If I am not wrong then while we are at mathematics the statements like "$A$ is greater than $B$" or "$A$ is lesser than $B$" are meaningless unless and until we have defined what exactly we mean by "greater" or "smaller". Hence we need to define order relations . An order relation $R$ is defined as a relation on a set $A$ which has the following properties:
- If $x,y$ belong to $A$ then either $xRy$ or $yRx$;
- For no $x$ belonging to $A$, $xRx$.
- $R$ is transitive.
Now let us consider the two positive real numbers $x$ and $y$. Now let $x$ be greater than $y$ , with the order relation being defined as $a>b$ if $a$ lies to the right of $b$ on the number line. So $\frac{x}{y}>1$. Now this means $\frac{-x}{-y}>1$. But as per the definition of order relation $-y$ lies to the left of $-x$. So will I be right in concluding that these two are completely different order relations.
At the same time, I have heard that the whole set of complex numbers is not an ordered field.(I say , I have heard cause I don't have anything to prove or disprove). Can anyone provide me a proof that we can never define a relation on $\Bbb C$ in such a way that it obeys the conditions satisfied and required by an order relation?