Thread: Circles and Their Diameters

I'm new to this forum (and all forums outside of the world of r/c helicopters) and I went through the listed topics looking for a discussion about circles and their diameters and did not find one. If there is such a thread please point me in that direction.

I have been troubled for a long time about several things in math but particularly about circles and the math used to describe them. Pi is used to represent the relationship between a circumference and its diameter. But in spite of the fact that there are circles all around us with finite circumferences and diameters it doesn't appear that we can compute finite results without approximations.

Is this a failure of our system of mathematics that will be left to be corrected in the future with a new system that will be able to describe a circle more in line with our familiarity with them?

Note: I am not a mathematician. I work in manufacturing management. My degrees are in philosophy and psychology but I did study math through calculus.

Originally Posted by Ots

I'm new to this forum (and all forums outside of the world of r/c helicopters) and I went through the listed topics looking for a discussion about circles and their diameters and did not find one. If there is such a thread please point me in that direction.

I have been troubled for a long time about several things in math but particularly about circles and the math used to describe them. Pi is used to represent the relationship between a circumference and its diameter. But in spite of the fact that there are circles all around us with finite circumferences and diameters it doesn't appear that we can compute finite results without approximations.

Is this a failure of our system of mathematics that will be left to be corrected in the future with a new system that will be able to describe a circle more in line with our familiarity with them?

Note: I am not a mathematician. I work in manufacturing management. My degrees are in philosophy and psychology but I did study math through calculus.

A circle is the set of all points in the plane equidistant from some fixed point. The distance is called the radius and the fixed point is called the center of the circle.

The circumference of a circle is where is the radius. The area is . These facts are proved rigorously in calculus. They are exact relationships and not approximations. The formula for the circumference actually serves as a definition for once you have shown that the ratio of the circumference to the radius is a constant. This is done using methods of calculus.

No other system is either needed or desired. These basic facts have been known since the time of the ancient Greeks.

DrRocket, thank you - I understand what you've said, and in a 'pure' sense there's really nothing more to it.

But I think there's more going on here. There seems something inadequate with telling me that my one foot diameter circle is pi feet in circumference and you can't give me an exact decimal representation of that number. I mean, every added decimal place to the decimal representation of pi makes the circumference bigger even though the circle doesn't grow. I realize as well that this isn't the only instance of this type of difficulty.

I just see a problem here and it makes me wonder if it doesn't reflect a flaw in our system of mathematics even given the 'pure' explanation above.

it just means our number system doesn't work in computing that relationship. that's all. Pi is an exact value that IS irrational, so it can't be represented in our rational based system.

Originally Posted by Arcane_Mathematician

it just means our number system doesn't work in computing that relationship. that's all. Pi is an exact value that IS irrational, so it can't be represented in our rational based system.

You know - I think this is exactly what I'm driving at, and in a somewhat clumsy fashion, that it doesn't work like you would think that it should work. I mean we're talking about a circle and its diameter! And our math can't do that in a real simple way. It has to resort to an irrational number for such a simple thing.

I suspect that there will come a time, lots of years from now, when people will laugh at our difficulties with such trivial matters. I think it will take a completely new system though.

No, I doubt that. We just won't care about irrational numbers, as they are prolific and EVERYWHERE.

Pi is an exact value that IS irrational, so it can't be represented in our rational based system.

FALSE

Originally Posted by clearwar

Pi is an exact value that IS irrational, so it can't be represented in our rational based system.

FALSE

TRUE. Look it up.

Originally Posted by Ots

DrRocket, thank you - I understand what you've said, and in a 'pure' sense there's really nothing more to it.

But I think there's more going on here. There seems something inadequate with telling me that my one foot diameter circle is pi feet in circumference and you can't give me an exact decimal representation of that number. I mean, every added decimal place to the decimal representation of pi makes the circumference bigger even though the circle doesn't grow. I realize as well that this isn't the only instance of this type of difficulty.

I just see a problem here and it makes me wonder if it doesn't reflect a flaw in our system of mathematics even given the 'pure' explanation above.

The reason that there is no exact decimal representation for is because one does not exist. Nor is there an exact decimal representation for .

This is not a failing of mathematics, it is just a characteristic of irrational real numbers. is not only irrational, it is transcendental.

There is no more going on. This is not subtle. It is well-known and has been studied for a couple of thousand years.

Adding terms to a decimal approximation for does not make the circumference of anything grow. It only makes the calculated number closer to the exact number for the circumference. There is no difficultly in the mathematics, only in the ability to calculate using rational approximations for irrational numbers.

Perhaps I expected more from this forum.

I suppose we happen to be more realistic and rational than you thought?

Maybe, but I really didn't expect the 'this is the way it is, it's been that way for two thousand years and we don't need to change it' kind of attitude. Seems a little redneck.

I guess I was thinking that this thread would generate more of the "inadequacies" that exist in our math system. Like yeah, that pi thing - we don't even have a formula to compute it. What about this other thing - you know, where 1/x^2 converges but 1/x doesn't - let's talk about that, etc.

Like a thread where counter-intuitive things would be discussed perhaps.

They both converge on 0, it just so happens that the area under the latter doesn't converge to a number as x becomes arbitrarily large.

We can go on and on about counter-intuitive things, but you really must ask, what use would they be beyond one special situation?

Originally Posted by Ots

Perhaps I expected more from this forum.

Perhaps you need to learn enough on your own to recognize a complete answer when you see one.

Originally Posted by DrRocket

Originally Posted by Ots

Perhaps I expected more from this forum.

Perhaps you need to learn enough on your own to recognize a complete answer when you see one.

I'm sure you're right DrRocket. In fact I'm sure you're always right.

Originally Posted by Arcane_Mathematician

They both converge on 0, it just so happens that the area under the latter doesn't converge to a number as x becomes arbitrarily large.

We can go on and on about counter-intuitive things, but you really must ask, what use would they be beyond one special situation?

Probably no use at all. I think starting this thread was a bad idea. But I do appreciate your responses. Thanks.

Originally Posted by clearwar

Pi is an exact value that IS irrational, so it can't be represented in our rational based system.

FALSE

You are completely and totally wrong. You could not possibly be more wrong.

Pi is a well-defined number. It comes up in many places in mathematics. It is has been rigorously proved that not only is pi irrational it is transcendental. That means it is not a root of any polynomial with integer coefficients.

Among other things, irrationality means that the decimal representation of pi is neither terminating nor repeating. There are algorithms for approximating pi to any number of decimal places that one might desire, but of course none of these approximations are exact. In fact, the digits in the decimal representation for pi are sometimes used as pseudo-random numbers since there seems to be no discernible pattern.

Originally Posted by Ots

Maybe, but I really didn't expect the 'this is the way it is, it's been that way for two thousand years and we don't need to change it' kind of attitude. Seems a little redneck.

I guess I was thinking that this thread would generate more of the "inadequacies" that exist in our math system. Like yeah, that pi thing - we don't even have a formula to compute it. What about this other thing - you know, where 1/x^2 converges but 1/x doesn't - let's talk about that, etc.

Like a thread where counter-intuitive things would be discussed perhaps.

What do you mean by "1/x^2 converges but 1/x doesn't" ? Both converge to zero as x grows without bound.



While which grows without bound as t becomes large.

The problem is that you apparently do not understand mathematics or what "the system" really is.

"The system" is really nothing more or less than some rather simple axioms and rules of logic that permit one to draw inferences. The axioms, known commonly as the Zermelo Frankel axioms plus choice may seem a bit foreign at first, but basically they provide basic set theoretical notions and permit the construction of the natural numbers. From there one one is working with the implications of these axioms, and essentially all of mathematics follows. http://en.wikipedia.org/wiki/Zermelo...kel_set_theory

There are no issues of the sort that you seem to imply with "the system". The issues that do exist lie at the very foundations and relate to work of Godel who showed that the axioms contain theorems that are true but not provable within the axioms themselves. Another issue is that it is not possible to prove within the axioms themselves that the axioms of arithemetic are self-consistent although there are transcendental proofs.

But there are no problems with things like pi or with 1/x or 1.x^2.

You need to learn some mathematics so that you can both understand what mathematics says and formulate cogent questions.

