# Terminology for $x=-x$ when x is a positive integer

My proof by contradiction ends with $x=-x$ when x is a positive integer. What is the correct terminology for why this is a contradiction?

Right now it says "This yields a contradiction since a positive integer cannot equal the negative value of itself", but I'm sure there's a theorum or something for this.

Thank you!

• You should probably give some context as to what the proof is, and what you tried. – Mark Pineau Apr 14 '17 at 11:17
• What does the $/$ mean? – kingW3 Apr 14 '17 at 11:17
• Did you mean $x\ne -x$? If you did then that isn't necessarily a contradiction unless $x=0$. – Laars Helenius Apr 14 '17 at 11:19
• @LaarsHelenius: You mean "unless $x\ne 0$", right? – Henning Makholm Apr 14 '17 at 11:22
• In some cases, $x = -x$ and not just for 0. In a field of characteristic 2 for example. – badjohn Apr 14 '17 at 11:25

Trichotomy Law: If $a,b\in\mathbb{R}$ then exactly one of the following holds $$(i)a>b\quad (ii)a=b\quad (iii)a<b$$
In your case, we have $-x<x$ since $x$ is positive integer. Thus, by the above law, $-x\neq x$. This is in contradiction to the one you obtained $-x=x$.
Assume $x\ne0$ and $x=-x$. Since $x$ is not $0$, we may divide by $x$ to achieve: $1=-1$. This is a falsity. Therefore we conclude $x=0$, which is not a positive number.
Just note that $x=-x$ is equivalent to $x+x = 0$, equivalent to $x = 0$.