Pi expressed by different bases

Discussion in 'Recreational Math' started by David W. Cantrell, May 3, 2006.

  1. Sure (unless you insist that your bases must be integers).

    In a certain base, which is a bit larger than 3, pi = 10 exactly.

    Cheers,
    David
     
    David W. Cantrell, May 3, 2006
    #1
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  2. David W. Cantrell

    Tarantella Guest

    This is a question from someone with
    hardly any math knowledge.

    Since the ratio Pi cannot be expressed
    wholly in the usual numerical system "base 10",
    can it be wholly expressed by choosing a
    different base?

    Thanks,
     
    Tarantella, May 3, 2006
    #2
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  3. Not with an integral basis.

    If b>1 is an integer, then the expression

    (d_n)(d_{n-1})...(d_1)(d_0).(e_1)(e_2)...(e_m)

    where 0<= d_i < b
    0<= e_i < b

    in "base b" corresponds to the real number

    d_n*b^n + d_{n-1}*b^{n-1} + ... + d_1*b + d_0 + e_1/b + e_2/b^2 +... +e_m/b^m.

    We can write this as a single fraction by multiplying through by b^m,
    so we would obtain

    (d_n*b^{n+m} + d_{n-1}*b^{n+m-1} + ... + d_0*b^m + e_1*b^{m-1} + ... + e_m)
    ---------------------------------------------------------------------------
    b^m

    and in particular, if the expression terminates then it must represent
    a rational number since both numerator and denominators are
    integers. It is known that pi is not rational (in fact, it is not even
    algebraic), so it cannot be thus expressed.

    In general, for b an integer greater than 1, the only numbers that can
    be expressed with a terminating "base b" expansion are those that can
    be expressed as n = p/q, where p and q are integers, q is not zero,
    and the prime divisors of q are all prime divisors of b.

    (So, for base 10, only fractions that can be written with a
    denominator which can be written as 2^a*5^b, a,b>=0, have terminating
    decimal expansion).

    However, for any b>0, a number expressed in base b is rational if and
    only if its "base b expansion" is eventually periodic (eventually
    repeats).


    One can define a similar notion of "base b" for nonintegral values of
    b, though it is a much more complicated affair; normally the
    conditions we put on b to make this sensible also preclude pi from
    having a terminating "base b expansion".

    --
    ======================================================================
    "It's not denial. I'm just very selective about
    what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
    ======================================================================

    Arturo Magidin
     
    Arturo Magidin, May 3, 2006
    #3
  4. Pi is irrational, so it cannot be wholly expressed in any integer base.
     
    Frank J. Lhota, May 4, 2006
    #4
  5. David W. Cantrell

    Tarantella Guest

    Thanks for all the replies.
     
    Tarantella, May 5, 2006
    #5
  6. David W. Cantrell

    Spaz Guest

    But a circle has a finite circumference, a finite diameter, and a finite
    area. So how can pi be irrational? Must not have been calculated right in
    the first place.
     
    Spaz, May 7, 2006
    #6
  7. David W. Cantrell

    Dennis Marks Guest


    You don't understand what an irrational number is. It means that you can't
    express the number as a ratio of 2 integers. Any circle that has an integer
    diameter will have an irrational circumference. It has nothing to do with
    finite.

    --
    Dennis

    Disclaimer: The above is my opinion. I do not guarantee it. Be sure to back
    up any files involved and use at your own risk. Batteries not included. Not
    for internal use. Don't run with knives.
     
    Dennis Marks, May 7, 2006
    #7
  8. So? Irrational numbers are still finite. They just cannot be expressed
    with a repeating decimal expansion.

    What does one thing have to do with the other?


    --
    ======================================================================
    "It's not denial. I'm just very selective about
    what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
    ======================================================================

    Arturo Magidin
     
    Arturo Magidin, May 8, 2006
    #8
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