Ordering on complex numbers

So I just learned that complex numbers cannot be ordered from Total ordering on complex numbers.

Now, does this mean that 4 - 3i < 4 + 4i isn't true? I haven't really seen any concrete examples and I don't even get what the proof means, but I know it makes sense - I just can't relate to it.

In short, what are some concrete examples of this theorem? What is also a walkthrough of the theorem?

Solution 1:

You can order the complex numbers, just not in a way compatible with multiplication in the way we want it to be.

An ordered field is not just a field that is ordered, it's one that has compatibility with the multiplication and the ordering. One of the conditions is that if $a<b$ and $c>0$, then $ac<bc$, and another is that if $a>0$ and $b>0$, then $ab>0$.

This solution at the problem you linked to gives the elementary details of showing that no total ordering is consistent with the multiplication of $\Bbb C$.

But if you don't care about the compatibility of the order with multiplication, then you can order it as a set however you like. One example ordering could be ordering lexicographically by real and imaginary parts.

Solution 2:

It appears like you are considering the product ordering in your example, i.e. $a + bi \leq c + di$ if and only if $a\leq c$ and $b\leq d$. This is a perfectly fine partial ordering, but it is (1) not total, i.e. not all complex numbers are comparable: $1-i \not\leq -1+i$ and $-1+i\not\leq 1-i$ and (2) not compatible with multiplication: $i\geq 0$, but $i^2 \not\geq 0$.