This not a matter of "redneck thinking" or anything of the sort. It is a matter of knowing what one is talkiing about. Mathematics is a discipline that requires a tremendous amount of imagination, but it also requires a great deal of disciplined logic. [/tex]

It's over DrR.

I thought I'd jump in here with a quick post since Ots threw in the towel.

I have a probability question that involves pi.

I once saw a chart that had random facts about pi, and one of them mentioned a sequence of repeating nines (like five in a row) that occurred within the first 10,000 digits or something.

I thought to myself "Since pi goes on forever, would the probability of finding any given sequence of numbers be 1" Say we wanted to search for one hundred 7's in a row, well we might not find the sequence within the first five million digits, and we might not find the sequence within the first trillion digits either, but since pi goes on forever, can I say with absolute certainty that a sequence of one hundred 7's WILL OCCUR somewhere, since an infinitely long number allows for an infinite possibility of sequences.

Can I make this statement and would it be correct.

that large of a sequence, highly unlikely.

What abut replacing D with infinite in C = Pi*D formula

Originally Posted by SteveC

I thought I'd jump in here with a quick post since Ots threw in the towel.

I have a probability question that involves pi.

I once saw a chart that had random facts about pi, and one of them mentioned a sequence of repeating nines (like five in a row) that occurred within the first 10,000 digits or something.

I thought to myself "Since pi goes on forever, would the probability of finding any given sequence of numbers be 1" Say we wanted to search for one hundred 7's in a row, well we might not find the sequence within the first five million digits, and we might not find the sequence within the first trillion digits either, but since pi goes on forever, can I say with absolute certainty that a sequence of one hundred 7's WILL OCCUR somewhere, since an infinitely long number allows for an infinite possibility of sequences.

Can I make this statement and would it be correct.

You could make that statement if the sequence of digits in the decimal expansion of pi were truly random. However, there is not even a good definition of what that means, although there are some attempts to make one.

The existence of any finite sequence within an infinite string of random numbers follows from the fact that anything with a non-zero probability has a probability of one of happening in infinitely many trials.

However, you cannot make such a statement with absolute certainty since you cannot rigorously invoke probability theory.

That is one problem with probability theory. It is a very elegant theory, a beautiful application of the theory of measure and integration. However, it is nearly impossible to verify the basic assumptions in a problem that is originally stated in a situation that does not directly invoke probability theory and its basic premises from the start. Most applications involve use of probabilistic models as a means of compensating for ignorance of initial conditions, as in gambling scenarios. The only truly probabilistic systems of which I am aware come from quantum theory.

The digits of pi may appear to be random, but one can construct and algorithm for determining what they are and that algorithm is completely deterministic.


BTW you clearly need more than just an infinite (non-terminating) decimal expansion to even begin to use your logic, as you can see from the infinite decimal expansion for 1/3 =0.33333333333..... which clearly does not contain any sequence of, for instance 5's.

BTW you clearly need more than just an infinite (non-terminating) decimal expansion to even begin to use your logic, as you can see from the infinite decimal expansion for 1/3 =0.33333333333..... which clearly does not contain any sequence of, for instance 5's.

Absolutely, that why I am only talking about pi, which has the characteristic of randomness to it.

You could make that statement if the sequence of digits in the decimal expansion of pi were truly random. However, there is not even a good definition of what that means, although there are some attempts to make one. The existence of any finite sequence within an infinite string of random numbers follows from the fact that anything with a non-zero probability has a probability of one of happening in infinitely many trials.

So basically what your saying is that my statement would be true for a number that is random, but I cannot say it with absolute certainty in regards to pi, since we can't prove that pi is random. But why the heck not, there doesn't seem to be any logical repetitions or repeating sequences anywhere in pi that anyone has found. How does one even go about proving a number is random anyway.

I still don't understand how my statement would invoke probability theory, for the simple reason that I don't even know what probability theory is. I'll check it out.

The digits of pi may appear to be random, but one can construct and algorithm for determining what they are and that algorithm is completely deterministic.

Wait a second, I think I follow you now. You are saying that since pi can be generated by carrying out long algorithms, this makes it not so random after all, since its follows the commands of the algorithm, and never violates it.

like the golden ratio and the Fibonacci numbers.

Originally Posted by DrRocket

What do you mean by "1/x^2 converges but 1/x doesn't" ? Both converge to zero as x grows without bound.



While which grows without bound as t becomes large.

The problem is that you apparently do not understand mathematics or what "the system" really is.[/tex]

LOl!

Unless of course he means the series, in which case he should've said so.

Originally Posted by Chemboy

Unless of course he means the series, in which case he should've said so.

The series and the integrals that I listed are closely related.

Originally Posted by Arcane_Mathematician

I suppose we happen to be more realistic and rational than you thought?

Or rude and more condescending.
I am sure the OP really appreciated DrR explaining what a circle was

Originally Posted by Ots

Originally Posted by DrRocket

Originally Posted by Ots

Perhaps I expected more from this forum.

Perhaps you need to learn enough on your own to recognize a complete answer when you see one.

I'm sure you're right DrRocket. In fact I'm sure you're always right.

Oh he is, believe me he is :wink: especially when he is wrong,

Originally Posted by Ots

I'm new to this forum (and all forums outside of the world of r/c helicopters) and I went through the listed topics looking for a discussion about circles and their diameters and did not find one. If there is such a thread please point me in that direction.

I have been troubled for a long time about several things in math but particularly about circles and the math used to describe them. Pi is used to represent the relationship between a circumference and its diameter. But in spite of the fact that there are circles all around us with finite circumferences and diameters it doesn't appear that we can compute finite results without approximations.

Is this a failure of our system of mathematics that will be left to be corrected in the future with a new system that will be able to describe a circle more in line with our familiarity with them?

Note: I am not a mathematician. I work in manufacturing management. My degrees are in philosophy and psychology but I did study math through calculus.

Part of the problem is "circles all around us with" when in fact the are not all around (lol) us, nobody has ever seen one nor will anyone ever see one, what we do see is approimations of circles.
If you look at them under a microscope you will see they are ot circular, they have very jagged edges!!

I had 'thrown in the towel' on this thread, Esbo, but since your posts aren't wrapped in criticism (which I appreciate) let me try to make some sense out of this.

I know the round things we see everyday aren't the same as circles defined mathematically. But they are familiar sights unlike lots of other stuff in math. And I guess its always bothered me how we have to describe these everyday things with the math that's available to us. I suspect that there will come a time when they will be able to be described mathematically in ways that don't get burdened with elusive things like pi.

Disclaimer: Note that it's just a 'suspicion' of mine - it may never happen. I don't want anyone to get too excited here.

And then there are other things, and let me try and say this right, that just bug me and make me more suspicious, like the story earlier of 1/x and its cousin, 1/x^2. They kind of do the same thing, one's just's a bit quicker about it. But if they are inf series and integral tested for convergence we are to believe that one converges and the other doesn't. This appears to me to be a situation where it's not so much that 1/x doesn't converge, but the technique of integration gets wierd when we get close to messing with 1/x. It had to get its own definition here instead of being able to play along like its x^n "n not equal to -1" cousins.

There are a lot of these kinds of examples in math and when you sit back and look at it and makes you wonder: isn't there a better way to do all this stuff without all of these wierdo things going on?

Maybe not.

Anyway, I was just expressing this perspective here in this forum just to see, I guess, if others felt the same and maybe had other interesting examples.

Originally Posted by Ots

I had 'thrown in the towel' on this thread, Esbo, but since your posts aren't wrapped in criticism (which I appreciate) let me try to make some sense out of this.

I know the round things we see everyday aren't the same as circles defined mathematically. But they are familiar sights unlike lots of other stuff in math. And I guess its always bothered me how we have to describe these everyday things with the math that's available to us. I suspect that there will come a time when they will be able to be described mathematically in ways that don't get burdened with elusive things like pi.

Disclaimer: Note that it's just a 'suspicion' of mine - it may never happen. I don't want anyone to get too excited here.

And then there are other things, and let me try and say this right, that just bug me and make me more suspicious, like the story earlier of 1/x and its cousin, 1/x^2. They kind of do the same thing, one's just's a bit quicker about it. But if they are inf series and integral tested for convergence we are to believe that one converges and the other doesn't. This appears to me to be a situation where it's not so much that 1/x doesn't converge, but the technique of integration gets wierd when we get close to messing with 1/x. It had to get its own definition here instead of being able to play along like its x^n "n not equal to -1" cousins.

