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 . A 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 on the number line. so X/Y>1. now this means (-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 C in such a way that it obeys the conditions satisfied and required by an order relation?