Can we characterize the square root of a prime number?

Discussion in 'General Math' started by Doug Wedel, Aug 25, 2008.

  1. Doug Wedel

    Doug Wedel Guest

    Is there anything we can say with certainty such as: the square root of a
    prime number is _always_ a transcendental?
     
    Doug Wedel, Aug 25, 2008
    #1
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  2. Doug Wedel

    fishfry Guest

    Yes. With certainty we can say that the square root of a prime number is
    NEVER transcendental. It's always algebraic, being a root of the
    equation x^2 - p = 0.
     
    fishfry, Aug 25, 2008
    #2
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  3. The square root of a prime, or any rational number for that matter, is
    never a transcendental. But you could say the square root of a prime
    is always irrational.
     
    Barry Schwarz, Aug 25, 2008
    #3
  4. I am curious as to why you might ever believe that an algebraic
    function
    evaluated with an algebraic argument might *ever* be transcendental???

    You seem not to understand the meaning of transcendental.
     
    Pubkeybreaker, Aug 26, 2008
    #4
  5. Doug Wedel

    Doug Wedel Guest

    I'm sure you are detecting some "black hole" in my understanding of these
    concepts.
    I was only trying to imagine what "qualities" the square root of a prime
    number might have.
    I have actually learned a lot from this exchange--I have learned for example
    from your
    post that my question itself was some kind of oxymoron.

    I wonder if someone could clearly state for me the oxymoron in my question?


    I am curious as to why you might ever believe that an algebraic
    function
    evaluated with an algebraic argument might *ever* be transcendental???

    You seem not to understand the meaning of transcendental.
     
    Doug Wedel, Aug 30, 2008
    #5
  6. You asked

    Is there anything we can say with certainty such as: the square
    root of a prime number is _always_ a transcendental?

    A square root of a prime number p would be an x such that

    x^2 - p = 0.

    Now, a transcendental number is one that is not the root of any
    polynomial equation with integral coefficients. So such an x cannot
    possibly be transcendental. It may be that you simply used the wrong
    word and meant to ask

    Is there anything we can say with certainty such as: the square
    root of a prime number is _always_ an irrational number?

    And we can indeed say that with certainty.
     
    Frederick Williams, Aug 30, 2008
    #6
  7. I already did. Your question simply shows that you do not know the
    definition of
    'transcendental'. There is nothing wrong with that.

    However, as a general principle, one should NOT use terminology for
    which
    one does not know the definition! There is something wrong with that.

    Mathematics is a language in which it is possible to state precisely
    what
    is intended. If one wants to discuss math, one MUST learn the
    language.
    Learn to say precisely what you mean. If you want to know something
    about the properties of sqrt(p) for prime p, then ASK THAT. Don't
    try to throw in terminology that you don't understand. It just makes
    you
    look silly.

    I assume that when someone uses the word 'transcendental', that they
    know
    what it means.
     
    Pubkeybreaker, Sep 2, 2008
    #7
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