Ughh. Think I'm gonna hurl dude.
Proof that pi is rational - lol
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kriminal99
Registered Senior Member
Why I think mathematicians have problems with philosophy and philosophy style thinking:
http://bbsonline.cup.cam.ac.uk/Preprints/OldArchive/bbs.halford.html
To sum the theory above, basically the complexity of a mental task is based on the number of relations it requires you to process at once. It claims that humans are only capable of recognizing 4 relations at once, so for example if you saw 4 similar looking rocks you could recognize them as the number four, but if you saw 9 you could not without counting (segementation) or placing them in a formation that you recognized as a larger number (like the six dots in formation on a dice.. this is what they call chunking)
If we were capable of making 9 relations at once you could see a pile of 9 rocks and instantaneously be like "look! 9 rocks!"
So basically the idea is mathematicians make heavy use of procedures like segmentation and chunking which allow us to break down ideas which would otherwise require us to consider many things at once, and then process them easily. The means of doing this really are mathematics itself.
Philosophers on the other hand A) Have no obvious or currently available means of breaking down the ideas they have to deal with, so have to instead have to discover them on their own or process more things at once B) Cannot rely on chunking unless the everything about what is being chunked is completely understood because not making mistakes in reasoning is the whole point of this type of thought C) Philosophy REQUIRES you to consider many more factors of the same problem.
So basically mathematician's minds turn to a pile of goo when considering anything outside of the norm, and since everything is so easy in the mathematics that is commonly accepted and it makes them feel smart they refuse to do it.
Edit:
Come to think of it... We might say that 1/3 as a decimal is better defined than an infite sequence of rational numbers. Perhaps because the algorithm to construct .33 repeating is so simple and obviously unchanging for as long as we want to do it.
If we were beings capable of making many relations at once, perhaps we would see a sequence such as pi exactly the same way. IE we would percieve the entire repeating rational term at once and it would just repeat over and over or something... funny to think about
For that matter if we were capable of infinite relations at once, we wouldn't need math at all. We would have a base infinity number system.
http://bbsonline.cup.cam.ac.uk/Preprints/OldArchive/bbs.halford.html
To sum the theory above, basically the complexity of a mental task is based on the number of relations it requires you to process at once. It claims that humans are only capable of recognizing 4 relations at once, so for example if you saw 4 similar looking rocks you could recognize them as the number four, but if you saw 9 you could not without counting (segementation) or placing them in a formation that you recognized as a larger number (like the six dots in formation on a dice.. this is what they call chunking)
If we were capable of making 9 relations at once you could see a pile of 9 rocks and instantaneously be like "look! 9 rocks!"
So basically the idea is mathematicians make heavy use of procedures like segmentation and chunking which allow us to break down ideas which would otherwise require us to consider many things at once, and then process them easily. The means of doing this really are mathematics itself.
Philosophers on the other hand A) Have no obvious or currently available means of breaking down the ideas they have to deal with, so have to instead have to discover them on their own or process more things at once B) Cannot rely on chunking unless the everything about what is being chunked is completely understood because not making mistakes in reasoning is the whole point of this type of thought C) Philosophy REQUIRES you to consider many more factors of the same problem.
So basically mathematician's minds turn to a pile of goo when considering anything outside of the norm, and since everything is so easy in the mathematics that is commonly accepted and it makes them feel smart they refuse to do it.
Edit:
Come to think of it... We might say that 1/3 as a decimal is better defined than an infite sequence of rational numbers. Perhaps because the algorithm to construct .33 repeating is so simple and obviously unchanging for as long as we want to do it.
If we were beings capable of making many relations at once, perhaps we would see a sequence such as pi exactly the same way. IE we would percieve the entire repeating rational term at once and it would just repeat over and over or something... funny to think about
For that matter if we were capable of infinite relations at once, we wouldn't need math at all. We would have a base infinity number system.
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Wait. I thought mathemeticians were on the cutting edge of abstract, "out-there", multidimensional, outside the norm thinking.
kriminal99
Registered Senior Member
superluminal said:
Im not sure when your claiming I said all of that exactly, but your misreading what I'm saying. EVERYONE's mind turns to goo when dealing with philosophy occasionaly, but many mathematicians are so used to being able to turn seemingly complex situations into simple math and figuring things out that the prospect of having to deal with something where the way is not clear scares them. Especially when it attacks the methods that they already have.
