prove by contradiction: there is no greatest negative rational number?

I know you have to keep dividing by 2 and prove that it is still negative but I don't know how to prove it as a full proof

2 Answers

  • You basically have it already in your update. You just have to write it down a little more formally.

    Assume that there is a greatest negative rational number r.

    The number is of the form r = -p/q, where p and q are positive integers.

    Divide that number by 2 to get -p/2q.

    That new number is rational and negative since 2q is a positive integer and that doesn't affect the sign or the rationality.

    But let's compare -p/q and -p/2q.

    Take the absolute value of each:

    |-p/q| = p/q

    |-p/2q| = p/2q

    If we compare these we see:

    1(p/q) > ½(p/q)

    1 > ½

    The absolute value is another way of describing the distance from the number to zero. Since -p/q is further from zero and they are both negative, that means -p/2q is closer to zero and thus greater.

    But this contradicts our initial statement that r = -p/q is the largest negative rational number.

    QED: There is no negative rational number greater than all other negative rational numbers.

  • Easy. Let X be that greatest negative rational number. But then (1/2)x is greater, and is also rational. Contradicting the assumption.

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