physics would rather think they know, than find out what they don't know.
Bunch of unimaginative chickenshits.
AlphaNumeric doesn't even know how "describes" differs from "represents".
Yo, describe means "trace".
You can't trace the extent or surface of a crouton that separates it from its enviornment, to any degree greater than .9r, any more than you can trace the extent or surface of any of its components.
M-theroy my butt. Like I don't know a BS infinitesimal when I see one.
Of course, getting a PhD is about understanding what is already accepted. So definately don't think of anything new, until you've got the paper in hand.
Why is it so vital that 0.9... = 1? Why is it so vigourously defended?
What is at stake?
Because they would go gahgah, if they let go of it.
They would be faced with an infinity they wouldn't know what to do with and their finite measurements would always have to come with an * (*approximation).
Mathematics and physics would rather think they know, than find out what they don't know.
Bunch of unimaginative chickenshits.
AlphaNumeric doesn't even know how "describes" differs from "represents".
Yo, describe means "trace".
You can't trace the extent or surface of a crouton that separates it from its enviornment, to any degree greater than .9r, any more than you can trace the extent or surface of any of its components.
M-theroy my butt. Like I don't know a BS infinitesimal when I see one.
Of course, getting a PhD is about understanding what is already accepted. So definately don't think of anything new, until you've got the paper in hand.
QUOTE (NoCleverName+Mar 7 2007, 02:38 AM)
This is far easier to understand when you realize that written numbers are just symbols for values that exist outside the writing system. In the "decimal" symbolic system, the value 1/3 cannot be expressed as a finite symbol (it's an infinitely long writing). Similarly, the 0.9r is an alternative way of saying "1" in the same decimal writing system. It's just a quirk of the symbolic writing system, not mathematics. The values are still the values ... sort of like objects remain objects, even if you don't have words to describe them.
I don't think that's entirely correct. The original form and intent needs to be maintained when problems can result or the answers become indeterminant.
Basically, if there's no memory of where something came from, you can't tell where it's intended to go.
For example, if someone writes pi as 3.1415... and goes ahead and uses this to calculate something, the true value can never be recovered from this.
The same goes for the value .99999.....
Let's say that an equation was truly sensitive to whether or not a value was precisely 1 or whether or not it was infinitesimally different. In this case you'd need to allow for the existance of .99999..... and not fold it up into 1 or you'd get an error in the answer as you could never detect for a value slightly less than 1.
This is the same as some equations not being valid when a variable is precisely some value. We can often get around this with calculus but that requires a memory of what the original problem was. If this information isn't retained accurately, then the valid solution can't be recovered.
How can we represent an infinitesimal value in calculus for example if we just round them out?
Let's say we want to differentiate x^2, we substitute:
x'=((x+h)^2-x^2)/h= x+2xh+h^2-x^2 = (2xh + h^2)/h = 2x+h
Now normally we could throw away h^2 and say the result is 2x, but if we have a function very sensitive to h, then we can't throw it away. In the case of the equality function it's very sensitive to h.
In this case the equality function is infinitely sensitive to values approaching 0 and the different between an infinitesimal and a perfect 0 aren't enough for an equality equation to be true because the slope of equality is always higher than the slope of that being measured - there is no point where you can "slide in" soft enough to go overlooked by equality. It's all or nothing.
So when .9r is expanded, it always remains unequal to 1 no matter how far you go ... and even if you take it out further it still remains less than 1 etc. because equality is a test that requires the values to be identical and they never are. The limit of this indentity test is always false.
One group of people are looking at it from the perspective of an identity while the others are trying to blur things out a bit first and then say if it looks close enough, it can always be treated identically but it's obviously not the case from many examples I've shown above.
The truth is though that this same issue creates many problems in mathematics elsewhere and results in many cases where formulas only apply within constraints because the underlying methodology from which they were derived, and the limits those impose go overlooked.
I don't think that's entirely correct. The original form and intent needs to be maintained when problems can result or the answers become indeterminant.
Basically, if there's no memory of where something came from, you can't tell where it's intended to go.
For example, if someone writes pi as 3.1415... and goes ahead and uses this to calculate something, the true value can never be recovered from this.
The same goes for the value .99999.....