Ok.
kriminal99:
Reading this thread, it seems that you are railing against the standard mathematical definition of the term "rational number".
Using that definition, all mathematicians agree that pi is irrational.
Using the kriminal99 definition of "rational number" (whatever that may be), it seems pi is rational. Ok. Fine. You'll generate a lot of confusion by using your own definition of the term, and you'll be sadly misunderstood by all mathematicians when you use it, but I guess that's a price you're willing to pay.
It seems this is a non-argument.
Reading this thread, it seems that you are railing against the standard mathematical definition of the term "rational number".
Using that definition, all mathematicians agree that pi is irrational.
Using the kriminal99 definition of "rational number" (whatever that may be), it seems pi is rational. Ok. Fine. You'll generate a lot of confusion by using your own definition of the term, and you'll be sadly misunderstood by all mathematicians when you use it, but I guess that's a price you're willing to pay.
It seems this is a non-argument.
HallsofIvy
Registered Senior Member
"Actually we didn't do any of that stuff and dx was clearly indicated as an infintessimally small value."
Then either it was a realy bad class or it was doing "non-standard" analysis (very strange for an introductory calculus class)- no introductory calculus text I have ever seen used "infinitessmals". Did your text or teacher DEFINE "infinitesmally small"?
"Oh and by the way the link you provided does not clearly indicate that infinite summations of rational numbers are not rational. "
No one has said they are not. .3+ .03+ .003+ .0003+ ... is an infinite series that clearly has a rational sum. What has been said is that the sum is not NECESSARILY rational- it can be either rational or irrational. However, since you have refused to accept the standard definition of "rational" number, I don't see how you can say that a number is or is not rational until you have told us what YOU mean by a rational number!
(The argument in your original post shows that there exist rational numbers arbitrarily CLOSE to pi, not that pi itself is rational. That's true of any number, rational or irrational.)
Then either it was a realy bad class or it was doing "non-standard" analysis (very strange for an introductory calculus class)- no introductory calculus text I have ever seen used "infinitessmals". Did your text or teacher DEFINE "infinitesmally small"?
"Oh and by the way the link you provided does not clearly indicate that infinite summations of rational numbers are not rational. "
No one has said they are not. .3+ .03+ .003+ .0003+ ... is an infinite series that clearly has a rational sum. What has been said is that the sum is not NECESSARILY rational- it can be either rational or irrational. However, since you have refused to accept the standard definition of "rational" number, I don't see how you can say that a number is or is not rational until you have told us what YOU mean by a rational number!
(The argument in your original post shows that there exist rational numbers arbitrarily CLOSE to pi, not that pi itself is rational. That's true of any number, rational or irrational.)
kriminal99 said:
No epsilon-delta definition definition of the limit then, eh? (I assume you had limits of some vague form at least)
I said elsewhere that you haven't taken what I consider a 'real math course', and this just reaffirms that. If this wasn't a non-standard analysis class (which would be remarkable for an intro class), your calculus class was a crappy one. Did you have a text?
Just to point out that kriminal99 may be being misled by a belief that "rational" in mathematics has the same meaning as it does outside maths, ie "understandable, logical". In fact the root of "rational" in mathematics is the "ratio" part. It is "ratio"-nal, ie a number that can be expressed as a ratio, which of course is one integer divided by another.
I suggest you actually bother to read what was written, and not interpret it wrongly to suit your own needs.kriminal99 said:
A number that can ONLY be described as the sum of an INFINITE series of rational numbers (like PI) is, BY DEFINITION, IRRATIONAL.
This is NOT begging the question.
It is NOT a circular argument.
It is a DEFINITION.
Please indicate where in my original post I said that an "infite summation of rational numbers is not rational because an infinite sequence of rational numbers is not rational"
That indeed would be circular reasoning HAD I DONE SO.
But don't worry - I don't expect an apology from you.
kriminal99
Registered Senior Member
Why would I apologize? Your still missing the point. Thats YOUR arbitrary definition of a rational number which you have given no argument for.
IE someone can still hold that a rational number can be written as p/q and still allow for infinite summations of rational numbers. As I said earlier in the thread, to rule this out a definition would have to say "constructively". An infinite summation has a value, but you don't know what it is yet. If you don't know what it is yet, how can you know if you can write it as p/q.