There are a lot of these kinds of examples in math and when you sit back and look at it and makes you wonder: isn't there a better way to do all this stuff without all of these wierdo things going on?

Maybe not.

Anyway, I was just expressing this perspective here in this forum just to see, I guess, if others felt the same and maybe had other interesting examples.

There is nothing "weird" going on. It is all perfectly logical. It is based on a very few axioms. basically the existence of the natural numbers and some elementary concepts involving sets.

What is missing is sufficient study and comprehensin on your part to understand the logic.

There are some things that you need to learn to understand what is going on. There are two very thin books that would do you a lot of good. One is Foundations of Analysis by Landau and the other is Naive Set Theory by Halmos. Those will show the basis on which all of modern mathematics is built.

Once you have that under you belt, try reading an introductory book on real analysis. Two good ones are Elements of Real Analysis by Bartle and the classic Principles of Mathematical Analysis by Rudin.

The fundamental problem is that at this stage it is abundantly clear that you have no idea what you are talking about. You need to get enough grounding in the basic theory to be able to pose a sensible question.

No, "common sense" won't get you there. What you think is "common sense" is actually nonsense.

Wow, Rocket. You really don't get it.

Originally Posted by Ots

Wow, Rocket. You really don't get it.

I don't think he is capabble of discussing anything outside the narrow view he got through wrote learning the subject. It means he has to think for hiself, and that I think purs him on shakey ground :wink:

I have noticed he is unable to give an answer on anything (unless it has already been given in the thread) and his ty[ical response is make out he understands it but you need to learn it yourself!! Typical of a fake.

Apparently so.

esbo and Ots: Your comments are unpleasant and inaccurate in equal measure. Please do not do this here.

I am leaving your posts up for now, so that your "attackee" (DrRockect) can respond, which I am confident he will do with his usual dignity and restraint.

But be warned - any further such posts will be removed sine die

Moderator

I think it best if you remove this thread completely. If you feel Dr. Rocket's responses were appropriate then there are lots of problems here that you aren't addressing.

I don't think he is capabble of discussing anything outside the narrow view he got through wrote learning the subject. It means he has to think for hiself, and that I think purs him on shakey ground

I have noticed he is unable to give an answer on anything (unless it has already been given in the thread) and his ty[ical response is make out he understands it but you need to learn it yourself!! Typical of a fake.

Man, its too bad that people who have the dilligence and discipline to read textbooks are labeled as fakes. Those who think the most, and those who have most active minds, are usually those who actually choose to read such books, and thoroughly understand the concepts that lie within them.

EDIT: Ots, my comment was directed towards esbo, not you. I have no problem at all with you. The fact that you started a thread indicates your desire to discuss math, learn concepts, etc. The problem here is that people who understand math at a very high level become irked when others ask questions that are somewhat illogical. I am definitely not some guy on the internet who wants to make you feel stupid, and to tell you the truth, I don't think Dr. Rocket is either. In fact, he recommended several good books for you to consider reading that give you a solid foundation in math, if you so desire.

Ots, responding to a mathematical question using one's knowledge of mathematics is appropriate. In fact, were you to ask the top mathematicians of today, many if not all of them would agree completely with Dr. Rocket. Your inability to accept what is standard is not our fault, nor will it ever be on us to accomadate the laws and regulations of mathematics for you. Oh, and just fyi on the integration deal, the general power rule is and if , we get which is undefined which means that it won't work for ANY kind of area under that curve That's the only reason it's special, and that kind of exception to the rule is everywhere, there's always an exception.

Originally Posted by SteveC

I don't think he is capabble of discussing anything outside the narrow view he got through wrote learning the subject. It means he has to think for hiself, and that I think purs him on shakey ground

I have noticed he is unable to give an answer on anything (unless it has already been given in the thread) and his ty[ical response is make out he understands it but you need to learn it yourself!! Typical of a fake.

Man, its too bad that people who have the dilligence and discipline to read textbooks are labeled as fakes. Those who think the most, and those who have most active minds, are usually those who actually choose to read such books, and thoroughly understand the concepts that lie within them.

Ain't that the truth. People with a passion to learn are disregarded by the willfully ignorant, Go figure

It's a common expectation of esbo by now to simply attack the standing of anyone who makes sense.

Thanks for input guys. I really do appreciate it. I am a member of a number of forums and I have to say I've not had an experience quite like this one. I tend to communicate in somewhat of a casual fashion and I think I'm at fault here as it's clearly not working well.

When I say things like "cousins of 1/x" and "gets wierd when we get close to messing with 1/x" I am making casual reference to what you are feeding back to me in a rigorous way. I could have stated it rigorously, but I didn't think it was necessary as you guys are clearly all up to speed.

But that's where I think I went wrong. You don't know me. I'm new to this forum. You don't know what I know or don't know, and my casual references could and probably have been interpreted in a fashion different from what I intended. I thought by saying things as I did it would be clear that I'm familiar with the subject matter at hand but I can see now that it did just the opposite.

My apologies to all (you too DrR).

Thanks for input guys. I really do appreciate it. I am a member of a number of forums and I have to say I've not had an experience quite like this one. I tend to communicate in somewhat of a casual fashion and I think I'm at fault here as it's clearly not working well.

When I say things like "cousins of 1/x" and "gets wierd when we get close to messing with 1/x" I am making casual reference to what you are feeding back to me in a rigorous way. I could have stated it rigorously, but I didn't think it was necessary as you guys are clearly all up to speed.

But that's where I think I went wrong. You don't know me. I'm new to this forum. You don't know what I know or don't know, and my casual references could and probably have been interpreted in a fashion different from what I intended. I thought by saying things as I did it would be clear that I'm familiar with the subject matter at hand but I can see now that it did just the opposite.

My apologies to all (you too DrR).

Excellent post sir.

Originally Posted by Guitarist

esbo and Ots: Your comments are unpleasant and inaccurate in equal measure. Please do not do this here.

I am leaving your posts up for now, so that your "attackee" (DrRockect) can respond, which I am confident he will do with his usual dignity and restraint.

But be warned - any further such posts will be removed sine die

Moderator

Obviously you are entitled to your opinion, howerever mine is was that Rockets answer was rude and unnhelpful and basically just his opinion.
I think he has a valid point that were cannot calculate Pi accurately.

For Dr Rocket to say the answer is PI is not very helpful, especially when the Question was "What is PI?".

Originally Posted by Ots

Thanks for input guys. I really do appreciate it. I am a member of a number of forums and I have to say I've not had an experience quite like this one. I tend to communicate in somewhat of a casual fashion and I think I'm at fault here as it's clearly not working well.

When I say things like "cousins of 1/x" and "gets wierd when we get close to messing with 1/x" I am making casual reference to what you are feeding back to me in a rigorous way. I could have stated it rigorously, but I didn't think it was necessary as you guys are clearly all up to speed.

But that's where I think I went wrong. You don't know me. I'm new to this forum. You don't know what I know or don't know, and my casual references could and probably have been interpreted in a fashion different from what I intended. I thought by saying things as I did it would be clear that I'm familiar with the subject matter at hand but I can see now that it did just the opposite.

My apologies to all (you too DrR).

explanations are always helpful. Rigorous explanations aren't always necessary, but having some mathematical content or at least a little mathematical explanation will always clarify as to what you mean. You are right though, you are new and we have no idea if you know what you are talking about or just being silly like some other members that join, seemingly just to upset people, and it's easy to confuse those who are, how do you say, "too familiar" with a topic for those who know absolutely nothing about it.

SteveC and A-Math: thanks. I will try to do a better job as we go forward.

Esbo - dude, try and keep it positive. It works better when we do so. Really.

Originally Posted by Ots

SteveC and A-Math: thanks. I will try to do a better job as we go forward.

Esbo - dude, try and keep it positive. It works better when we do so. Really.

I was trying to be positive, I thought your question was a perfectly valid and reasonable one. I don't think 'come back when you know when you are talking about' is a very positive response!!

I understand.

And the real issue is we don't know yet what you understand, but I'm sure we will soon

esbo, we already know what you understand, and know when to avoid your ramblings like the plague.

