Should the inequality operators $>$ and $<$ be defined between positive and negative integers?

Question

These operators aren't defined between complex numbers. I think that the reason is that Complex numbers represent co-ordinates. So, we can't say whether $(6,4)$ is greater or smaller than $(-8,1)$.

But why are they defined between positive and negative integers? I think negative numbers simply represent the opposite of what positive numbers represent. And, opposite doesn't mean less. They just represent the other direction. How can we say that $5$ units North is 'greater than' $3$ units South? They're different things.

Answer 1

The reason you want $-3<5$ is so that you can say $(10-3)<(10+5)$, i.e. if $a<b$ then $a+c<b+c$, for all integers $a,b,c$.

If we don't allow comparisons between positives and negatives, this rule would only hold sometimes. You would have $3<15$ and $(3-20)<(15-20)$, but not $(3-5)<(15-5)$.

Answer 2

The integers have a sense of an order since they have a relative position one of each other, that is, an integer is either "to the left", "to the right" or equal to another integer. This can give us the idea of an ordered set and moreover is "independent" of the idea of a number being postiive or negative.

This is not the case with the complex numbers, where there is not a clear way to relate two vectors in an "ordered" way. To be more precise, one can show that there can not be a total order over $\Bbb C$ that respects the field operations (addition and multiplication). This anomaly is basically a consequence of the relation $i^2 = -1$.