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Math Help - Absolute value proofs

  1. #1
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    Absolute value proofs

    I need to prove the implication,

    |a| < |b| \Rightarrow a^2 < b^2 I know that x^2 \mbox{is equal to} |x|^2 but how can I use that in a formal proof?

    Also I need to show that  |a|<|b| \Leftrightarrow  a^2 < b^2

    Once again all I can think of is the thing I wrote above. Thank you!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by meg0529 View Post
    I need to prove the implication,

    |a| < |b| \Rightarrow a^2 < b^2 I know that x^2 \mbox{is equal to} |x|^2 but how can I use that in a formal proof?
    note that an alternate definition for |x| is \sqrt{x^2}. also note that the square root function is a strictly increasing function, that is, for positive x and y, x < y \implies \sqrt{x} < \sqrt{y}.

    Now, to prove our implication, we can use the contrapositive: assume a^2 \ge b^2, then we have \sqrt{a^2} \ge \sqrt{b^2}. But that means |a| \ge |b|.

    Also I need to show that  |a|<|b| \Leftrightarrow  a^2 < b^2

    Once again all I can think of is the thing I wrote above. Thank you!
    in light of what i did before, try this one
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    note that an alternate definition for |x| is \sqrt{x^2}. also note that the square root function is a strictly increasing function, that is, for positive x and y, x < y \implies \sqrt{x} < \sqrt{y}.

    Now, to prove our implication, we can use the contrapositive: assume a^2 \ge b^2, then we have \sqrt{a^2} \ge \sqrt{b^2}. But that means |a| \ge |b|.

    in light of what i did before, try this one

    This might be really stupid, but by proving using contrapositive A -> B are we not proving B->A using inverse?
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  4. #4
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    Why not just apply simple rules?
    \begin{gathered}<br />
  \left| a \right| < \left| b \right|\, \Rightarrow \,\sqrt {a^2 }  < \sqrt {b^2 } \, \Rightarrow \,a^2  < b^2  \hfill \\<br />
   \hfill \\<br />
  a^2  < b^2 \, \Rightarrow \,\left| a \right|^2  < \left| b \right|^2 \, \Rightarrow \,\left| a \right| < \left| b \right| \hfill \\ <br />
\end{gathered}
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