I was just responding to esbo.

So anyway, moving right along, let's go back to the main topic.

Here's what I understand. Pi is the ratio between the circumference of a circle and its diameter. So pi = C/D is the formula for pi. I further understand that there have been, over the years, lots of different formulae used for computational purposes, but there is no formula that yields an exact answer other than when pi is represented symbolically. That is pi exactly. It is not unique, as a formula for pi, but it is exact. Other exact formulae exist as well but as far as I know none yields an exact decimal representation of pi. This is because pi is a member of the irrational numbers which are part of our real number system and therefor does not lend itslf to an exact decimal representation, similar to other irrational numbers.

Any corrections so far? 'Nitpicking' is OK!

I think we're all on the same page here, yeah. the exact circumference of a circle divided by it's exact diameter would, indeed, be pi.

Originally Posted by Arcane_Mathematician

I think we're all on the same page here, yeah. the exact circumference of a circle divided by it's exact diameter would, indeed, be pi.

Hope no one was under the impression that it was the approximate circumference divided by the approximate diameter.

Originally Posted by salsaonline

Originally Posted by Arcane_Mathematician

I think we're all on the same page here, yeah. the exact circumference of a circle divided by it's exact diameter would, indeed, be pi.

Hope no one was under the impression that it was the approximate circumference divided by the approximate diameter.

I think maybe some people are dealing with the approximate diameter of an approximate circle to get an approximate (pumplkin) pie.

So, my circle of one foot diameter has the exact circumference of pi feet.

This isn't much different than my square with one foot sides having a diagonal of 2^1/2 feet I suppose, but circles are more important than squares! Lol!

So, the math is clear but the result is goofy. Pi feet. And no one can give me the formula so I can figure out the length of my circumference. I can only be given a formula that will approximate it.

Now that the facts are laid out (please correct any of the above if I have not said something correctly), I form the following judgement: I think there is a failure here. I am unable to represent the circumference of my circle with other than an approximation or the symbol pi.

Now, we can live with it (obviously). But it just seems like there has to be a better way.

Invent your own universe then with its own unique system of logic. Barring that, I think you're stuck with the geometry we're given.

Originally Posted by Ots

So, my circle of one foot diameter has the exact circumference of pi feet.

Precisely

Originally Posted by Ots

This isn't much different than my square with one foot sides having a diagonal of 2^1/2 feet I suppose, but circles are more important than squares! Lol!

So, the math is clear but the result is goofy. Pi feet. And no one can give me the formula so I can figure out the length of my circumference. I can only be given a formula that will approximate it.

You are missing the point. THE formula for the circumference is .
Pi is a perfectly good real number.

Originally Posted by Ots

Now that the facts are laid out (please correct any of the above if I have not said something correctly), I form the following judgement: I think there is a failure here. I am unable to represent the circumference of my circle with other than an approximation or the symbol pi.

Now, we can live with it (obviously). But it just seems like there has to be a better way.

You are again missing the pont. You are stuck in the paradigm of the ancient Greeks who where upset when it was proved that is not a ratinal number. It is simply the case that not all numbers are rational numbers. In fact most real numbers are not rational.

If we had only the rational numbers to work with then most of algebra, caluculus, trigonometry, and modern mathematics would simply not work.

Irrational and complex numbers play a major role in electrical circuit analysis, and without them you would not be discussing this on the internet.

Let's try this. Consider these two sentences:

1) five = 5

2) pi =

Now, you are probably more confortable with 1) than with 2). You shouldn't be. They are both correct, and they are both equally abstract. Pi is just as valid a real number as is five. Just like five, pi has a symbol that is used to designate it in common useage. One symbol is written "5" and the other symbol is written . But both numbers are valid and both symbols are valid. You are simply psychologically attuned to be comfortable with "5". Work on it a bit and you will also become comfortable with .

I wasn't clear about the formula. I was refering to a formula such that I could compute pi. It just so happened in this case that pi is also the circumference. My point being there is no formula with which to compute pi in decimal terms. Just ones that approximate doing so.

And I have no problem with the math as such. It is what it is, be it pi, 5, or (-4)^1/2 = 2i. I'm standing back from it all and wondering, philosophically, just why is it that a circle and its diameter have such a curious relationship. Elegant solutions are often simple and this one seems 'burdened.' I really don't have a problem at all with doing this math or anything else I've done.

The point I was trying to make earlier in this thread, and did a terrible job of it, has to do with integration definitions. Integration is defined for all x^n except n = -1 by using the formula x^n+1/n+1. The reason x cannot equal -1 is obvious.

So the integral of 1/x was defined as ln x + C (right now I'm going to ignore the integration constant as it's not relevant). So here we have this cool formula and we can integrate to our hearts' content and the one case where the formula's not defined we use ln x instead. All fine and dandy.

We then learn to test for convergence of a series by using the integral test (and other things too, but just follow me here). Ok, so we test various series for convergence and we find that 1/x^3 is convergent. We find that 1/x^2 is convergent. But we find, due to the definition learned earlier, that 1/x is not convergent. ln x grows without bound as x increases.

Just like before, the math is the math and that's that.

But if you step back and think about it, 1/x^3, 1/x^2 and 1/x all approach the x axis as x grows. And all 1/x^n converges except when n = 1.

It is not convergent because there had to be a special definition for integrating 1/x. So, think about this family of curves. They all approach the x axis as x increases. They all test positive for convergence - except the one that had to have the special definition.

I hope you get my point. The math is what it is. I accept that. And when I integrate 1/x I get ln x like everyone else. But I look back at that convergence test and say to myself that 1/x is considered to not converge not because it graphically does anything radically different from the other 1/x^n's, but because it had to get tagged with that special definition to avoid zero in the denominator of the integration formula.

So, (and I'm laughing as I say this), I think 1/x is as covergent as any of the others. It just doesn't test out that way!

I hope you can at least see where I'm coming from. I'm not confused by the math - I just think things came out certain ways because of other problems. So it gave us some goofy outcomes here and there.

The test for divergence is, more specifically, about the sum of the series, the integral, not the whether or not the function converges on a value. Likewise, the integral of the function f(x)=5 is divergent, even though the function is constantly 5. The test has nothing to do with how the function behaves, but how the infinite sum of the values of the function behaves as the function is allowed to become arbitrarily large.

Originally Posted by Ots

I wasn't clear about the formula. I was refering to a formula such that I could compute pi. It just so happened in this case that pi is also the circumference. My point being there is no formula with which to compute pi in decimal terms. Just ones that approximate doing so.

And I have no problem with the math as such. It is what it is, be it pi, 5, or (-4)^1/2 = 2i. I'm standing back from it all and wondering, philosophically, just why is it that a circle and its diameter have such a curious relationship. Elegant solutions are often simple and this one seems 'burdened.' I really don't have a problem at all with doing this math or anything else I've done.

The point I was trying to make earlier in this thread, and did a terrible job of it, has to do with integration definitions. Integration is defined for all x^n except n = -1 by using the formula x^n+1/n+1. The reason x cannot equal -1 is obvious.

So the integral of 1/x was defined as ln x + C (right now I'm going to ignore the integration constant as it's not relevant). So here we have this cool formula and we can integrate to our hearts' content and the one case where the formula's not defined we use ln x instead. All fine and dandy.

We then learn to test for convergence of a series by using the integral test (and other things too, but just follow me here). Ok, so we test various series for convergence and we find that 1/x^3 is convergent. We find that 1/x^2 is convergent. But we find, due to the definition learned earlier, that 1/x is not convergent. ln x grows without bound as x increases.

Just like before, the math is the math and that's that.

But if you step back and think about it, 1/x^3, 1/x^2 and 1/x all approach the x axis as x grows. And all 1/x^n converges except when n = 1.

It is not convergent because there had to be a special definition for integrating 1/x. So, think about this family of curves. They all approach the x axis as x increases. They all test positive for convergence - except the one that had to have the special definition.

I hope you get my point. The math is what it is. I accept that. And when I integrate 1/x I get ln x like everyone else. But I look back at that convergence test and say to myself that 1/x is considered to not converge not because it graphically does anything radically different from the other 1/x^n's, but because it had to get tagged with that special definition to avoid zero in the denominator of the integration formula.

