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Thread: A Theorem on The Euclidean circle

  1. #1 A Theorem on The Euclidean circle 
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    Pi is generally defined as the ratio of a circle's circumference to its diameter. It has been proven via abstract algebra that pi is transcendental, which implies that it can not be the solution of a polynomial with rational coefficients (and therefore not algebraic).

    It must therefore follow that for any given circle, either the circumference or the diameter is transcendental, if not both.

    For if one divides or multiplies a transcendental number by a real number greater than zero the result must be transcendental.


     

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  3. #2 Re: A Theorem on The Euclidean circle 
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    Quote Originally Posted by Ellatha
    Pi is generally defined as the ratio of a circle's circumference to its diameter. It has been proven via abstract algebra that pi is transcendental, which implies that it can not be the solution of a polynomial with rational coefficients (and therefore not algebraic).

    It must therefore follow that for any given circle, either the circumference or the diameter is transcendental, if not both.

    For if one divides or multiplies a transcendental number by a real number greater than zero the result must be transcendental.
    Wrong.

    Edit: The next-to-last sentence is not wrong, just utterly trivial. The logic below addresses the statement that "either the circumference or the diameter is transcendental, but not both"which is what I had originally thought the OP had said." The fact that the circumference and diameter cannot both be algebraic is evident since the ratio of the two, is transcendental. Nevertheless the last sentence, "For if one divides or multiplies a transcendental number by a real number greater than zero the result must be transcendental" is completely wrong".

    What is true is that algebraic numbers form a field, and from that one knows that
    the product of a transcendenntal number and an algebraic number is transcendental number is transcendental. But the prodect of two transcendental numbers can quite easily be algebraic.

    Now consider the set of all real numbers of the form where and let be the set of those for which is algebraic. Clearly and is countable since the set of algebraic numbers is countable.

    Since is transcendental it follows from the remark above that every element of is transcendental. Thus contains all positive real algebraic numbers. But the cardinality of is c, the cardinality of the real numbers, and so is the cardinality of the relative complement of in . Therefore there are elements in that are transcendental, and hence circles with diameters and circumferences that are both transcendental. In fact there must be uncountably infinitely many such elements.


     

  4. #3  
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    This post is less about transcendental numbers than it is on the properties of a circle. If not we have a number C that is not transcendental, the circumference of a circle, than it follows that the following holds true:



    Where d is transcendental, the diameter of the circle.

    Furthermore, if we assume that d is not transcendental, than the following holds true:



    Where C is transcendental.

    In the final case, let us assume that d and C are transcendental:



    There exists at least one case of d and C both being transcendental (by the fact that pi times e is transcendental [I know not the most rigorous proof]).

    Therefore, my theorem is the following:

    "In any given circle, in Euclidean space, either a circle's diameter or circumference is transcendental, if not both."
     

  5. #4  
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    Quote Originally Posted by Ellatha
    Therefore, my theorem is the following:

    "In any given circle, in Euclidean space, either a circle's diameter or circumference is transcendental, if not both."
    OK, I had misread this as "In any given circle, in Euclidean space, either a circle's diameter or circumference is transcendental, but not both."

    The statement that "In any given circle, in Euclidean space, either a circle's diameter or circumference is transcendental, if not both." Is true and an utterly trivial consequence of the facts that is transcendental and that the algebraic numbers form a field.

    The misinterpreted statement, though false, involves more interesting mathematics in showing that it is not true.


    However, your final assertion in the OP thar "For if one divides or multiplies a transcendental number by a real number greater than zero the result must be transcendental" is totally wrong and inconsistent with your trivial "theorem".
     

  6. #5  
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    Quote Originally Posted by DrRocket
    However, your final assertion in the OP thar "For if one divides or multiplies a transcendental number by a real number greater than zero the result must be transcendental" is totally wrong and inconsistent with your trivial "theorem".
    I should have said the following: a product is the result of multiplying two numbers, x and y, yielding a third z.



    If there exists at least one transcendental number in the above equation, at least one other must exist. I.e., if x is transcendental, than y or z must be as well. This is assuming that zero does not exist in the above equation, for if it did there would have to exist at least one other zero.

    The expression is a case of the two transcendental numbers existing without one being z.

    The relationship between a Euclidean circle's diameter and circumference is the following:



    Because one transcendental number exists, pi, at least one another must exist in the equation.
     

  7. #6  
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    Quote Originally Posted by Ellatha
    I should have said the following: a product is the result of multiplying two numbers, x and y, yielding a third z.



