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Math Help - Finding rational and irrational between two numbers

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    Finding rational and irrational between two numbers

    My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of \frac{1}{2}n and write the irrationals in the form of \frac{1}{\sqrt{2}}n, where n is rational.

    Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?
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  2. #2
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    Quote Originally Posted by Jameson View Post
    My professor assigned a proof that between any two real numbers there exists both a rational and an irrational number. She told us to think about writing the rationals in the form of \frac{1}{2}n and write the irrationals in the form of \frac{1}{\sqrt{2}}n, where n is rational.

    Any ideas on how to write n in terms of two numbers, say a and b, such that n times its respective constant falls between a and b?
    Scrolling down this link might help get the ball rolling ...?
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    Here is a proof that I like very much. I first saw in it in Martin Davisí Applied Nonstandard Analysis.
    LEMMA: x - y > 1 \Rightarrow \quad \left( {\exists n \in Z} \right)\left[ {y < n < x} \right].
    The proof of the is straight forward using properties of the floor function.
    \left\lfloor y \right\rfloor  \le y < \left\lfloor y \right\rfloor  + 1 \Rightarrow \quad y < \left\lfloor y \right\rfloor  + 1 \le y + 1 < x. Note that \left\lfloor y \right\rfloor  + 1 is an integer.

    Suppose that {a < b} then \left( {\exists J \in Z} \right)\left[ {\frac{1}{{b - a}} < J} \right]. Because Jb - Ja > 1 \Rightarrow \quad \left( {\exists K \in Z} \right)\left[ {Ja < K < Jb} \right].
    But this means a < \frac{K}{J} < b or there is a rational between a & b.

    If c < d \Rightarrow \quad c\sqrt 2  < d\sqrt 2 , then \left( {\exists r \in Q\backslash \{ 0\} } \right)\left[ {c\sqrt 2  < r < d\sqrt 2 } \right].
    This means that {c < \frac{r}{{\sqrt 2 }} < d}.
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