If you take the view that it doesn't have a value until you know it, then one might also be inclined to say it isn't even a number.
My definition of a rational is a number that can be written as p/q. Can, as in can if you knew what the number was. Actually they were saying that infinite summations of rationals are definitely irrational not that they could be either. And actually the ONLY type of number you can't constructively (as in right now) write as p/q are infinite summations, and the ONLY reason you can't is because the number hasn't finished being formed yet. There is no such thing as a number that isn't being continually generated or altered that you can't write as p/q.
And that arbitrarily close is the same as equal to was the point of the .333..*3 = .999999.. = 1 = 1* (1/3) bit.
Btw your argument and the other people's that are arguing with me are totally different.
Crappy why? Based on what argument? That you said so? That is hardly reason for concern. If you people are going to continue posting here, you really should learn how to reason correctly. I am not you. I do not start out with your beliefs. If you want to communicate a belief to me, then you must give the argument for it. If you believe it just because someone told you so, then don't expect to be able to convince anyone who isn't as easily herded as you are.
We did limits of riemann sums as n went to infinity, and thats how we started on integrals. dx was specifically explained to mean a infintessimal value although now that you mention I vaguely remember the guy mentioning something about them being a source of confusion in the math community or something like that. We had a textbook but I dont buy them. The one my roomate has is the same one I think used for like 3 calculus classes and its edwards and penny early transcendental calculus.
Not really. Saying a rational number is one that you can write as p/q is ambiguous and does not rule out infinite summations of rational numbers. The question isn't what is the convention, but whether or not their beliefs are consistent or self contradicting. These guys are trying to claim that rational numbers must be able to be written, right now as in constructively , as p/q. Im saying that rational numbers have not been defined as such, and if they were it would introduce other problems with all else remaining the same. Most obviously, it would contradict the basics of arithematic which dictate that adding rational numbers results in another rational number... If your admitting that math is self - contradicting even within this small area, then we are not really arguing.
IE someone can still hold that a rational number can be written as p/q and still allow for infinite summations of rational numbers. As I said earlier in the thread, to rule this out a definition would have to say "constructively". An infinite summation has a value, but you don't know what it is yet. If you don't know what it is yet, how can you know if you can write it as p/q.
If you take the view that it doesn't have a value until you know it, then one might also be inclined to say it isn't even a number.
HallsofIvy said:
My definition of a rational is a number that can be written as p/q. Can, as in can if you knew what the number was. Actually they were saying that infinite summations of rationals are definitely irrational not that they could be either. And actually the ONLY type of number you can't constructively (as in right now) write as p/q are infinite summations, and the ONLY reason you can't is because the number hasn't finished being formed yet. There is no such thing as a number that isn't being continually generated or altered that you can't write as p/q.
And that arbitrarily close is the same as equal to was the point of the .333..*3 = .999999.. = 1 = 1* (1/3) bit.
Btw your argument and the other people's that are arguing with me are totally different.
shmoe said:
Crappy why? Based on what argument? That you said so? That is hardly reason for concern. If you people are going to continue posting here, you really should learn how to reason correctly. I am not you. I do not start out with your beliefs. If you want to communicate a belief to me, then you must give the argument for it. If you believe it just because someone told you so, then don't expect to be able to convince anyone who isn't as easily herded as you are.
We did limits of riemann sums as n went to infinity, and thats how we started on integrals. dx was specifically explained to mean a infintessimal value although now that you mention I vaguely remember the guy mentioning something about them being a source of confusion in the math community or something like that. We had a textbook but I dont buy them. The one my roomate has is the same one I think used for like 3 calculus classes and its edwards and penny early transcendental calculus.
James R said:
Not really. Saying a rational number is one that you can write as p/q is ambiguous and does not rule out infinite summations of rational numbers. The question isn't what is the convention, but whether or not their beliefs are consistent or self contradicting. These guys are trying to claim that rational numbers must be able to be written, right now as in constructively , as p/q. Im saying that rational numbers have not been defined as such, and if they were it would introduce other problems with all else remaining the same. Most obviously, it would contradict the basics of arithematic which dictate that adding rational numbers results in another rational number... If your admitting that math is self - contradicting even within this small area, then we are not really arguing.
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Since we all know my mathematics is limited, I am not joining this debate but wanted to post a note as an observer.