Let's say that an equation was truly sensitive to whether or not a value was precisely 1 or whether or not it was infinitesimally different. In this case you'd need to allow for the existance of .99999..... and not fold it up into 1 or you'd get an error in the answer as you could never detect for a value slightly less than 1.
This is the same as some equations not being valid when a variable is precisely some value. We can often get around this with calculus but that requires a memory of what the original problem was. If this information isn't retained accurately, then the valid solution can't be recovered.
How can we represent an infinitesimal value in calculus for example if we just round them out?
Let's say we want to differentiate x^2, we substitute:
x'=((x+h)^2-x^2)/h= x+2xh+h^2-x^2 = (2xh + h^2)/h = 2x+h
Now normally we could throw away h^2 and say the result is 2x, but if we have a function very sensitive to h, then we can't throw it away. In the case of the equality function it's very sensitive to h.
In this case the equality function is infinitely sensitive to values approaching 0 and the different between an infinitesimal and a perfect 0 aren't enough for an equality equation to be true because the slope of equality is always higher than the slope of that being measured - there is no point where you can "slide in" soft enough to go overlooked by equality. It's all or nothing.
So when .9r is expanded, it always remains unequal to 1 no matter how far you go ... and even if you take it out further it still remains less than 1 etc. because equality is a test that requires the values to be identical and they never are. The limit of this indentity test is always false.
One group of people are looking at it from the perspective of an identity while the others are trying to blur things out a bit first and then say if it looks close enough, it can always be treated identically but it's obviously not the case from many examples I've shown above.
The truth is though that this same issue creates many problems in mathematics elsewhere and results in many cases where formulas only apply within constraints because the underlying methodology from which they were derived, and the limits those impose go overlooked.
Some more examples of how .9r can't be treated identically to 1.
As we expand .9r we get
.9
.99
.999
etc.
The function grows as (10^n-1)/10^n.
If this relationship ever became 1 then:
10^n-1=10^n
Which is obviously always false.
Or again, using calculus, if we take the function:
f(x)={ 0 : x!=0 ; 1 : x=0)
And if we try to differentiate it at 0, the value is discontinuous, so there is no rational limit to the function and again infinitesimals make all the difference. We could eyeball values and say it approaches +infinity from the left and - infinity from the right, but without a precise specification for this it can't be reliably duplicated and the requirements for one mode of analysis might not remain the same for another so even with specific rules for this you might arbitrarily rule out other valid needs to analyze it differently (for example as an ideal cutoff filter with complex rotations along the transient). The general rule is that you just don't differentiate functions at discontinuities without risking strange artifacts - that's one of the tradeoffs you get when you decide to create rules, you have to live by them for them to remain in place.
As we expand .9r we get
.9
.99
.999
etc.
The function grows as (10^n-1)/10^n.
If this relationship ever became 1 then:
10^n-1=10^n
Which is obviously always false.
Or again, using calculus, if we take the function:
f(x)={ 0 : x!=0 ; 1 : x=0)
And if we try to differentiate it at 0, the value is discontinuous, so there is no rational limit to the function and again infinitesimals make all the difference. We could eyeball values and say it approaches +infinity from the left and - infinity from the right, but without a precise specification for this it can't be reliably duplicated and the requirements for one mode of analysis might not remain the same for another so even with specific rules for this you might arbitrarily rule out other valid needs to analyze it differently (for example as an ideal cutoff filter with complex rotations along the transient). The general rule is that you just don't differentiate functions at discontinuities without risking strange artifacts - that's one of the tradeoffs you get when you decide to create rules, you have to live by them for them to remain in place.
I am no student of math; but nevertheless, one wonders where reality begins and ends with this problem. Is the difference between .9r and 1 real, or is it just a side effect of symbolic manipulation? The 1/3 + 1/3 + 1/3 = 1 yet arguing .3r + .3r +.3r != 1 would seem to speak to a quirk of manipulation, not reality. And justify how you can get from .3r to 1/3? I'll answer that: by stating that 1/3 is the algorithm that will result in .3r when using the decimal representation system.
After all, in a computer decimal arithmetic is not really calculation, it's symbol manipulation. The only "calculation" occurs later in analysis of error.
Pi has no finite writing in any symbolic system save the Greek letters, yet it is nevertheless real ... but it can be better represented by the algorithm that "determines" or "computes" it. We are forced to employ this algorithm to transform pi into something suitable for calculation with other "numbers". So is pi a number or a process?