So, (and I'm laughing as I say this), I think 1/x is as covergent as any of the others. It just doesn't test out that way!

I hope you can at least see where I'm coming from. I'm not confused by the math - I just think things came out certain ways because of other problems. So it gave us some goofy outcomes here and there.

No, you are very confused about the math. You simply don't recognize how confused you are.

Let's start with definite integrals.



is defined in a perfectly rigorous manner for a general class of functions. For the case of the ordinary Riemann integral of elementary calculus this class of functions are those that are Riemann integrable, and it is proved in more advanced classed that these are precisely the functions for which the set of discontinuities has Lebesgue measure 0.


In particular is perfectly well defined for any value of x. There is no problem whatever with the integral being defined or existing.

It turns out from the fundamental theorem of calculus that if but that formula simply does not make sense or apply if .

The integral is not defined by formulas, but rather it is defined as a limit of expressions that mimic "area".

Now, when you look at an integral like the issue is not whether as but rather it is a question of whether decays to fast enough for the area under the curve to remain finite, despite the fact that you are integrating over the entire right half of the (infinite) real line.

There is no special definition for . It is just a mathematical fact that it does not decay fast enough for the integral to converge when you integrate over the entire real line, and it is just a fact that the indefinite integral is not a rational function of x. It turns out to be a very important function, the natural logarithm. In fact the DEFINITION of the natural logarithm is

and it is the natural logarithm that is being defined, NOT the integral.

You seem to have a deep problem in that you think you understand things that you do not understand at all.

Mathematics is not about formulas. It is about concepts and order. You need to read some of those books that I named earlier. Your misconceptions regarding mathematics are getting in the way of your learning the subject.

Although it's perfectly fine, from a mathematical standpoint, to define

,

I've always found this definition slightly dishonest from a pedagogical viewpoint. I mean, we're all introduced to ln(y) as the inverse function of e^x. So I think it makes sense to stick with that definition and derive the above formula as a consequence. I think the reason people don't want to do this is because you'd have to go through the process of verifying that the inverse function of a differentiable function is differentiable. Well, to those concerned with such things, I say: man up!

Originally Posted by salsaonline

Although it's perfectly fine, from a mathematical standpoint, to define

,

I've always found this definition slightly dishonest from a pedagogical viewpoint. I mean, we're all introduced to ln(y) as the inverse function of e^x. So I think it makes sense to stick with that definition and derive the above formula as a consequence. I think the reason people don't want to do this is because you'd have to go through the process of verifying that the inverse function of a differentiable function is differentiable. Well, to those concerned with such things, I say: man up!

Actually the very first time I saw the logarithm was in high school and it was introduced as the area under !/x, and we did not have the benefit of calculus. Only later did we get to the exponential.

Yoiu can, of course, go at it from either direction.

From a pedagogical point of view I think the best approach depends on the audience. Doing the exponential first is more satisfying with a sophisticated audience. But you have to start with the exponential function, not where e is some number "multiplied times itself x times". That means defining the exponential with the power series, which is perhaps a bit tough on the average freshman.

Hmm, good point.

That's one thing I don't like about math sometimes. Very often you end up going through all kinds of silly gymnastics to introduce a topic simply because your audience isn't sophisticated enough to take the most direct route.

DrR - I think you've just re-stated what I said before (plus some) and for some reason you feel the need to include critical remarks in your responses which have no place in forums - but if you haven't learned that by this age you probably never will.

My main point is that there are some curious things in math, even if you feel otherwise, and I find it interesting to discuss them.

Originally Posted by Ots

DrR - I think you've just re-stated what I said before (plus some) and for some reason you feel the need to include critical remarks in your responses which have no place in forums - but if you haven't learned that by this age you probably never will.

No, I idid not just re-state things that you had already said.

I tried to help you. You quite clearly have a set of misconceptions that is getting in the way or your ability to understand and learn mathematics.

Your misconceptions can be overcome, but only if you want to overcome them. The first step is to recognize that you need to learn some things.

Apparently you are not interested in learning those things that you need to know and developing the necessary skills and insights.

You dampen the spirit of participation.

Ots, I think I understand what you're getting at.

I think you're saying this: we know that for , we have a nice family of antiderivatives for x^n, namely



But something weird happens at n = -1: namely, this ln(x) function pops up all of a sudden. So what gives, right? It almost appears like there's some sort of discontinuity in the family of anti-derivatives for x^n.

But this can be explained rather easily. First things first though: If f_n(x) are any collection of functions with anti-derivatives F_n(x), then we can always find a family of anti-derivatives G_n(x) that is "singular" at n=-1 simply by defining G_n(x) = F_n(x)+1/(n+1). So there is nothing surprising about having a family of anti-derivatives with this kind of singularity. But maybe we can gain some ground by subtracting off 1/(n+1) whenever we encounter this type of problem:

Let us therefore consider instead the family of antiderivatives



These are still anti-derivatives for x^n. But now we can actually take the limit as n-->-1, at least when x > 0. Let us do that now:

Since x > 0, we can write . We then want to take the limit of



as (n+1)-->0. Expanding it is easy to see that this limit is equal to y. But means that .

Going back to the expressions



we see that these are nothing less than



So for n=-1, we do indeed get the integral expression for ln(x).

Hope this clears up what's going on "behind the scenes" of these concepts.

Originally Posted by Ots

You dampen the spirit of participation.

Sorry if your feelings are hurt.

But there are probably more innocent lurkers in a forum like this than there are active participants. Some of them are actually interesting in learning something and they deserve to see accurate information.

If that requires correcting inaccurate statements and incoherent babble then so be it.

IF it were not for people like that then there would be no point whatever in addressing posts from esbo, for instance.

Salsaonline - that's quite nice. I will need a bit of time to digest it.

And your really close to what I was getting at. My point wasn't so much about what happens to the family of antiderivatves when we get to 1/x, but because of what happens, the family of series 1/x^2, 1/x^3, etc., all converge except 1/x according to the integral test.

So I'm sitting there looking at this family of curves, graphically. They all do the same thing: approach the x axis as x grows. But 1/x doesn't converge. Why? Because of how 1/x had to be treated with respect to integration.

So, 1/x^2 converges but 1/x grows without bound (the area under the curve). Baloney.

DrR mentioned that the area didn't decay fast enough compared to the others. I don't buy it 'cuz it's not time based.

What's happening here is poor old 1/x got tagged as not converging because of how the integral of it got defined.

This is what it seems like to me.

DrR - please, my feelings aren't hurt. I just think it most appropriate to respond objectively to the subject matter at-hand and not make personal attacks on the contributors. That encourages participation!

Originally Posted by Ots

Salsaonline - that's quite nice. I will need a bit of time to digest it.

And your really close to what I was getting at. My point wasn't so much about what happens to the family of antiderivatves when we get to 1/x, but because of what happens, the family of series 1/x^2, 1/x^3, etc., all converge except 1/x according to the integral test.

So I'm sitting there looking at this family of curves, graphically. They all do the same thing: approach the x axis as x grows. But 1/x doesn't converge. Why? Because of how 1/x had to be treated with respect to integration.

So, 1/x^2 converges but 1/x grows without bound (the area under the curve). Baloney.

DrR mentioned that the area didn't decay fast enough compared to the others. I don't buy it 'cuz it's not time based.

What's happening here is poor old 1/x got tagged as not converging because of how the integral of it got defined.

This is what it seems like to me.

DrR - please, my feelings aren't hurt. I just think it most appropriate to respond objectively to the subject matter at-hand and not make personal attacks on the contributors. That encourages participation!

I did make any personal attack on you.

I simply pointed out that you don't know what you are talking about and are apparently determined to not learn mathematics.

The objective answers to your questions have been posted --- many times.

There is NO issue with the definition of the integral, or the reason why the logarithm is unbounded.

There is nothing whatever wrong with the mathematics. And there is nothing wrong with the integral of .

What is amiss is your understanding of what an integral is and why the integral of diverges. This has nothing whatever to do with the "integral test", since we are talking about an integral in the first place. The "integral test" is a test for the convergence of series, and in fact the integral of is used to prove that the harmonic series diverges. You need to know about logarithms and BEFORE you can apply the integral test in that situation.

