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Normal Numbers

  1. Feb 24, 2005 #1


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    This is all really hard stuff, and speculation.

    Normal numbers are irrational numbers that have the property that all the digits in their decimal expression are equally distributed.
    An example of a normal 10-number might be:

    Clearly whether a number is normal can depend on the base that it is represented in, so it makes sense to refer to b-normal numbers where b is the base.

    An absolutely normal number is a number that is normal in any fixed base.

    I was wondering about this stuff a while ago, and the following questions came to me - insightful comments would be appreciated:
    If a number is [itex]p[/itex]-normal for all primes, is it necessarily absolutely normal?
    If a number is [itex]n^i[/itex]-normal for some whole number [itex]n[/itex] is it absolutely normal?
    If a number is [itex]p[/itex]-normal and [itex]q[/itex]-normal is it also [itex]pq[/itex]-normal? What about the converse?

    What about more exotic bases - like negative bases or base factorial? I'd like to call a number !-normal if its base factorial expansion has a distribution so that all digits have zero density, in all of it's base factorial expansions. (Phrased that way to cover 0.1,2,3,4,5,6,7,8,9,10,11,12,13...=1.0,0,0... (base factorial) and it's cousins.) Base factorial seems particularly interesting because it should dovetail well with the Taylor-series-type stuff we have for caculating constants.
  2. jcsd
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