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

1. ## 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 $*$?

2. ## Re: Is this a valid identity element? A dilemma

Originally Posted by x3bnm
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.

3. ## Re: Is this a valid identity element? A dilemma

Originally Posted by Plato
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.