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Convince a classmate that division is not associative.

  1. May 26, 2007 #1
    1. The problem statement, all variables and given/known data
    How would you convince a classmate that division is not associative?
    By "associative" the book aims at the Associative properties of multiplication and addition.

    Is this equation correct?
    a/(b/c) = (a/b)/c

    2. Relevant equations
    a + (b + c) = (a + b) + c
    a(bc) = a(bc)

    a/(b/c) = (a/b)/c (?)

    3. The attempt at a solution
    The parenthesis' are top priority, and the arithmetic in it should be done first.
    - This means that a/(b/c) yields a/<new number>.
    - This means that (a/b)/c yields <new number>/c.
    -- This means a/<new number> does _not_ equal <new number>/c.

    Or can it?

    I would like to solve it algebraically, but I don't know how. Is it possible to solve it with algebra? I mean, does it take more advanced mathematics or can I simply use basic algebra as a tool for solving this?

    This is not homework actually, I'm trying to learn math by myself with the book Algebra and Trigonometry (Wesley 2007).

    Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 26, 2007 #2

    honestrosewater

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    Gold Member

    Well, the associative property claims to hold for all values of a, b, and c. So you could just find one set of values for which it doesn't hold, yes?
     
  4. May 26, 2007 #3
    Yes, that is correct. But are answers like that accepted? What if I would like to prove it without trial and error (if that's the correct expression)?
     
  5. May 26, 2007 #4

    honestrosewater

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    Gold Member

    Yes, counterexamples are perfectly acceptable.

    You could try to prove from the axioms

    There exists some a, b, and c such that a/(b/c) != (a/b)/c​

    where "!=" means "not equal", sure. Counterexamples are one way to do that.

    By the bye, proving things most ways involves a bit of trial and error. :wink:
     
  6. May 26, 2007 #5
    Its called indirect proof or proof by contradiction, and its a perfectly legitamate form of proof. Assume to the contrary (division is associative) and work until you reach something that cant be true
     
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