Why it is necessary to consider positive $ \ x,y \ $ in $ F\ $?

Question

If $ F $ is an ordered field , then for any positive elements $x $ and $y$ in $F$ there is a natural number $ n$ such that $ nx>y \ $ i.e.,
\begin{eqnarray*} \underbrace{x+ \cdots+ x}_{n \ \text{times}}>y . \end{eqnarray*} This is the Archimedian property.

But my question is -

Why it is necessary to consider positive $ \ x,y \ $ in $ F\ $?

Because since $ F $ is ordered field , there can be other orders except positivity.

So I think $x,y$ to be positive is not necessary.

Help me to clear my confusion.

Answer 1

You can modify the definition by stating the property as

for all $x,y\in F$, if $x\ne0$ then there exists an integer $n$ such that $nx>|y|$.

The meaning of $|y|$ is the usual one: $|y|=y$ if $y\ge0$ and $|y|=-y$ if $y<0$.

Exercise: prove that this is equivalent to the standard statement of the property.

Answer 2

Simply because $n$ is positive.