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My friend recently told me that $\frac{-x}{x}$ could not be simplified any further. Is he correct or could it be simplified such that the answer isn't undefined when you x=0?

Joe Torraca
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    sometimes this is called signum, abbreviated sgn, with $\operatorname{sgn} x = 1$ if $x>0,$ $\operatorname{sgn} x = 0$ if $x=0,$ and $\operatorname{sgn} x = -1$ if $x<0.$ Related to the Heaviside function http://en.wikipedia.org/wiki/Heaviside_step_function – Will Jagy Sep 07 '14 at 21:17
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    It's $-1$ when $x \ne 0$, and undefined when $x = 0$. Usually, people will say that it can be simplified to $-1$, and write $-x/x = -1, x \ne 0$. – Dave Sep 07 '14 at 21:17
  • As a member of the field ${\mathbb F}(x)$ of rational functions in indeterminate $x$ over field $\mathbb F$, it is equal to $-1$. – Robert Israel Sep 07 '14 at 21:23
  • Oh, I automatically inserted absolute values in the denominator, not really there. That's life... – Will Jagy Sep 07 '14 at 21:56

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Well this is how it is $$- \frac{x}{x} = \begin{cases} -1 & \text{if} & x \neq 0 \\ \text{undefined} & \text{if} & x = 0 \end{cases}$$

Sheheryar Zaidi
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