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Thread: Number of Rational vs irrational numbers

  1. #1
    Senior Member TriKri's Avatar
    Nov 2006

    Number of Rational vs irrational numbers

    I know there exists an infonite number of irrational numbers between each pair of rational numbers, still there exists an infinite number of rational numbers between each pair of irrational numbers. So, there must be an infinite number of both rational and irrational numbers, within any given interval.

    But is it possible to calculate the ratio between the quantities? Dividing one infinity it by another infinity?
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  2. #2
    MHF Contributor

    Aug 2006
    There have been volumes written on this question.
    Basically, two sets are equipotent if there is a one-to-one correspondence between them. Consider the set of counting numbers: N={0,1,2,3…}. Any set equipotent with N is said to be countable. This collection of sets includes the set of integers and the set of rational numbers. However, the interval [0,1] can be shown to be uncountable. Therefore the set of irrationals is uncountable. Any set equipotent with [0,1] is said to have the power of the continuum. These sets include the irrationals, the reals, the complex, etc. The continuum hypothesis states the there is no set with cardinality strictly between N and R. Most mathematicians take the continuum hypothesis as an axiom.

    So, to your question. Any talk of ‘ratios’ at this level of abstraction is quite impossible.
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  3. #3
    Global Moderator

    Nov 2005
    New York City
    If you are familar with countability and set theory here is a more elegant version of the diagnol argument which shows the reals are uncountable!

    Click Heir
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  4. #4
    Nov 2006
    if u are familiar with the idea of sections of rational numbers and the theorems regarding them like dedekinds theorem and weirstrass theorem , then u will see that we can easily prove that there are infinity of rationals and irrationals
    and here when we say infinite we mean that if u say i have 'n' rationals in an interval i can show u one more rational no(quie easily the mean of smallest and greatest rational)..
    so talking about the ratio of the number of these nos in abstract
    also rational form a countable set where as irrationals do not form a countable set
    they form uncountable set
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