    If there exists at least one transcendental number in the above equation, at least one other must exist. I.e., if x is transcendental, than y or z must be as well.
    A trivial consequence of the fact that the algebraic numbers form a field and hence a ring.


    Quote Originally Posted by Ellatha
    This is assuming that zero does not exist in the above equation, for if it did there would have to exist at least one other zero.
    An unnecessary qualification. See above. More simply, just not that if one of x,y or z is 0 then two of them must be 0.

    Quote Originally Posted by Ellatha
    The expression is a case of the two transcendental numbers existing without one being z.
    This sentence is nonsensical. Gibberish.

    Quote Originally Posted by Ellatha
    The relationship between a Euclidean circle's diameter and circumference is the following:

    yep

    Quote Originally Posted by Ellatha
    Because one transcendental number exists, pi, at least one another must exist in the equation.
    Obviously. This is so obvious that it is not worth stating.

    The heart of the matter is that the algebraic numbers form a field. You seem to have completely missed that point. It takes some work to prove that fact.

    http://en.wikipedia.org/wiki/Resultant
     

  8. #7  
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    Quote Originally Posted by DrRocket
    A trivial consequence of the fact that the algebraic numbers form a field and hence a ring.
    Your point?

    Quote Originally Posted by DrRocket
    An unnecessary qualification. See above. More simply, just not that if one of x,y or z is 0 then two of them must be 0.
    This makes no sense.

    Quote Originally Posted by Ellatha
    This sentence is nonsensical. Gibberish.
    It makes perfect sense: a transcendental number exists in the equation, x, however it just so happens that the other transcendental number is not z, therefore making it y. The formula for the circumference of a circle does not allow this because the transcendental number that exists, pi, is fixed as y.

    Quote Originally Posted by DrRocket
    Obviously. This is so obvious that it is not worth stating.
    It is simple, but not obvious.

    Quote Originally Posted by DrRocket
    The heart of the matter is that the algebraic numbers form a field. You seem to have completely missed that point. It takes some work to prove that fact.

    http://en.wikipedia.org/wiki/Resultant
    Stop telling me what my thread is about and what I may have missed. It is my thread, therefore the basis of discussion is dictated by what I write, rather than what I do not write.
     

  9. #8  
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    Quote Originally Posted by Ellatha
    Stop telling me what my thread is about and what I may have missed. It is my thread, therefore the basis of discussion is dictated by what I write, rather than what I do not write.
    The discussion in any thread follows the dictates of logic and the subject matter, not the whims and delusions of the OP.

    The basic problem appears to be that you fail to understand the mathematics involved in the issues raised by your post. Thus, that which you did not write is both more interesting and more germane than that which you did write.

    Too bad. Maybe you should apply your (self-perceived) enormous intellect and actually learn some relevant mathematics.

    In this case the relevant subject matter is that of the field of algebraic numbers. The link that I provided represents just the tip of the iceberg in a large body of mathematical theory.
     

  10. #9  
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    I'm way smarter than Newton ever was. Got a heart attack yet? Ha.
     

  11. #10  
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    Quote Originally Posted by Ellatha
    I'm way smarter than Newton ever was. Got a heart attack yet? Ha.
    Then again I ask "Why do you say such stupid things ?"
     

  12. #11  
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    Quote Originally Posted by DrRocket
    Then again I ask "Why do you say such stupid things ?"
    You have not been able to identify anything "stupid." 90% of the time you merely say "you are correct, but your statements are utterly trivial."

    And yet mathematical precision is often binary: one is either correct or incorrect, rigorous or intuitive, trivial or relevant, right or wrong, etc... yet you seem to have no problem applying as many adjectives to these conceptions as you'd like: saying you are utterly, completely, fully, without a doubt wrong is no different from saying you are wrong. So why do you do so? Because it makes you feel good.

    Perhaps you can apply your (self perceived) enormous understanding of mathematics to solving some of the more sublime problems rather than trying to classify trivial and non-trivial material.
     

  13. #12  
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    Quote Originally Posted by Ellatha
    I'm way smarter than Newton ever was. Got a heart attack yet? Ha.
    !!!!!!! you are the second person I have heard saying this lol.
    ------------------




    "Mathematicians stand on each other's shoulders."- Carl Friedrich Gauss


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    Quote Originally Posted by Ellatha
    I'm way smarter than Newton ever was. Got a heart attack yet? Ha.
    Ellatha doesn't believe that, nor does he believe he is smarter than Pascal (which I saw him say in another thread)-as he made quite clear with the "Got a heart attack yet?" remark he is merely trying to provoke a response (and gain attention?).

