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Math Help - Rationals and irrationals - properties

  1. #1
    Newbie Femto's Avatar
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    Rationals and irrationals - properties

    Hey guys!

    So I have two questions which are similar, but not the same. The first asks me to prove that between any two distinct rational numbers there exists an irrational number - I haven't managed to do this. The question after however, which asks me to show that between any two real numbers there exists an irrational number, I've had an attempt at but I'm not sure whether the proof is adequate enough - it is written below (please bear in mind that I have already proven that between any two real numbers lies a rational number, so the first statement of my proof follows from that theorem, and in addition that \sqrt2 \notin \mathbb{Q}):

    Consider \dfrac{a}{\sqrt2} < \dfrac{p}{q} < \dfrac{b}{\sqrt2} where p,q \in \mathbb{Z} with q\not= 0 and a,b \in \mathbb{R}. That is, by definition, \dfrac{p}{q} \in \mathbb{Q}. Multiplying this inequality by \sqrt2 > 0 means that the inequality still holds, hence a < \dfrac{p\sqrt2}{q} < b and therefore it follows that between any two real numbers there lies an irrational number as  \dfrac{p\sqrt2}{q} \notin \mathbb{Q}.
    Last edited by Femto; November 1st 2012 at 06:19 AM.
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  2. #2
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    Re: Rationals and irrationals - properties

    some things you need to add, for clarity:

    since a,b are assumed distinct, without loss of generality you may assume a < b (or else switch them, a standard tactic).

    you should show why p√2/q is not rational (it's not hard, and only takes a line or two).

    p needs to be non-zero, or else your argument fails. for example, what if a = -1/n, and b = 1/n, where n is a VERY large positive integer (like 3 billion)? this is a rather serious defect.

    what can you do about this?
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  3. #3
    Newbie Femto's Avatar
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    Re: Rationals and irrationals - properties

    Quote Originally Posted by Deveno View Post
    some things you need to add, for clarity:

    since a,b are assumed distinct, without loss of generality you may assume a < b (or else switch them, a standard tactic).

    you should show why p√2/q is not rational (it's not hard, and only takes a line or two).

    p needs to be non-zero, or else your argument fails. for example, what if a = -1/n, and b = 1/n, where n is a VERY large positive integer (like 3 billion)? this is a rather serious defect.

    what can you do about this?
    Hmm, first of all thanks very much for your input; I really appreciate it.

    I understand your points apart from the last one - sorry I don't follow. I sort of understand that p must be non-zero but how does the following argument where a = -1/n and b = 1/n relate?

    Also, I've just proved that between two distinct real numbers there lies an irrational number. But how do I prove that between two distinct rational numbers there lies an irrational number? I thought that surely if all rational numbers are real then haven't I kind of just proved that already?
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    Re: Rationals and irrationals - properties

    Quote Originally Posted by Femto View Post
    Also, I've just proved that between two distinct real numbers there lies an irrational number. But how do I prove that between two distinct rational numbers there lies an irrational number? I thought that surely if all rational numbers are real then haven't I kind of just proved that already?
    You have done that.
    Between any two numbers there is a rational number.
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  5. #5
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    Re: Rationals and irrationals - properties

    how do i answer this question about rational and irrational numbers?








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