Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Normal numbers

  1. Mar 25, 2005 #1

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    I was just wondering if it was possible to prove anything about the normality of the number:

    [tex]\sum_{x=0}^{\infty} \left((P(x) \mod b)\left(b^{-x}\right)\right)[/tex]

    Where P(x) is a Polynomial with integer coefficients and b is the base of decimal representation. Is anything even known for simple polynomials such as P(x) = x^2?
     
  2. jcsd
  3. Mar 25, 2005 #2

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    that looks like a pretty normal number to me.
     
  4. Mar 25, 2005 #3

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    Well thinking about it is fairly obvious that for P(x) = x^a it is normal to base b and I would imagine not too difficult to prove rigourously. For that matter P(x) = x^a + c would seem to always be normal to base b as well.

    Hmm, just a matter of curiosity I suppose, I've always been interested in the normality of numbers since I first heard about it.
     
  5. Mar 25, 2005 #4

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    Am I being stupid here?

    Have I simply defined a rational number :confused:, anyone?
     
  6. Mar 25, 2005 #5

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    Looks that way. P(x) = P(x+k*b) mod b for all integers k, so you have a repeating decimal.
     
  7. Mar 26, 2005 #6

    Zurtex

    User Avatar
    Science Advisor
    Homework Helper

    Hmm, o.k fair enough, but lets suppose I start here in thinking about normal numbers. The way I've defined rational numbers here doesn't stray too far from being able to define all real numbers. So does anyone know if this defines all rational numbers or at least how I would start about proving if it does?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Normal numbers
  1. Normal Numbers (Replies: 0)

  2. NH normal => H normal (Replies: 2)

  3. Normal Groups (Replies: 15)

Loading...