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Can π or e ever be rational?

  1. Jan 4, 2005 #1
    Hello,

    I doubt it, but is there any number system (base) where π or e run out of digits?
     
  2. jcsd
  3. Jan 4, 2005 #2
    Nope.

    It is known that [tex]\pi[/tex] and [tex]e[/tex] are irrational (they cannot be expressed as the quotient of two integers). Given this, in an arbitrary base [tex]b[/tex], if they could be expressed in a finite number of digits, say

    [tex]\pi=a_1b^na_2b^{n-1}...a_nb^0a_{n+1}b^{-1}...a_{m+n-1}b^{-m+1}a_{m+n}b^{-m}, a_i\in\mathbb{N}[/tex]

    then
    [tex]\pi=\frac{a_1b^{n+m}+...+a_{n+m}b^0}{b^m}[/tex]

    which is a quotient of integers and thus rational, a contradiction. Therefore irrational numbers like [tex]\pi[/tex] and [tex]e[/tex] cannot be expressed in a finite number of digits in any base system.

    The proofs of irrationality are slightly tricky, you can see them

    here (irrationality of [tex]\pi[/tex])
    http://www.mcs.csuhayward.edu/~malek/Mathlinks/Pi.html

    and here (irrationality of [tex]e[/tex]).
    http://en.wikipedia.org/wiki/Proof_that_e_is_irrational

    -rachmaninoff
     
  4. Jan 4, 2005 #3
    We can create a number system where [itex]\pi=1[/itex], but that's not really worth it.
     
  5. Jan 4, 2005 #4

    matt grime

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    PLease note this is written with tongue firmly in cheek, but suerly everyone knows that 2pi is actually 1. (joke from some famous mathematician about Fourier analysis, wish I could remember who)
     
  6. Jan 5, 2005 #5

    Galileo

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    Can we really?
    Changes our base won't accomplish anything. The property of numbers is independant from a base choice. Writing 3.14... is only a representation of the mathematical number pi, which is defined by a geometric ratio. How could pi equal 1?
     
  7. Jan 5, 2005 #6

    matt grime

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    in base pi he means, then pi has the representation 1 in this radix.
     
  8. Jan 5, 2005 #7
    In base pi wouldn't pi be 10 and not 1?
     
  9. Jan 5, 2005 #8

    matt grime

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    ok, 10, then
     
  10. Jan 5, 2005 #9

    Janitor

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    By the way, last week I ran into the name 'Doran Shadmi' at another website.
     
  11. Jan 5, 2005 #10

    krab

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    pi cannot be 1 as long as 1 represents the multiplicative identity element, and pi the ratio of circumference to diameter. Base cannot change that.
     
  12. Jan 6, 2005 #11

    matt grime

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    Yep, off topic but amusing. He posts now under lama at scienceforums.net, doron being now banned, and has reached the stage of having his threads instantly locked. He also accused me of folowing him there just to discredit him until someone pointed out my joining date was 4 months earlier than his. He's still asking for detailed responses and then not appreciating them.
     
  13. Jan 6, 2005 #12

    Janitor

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  14. Jan 6, 2005 #13

    Hurkyl

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    All right, break it up.
     
  15. Jan 6, 2005 #14

    Janitor

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    HE STARTED IT. :frown:
     
  16. Jan 7, 2005 #15

    JesseM

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    A non-whole-number base wouldn't make much sense...what would the digits be? If you just use the digits 0,1,2,3 then there can be totally different representations of the same number--pi^2 (about 9.869604401) could be represented as 100, but it could also be represented as 30.1102... Only whole number bases have the nice property that the largest possible number of N digits before the decimal is equal to the smallest possible number of N+1 digits before the decimal, like how 99.9999... (largest number in base 10 with two digits before the decimal) is equal to 100.000... (smallest number in base 10 with three digits before the decimal).

    As for the original post, it should be noted that the real definition of "rational" is that a number can be written as a fraction involving two integers, like 5/7...it's a consequence of this definition that all rational numbers will have terminating or repeating decimal expansions in whole-number-bases, but that isn't the actual definition, so even if you made up a weird non-whole-number-base where an irrational number had a terminating decimal expansion, that still wouldn't make the number "rational".
     
    Last edited: Jan 7, 2005
  17. Jan 7, 2005 #16

    Zurtex

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    It's not that difficult to think of really. I mean choosing Natural Numbers to navigate the real number line although makes sense to us doesn't have any significance on the real number line.

    Consider you had a system where 1 is the equivalent of Pi, 2 is the equivalent of 2Pi and so on and so forth. I mean if you really think about it how do you define integers in the real number line other than arbitrary equally spaced points and that there are countably infinitely many of them?
     
  18. Jan 7, 2005 #17
    would you all mind helping me out?
     
  19. Jan 7, 2005 #18

    Zurtex

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    It usually helps if you post your problem, I think you may have posted on the wrong thread.
     
  20. Jan 7, 2005 #19
    I think that pretty much says it all right there. Any attempt to treat the ratio of a geometric circle as though its just some kind of arbitrary abstract number that we can call whatever we please is going to be pretty futile.

    We didn't make up the concept of pi. Pi is a quantitative property of 3-dimensional space. While many modern mathematicians have become totally lost in the absurdity of pure abstractions, others recognize that there really is an ontological basis behind the idea of number.

    I think this should be obvious when we talk about transmitting the number pi out into space as a signal to extraterrestrials.

    Pi is an ontological property of our universe. Every intelligent race of beings that makes it to the technological level will discover the number Pi. Pi has reality! As do all the other numbers actually. Unfortunately this type of thinking is frowned up on by modern abstract mathematics. Cantor's formal introduction of the idea of an empty set as a way to guarantee the purity of numbers a mere couple hundred years ago put to death any chance for the mathematical community to have a complete ontological understanding of numbers.

    On the bright side, I believe that there will come a day when the ontological aspect of number will be fully appreciated and mathematics will finally become a true science. The human race is still in its infancy so there's plenty of time for us to wake up and erase the errors of the past. :approve:
     
  21. Jan 7, 2005 #20

    matt grime

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    pi is only special, and commented on because we hacve chosen our geometry in such a way that circles are special. To say that we live in a 3-d world, and let's for the sake of argument accept that, is it not surprising that we didn't pick the ratio between the volume of a sphere and the cube of its radius?

    Of course, you need to prove to use that there is such a thing as a perfect circle in existence before you claim pi is ontological.

    Why do you insist on introducing Cantor in such a spurious manner? Cantor came well after the formalization of the real numbers, and the empty set has nothing to do with this.
     
    Last edited: Jan 7, 2005
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