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Math Help - Is this a valid identity element? A dilemma

  1. #1
    Senior Member x3bnm's Avatar
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    Is this a valid identity element? A dilemma

    There is an operation on a set \mathbb{R} (real numbers) such that:

    x * y = \sqrt{x^2 + y^2}

    Does \mathbb{R} have an identity element with respect to operation *?


    Now I find the identity element like this:

    \begin{align*}x * e =& \sqrt{x^2 + e^2} \\ x =& \sqrt{x^2 + e^2}......................\text{[by definition of identity element]} \\ x^2 =& (x^2 + e^2) \\ e^2 =& 0 \\ e =& 0 \end{align*}

    So now x * e = x * 0 = \pm x

    Is this e = 0 a valid identity element for this operation?

    Because an identity element has a property that it doesn't change the original element when used with the operation * on set \mathbb{R}.

    x * e = \sqrt{x^2 + 0^2} can be -x The original element is modified.

    But it's also possible that x * e = +x. The original element is left untouched. I'm confused by these two situations.

    So my question is:
    Is this e = 0 a valid identity element of \mathbb{R} with respect to *?
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  2. #2
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    Re: Is this a valid identity element? A dilemma

    Quote Originally Posted by x3bnm View Post
    There is an operation on a set \mathbb{R} (real numbers) such that:
    x * y = \sqrt{x^2 + y^2}
    Does \mathbb{R} have an identity element with respect to operation *?
    There is no e such that (-1)*e=-1.
    So the answer is no.
    Thanks from x3bnm
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  3. #3
    Senior Member x3bnm's Avatar
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    Re: Is this a valid identity element? A dilemma

    Quote Originally Posted by Plato View Post
    There is no e such that (-1)*e=-1.
    So the answer is no.
    Thanks Plato for answer. I didn't think of that way. Again thanks.
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