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Math Help - Check whether root 17 is rational or irrational?

  1. #1
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    Check whether root 17 is rational or irrational?

    Prove whether root 17 is rational or irrational.
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  2. #2
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    The square root of any positive non-square integer is irrational.
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  3. #3
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    Rational or irrational

    Thanks a lot friend... but i need this to be proved by proof by contradiction.. Anybody there to help me?
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    The square root of any positive non-square integer is irrational.
    Proof:
    Suppose that n is a positive non-square integer.
    If \sqrt n were rational then the set T = \left\{ {k \in \mathbb{Z}^ +  :k\sqrt n  \in \mathbb{Z}^ +  } \right\} is not empty.
    Let j be the first in T. We see j \ne 1 because n is a on-square.
    That means that j > 1. Use the floor function: 0 < \sqrt n  - \left\lfloor {\sqrt n } \right\rfloor  < 1\, \Rightarrow \,0 < j\sqrt n  - \left\lfloor {\sqrt n } \right\rfloor j < j.
    That is a contradiction. Do you see it?
    Is j\sqrt n  - \left\lfloor {\sqrt n } \right\rfloor j a positve integer?
    What is \left( {j\sqrt n  - \left\lfloor {\sqrt n } \right\rfloor j} \right)\left( {\sqrt n } \right)?
    Last edited by Plato; March 11th 2009 at 10:53 AM. Reason: Fix LaTeX
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  5. #5
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    I have a similar question and not starting a new thread since they may be related:

    Suppose that a and b are positive non-square integers.

    \sqrt a + \sqrt b = c
    where c is a rational number. No such a, b, c exists.

    How can we prove this?

    -O
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