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What is pi?

  1. Jun 30, 2007 #1
    Hi all
    In almost all the formulae related with symmetry, we use pi. For example, in case of circle, we use pi, and in case of sphere we use 4*pi.
    e.g. E (electric field)= (q * qo) * 1/(4*pi*eps* r^2)
    A (circle)= pi*r^2
    A (sphere)=4*pi*r^2

    My question is: How does pi signify symmetry? Why is the value of pi what it is, and what does this value (22/7) represent? How was this value originally found?

    regards
    Mr V
     
  2. jcsd
  3. Jun 30, 2007 #2

    morphism

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  4. Jun 30, 2007 #3
    Pi could be said to be the convergence of series that we obtain out of some integrals.
     
    Last edited: Jun 30, 2007
  5. Jun 30, 2007 #4

    cristo

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    pi is the ratio of the circumference of a circle to its diameter.
     
  6. Jun 30, 2007 #5
    I purposefully avoided that definition to stay away from any physical interpretation of pi. Modern analysis would define the concept of integration before circumference, making the definition in question a result rather than a definition in itself.
     
  7. Jun 30, 2007 #6
    What exactly does this mean?
     
  8. Jun 30, 2007 #7

    cristo

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    But pi is a fundamental constant of nature (along with G, c, etc), and thus is a physical quantity. The value of this constant is determined by the ratio of the circumference of a circle to its diameter, for any circle.

    Besides, how were you taught about pi? I definitely didn't know how to integrate before the concept of pi was introduced. It is introduced in geometry classes in school.
     
  9. Jun 30, 2007 #8
    Thanks for your replies

    Mr V
     
  10. Jun 30, 2007 #9

    It simply means that pi's definition is derived out of the concept of integration; that is the function [tex]f(x) = \sqrt{r^{2} - x^{2}} [/tex], whether we consider its circumference or area, both defined in terms of integral nonetheless, pi is defined within those limits. Out of the integrals that define pi, we can derive series that converge towards pi as defined.
     
  11. Jun 30, 2007 #10
    But the introduction of pi in school is the presentation of a result: that the circumference of any circle is directly proportional to its radius. Yes pi is the ratio of the circumference to the diameter, however the "origin" of pi, how it is discovered and defined, is not there.
     
  12. Jun 30, 2007 #11

    cristo

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    Erm... the fact that the ratio of the circumference to the diameter of a circle is equal to some constant for all circles was known long, long before calculus was even invented. The exact value of the constant may have been calculated by taking limits of integrals, but the OP asked for the origin of this constant. It most certainly originates from the definition I gave.
     
  13. Jun 30, 2007 #12
    Yes I am very well conscious that it was known way before. However, I believe you reckon that modern analysis redefined all geometry known before its elaboration - and modern analysis wants pi to be derived out of the concept of integration. It's as if all mathematics prior to analysis had never existed. Before analysis, the proportionality of circumference to radius was known because of our physical interpretation of distances and areas, but with analysis, this proportionality is a result independent of physical reality.
     
    Last edited: Jun 30, 2007
  14. Jul 1, 2007 #13
    Pi

    From a geometric point of view, Pi is what is it due to how unit area is defined.
    It's just so happens that on planet earth, humans decided to define a unit of area as a little square box with sides of unit length.

    In another corner of the galaxy, some other intelligent life form might have decided to define unit area as a little circle with unit diameter, in which case their circles would have an area of 4*r^2, rather than Pi*r^2
     
  15. Jul 1, 2007 #14
    Good Heavens! If anybody took Plane Geometry like in the old days, you would know that Euclid defined that. The critical fact is that it relates all circles to their diameters, and thus is a justified constant.

    Werg22:Before analysis, the proportionality of circumference to radius was known because of our physical interpretation of distances and areas, but with analysis, this proportionality is a result independent of physical reality.

    Actually the Greeks recognized that the "perfect circle," was an abstraction, which they termed an "Ideal." And there has been a long belief in the existance of Ideals apart from physical forms. Decartes wrote at length on this, and even Godel was a believer in Platonism. Godel I quote:

    Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.
    It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e., in the latter case the actually occurring sense perceptions.
    http://www.friesian.com/goedel/chap-2.htm
     
    Last edited: Jul 1, 2007
  16. Jul 1, 2007 #15
    Yes Euclid did define the ratio but he did so because, at the peril of repeating myself, because of the physical interpretation of what circumference and diameter are. As I said, it's as if anything prior to analysis never existed, every definition that we now have has to be defined by analysis and nothing else.
     
  17. Jul 1, 2007 #16
    You got in there before I could finish: Godel I quote:

    Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.
    It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e., in the latter case the actually occurring sense perceptions. http://www.friesian.com/goedel/chap-2.htm

    WErg22: As I said, it's as if anything prior to analysis never existed, every definition that we now have has to be defined by analysis and nothing else.

    I would love to know who thinks that? Can you name a single outstanding mathematican who says that? I don't think it was Godel.
     
    Last edited: Jul 1, 2007
  18. Jul 1, 2007 #17

    rcgldr

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    ... but that definition is mathematical, not physical. Definition of circle and diamter are mathematical. There are some physical objects which approximate a circle, but they're not perfect.

    Defining a series to represent pi doesn't mean that pi is defined by the series. I thought the goal of most sciences is to use the simplest definitions as long as they are accurate.
     
  19. Jul 1, 2007 #18
    Jeff Reid: Defining a series to represent pi doesn't mean that pi is defined by the series. I thought the goal of most sciences is to use the simplest definitions as long as they are accurate.

    Sounds good to me! However, I suppose if you are working in analysis, you might get your own set of ideas, but if you were in Algebra, maybe you have different ideas.
     
  20. Jul 1, 2007 #19
    robert Ihnot: In the ideal circle pi is the ratio between circumference and diameter. However, what is circumference? The Greeks only had physical means to define it. They may have believed that a the circle was ultimately an abstract idea, but did not have any tool to set that intuition into stone. All that I am saying is that I am convinced that the starting point shouldn't be the ratio itself but rather what circumference (or area) is and how is it mathematically represented - pi arises from that starting point. However I realize this is a egg-chicken question and I myself admit there is some room for debate.
     
  21. Jul 1, 2007 #20
    I once heard in Physics that if there are two competing theories to use with both giving satisfactory results, then prefer the simplest.

    Then again some will argue that anything out of Physics could not be valid here, and perhaps the more complicated, advanced, difficult theory should be chosen. After all, otherwise maybe too many people would pass math.
     
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