The theory of the integral, in the normal development, is NOT based on anti-derivatives. It is based on Riemann sums, or in more advanced work on the general theory of measure and integration. Only after the integral is developed, and one proves the fundamental theorem of calculus, do anti-derivatives come into the picture.

The fact that the integral of 1/x diverges and you "don't buy it" is simply proof that you 'don't get it". The divergence is an inevitible consequence of logic, and it is NOT a definition.

There is NOTHING different about how 1/x is treated in the integral as compared to . That is the whole point. The techniques of integration treat all integrable functions in the same way. Some of them result in convergent integrals and some don't. doesn't converge either.

The graphs of and DO NOT "do the same thing". The graph of always below the graph of for and that is why one integral converges and the other diverges. This is what is meant by "decays faster" than . It has nothing to do with time. Note that neither one has a convergent integral on [0.1].

You need to learn some basic mathematics, and that mathematics is well presented in the books that I suggested miuch earlier in this thread.

Until you get some basic grounding in mathematics what you have been told by competent mathematicians (salsaonline and I are both real mathematicians) will just go totally over your head. That is precisely what has happened so far.

Actually, the above is a pretty good example of a personal attack.

And your approach and style keep people from posting here I'm sure.

And it's a shame, DrR, as your knowledge is clearly valuable in a forum like this. It's too bad you can't seem to adopt a more supportive approach. I'm sure the forum would be a more active site if your style were different.

Originally Posted by Ots

Actually, the above is a pretty good example of a personal attack.

And your approach and style keep people from posting here I'm sure.

And it's a shame, DrR, as your knowledge is clearly valuable in a forum like this. It's too bad you can't seem to adopt a more supportive approach. I'm sure the forum would be a more active site if your style were different.

Go back and read the thread from the start.

You might note your "redneck" comments.

Ots, DrRocket is taking the subject matter more seriously than you might think, and that's a big contributor to the posting style and behavior you're noting. The 'personal attacks' appear after the same posting style was adopted by you and others. Trust me, it's not a personal issue nor a blatant ad hominem, it's more a way to ensure that the proper method for doing math and the proper explanations are the ones given precedence, instead of the ideals of someone who, in all right, may not really know what the hell they're taking about.

Originally Posted by Ots

And then there are other things, and let me try and say this right, that just bug me and make me more suspicious, like the story earlier of 1/x and its cousin, 1/x^2. They kind of do the same thing, one's just's a bit quicker about it. But if they are inf series and integral tested for convergence we are to believe that one converges and the other doesn't. This appears to me to be a situation where it's not so much that 1/x doesn't converge, but the technique of integration gets wierd when we get close to messing with 1/x. It had to get its own definition here instead of being able to play along like its x^n "n not equal to -1" cousins.

Aha, I think I finally see what your objection is. Okay, first off, your suspicion is unfounded. But let's explore why.

It seems like what troubles you is that when we take the anti-derivative of x^n, for integers n not equal to -1, we have this nice polynomial expression. But when we take the anti-derivative of x^{-1}, we suddenly get a completely different kind of function.

This leads you to suspect that we're giving a whole different definition for



This is not the case: has a definition (loosely interpreted as the area under the graph of f(x)) that makes no reference to anti-derivatives.

Still, you may still wonder why the anti-derivative of 1/x is so unique.

It's actually no surprise that logarithms appear when we discuss things like powers of x, because to even define an arbitrary real number we have to use the natural logarithm. That is, by definition,



It's easy to see that this coincides with the definition for x^n when n is an integer. But we can't interpret expressions like as multiplying x by itself -many times. But we can give a meaning to such expressions using the logarithm.

(Note: You might think I'm cheating, since I'm taking e to different powers. But e^x is just short-hand for



which involves only integer powers of x.)

So when we say that the anti-derivative of x^r is x^(r+1)/(r+1), remember that for arbitrary reals r, we have to write this expression as

Salsaonline - this is the real issue:

What's happening here is poor old 1/x got tagged as not converging because of how the integral of it got defined.

The issue is concluding that the series 1/x, when tested for convergence, does not converge.

Originally Posted by Ots

Salsaonline - this is the real issue:

The issue is concluding that the series 1/x, when tested for convergence, does not converge.

You can show that the series



diverges without invoking the divergence of



See this Wiki article for several proofs that do not rely on the integral at all.
http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

This has nothing to do with the integral test for divergence of a series. In fact using these results it is trivial to show that diverges since it dominates the series .

So, one more time. There is nothing weird, or strange about . It is the DEFINITION of one of the most fundamental functions in mathematics, the natural logarithm. Note that the function ln is defined by this expression which is in terms of an integral but the integral exists quite independently. The integral is defined in terms of Riemann sums involving the function 1/x which is treated no differently than any other function.

Well, that's no fun!

I can't make any contribution other than to say I see the same problem as you; the inability to get a rational answer to a problem that should provide one (as the diameter of a circle is rational). It does seem to be a shortcoming...

Originally Posted by Olly

I can't make any contribution other than to say I see the same problem as you; the inability to get a rational answer to a problem that should provide one (as the diameter of a circle is rational). It does seem to be a shortcoming...

The diameter of a circle may be rational. It can be any real number.

The RATIO of the circumference of a circle to the diameter is a single fixed number, independing of the size of the circle. That number is pi. Pi is not rational. That is just the way that it is. It is a beautiful thing.

Dr. Rocket,

I'm well aware that pi is irrational. However, the definition of a rational number is one that we can be written as the ratio of two numbers. If we measure the circumference of a two-dimensional manifold in the form of a circle, and than its diameter, and than divide the circumference by the diameter, we recieve pi. I assume the irrationality of pi is somehow a byproduct of calculus (with regard to euclidean geometry). Am I correct here?

Originally Posted by Ellatha

Dr. Rocket,

I'm well aware that pi is irrational. However, the definition of a rational number is one that we can be written as the ratio of two numbers. If we measure the circumference of a two-dimensional manifold in the form of a circle, and than its diameter, and than divide the circumference by the diameter, we recieve pi. I assume the irrationality of pi is somehow a byproduct of calculus (with regard to euclidean geometry). Am I correct here?

A rational number is the ratio of two integers, not just two numbers.

A circle is not a two-dimensional manifold. It is a one-dimensional manifold. The dimension of a manifold is determined by the Euclidean space that the manifold "looks like" locally and a circle locally looks like a line.

If you take the circumference of a circle (not just any one-dimensional manifold) and divide by the diameter you get pi. This is due to Euclidean geometry in two-dimensions, and the definition of a circle. Perhaps itis a bit easier to deal with the area, which is in a Eudidean geometry but not in non-Eudlidean geometries (say on the surface of a sphere).

I don't know that I would say that the irrationality of pi is a by-product of calculus. It can be proved using only the methods of elementary calculus though. However, pi is not only irrational, it is transcendental. That means that it is not a zero of any polynomial with rational coefficients. on the other hand, is irrational but it is also algebraic, a zero of the polynomial . The proof that pi is transcendental is quite a bit more complicated.

It is because pi is transcendental that it is not possible to construct a line segment of length pi (given a length defined to be "1"), given only an unmarked straight edge and a compass. It is quite possible to construct . You just construct an isosceles right triangle, with the two sides of length "1" and the hypotenuse is .

A circle is not a two-dimensional manifold. It is a one-dimensional manifold. The dimension of a manifold is determined by the Euclidean space that the manifold "looks like" locally and a circle locally looks like a line.

I was under the impression that a manifold's dimension is dependant on the dimension of the plane that it could be put on. For example, a sphere can be drawn on a piece of paper (the paper representing the standard [x,y] coordinate plane), and thus was as two-dimensional manifold. The circle that I stated previous could have been anything that was three-dimensional that takes the form of a circle when graphed in euclidean space (such as, once again, a sphere).

I was always under the impression the ratio of the circumference of a circle to its radius was twice pi? I don't know much about non-euclidean geometry other than something a friend of mine told me about convex optimization and hyper and half planes.

I understand now though--a circle will never have an integer circumference and diameter (and thus radius). Actually, I believe the length of a circle's circumference might actually always be an infinitisemal as well. This is because whenever you multiply a number by a particular number it will always increase by that percentage. Thus, when we multiply a circle's diameter by 3.14159.... its circumference will be 314.159...% greater and yield an infinitisemal value. I'm really not too sure here.