After all, in a computer decimal arithmetic is not really calculation, it's symbol manipulation. The only "calculation" occurs later in analysis of error.
Pi has no finite writing in any symbolic system save the Greek letters, yet it is nevertheless real ... but it can be better represented by the algorithm that "determines" or "computes" it. We are forced to employ this algorithm to transform pi into something suitable for calculation with other "numbers". So is pi a number or a process?
QUOTE (NoCleverName+Mar 7 2007, 04:06 AM)
I am no student of math; but nevertheless, one wonders where reality begins and ends with this problem. Is the difference between .9r and 1 real, or is it just a side effect of symbolic manipulation? The 1/3 + 1/3 + 1/3 = 1 yet arguing .3r + .3r +.3r != 1 would seem to speak to a quirk of manipulation, not reality. And justify how you can get from .3r to 1/3? I'll answer that: by stating that 1/3 is the algorithm that will result in .3r when using the decimal representation system.
After all, in a computer decimal arithmetic is not really calculation, it's symbol manipulation. The only "calculation" occurs later in analysis of error.
Pi has no finite writing in any symbolic system save the Greek letters, yet it is nevertheless real ... but it can be better represented by the algorithm that "determines" or "computes" it. We are forced to employ this algorithm to transform pi into something suitable for calculation with other "numbers". So is pi a number or a process?
Good questions. Here's how I see it.
For most things, if it's close enough, that's fine. Horseshoes and handgrenades are like this
Yes, 1/3 and pi represent processes as well as .3r etc.
These are finite representations of infinite processes. They exist as finite values only symbolically and not when they're expanded into an infinite form.
The number 1 doesn't represent a process though because it's a fundamental unit from which the others are built. We have to select something to remain absolute so we can build relationships upon it and 1 is the symbol for unity in multiplication versus the 0 for unity in addition. Division and subtraction are the inverses of these and only apply when such inverses are finitely computable (for example, you can't divide by 0, simply by definition, though some could estimate what a value divided by 0 would be, for example x/x=1, except when x=0, in which case you can generally still assume the answer is 0, but not always)
For example, we might say that 1/0=infinity, but think about this for a second ... is that actually true - even if we had someone subtracting 0 from 1 forever, would they ever be able to reduce 1 at all? Realistically, no, unless that was the amount of time it took that person to think of a better way to represent the ratio 1/0. So we can't really know if the vague notion of "infinity" is even correct in this case.
You can't redefine 0 or 1 as processes. They can only be inputs and outputs and not functions.
For example, y=x/3 is a function. 3/3 is 1 and an exact result that can be computed by definite in finite time, but 2/3 is an intermediate value that represents an irreducible computation. It's a symbolic representation of an infinite computation.
Once again, you could subtract 2 from 3, get 1, then rescale things so you have 10, then subtract 9 and get 1 again and then rescale by 10 etc. etc. etc. and there would always be an error in the answer, not much different than trying to subtract 0 from 1 forever, except in this case the relative weightings of the digits decreases, so it does approach a fixed value but that fixed value still has an infinite representation and can't be symbolically manipulated directly in that infinite format, because any finite quantization of it leads to errors and approximations. So you can write 2/3 and be precise and not find errors, but you can't try to treat it the same as .666666..... without introducing errors (notice everytime people write it, they always ignore the last digits and you can see when they manipulate this value they either have to show a limited example where the importance of the least significant digits becomes undeniably trivial or they mask the importance of them by no recognizing every time they shift the value left the errors grow in significance).
For example, 10*.9r is not the same as 9+.9r, because any errors in the 10*.9r are now 10 times as significant as the errors in .9r so the error terms no longer match. Now if we just say the errors are 0 so that 10*0=0 then the problem is simply ignored.
To work with this correctly an error term should be used and the boundary for the error analyzed as the expansion continues so that you can be certain the error does not grow to be more than infinitesimal, but this requires that you specify how digits the expansion occurs to and then the manner in which one expansion occurs versus a different one can result in the error never approaching zero.
As long as some deterministic process can be specified that reduces their evaluation algorithmically (which by definition cannot take an infinite number of steps) to another finite representation, then we can bypass the need for any infinities to exist and everything remained a deterministic value.
But infinity is inherently nebulous and ill posed to work with in many processes.