I believe the issue is one of symantics and that most here are right in that Kriminal99 seems to be forming his own definition.
I get the impression he does so because he sees the existance of a perfect circle as requiring a rational number for pi since a circle couldn't be formed otherwise.
It seems to me that is incorrect.
Pi is the ratio of two numbers. Those two numbers, if based on the existance of a perfect circle, must indeed be rational and have a finite number of decimals, etc.
That would seem to be required or one must consider there is no such thing as a perfect circle.
Have I got this even close?
I believe the issue is one of symantics and that most here are right in that Kriminal99 seems to be forming his own definition.
I get the impression he does so because he sees the existance of a perfect circle as requiring a rational number for pi since a circle couldn't be formed otherwise.
It seems to me that is incorrect.
Pi is the ratio of two numbers. Those two numbers, if based on the existance of a perfect circle, must indeed be rational and have a finite number of decimals, etc.
That would seem to be required or one must consider there is no such thing as a perfect circle.
Have I got this even close?
I suggest you look at the proofs of PI's irrationality provided by: Lambert 1761; Legendre 1794; Hermite 1873; Nagell 1951; Niven 1956; Struik 1969; Königsberger 1990; Schröder 1993; Stevens 1999; Borwein and Bailey 2003, pp. 139-140.
The one on THIS page is by Niven.
It starts with the assumption that pi=a/b (the requirement for rationality). And disproves its possibility.
PI does have a value - it is equal to the ratio of the circumference of a circle to its diameter.kriminal99 said:
The fact that we can not physically write the number in decimal form is irrelevant.
The same for "e", square root of 2 etc. They all have values by definition.
And thus I'm sure the link above will happily correct your understanding of whether or not pi is rational or irrational.kriminal99 said:
No. The only reason you can't is because it is IRRATIONAL. The number PI has finished being formed. It has been defined. It needs nothing more.kriminal99 said:
The fact you can't write it out in decimal, or Hex or binary is irrelevant. The number exists.
Furthermore, you can not "constructively" write a fraction with A and B both as a 10^(10^100) digit integer. But this would still qualify as a rational number.
Your definition is flawed.
PI is indeed based on a perfect circle.MacM said:
But unfortunately there is no such thing as a "perfect" circle in anything other than concept.
The only way you can define a perfect circle is by using PI.
Thus your argument is unfortunately flawed.
Sarkus said:
Thanks. I see why my statement is absolutely false. Pi is irrational since a perfect cirlce requires either the diameter or the circumferance to be irrational.
If you take a rational, finite diameter - i.e. 3.000000000n and multiply that by Pi, does that not then make the new circle circumferance become a rational number?
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No.MacM said:
To be rational, it must be possible to state it as A/B, okay.
If you assume Pi = X/Y and then multiply it by 3 you get 3X/Y.
3X/Y is the same algebraic form as A/B - i.e. A=3X and B=Y.
So if PI does not equal A/B (see proof posted above) it can not = 3X/Y.
Once again: it's not an arbitrary definition. It's the mathematical definition. There doesn't need to be an "argument" for a defined term.kriminal99 said:
Of course, we're all arguing with kriminal99 because his thread says he provides a "proof that pi is rational", which we all know (knowing the actual definition of "rational") to be nonsense. But since he doesn't mean "rational" (since he doesn't know what it means) the argument kind of falls by the wayside. kriminal99 doesn't seem to be disagreeing that pi cannot be represented by one integer divided by another.
Incidentally, pi is more than just irrational. It is also transcendental - which means that it cannot be represented as the root of an equation. sqrt(2) is irrational, but it is not transcendental because it is one of the roots of the equation x<sup>2</sup> - 2 = 0. There is no equivalent for pi (or e).
"Saying a rational number is one that you can write as p/q is ambiguous and does not rule out infinite summations of rational numbers."
I think you're missing the spirit of the definition. I'd guess it can be shown that rational numbers may consist of some finite summation of other rational numbers. Like 1/4 = 1/8 + 1/8 or 1/4 = 1/16 + 1/16 + 1/16 + 1/16. If however, you have to involve an infinite summation... wouldn't that mean the number is irrational? Maybe I've missed something.
