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Homework Help: Inequality - Proof that √(a^2)<√(b^2) does not imply a<b

  1. Aug 22, 2014 #1
    Hi everyone!

    First of all thank you for a great forum! I downloaded the app and find it ingenious!

    The problem stated above is from "3000 Solved Problems in Calculus".

    The book solves this problem simply by stating: "No. Let a=1 and b=-2".

    However, I am curious to know if it is possible to provide a more algebraic proof, or of this is the only way to prove it. As I really cannot provide any attempts of my own, I will just ask if anyone know what topic I have to study in order to find the answer. If I fail after that, I will return to you again.

    (Maybe it should be mentioned that I am interested in doing Calculus 1,but I want the fundamentals in order first.)

    Yours sincerely,
  2. jcsd
  3. Aug 22, 2014 #2


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    I am puzzled by this. Are you saying that this proof is too simple and you want a harder proof?

    The problem is to show that "if [itex]\sqrt{a^2}< \sqrt{b^2}[/itex] then a< b" is NOT true. A standard way to show that an "if-then" statement is not true is to give a "counter example". While no number of "examples" will show that such a statement is true one example in which it does not work is enough to show that it is NOT true.
  4. Aug 22, 2014 #3
    Ah, cool! I see your point. I'm not really used to proofs at all. Thank you very much for the quick reply!
  5. Aug 22, 2014 #4

    Ray Vickson

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    Just to be clear: what the book gave was not a "proof", it was a "disproof". Even if you do not know how to do proofs you may be able to do disproofs, as they are very different concepts.
  6. Aug 22, 2014 #5
    I see. Thank you for the clarification!
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