    I met Ellatha on another forum (centered around zoology); we had many unfriendly discussions, and I found him extremely annoying. I and many others had similar problems with those criticisms I see here-his self perceived enormous intellect, arrogance, etc.

    Through some pm discussions that eventually led to im discussions (mostly focused on topics in threads-as the forums were somewhat childish I found this to be a great relief-discussing topics with someone else interested in reading and studying journal publications is definitely more productive) I have learned a fair bit about him.

    For one thing, apparently he liked reading my posts and thus acted like an asshole to provoke responses.
    Another fact is that I discovered he is also very interested in IQ and liked discussing it; from these it is easy to gather that he doesn't actually believe himself to smarter than these historical geniuses.

    Perhaps most surprising is that he stated he has low self esteem and feelings of a total lack of self worth.
    He was apparently surprised that people repeatedly referred to him as arrogant, or believe that he thinks himself to be such a genius. He asked for why and I failed to provide much of an answer-before viewing this forum it had been quite some time before I'd seen discussions where he made it clear and thus I couldn't provide many anecdotes. The manner of his replies however in general do give this impression, which is quite an oddity considering the presumed validity of claims of the inverse of arrogance.
    Although, I do know that he thinks many others are idiots-and I believe this attitude is perhaps the root of the issue; it is extremely negative in nature.

    The motive behind this post is that I would like to avoid seeing a continual degeneration of topics and discussion, and hopefully try and direct things in a more productive manner as I know some of this posters history and have seen these disingenuous patterns before.

    P.S.
    In the 0.999=1 I thought I made it quite clear that "I may have even forgotten" (that was 7 years ago), I just didn't think it was really taught (despite the fact that the necessary knowledge to reach that conclusion is). Could depend on the state-for the most part in Texas if it isn't on the "TAKS" test, it isn't taught at all, and I suspect the case is similar for other states and their standardized tests.
     

  15. #14  
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    Quote Originally Posted by C.Elrod
    Quote Originally Posted by Ellatha
    I'm way smarter than Newton ever was. Got a heart attack yet? Ha.
    Ellatha doesn't believe that, nor does he believe he is smarter than Pascal (which I saw him say in another thread)-as he made quite clear with the "Got a heart attack yet?" remark he is merely trying to provoke a response (and gain attention?).

    I met Ellatha on another forum (centered around zoology); we had many unfriendly discussions, and I found him extremely annoying. I and many others had similar problems with those criticisms I see here-his self perceived enormous intellect, arrogance, etc.

    Through some pm discussions that eventually led to im discussions (mostly focused on topics in threads-as the forums were somewhat childish I found this to be a great relief-discussing topics with someone else interested in reading and studying journal publications is definitely more productive) I have learned a fair bit about him.

    For one thing, apparently he liked reading my posts and thus acted like an asshole to provoke responses.
    Another fact is that I discovered he is also very interested in IQ and liked discussing it; from these it is easy to gather that he doesn't actually believe himself to smarter than these historical geniuses.

    Perhaps most surprising is that he stated he has low self esteem and feelings of a total lack of self worth.
    He was apparently surprised that people repeatedly referred to him as arrogant, or believe that he thinks himself to be such a genius. He asked for why and I failed to provide much of an answer-before viewing this forum it had been quite some time before I'd seen discussions where he made it clear and thus I couldn't provide many anecdotes. The manner of his replies however in general do give this impression, which is quite an oddity considering the presumed validity of claims of the inverse of arrogance.
    Although, I do know that he thinks many others are idiots-and I believe this attitude is perhaps the root of the issue; it is extremely negative in nature.

    The motive behind this post is that I would like to avoid seeing a continual degeneration of topics and discussion, and hopefully try and direct things in a more productive manner as I know some of this posters history and have seen these disingenuous patterns before.

    P.S.
    In the 0.999=1 I thought I made it quite clear that "I may have even forgotten" (that was 7 years ago), I just didn't think it was really taught (despite the fact that the necessary knowledge to reach that conclusion is). Could depend on the state-for the most part in Texas if it isn't on the "TAKS" test, it isn't taught at all, and I suspect the case is similar for other states and their standardized tests.
    Let's not turn this thread into an ongoing analysis of Ellatha. But in order to make the important points regarding the nature of mathematics -- what is important, what is deep, what is trivial, which concepts lead to insight -- we apparently need to clear the air with regard to some of his statements and odd behavior.

    You have no monopoly on unsolicited PMs from Ellatha, or a lengthy history of interchanges in forum threads.

