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Can pi be valued mathematicly?

  1. Nov 9, 2006 #1
    in calculus class i have been told that pi is irational, so then i realized that pi is not valued by actually measuring by hand the diameter and perimeter, since it would not be a pure number...
    1. so how the hack is pi measured?
    2. i want a proof of it being irational.


    crap, sorry wrong forum, admin, could you please move it to general math? thank you
     
    Last edited: Nov 9, 2006
  2. jcsd
  3. Nov 9, 2006 #2
    first off its Pi and its the ratio of a circles circumferance to its diameter, i think.
     
  4. Nov 9, 2006 #3

    NateTG

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  5. Nov 9, 2006 #4
    And if you want a way to solve it by using math that may seem more obvious to you: take the integral of a quarter of a unit circle. With a known relationship between the radius and area, you can then use this to determine the ratio between the area and its circumference. Voila! Pi!

    This obviously isn't the normal/best way, but it may seem like a more obvious solution to you.
     
  6. Nov 9, 2006 #5

    arildno

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    Eeh, what is impure about pi??
    Is purity of a number related somehow to repeating decimal sequences? :confused:

    2. Euler did a clever proof of that back in the 18th century, whereas Lindemann furnished a rather trivial of the transcendence of pi a century later.

    The band-width on this site is too small to contain both or any one of those proofs..:smile:
    (or possibly not..)
     
  7. Nov 9, 2006 #6

    matt grime

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    pure? impure? what is that? i think you have a misapprehension about real numbers here. in particular you seem to have the idea that we find numbers by 'measuring' things in real life. real life and mathematical abstractions are not the same thing.


    get a number theory text book then. I think that le veque proves the irrationality of pi. transcendence is harder.
     
  8. Nov 9, 2006 #7

    arildno

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    Also remember that defining numbers in terms of the denary/decimal system is not the only way to define a number.

    For example, a linear equation of one variable with one solution may be regarded as defining the so-called "x".
    We could, for example, use the equation x-2=3 as the definition of the number "5".
     
  9. Nov 9, 2006 #8

    Office_Shredder

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    It's actually a crappy way of defining numbers, since you need infinite accuracy to add any two (rational even) numbers and be sure you got the right answer
     
  10. Nov 9, 2006 #9

    StatusX

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    Don't fall into the same trap. It makes no sense to talk of the accuracy of adding two decimal expansions, unless you decide to terminate the sum at some point, which could only be for practical reasons. The sum of two decimal numbers, even irrational ones, is perfectly well defined and exact.
     
  11. Nov 9, 2006 #10

    Office_Shredder

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    status, that's what I mean. The OP seems to think that pi isn't a real number because of the inability to write it down.

    Obviously if you know the full number, you can add the decimals together and know what the new decimal is supposed to be. But since we're talking about pi here, good luck knowing the decimal expansion in the first place ;)
     
  12. Nov 9, 2006 #11

    arildno

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    Eeh?:confused:
    I can define numbers any way I like.
    For example, the number Johnny is defined as follows:
    [tex]Johnny=\sum_{i=1}^{\infty}\frac{1}{n^{2}}[/tex]
     
  13. Nov 9, 2006 #12

    arildno

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    What does that mean?
    Why would you do that?
    Signifying what, exactly?
    Who cares about the decimal expansion of [itex]\pi[/itex]???
    [itex]\pi[/itex] would remain perfectly well defined, even if hadn't had the intelligence to deduce that it had to be slightly greater than the denary number 3.
     
    Last edited: Nov 9, 2006
  14. Nov 9, 2006 #13

    Office_Shredder

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    arildno, the original poster claimed that pi, by being irrational, is somehow less pure. I extrapolated that he was comparing irrationals to rationals in general (certainly not an unreasonable leap), and thinking rational numbers were more pure than irrationals somehow. The main reason for this line of thinking is that you can't write down the decimal expansion of an irrational number, so it doesn't feel as real.

    So I responded by saying defining a number by its decimal expansion is a bad way of doing things, since decimal truncation (which is absolutely necessary to write the number down) means you can't accurately add two numbers together unless you have knowledge from elsewhere as to what the two numbers are intended to be in their entirety
     
  15. Nov 9, 2006 #14

    CRGreathouse

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    This gives another (very slowly converging) method for approximating pi:

    [tex]\pi=\sqrt{6\sum_{n=1}^\infty\frac{1}{n^2}}[/tex]
     
    Last edited: Nov 9, 2006
  16. Nov 9, 2006 #15
    There's one famous product series... To prove that Pi is irrational you should look into these type of series.
     
  17. Nov 10, 2006 #16
    Hrrm... I haven't yet gotten to infinite sequences and series, so I cannot contribute calculations.

    What I do know (or think I know) is that [tex] \pi [/tex] is an irrational (trancendental, too) number with an infinite decimal expansion.

    The fact that for a circle of diameter 1 that the circumference is:

    [tex]Circumference = 2{\pi}r [/tex]
    [tex]= {\pi}D [/tex]
    [tex]= {\pi}(1) [/tex]
    [tex]= \pi [/tex]

    Means that the circumference is finite, but pi is still infinite.

    Though I guess it depends on your definition of what finite is. If you're using a physical circle, I suppose the circumference is actually an infinite connection of dimensionless points in space. Likewise, you could consider it to be a finite number of atoms.
     
    Last edited: Nov 10, 2006
  18. Nov 10, 2006 #17
    Pi is finite.... it has an infinite decimal expansion, but that certainly doesn't make the number itself infinite.
     
  19. Nov 10, 2006 #18
    Thank-you for clarifying that.
     
  20. Nov 10, 2006 #19
    Hi Tuvia,

    As far as I know, and proved:
    1. There are many ways to calculate it. (Hint: If Taylor Series is to be used, it must be expanded in a certain way to ensure convergence.)
    2. Pi has an infinite number of decimals. Examine some of the ways of finding its value and you can check this. This means that it is irrational.
    3. 22/7 is a historical value used as an approximation to the Measured value. It is of course irrational. (Hint: When you say an irrational number; this means that there is some way to describe this number exactly, but, Only, it has a decimal form that contains and infinte 'number' of digits.)
    4. Pi is an arc angle, i.e. it is angle in radians that corresponds to 180 degrees. It's main definition is: "The ration between the circumference and the diameter of a (any) circle."

    But, what I can find an astonishing question:
    Is the value of Pi is the origin in geometry (may be in algebra)? Or, it is a consequence of another geometrical property?..;)............ It deserves to check!!!!!!!

    Welcome for any (request for) clarification.

    Amr Morsi.
     
  21. Nov 10, 2006 #20

    HallsofIvy

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    No, THAT does not mean it is irrational. 0.33333... has an infinite number of decimal places but it is rational.

    Again, no. There exist many (in fact, "almost all") rational numbers that have infinite decimal expansions. All such are eventually repeating where irrational numbers are not.

    I don't follow this. Of course, the "origin" of Pi was in geometry. It's definition was just what you said in "4". Now, of course, it is just a number that has many different applications.

    You mean you won't accept any clarification by someone else?:rolleyes:
     
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