Originally Posted by Ellatha

A circle is not a two-dimensional manifold. It is a one-dimensional manifold. The dimension of a manifold is determined by the Euclidean space that the manifold "looks like" locally and a circle locally looks like a line.

I was under the impression that a manifold's dimension is dependant on the dimension of the plane that it could be put on. For example, a sphere can be drawn on a piece of paper (the paper representing the standard [x,y] coordinate plane), and thus was as two-dimensional manifold. The circle that I stated previous could have been anything that was three-dimensional that takes the form of a circle when graphed in euclidean space (such as, once again, a sphere).

A manifold is an object that is locally isomorphic to some Euclidean space. It is the dimension of that Eudlidean space that determines the dimension of the manifold. So. a circle is one-dimensional. A sphere is 2-dimensional. etc.

The dimension of a manifold is independent of any space in which one might find it embedded. You can embed a circle in 2-space or any n-space with n>1.

When you say "plane" that implies a 2-dimensional plane. If you mean just any old Euclidean space then sometimes that is called a "hyperplane".

I don't understand the last sentence.

I was always under the impression the ratio of the circumference of a circle to its radius was twice pi? I don't know much about non-euclidean geometry other than something a friend of mine told me about convex optimization and hyper and half planes.

You are correct. I thought "diameter" and typed "radius". I have edited the post to fix that error. Thanks.

You probably know more about non-Euclidean geometries than you think you do. The sphere, with lines being great circles is a non-Euclidean geometry. All lines intersect in that case.

I understand now though--a circle will never have an integer circumference and diameter (and thus radius). Actually, I believe the length of a circle's circumference might actually always be an infinitisemal as well. This is because whenever you multiply a number by a particular number it will always increase by that percentage. Thus, when we multiply a circle's diameter by 3.14159.... its circumference will be 314.159...% greater and yield an infinitisemal value. I'm really not too sure here.

No. That doesn't work. In most useage there really is no such thing as an infitesimal, no matter how often you hear the idea used, and physicists use it quite a bit. It is a help in motivating some derivations, but rigorous mathematics avoids that sort of thing, with one exception.

There is a way to make infitesimals rigorous. It is found in the work of Abraham Robinson and is called "non-standard analysis". But to do that you have to use some relatively sophisticated mathematics and construct a new number system -- the non-standard real numbers. There are some generalizations of this work as well and some relatively strange number systems that result. But unless and until you get some pretty advanced education in mathematics it is best to avoid that stuff. Even if you get that advanced education you probably won't use those methods unless you become a specialist in non-standard analysis.

In the case of the diameter and circumference of a circle the relationship is quite simple. The two are proportional and the ratio is . If you multiply the diameter by 12 the circumference also increases by a factor of 12. No infitesimals anywhere, no matter what factor you use to multiply the diameter.

Originally Posted by DrRocket

So. a circle is one-dimensional. A sphere is 2-dimensional. etc.

this only applies to the circumference and surface of a circle or sphere, right? I mean, a solid sphere is 3-dimensional and a filled in circle is 2-dimensional, right? Or, as is entirely possible, am I HORRIBLY missing something here?

Well, Arcane, what follows is rigourously non-rigourous, but it may help to guide your (and other's) intuition.

Consider a line, any line, as it is usually understood. We will assume that this line can be infinitely sub-divided, in which case we will call it the real line, and denote it by for now. Then we will assume that each element in this sub-division corresponds to a real number.

We now ask, how many real numbers do I need to uniquely identify a point on this line. The answer is, of course 1. Let us call this line as 1-dimensional, by virtue of this fact. Accordingly, I might as well call this line , where the superscript refers only to the dimensionality of this line.

Let us now take a segment of this line, and join it head-to-tail. We instantly recognize this as a circle. Obviously, the same applies; any point on this circle can be uniquely described by a single real element, and accordingly I will call this geometric object as 1-dimensional.

Mathmen use the symbol for this character, and call it the 1-sphere.

Now let's try and think about the "2-dimensional line", or 2-line. What can this mean (if anything)?

Well, using the above, we may assume that this is the "line" that requires two numbers to uniquely describe a point. From which we infer that the "2-line" is the plane.

We may also infer, from the above, that the 2-sphere is, in some abstract sense, a head-to-tail "joining" of a part of this plane.

The 2-sphere is merely some sort of jazzed up plane, that is, it knows nothing about the area/volume it may or may not enclose. The same applies to any n-sphere.

With a grinding of gears, let's now consider the area enclosed by the 1-sphere as defined above. Intuition tells us, in this case quite correctly, that it is part of the "2-line" i.e the plane. This part of the plane is usually referred to as the "disk" . It is, in fact, the 2-ball.

Similarly, the "area" enclosed by the 2-sphere is a part of the 3-plane - the 3-ball, and so on. Obviously, this latter "area" is the volume enclosed by the 2-sphere, from which we conclude that, provided we are allowed to think of area as a 2-volume, then the n-sphere encloses an n + 1 volume.

Thing to think about is whether or not the converse is true; for any "n-volume", do I need to have an enclosing n - 1 sphere? Or, if you prefer, do we require that every n-ball be topologically closed?

Hint: No.

Originally Posted by Arcane_Mathematician

Originally Posted by DrRocket

So. a circle is one-dimensional. A sphere is 2-dimensional. etc.

this only applies to the circumference and surface of a circle or sphere, right? I mean, a solid sphere is 3-dimensional and a filled in circle is 2-dimensional, right? Or, as is entirely possible, am I HORRIBLY missing something here?

It is just termoinology.

A filled circle is called a disk.

A filled sphere is called a ball.

This extends to higher dimensions also. You only talk about the circle in the usual sense; i.e. the 1-manifold.

But one does talk about spheres in higher dimensions. A sphere in any dimension is just the set of points that are equidistant from a fixed point, and usually one talks about the unit sphere which is the points a unit distance from the origin. Distance is obtained from the usual Euclidean norm.

Similarly in higher dimensins the unit ball is the set of points within a distance of 1 from the origin. The boundary of the unit ball is the unit sphere.

This is pretty standard, but sometimes terminology gets a little loose, so if you are not sure what someone is talking about, just ask.


One other thing that you might run into. The unit circle in the complex plane is somtimes denoted as and called either the "circle group" or the 1-torus. This is sometimes seen in treatments of harmonic analysis. The reason is that it forms a group under complex multiplication which is important in the theory of Fourier series. If you take successive direct sums of this group you get successively higher dimensional groups , and since is just usual torus is called the n-torus.

Originally Posted by Guitarist

Thing to think about is whether or not the converse is true; for any "n-volume", do I need to have an enclosing n - 1 sphere? Or, if you prefer, do we require that every n-ball be topologically closed?

Hint: No.

Hint taken.

Thank you Guitarist and DrRocket.

That all explained a bit for me.

OK, took a break and read as suggested.

In the context of Landau's book, a rational number is a rational cut for which there exists a least upper number (not an element of the cut).

All other cuts are irrational numbers.

Dedekind's Fundamental Theorem says (basically) that every real number gives rise to one of two situations. The real number in question divides all real numbers into two classes and it's a member of one class or the other.

In the case of a rational number, per the above, it's not a member of the lower class but it is a member of the upper class. And it's the least upper class number.

An irrational number would seem, then, to be of the sort that is a member of and the greatest member of the first, or lower class. The second class is just greater than this number.

Have I got it right?

Originally Posted by Ots

OK, took a break and read as suggested.

In the context of Landau's book, a rational number is a rational cut for which there exists a least upper number (not an element of the cut).

All other cuts are irrational numbers.

Dedekind's Fundamental Theorem says (basically) that every real number gives rise to one of two situations. The real number in question divides all real numbers into two classes and it's a member of one class or the other.

In the case of a rational number, per the above, it's not a member of the lower class but it is a member of the upper class. And it's the least upper class number.

An irrational number would seem, then, to be of the sort that is a member of and the greatest member of the first, or lower class. The second class is just greater than this number.

Have I got it right?

Not quite. But you are close.