    I believe that he actually does think himself as smart or smarter than Pascal and Newton, at least on occasion. He seems to have manic cycles. How else might you assess someone who claims (unsolicited and unwelcome PM) to have several times scored at the top of the scale on standard IQ tests yet regularly posts nonsense and trivia with the impression that he thinks it both original and of depth ? Arrogant doesn't begin to describe it. There are other notable but less spactular examples of some combination of arrogance and delusion.

    That would be quite ignorable, except for the fact that forums such as this usually attract far more lurkers who simply want to read and learn than active participants. They deserve to understand what mathematics and science are really all about.

    I once believed Ellatha to have significant potential, and there is an outside chance that he does. I tried to help him learn mathematics. I stopped trying when he began his pattern of nonsensical posts and generally acting like an ass. He briefly reformed, and sent me a PM a couple of months ago to that effect. I then tried to help him again, but what he apparently really wanted was praise and approval for drivel. He quickly fell prey to recidivism and is now worse than ever. His posts have become either trivial, downright wrong or a combination of both, yet the tone is that of someone with superior knowledge offering instruction. A neophyte could easily be taken in. He compounds the offense with nonsensical attacks on anyone with the slightest criticism of the content of his posts -- witness his attacks on Guitarist.

    Since he apparently has little understanding of mathematics, most of his posts require exposure and debunking. He as gone from being annoying to being a menace to the innocent but naive.

    He fooled me once with repentance. Fool me once, shame on you. Fool me twice, shame on me.

    So, to put it bluntly, I have had it Ellatha. The only importance that I can attach to him is the necessity to see that misinformation and the lack of meaningful content in his posts is made abundantly clear to the most naive of lurkers. If that makes him squirm it is simply a bonus.

    End of diatribe. I do not intend to debate this further. With anyone, including most especially, Ellatha.

    .
     

  16. #15  
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    Quote Originally Posted by DrRocket
    Fool me once, shame on you. Fool me twice, shame on me.
    How very Wildean! Reminds when my eldest asked the meaning the word "cynic".

    Me (quoting): A cynic is a man who knows the price of everything and the value of nothing;

    Her: Hey Dad, that's not bad (for you);

    Me: It's Wilde;

    Her: I said not bad, not wild. Don't get carried away.
     

  17. #16  
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    This thread should be locked.

    Furthermore, everyone on this forum is placing too much weight on their ability to read the intentions and motives of other people; I have seen several errors in both C.Elrod and Dr.Rocket's posts that can only be attributed to a belief of some pseudo hidden intentions

    The only reason I said that I was smarter than Newton (I never said I was smarter than Pascal) was because the accusation of being smarter than Pascal was made against me before I said so. Infuriated, I did not choose to argue it, rather push it forward, making the claim regarding Newton. I find it even more ridiculous that my ideas are pushed aside on the basis of the argument that "You are not Pascal." By that reasoning no one would accomplish anything.

    The most important point of all: this is a science and mathematics forum, not a psychological analysis one. If one chooses to post in it, they should not have to be subject to read their motives or perceptions of themselves. Rather, the content should be discussed. Despite Dr. Rocket's claims of posting "nonsense" and other such matters, at specific instances the worst he has been able to come up with is an accusation of a trivial nature attributed to my posts, knowing full well that it is correct (this is what he would like students to learn anyway, so as you can see there is no real issue, contrary to what he would like me to believe).

    I fear that on any online forum I post in now there is no hope for discussing the issues that are supposed to be, rather I will receive another psychometric analysis from a bunch of cranks that pretend to know me (C.Elrod excluded). I'm gone.
     

  18. #17  
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    Ellatha has requested that his/her account be deleted. Since we do not do that (it would create problems with the database, etc.) I've simply terminated Ellatha's account. His/her posts and started threads will remain -deletion of these, even such a thing were feasible, would be unfair and insulting to the many people who spent time reading and responding to threads in which Ellatha participated or created.

    Simply put, that's not going to happen just because a single disgruntled members has a hissy fit (justified or not).

    Ellatha's account profile and posts, so far as I can tell, present no personal data about he/she. Indeed, even Ellatha's gender or sex is a mystery. So there is no good reason to want deletion of his/her account except to be spiteful to the persons with whom he/she disagreed.

    This thread also will remain open until good reason to close is presented.
     

  19. #18  
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    I have to agree with Ellatha on one thing: I think that making a public attempts at psychoanalyzing another forum member is in extreme distaste and should not be permitted.

    Moreover, publicly revealing negative information that you have obtained from private conversations with another forum member is, in my opinion, also highly distasteful.

    I believe it is possible to offer effective criticism without crossing this line.
     

  20. #19  
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    The time has come, the Walrus said......

    Thread locked
     

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