For example, is infinity+1> infinity or does infinity/infinity=1 or what about infinity+1/infinity=1 or does 1/0=infinity, how about 0/0=+/- infinity or maybe 1 etc.
The way to avoid these is to avoid using infinite representations. As long as you can remain within computable values, then mathematics works great, otherwise you have to create rules that can be arbitrary and unprovable simply because the assumptions could easily change from one calculation to the next and introduce unresolvabled paradoxes.
As long as the errors remain infinitesimally small, you can generally toss them away, as is often done in calculus but calculus acknowledges the problem as well and places restrictions on when the result is valid.
So yes, as long as you're able to keep the values distinctly seperated and round out the errors in the end, then it works fine but if the result is sensitive to infinitesimal differences then you have to maintain a way, during the calculations, to untangle them in the end or you're left with indeterminant results.
For example, if we take the arcsin of exactly 1, the result must be exactly pi/2. The 1 in this case cannot be anything over 1 or a result is impossible to calculate. On the other hand, the function is extremely sensitive to small changes below and, for example, the slope of the output between 1 and .99999 is 1000*2^0.5 or ~1,400. In cases with functions this sensitive, you can't leave unresolved error terms in place or they explode into things like trying to divide 1/0 etc.
So you have to keep track of which terms become most significant in the limits and track them as they approach to see which terms dominate. This isn't provided by the identity .9r=1 and so it can only be treated as an unprovable and questionably valid equation for limited uses only (to make it more explicit you could write .9r~=1) once you've thrown away the details of the process that generated the infinite sequence, you can no longer determine the relative signifance of the error term associated with it and I hope some of my previous posts demonstrated this.
Also, it may not actually be a quirk in the mathematics of infinity but represent a deeper understanding of mathematics that goes generally overlooked. Not all infinities (or infinitesimals) are created equal and transitions between them and finite values have many interesting properties.
After all, in a computer decimal arithmetic is not really calculation, it's symbol manipulation. The only "calculation" occurs later in analysis of error.
Pi has no finite writing in any symbolic system save the Greek letters, yet it is nevertheless real ... but it can be better represented by the algorithm that "determines" or "computes" it. We are forced to employ this algorithm to transform pi into something suitable for calculation with other "numbers". So is pi a number or a process?
Good questions. Here's how I see it.
For most things, if it's close enough, that's fine. Horseshoes and handgrenades are like this
Yes, 1/3 and pi represent processes as well as .3r etc.
These are finite representations of infinite processes. They exist as finite values only symbolically and not when they're expanded into an infinite form.
The number 1 doesn't represent a process though because it's a fundamental unit from which the others are built. We have to select something to remain absolute so we can build relationships upon it and 1 is the symbol for unity in multiplication versus the 0 for unity in addition. Division and subtraction are the inverses of these and only apply when such inverses are finitely computable (for example, you can't divide by 0, simply by definition, though some could estimate what a value divided by 0 would be, for example x/x=1, except when x=0, in which case you can generally still assume the answer is 0, but not always)
For example, we might say that 1/0=infinity, but think about this for a second ... is that actually true - even if we had someone subtracting 0 from 1 forever, would they ever be able to reduce 1 at all? Realistically, no, unless that was the amount of time it took that person to think of a better way to represent the ratio 1/0. So we can't really know if the vague notion of "infinity" is even correct in this case.
You can't redefine 0 or 1 as processes. They can only be inputs and outputs and not functions.
For example, y=x/3 is a function. 3/3 is 1 and an exact result that can be computed by definite in finite time, but 2/3 is an intermediate value that represents an irreducible computation. It's a symbolic representation of an infinite computation.
Once again, you could subtract 2 from 3, get 1, then rescale things so you have 10, then subtract 9 and get 1 again and then rescale by 10 etc. etc. etc. and there would always be an error in the answer, not much different than trying to subtract 0 from 1 forever, except in this case the relative weightings of the digits decreases, so it does approach a fixed value but that fixed value still has an infinite representation and can't be symbolically manipulated directly in that infinite format, because any finite quantization of it leads to errors and approximations. So you can write 2/3 and be precise and not find errors, but you can't try to treat it the same as .666666..... without introducing errors (notice everytime people write it, they always ignore the last digits and you can see when they manipulate this value they either have to show a limited example where the importance of the least significant digits becomes undeniably trivial or they mask the importance of them by no recognizing every time they shift the value left the errors grow in significance).
