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Homework Help: Absolute value proof

  1. Jun 10, 2010 #1
    1. The problem statement, all variables and given/known data

    If a,b are real numbers and b does not equal zero show that |a|=sqrt(a^2) and |a/b|=|a|/|b|.

    2. Relevant equations

    I know that |ab|=|a||b| and a^2 = |a|^2

    3. The attempt at a solution

    Attempt at showing that |a|=sqrt(a^2):
    |a|=sqrt(a^2)
    |a|^2=(sqrt(a^2))^2
    |a|^2=a^2
    a^2=a^2

    Not sure how to do the second part.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 10, 2010 #2

    Mark44

    Staff: Mentor

    So |a| = |a/b * b| = |a/b||b|. What can you conclude about |a/b|?
    It looks like you are assuming that |a| = sqrt(a^2) (which is what you need to prove), and concluding that a^2 equals itself.

    Instead of going about it this way, I would suggest using two cases: one with a >= 0 and the other with a < 0. Can you show for each case that sqrt(a^2) = |a|?
     
  4. Jun 10, 2010 #3
    It's reasonable to expect that the definition of the absolute value function would surface at some point.

    Your first attempt is invalid because you assumed the statement that you were trying to prove. If you reversed the steps, you would have to use the square root property, and it wouldn't work.

    Here's a sample of what you would do to prove that |ab| = |a|*|b|.
    Case: a > 0; b > 0.
    Then ab > 0 and |ab| = ab = |a|*|b|.

    Case : a > 0; b < 0 (or vice versa)...
    Case : a < 0; b < 0.

    Give that a shot.
     
    Last edited: Jun 10, 2010
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