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Math Help - Prove that absolute value of a equals square root of a squared

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    Prove that absolute value of a equals square root of a squared

    Hello, I am reviewing the precal section of a calculus textbook and one of the problems asks you to prove that

    Absolute value of a = the square root of a squared

    I have proved a few other properties of absolute values but for some reason this one is stumping me. Could anyone start me in the right direction? This is one of my first encounters with elementary proofs. Thank you!
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    Re: Prove that absolute value of a equals square root of a squared

    Quote Originally Posted by Ragnarok View Post
    Hello, I am reviewing the precal section of a calculus textbook and one of the problems asks you to prove that

    Absolute value of a = the square root of a squared

    I have proved a few other properties of absolute values but for some reason this one is stumping me. Could anyone start me in the right direction? This is one of my first encounters with elementary proofs. Thank you!
    It depends how you define the absolute value. If it's defined to be

    \displaystyle \begin{align*} |x| = \begin{cases} \phantom{-}x \textrm{ if } x \geq 0 \\ -x \textrm{ if } x < 0\end{cases} \end{align*}

    since it makes any negatives positive, it can be thought of as the distance from 0.

    Now if we remember that any square root has two possible answers, \displaystyle \begin{align*} \sqrt{x^2} = \pm x \end{align*}, that means the answer is something that is \displaystyle \begin{align*} x \end{align*} units away from \displaystyle \begin{align*} 0 \end{align*}, which is exactly how the absolute value is defined
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    Re: Prove that absolute value of a equals square root of a squared

    Thank you, that is the absolute value definition I am working with. I understand the theorem intuitively but I don't know how to prove it rigorously. In this case it looks so "obvious" to me that I don't know how one goes about proving it, and what we can assume. For example can I say that

    x squared = absoluteValue(x squared)

    without proving that? I apologize for not using formatting, I am on a phone right now and can't look it up.
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    Re: Prove that absolute value of a equals square root of a squared

    Quote Originally Posted by Ragnarok View Post
    Thank you, that is the absolute value definition I am working with. I understand the theorem intuitively but I don't know how to prove it rigorously. In this case it looks so "obvious" to me that I don't know how one goes about proving it, and what we can assume. For example can I say that

    x squared = absoluteValue(x squared)

    without proving that? I apologize for not using formatting, I am on a phone right now and can't look it up.
    Yes you can, since \displaystyle \begin{align*} x^2 \geq 0 \end{align*} for all \displaystyle \begin{align*} x \in \mathbf{R} \end{align*}. Therefore by definition of \displaystyle \begin{align*} |X| = X \textrm{ if } X \geq 0 \end{align*}, with \displaystyle \begin{align*} X = x^2 \end{align*}, we have \displaystyle \begin{align*} \left|x^2\right| = x^2 \end{align*}
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