For example, 10*.9r is not the same as 9+.9r, because any errors in the 10*.9r are now 10 times as significant as the errors in .9r so the error terms no longer match. Now if we just say the errors are 0 so that 10*0=0 then the problem is simply ignored.
To work with this correctly an error term should be used and the boundary for the error analyzed as the expansion continues so that you can be certain the error does not grow to be more than infinitesimal, but this requires that you specify how digits the expansion occurs to and then the manner in which one expansion occurs versus a different one can result in the error never approaching zero.
As long as some deterministic process can be specified that reduces their evaluation algorithmically (which by definition cannot take an infinite number of steps) to another finite representation, then we can bypass the need for any infinities to exist and everything remained a deterministic value.
But infinity is inherently nebulous and ill posed to work with in many processes.
For example, is infinity+1> infinity or does infinity/infinity=1 or what about infinity+1/infinity=1 or does 1/0=infinity, how about 0/0=+/- infinity or maybe 1 etc.
The way to avoid these is to avoid using infinite representations. As long as you can remain within computable values, then mathematics works great, otherwise you have to create rules that can be arbitrary and unprovable simply because the assumptions could easily change from one calculation to the next and introduce unresolvabled paradoxes.
As long as the errors remain infinitesimally small, you can generally toss them away, as is often done in calculus but calculus acknowledges the problem as well and places restrictions on when the result is valid.
So yes, as long as you're able to keep the values distinctly seperated and round out the errors in the end, then it works fine but if the result is sensitive to infinitesimal differences then you have to maintain a way, during the calculations, to untangle them in the end or you're left with indeterminant results.
For example, if we take the arcsin of exactly 1, the result must be exactly pi/2. The 1 in this case cannot be anything over 1 or a result is impossible to calculate. On the other hand, the function is extremely sensitive to small changes below and, for example, the slope of the output between 1 and .99999 is 1000*2^0.5 or ~1,400. In cases with functions this sensitive, you can't leave unresolved error terms in place or they explode into things like trying to divide 1/0 etc.
So you have to keep track of which terms become most significant in the limits and track them as they approach to see which terms dominate. This isn't provided by the identity .9r=1 and so it can only be treated as an unprovable and questionably valid equation for limited uses only (to make it more explicit you could write .9r~=1) once you've thrown away the details of the process that generated the infinite sequence, you can no longer determine the relative signifance of the error term associated with it and I hope some of my previous posts demonstrated this.
Also, it may not actually be a quirk in the mathematics of infinity but represent a deeper understanding of mathematics that goes generally overlooked. Not all infinities (or infinitesimals) are created equal and transitions between them and finite values have many interesting properties.
QUOTE (StevenA+Mar 7 2007, 04:52 AM)
The function grows as (10^n-1)/10^n.
If this relationship ever became 1 then:
10^n-1=10^n
Which is obviously always false.
Again, you're completely ignoring limits. What is the limit of that difference? It's obviously not negative and it's not positive, you can prove that (and I have). If it's not negative or positive, then it's zero.
Think about limits.
If this relationship ever became 1 then:
10^n-1=10^n
Which is obviously always false.
Again, you're completely ignoring limits. What is the limit of that difference? It's obviously not negative and it's not positive, you can prove that (and I have). If it's not negative or positive, then it's zero.
Think about limits.
QUOTE (StevenA+Mar 7 2007, 04:52 AM)
Or again, using calculus, if we take the function:
f(x)={ 0 : x!=0 ; 1 : x=0)
And if we try to differentiate it at 0, the value is discontinuous, so there is no rational limit to the function and again infinitesimals make all the difference.
f(x)={ 0 : x!=0 ; 1 : x=0)
And if we try to differentiate it at 0, the value is discontinuous, so there is no rational limit to the function and again infinitesimals make all the difference.
You talk about differentiation and continuity as if you know something about them but then you go and say that. Your f(x) is not differentiable. No ifs, no buts, no "but what about infinitesimals", it's not differentiable. A function must be continuous to be differentiable (it's a necessary but not sufficent condition) and so to say "But if we consider the derivative" is meaningless. If you insist on trying to extract information from something like that you'll prove anything you like!