I think you're missing the spirit of the definition. I'd guess it can be shown that rational numbers may consist of some finite summation of other rational numbers. Like 1/4 = 1/8 + 1/8 or 1/4 = 1/16 + 1/16 + 1/16 + 1/16. If however, you have to involve an infinite summation... wouldn't that mean the number is irrational? Maybe I've missed something.
Take any rational number. Now take another rational number. Now add them together. The sum will be rational right. Keep adding rational numbers and the sum will always be a rational number. We can conclude then that the sum of n rational numbers is always rational.
So 3 + 0.1 + 0.04 + 0.001 + 0.0005 is a sum of rational numbers and equals 3.1415 which is also a rational number.
But 3.1415 does not equal pi. Neither does 3.141592653589. Neither does any terminating sequence of decimal numbers. The only number which equals pi is the number 3.1415... (with an infinite number of decimals). If the decimal places terminate then it obviously isn't pi.
Now, can we write pi as a sum of rational numbers? Well, if pi had n decimal places (where n is not infinity obviously) then pi could be expressed as a sum of n rational numbers which would imply that pi is rational.
However, pi has an infinite number of decimal places and hence would require a sum of an inifnite number of rational numbers.
The problem arises when you sum to infinity. You say that the limit of 3 + 0.1 + 0.04 + ... as the sequence goes to infinity is pi, because this is just a sequence of rationals so pi must be rational. But the sequence consists of a infinite number of rational numbers. How on Earth can we even begin to think about adding an infinite number of things together. Any philosopher (and indeed a mathematician) should understand that strange things happen when you involve infinity, and actually infinity is generally avoided in mathematics because it doesn't make sense.
Try adding an infinite number of anything together. Say 1 + 1 + 1 + ... what does this equal? The sum has no limit and the sum is infinite. Is infinity a rational number? By your definition it is! becuase 1 is a rational number hence 1 + 1 + 1 + ... is rational. But now consider √2 + √2 + ... This is a sum of irrational numbers and the sum is infinity. Hence infinity is irrational. But haven't we just shown that infinity is rational?
This is all goobilygook if you are a mathematician and well versed in analysis, however I am trying to point out that once you involve infinity in your proof you can't rely on anything to make sense. This is why mathematicians have always avoided the use of infinity in their proofs or irrationality, simply because it doesn't make sense.
So if you are going to try to prove something using infinity, no-one is going to listen (or understand you for that matter) UNLESS you actually have counted to infinity.
Hope this helps you understand.
So 3 + 0.1 + 0.04 + 0.001 + 0.0005 is a sum of rational numbers and equals 3.1415 which is also a rational number.
But 3.1415 does not equal pi. Neither does 3.141592653589. Neither does any terminating sequence of decimal numbers. The only number which equals pi is the number 3.1415... (with an infinite number of decimals). If the decimal places terminate then it obviously isn't pi.
Now, can we write pi as a sum of rational numbers? Well, if pi had n decimal places (where n is not infinity obviously) then pi could be expressed as a sum of n rational numbers which would imply that pi is rational.
However, pi has an infinite number of decimal places and hence would require a sum of an inifnite number of rational numbers.
The problem arises when you sum to infinity. You say that the limit of 3 + 0.1 + 0.04 + ... as the sequence goes to infinity is pi, because this is just a sequence of rationals so pi must be rational. But the sequence consists of a infinite number of rational numbers. How on Earth can we even begin to think about adding an infinite number of things together. Any philosopher (and indeed a mathematician) should understand that strange things happen when you involve infinity, and actually infinity is generally avoided in mathematics because it doesn't make sense.
Try adding an infinite number of anything together. Say 1 + 1 + 1 + ... what does this equal? The sum has no limit and the sum is infinite. Is infinity a rational number? By your definition it is! becuase 1 is a rational number hence 1 + 1 + 1 + ... is rational. But now consider √2 + √2 + ... This is a sum of irrational numbers and the sum is infinity. Hence infinity is irrational. But haven't we just shown that infinity is rational?
This is all goobilygook if you are a mathematician and well versed in analysis, however I am trying to point out that once you involve infinity in your proof you can't rely on anything to make sense. This is why mathematicians have always avoided the use of infinity in their proofs or irrationality, simply because it doesn't make sense.
So if you are going to try to prove something using infinity, no-one is going to listen (or understand you for that matter) UNLESS you actually have counted to infinity.
Hope this helps you understand.
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