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B Real Probability

  1. Aug 27, 2016 #1
    So from what I've been reading rational numbers are a countable infinity, while the irrationals are an uncountable infinity. So the number of irrational numbers > the number of rational numbers. Irrational numbers can "normal irrationals" or transcendental numbers, or at least that is what I've read. This seems pretty intuitive, a random number would more likely be irrational than rational.

    So I was thinking, given a infinite random number generator would a given real be more likely to be represented as either a rational number or as a root of polynomial than transcendental? Or is this comparison impossible since I'd guess that transcendental numbers being a subset of an uncountable infinity are also an uncountable infinity? Or is this not true?

    Any information would be great, or where to start reading about set theory.
     
  2. jcsd
  3. Aug 28, 2016 #2
    The roots of rational polynomials are known as the algebraic numbers; this is a countable set. So, if a random number generator selected a real number uniformly between 0 and 1, say, it would select a transcendental number with probability 100%.
     
  4. Aug 28, 2016 #3
    Just read that on the wikipage that roots of rational polynomials are countable, I missed it during the first reading.

    Thanks.

    So question answered.

    So where's the proper place to start reading about set theory? My background only includes one proof based course on linear algebra, and I'm currently taking a course on ordinary differential equations.
     
  5. Aug 29, 2016 #4

    micromass

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    Get the book by Hrbacek and Jech. PM me if you want more information or help!
     
  6. Aug 30, 2016 #5
    I'll check it out! Much appreciated.
     
  7. Sep 19, 2016 #6

    Zafa Pi

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    A subset of an uncountable set is not necessarily uncountable, it could have just one element.
     
  8. Sep 19, 2016 #7

    FactChecker

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    But the compliment in the Real numbers of a countable set must be uncountable.
     
  9. Sep 19, 2016 #8

    Zafa Pi

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    True enough, but I was referring to BillhB's statement, "I'd guess that transcendental numbers being a subset of an uncountable infinity are also an uncountable infinity? Or is this not true?"
     
  10. Sep 19, 2016 #9

    FactChecker

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    Oh. Sorry. I missed that part.
     
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