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Quick question about Pi

  1. Oct 15, 2006 #1
    Hello to all,

    Quick question about Pi …

    If we can have a finite circle, how can Pi not be decimal ?




    VE
     
  2. jcsd
  3. Oct 15, 2006 #2

    Integral

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    Pi can be expressed as a decimal number, or a binary or in any other base you should choose.

    Now, perhaps you should try rephrasing your question.
     
  4. Oct 15, 2006 #3
    Thank you for your response Integral,

    what I mean by decimal is a number that has finite number of decimals.

    Does my OP make more sense ?


    VE
     
  5. Oct 15, 2006 #4
    I suppose you mean "rational" ?
     
  6. Oct 16, 2006 #5

    shmoe

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    It seems you should also have a problem with 1/3 being non-terminating then? Also sqrt(2) being the length of the diagonal of a 1x1 square?

    A number being "finite", like the circumference of a circle, has no bearing on whether it has a finite number of decimal digits or not.
     
  7. Oct 16, 2006 #6
    Hey !, thank's for your replies,

    You know what ?, I realize I have a lot of questions about how I comprehend mathematical abstractions.

    I know that many ratios of numbers like 1/3 also give an infinite decimal expansion and, yes, it gives me some difficulty to understand some of the math and how it relates to everyday life.

    Maybe I’m mixing-up different realms or concepts or rules that just don’t mix together. Here’s an example of my questioning … and reasoning…

    Couldn’t we say that a perfect 1 x 3 rectangle exists in the abstract realm ?, if so, then we should be able to divide this rectangle in three perfectly equal 1 x 1 squares , no ?

    Why then does the ratio 1/3 not give a finite decimal expansion ?

    Where is the flaw in this reasoning ? , what am I juggling with here that doesn’t jive ?




    Regards,

    VE
     
  8. Oct 16, 2006 #7

    shmoe

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    I'm more inclined to ask: why should it have a terminating decimal expansion?

    Don't confuse the ideas of a number that's "finite", like the length of a line segment on a piece of paper, and the idea of a number whose decimal expansion has only finite number of non-zero digits. A non-terminating decimal is not by an means an "infinite" number.
     
  9. Oct 21, 2006 #8

    Gib Z

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    In actual fact, we cant have a PERFECT circle in physical existence, only in theory, mathamatically. this is exaclty becuase pi is trancendental, meaning it isnt the root of any polynomials with ration coefficents, basically meaning u cant construct it. study it more, thats how they proved u cant square the circle. also, atoms will create a certain irregularity in the circle, so yea. but of course, they are good aproximations.
     
  10. Oct 21, 2006 #9
    valence, imagen this
    0,3*3=0,9
    0,33*3=0,99
    0,333*3=0,999
    so if 1/3 is finite amount of decimals then you wont get 1 when its multiplied with 3, you just get a long line of 9s. while a infinite amount of nines after each other equals one since
    x=0,9999....
    10x=9,9999...
    9x=9
    x=1

    and also you dont really get a excat value of a circles circumferens since you cant measure anything exact
     
    Last edited: Oct 21, 2006
  11. Oct 21, 2006 #10
    Hey guys, thank you for the replies…

    although I hadn’t mentioned it in the OP, the perfect circle in question certainly only exists in the abstract mathematical world. I agree that, even if we were to use the most sophisticated electronic microscope coupled with the most accurate laser cutter, we could not divide anything in perfectly equal parts.

    But we surely can represent a perfect circle or rectangle etc. in the math realm.

    I don’t have a problem with the infinite number of remainder of 3’s that come from 1/3, I have a problem putting together the fact that if we take the 1 x 3 rectangle as being whole, as being 1, and we divide it in three equal parts (in the math realm) making each part a 1 x 1 square with nothing left, then how come 1/3 always gives a remainder ?

    Is it because numbers used in different parts of the mathematical abstract realm such as geometry, calculus and basic arithmetic expressions, don’t represent the same ‘thing’?...

    Please explain,

    Regards,

    VE

    PS. I’ll get back to Pi later …
     
  12. Oct 21, 2006 #11
    On another planet, where the aliens have 12 fingers, earth's 0.333333 recurring is written as 0.4
     
    Last edited: Oct 21, 2006
  13. Oct 21, 2006 #12

    selfAdjoint

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    Yeah, but how do they write 1/5?

    But this is OT. The answer to the question is that pi is not a rational number, so there is NO basis in which its representation is a finite number of digits. The fact that pi is irrational was proved by Lambert in the eighteenth century, and you can find his and other proofs of it by googling on Pi irrational.

    Addendum: pi is not only irrational, but transcendental. This means it cannot be the root of any polynomial of finite degree. This fact was proved by Lindemann in the late nineteenth century using methods developed by Hermite to prove the same thing about e.
     
    Last edited: Oct 21, 2006
  14. Oct 21, 2006 #13

    CRGreathouse

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    Well, no rational (or even algebraic) base. Pi is 100 in base [itex]\sqrt\pi[/itex], for example.
     
    Last edited: Oct 21, 2006
  15. Oct 22, 2006 #14
    VE A decimal expansion is a shorthand form of writing a finite number as a sum of negative powers of some base. Thus in base 10 a decimal of length n after the period is simply shorthand for some integer divided by 10 to the n th power. For example 1.125 base 10 = 11/10 + 2/100 + 5/1000 = 1125/1000. Since a finite decimal expression is actually some intger divided by a power of 10 no reduced fraction thereof can contain a prime factor in the denominator other than a power of 2 or 5. Reduced fractions in which the denominator contains a prime factor which is not a factor of the base can never be writen in decimal form as a finite length decimal. On the other hand, .20 base 6 = 2/6 + 0/36 = 12/36 = 1/3 which is possible because 3 is prime factor of 6.
     
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