It's like me saying "I'm going to allow division by zero. With that I can prove maths wrong!! 0 = 0*1 = 2*0 => 2=1. Maths is wrong!!!".
Have I proven algebra wrong? No, because I made an invalid assumption mathematicians don't and that is what was wrong. Same goes for your logic. You ignore the fact your f(x) fails to meet the conditions to be differentiable but try anyway. There's a reason mathematicians defined differentiability a certain way, it gives consistent results! Not because they like to be boring but they like to be logical.
It's like me saying "I'm going to allow division by zero. With that I can prove maths wrong!! 0 = 0*1 = 2*0 => 2=1. Maths is wrong!!!".
Have I proven algebra wrong? No, because I made an invalid assumption mathematicians don't and that is what was wrong. Same goes for your logic. You ignore the fact your f(x) fails to meet the conditions to be differentiable but try anyway. There's a reason mathematicians defined differentiability a certain way, it gives consistent results! Not because they like to be boring but they like to be logical.
QUOTE (NoCleverName+Mar 7 2007, 04:52 AM)
one wonders where reality begins and ends with this problem. Is the difference between .9r and 1 real
Reality doesn't come into it. It's a maths problem. In reality can I have two numbers A and B such that A*B+B*A=0 ? No. But I can in maths because I'm interested in logical structure, not physical reality. (Google 'Grassman numbers').
QUOTE (StevenA+Mar 7 2007, 04:52 AM)
Not all infinities (or infinitesimals) are created equal and transitions between them and finite values have many interesting properties.
Yes, they do, shame your long rambles completely ignore the fact they've nothing do with this question.
QUOTE (Eric England+Mar 7 2007, 04:52 AM)
You can't trace the extent or surface of a crouton that separates it from its enviornment, to any degree greater than .9r, any more than you can trace the extent or surface of any of its components.
M-theroy my butt. Like I don't know a BS infinitesimal when I see one.
M-theroy my butt. Like I don't know a BS infinitesimal when I see one.
It's deranged ramblings like that which save me from having to give a serious reply to you. It's obvious to anyone you're just off in crazyland there.
QUOTE (gonegahgah+Mar 7 2007, 04:52 AM)
Otherwise you would appreciate that I have shown my counter arguments with new ways of talking about it and not just the same old counter arguments that people initially think of.
If you're so good at maths you should realise that there's no 'counter arguments' to a proven theorem. If there were, it would never have been able to be proved.
QUOTE (gonegahgah+Mar 7 2007, 04:52 AM)
I don't walk around with a chip on my shoulder. I can't help but feel that you do have a chip on your shoulder.
0.9r=1 threads 'push my buttons' more than other ones. It's the classic one which draws out the idiots. Just as people say "I don't accept quantum mechanics, it's not intuitive", plenty say "I don't accept 0.9r=1, it's not intuitive". The difference being 0.9r=1 isn't a matter of debate or experiment, it's a proven fact yet people stil ignore the proofs and then complain they're being 'bamboozled' because they haven't bothered to learn how to understand the proofs.
And just like every jackass who knows nothing about physics has a 2 cents opinion about M theory or Loop Quantum Gravity, the same (but worse) applies to this. For all a cranks talk about understanding logic etc, they completely ignore it here because 'it doesn't feel right' or they're too ignorant to understand how the underlying maths works. That's not a problem if they're willing to learn, but they aren't so people like StevenA continue to concoct vague nonsense hoping somehow it'll overturn the logic used to prove 0.9r=1. If he understood logic, he'd see it can't be overturned.
But no, the people who have the least information somehow think they know the most. Those of us who understand the underlying methods, who aren't bamboozled by it, somehow we're ALL wrong, includi
And just like every jackass who knows nothing about physics has a 2 cents opinion about M theory or Loop Quantum Gravity, the same (but worse) applies to this. For all a cranks talk about understanding logic etc, they completely ignore it here because 'it doesn't feel right' or they're too ignorant to understand how the underlying maths works. That's not a problem if they're willing to learn, but they aren't so people like StevenA continue to concoct vague nonsense hoping somehow it'll overturn the logic used to prove 0.9r=1. If he understood logic, he'd see it can't be overturned.
But no, the people who have the least information somehow think they know the most. Those of us who understand the underlying methods, who aren't bamboozled by it, somehow we're ALL wrong, includi