What is going on is a rather subtle construction and argument, the result of which is the construction of the real numbers using nothing but the rational numbers, which have in turn been constructed from nothing more than the Peano Axioms, a little set theory and basic logic. Thus the real numbers themselves are shown to result from a purely logical argument that assumes nothing more than the Peano Axioms. That it the point.

Once you have constructed the real numbers, a complete Archimedian field, you will never again look at them in terms of cuts. Cuts are just a means to an end.

The basic idea is that the main difference between the rationals and the real numbers is the least upper bound property. The reals have it and the rationals do not. Cuts are the way to get to the least upper bound property.

So, if you already had the reals (you don't yet, but let's look ahead), a cut would be simply all of the rationals strictly less than some real number R. R is, of course, the least upper bound for the cut. If R is rational the cut is called rational. If R is irrational the cut is called irrational. And a cut will eventually be identified with its least upper bound in the real numbers. But we can't define it this way yet, because we don't have the real numbers at our disposal.

Not having the reals available, the process proceeds by defining cuts as is done in Landau's book. Then you put an order on the cuts and show that there is a way to add, subtract, multiply and divide cuts (except the 0 cut) and eventually show that when you view cuts as numbers that the set of cuts has the least upper bound property and that the cuts, viewed as numbers actually comprise a complete Archimedian field -- the real numbers as you have known them since high school. So you go through this rather subtle argument to get there, but the intuitive idea is as in the preceding paragraph.

I'll need a bit of time to digest this.

Originally Posted by Ots

I'll need a bit of time to digest this.

Take all the time you need. This is not a timed event.

You might think about it this way.

You already know the answer. It is the real numbers.

When Dedekind was putting together the construction that you are studying, he also knew the answer, in that the real numbers had been around and used for quite a while. What was missing was just the tie to the basic axioms. So look at the construction in light of what you know the answer to be. That will make it seem less mysterious.

What you are seeing is, in part, a window on how mathematics is really developed. There is a misconception that mathematics is deductifve. It really is not. But the proofs that you see are deductive.

Mathematics is done by first guessing the right answer or the correct theorem, and then showing that the guess is right. What you see in the theorems is only the part that is showing that the guess is right.

Intuition is a large part of mathematics. First you have to understand what is going on at an intuitive and "gut" level. Then you formalize it and back up the intuition with rigorous proof. But you never leave the intuition totally behind.

Circle is a polygon with an infinite number of angles. The formula for calculating the area of a circle is the formula for calculating the area of a polygon with an infinite number of angles. If you submit a circle as a polygon with a very large but finite number of angles, one can calculate the number to be close to the number pi, but it will be rational. The number pi is irrational and it is necessary to calculate area of a circle.

"Not quite. But you are close.

What is going on is a rather subtle construction and argument, the result of which is the construction of the real numbers using nothing but the rational numbers, which have in turn been constructed from nothing more than the Peano Axioms, a little set theory and basic logic. Thus the real numbers themselves are shown to result from a purely logical argument that assumes nothing more than the Peano Axioms. That it the point.

Once you have constructed the real numbers, a complete Archimedian field, you will never again look at them in terms of cuts. Cuts are just a means to an end.

The basic idea is that the main difference between the rationals and the real numbers is the least upper bound property. The reals have it and the rationals do not. Cuts are the way to get to the least upper bound property.

So, if you already had the reals (you don't yet, but let's look ahead), a cut would be simply all of the rationals strictly less than some real number R. R is, of course, the least upper bound for the cut. If R is rational the cut is called rational. If R is irrational the cut is called irrational. And a cut will eventually be identified with its least upper bound in the real numbers. But we can't define it this way yet, because we don't have the real numbers at our disposal.

Not having the reals available, the process proceeds by defining cuts as is done in Landau's book. Then you put an order on the cuts and show that there is a way to add, subtract, multiply and divide cuts (except the 0 cut) and eventually show that when you view cuts as numbers that the set of cuts has the least upper bound property and that the cuts, viewed as numbers actually comprise a complete Archimedian field -- the real numbers as you have known them since high school. So you go through this rather subtle argument to get there, but the intuitive idea is as in the preceding paragraph."



The overall method employed by Landau is clear, but I'm looking for verification that my understanding of how he uses cuts is correct.

I think I have a correct understanding of a rational cut as it's a set of rational numbers that's less than some rational number R which is not a member of the set. Other conditions include that the set is not empty, every member of the set is smaller than every nonmember and that there is no greatest member.

The rational numbers are then represented by these rational cuts for which there exists a least upper number (not a member of the set).

He then says all other cuts are irrational.

Then he gets to the Fundamental Theorem. And here he makes a remark that "Every real number Z gives rise to two such divisions: one where H <or= Z, and the second one where H < Z. This second one appears to be the rational cut case as the least upper number is not a member of the set. The first case seems to be the irrational cut as there is no least upper number not a member of the set.

OK so far?

Originally Posted by Ots

Then he gets to the Fundamental Theorem. And here he makes a remark that "Every real number Z gives rise to two such divisions: one where H <or= Z, and the second one where H < Z. This second one appears to be the rational cut case as the least upper number is not a member of the set. The first case seems to be the irrational cut as there is no least upper number not a member of the set.

OK so far?

Cuts, once you have the reals are simply sets of the form If then the cut is rational and if is irrational then the cut is irrational.

In no case is the least upper bound and element of the cut. In the case of a rational cut there is a least upper bound in the rational numbers. In the case of an irrational cut the cut is bounded but, viewed solely in the context of the rational numbers, there is no least upper bound.

Hmmm...something's not sitting well with me on this. I'll post back in a day or so.

If you hate that you just can’t write pi 100% specific, maybe this will put your thoughts to rest. Pi can be calculated 100% precisely, using a range of different formulas. Some of them is (as pi) non-ending, so they might don’t help you in this case. The best one I can think of is this:

I assume you have herd of ln, the natural logarithm, before. 'i' may be unfamiliar to you, though. 'i' is the imaginary unit, which is equal to the square root of -1. I don’t think you quite understand the formula, but that doesn’t matter. The point is that an exact description of pi is possible (!), as you see above. It just isn’t if you want to write it numerically… Is this reply satisfying in any way?

Originally Posted by Ots

Maybe, but I really didn't expect the 'this is the way it is, it's been that way for two thousand years and we don't need to change it' kind of attitude. Seems a little redneck.

I guess I was thinking that this thread would generate more of the "inadequacies" that exist in our math system. Like yeah, that pi thing - we don't even have a formula to compute it. What about this other thing - you know, where 1/x^2 converges but 1/x doesn't - let's talk about that, etc.

Like a thread where counter-intuitive things would be discussed perhaps.

Maybe you should go to the philosophy forum.

Originally Posted by jammer

Originally Posted by Ots

Maybe, but I really didn't expect the 'this is the way it is, it's been that way for two thousand years and we don't need to change it' kind of attitude. Seems a little redneck.

I guess I was thinking that this thread would generate more of the "inadequacies" that exist in our math system. Like yeah, that pi thing - we don't even have a formula to compute it. What about this other thing - you know, where 1/x^2 converges but 1/x doesn't - let's talk about that, etc.

Like a thread where counter-intuitive things would be discussed perhaps.

Maybe you should go to the philosophy forum.

I think he has had an epiphany since he wrote that.

Originally Posted by jmd_dk

If you hate that you just can’t write pi 100% specific, maybe this will put your thoughts to rest. Pi can be calculated 100% precisely, using a range of different formulas. Some of them is (as pi) non-ending, so they might don’t help you in this case. The best one I can think of is this:

I assume you have herd of ln, the natural logarithm, before. 'i' may be unfamiliar to you, though. 'i' is the imaginary unit, which is equal to the square root of -1. I don’t think you quite understand the formula, but that doesn’t matter. The point is that an exact description of pi is possible (!), as you see above. It just isn’t if you want to write it numerically… Is this reply satisfying in any way?

Unfortunately to make sense of that formula you must pick a branch of the logarithm. Picking a branch of the logarithm is essentially choosing what you mean by the argument of a complex number -- writing it in polar form and deciding which angle to use (they are equivalent mod 2 pi). So basically you need to know about pi in order to make sense of ln. So, as a means of understand pi, that formula is rather circular.

Originally Posted by DrRocket

So, as a means of understand pi, that formula is rather circular.

No pun intended?