How do we say greater than or less than when talking about complex numbers?

This might appear to be fairly innocuous, but it is actually quite an interesting question. First of all, what do we want from an order?

A (total) order should satisfy the following properties for elements of a set X:

1. For any number a, a ≤ a
2. Transitivity: if a ≤ b and b ≤ c, then a ≤ c
3. Antisymmetric: if a ≤ b and b ≤ a, then a=b
4. Totality: a ≤ b or b ≤ a

Ok so that's fine and all, but the complex numbers form a field as well, so we want the order to interact correctly with the field structure. To make a field (F,+,x) and an order < into an ordered field, we need two more things:

1. If a < b then a + c < b + c
2. If 0 < a and 0 < b, then 0 < ab

Now, you can show that if you want all of these things to happen, you cannot have a total order on the complex numbers. However, people still want to compare complex numbers, so how does one go about this? Well, if we can map the complex numbers to the reals in some sensible way, one can compare real numbers in a straightforward way. So, a lot of the time, comparing complex numbers is done by comparing their absolute values.
There is no natural ordering of the complex numbers like there are for the reals. We cannot say z1 > z2 for z1, z2 complex numbers. We can however take the absolute values of z1, z2 and assert |z1| > |z2|
The real line is one dimensional so there is a natural ordering, but the complex plane is two dimensional so there is no such ordering - we can only consider absolute values, i.e distance between points
by
It helps to think of them as points on a 2-dimensional plane. so a+ib and c+id would denote two points with Cartesian coordinates (a,b) and (c,d).

Now can you say one point is greater than the other? It makes no logical sense. You can however compare how far each point is away from the origin, and that'll be a comparison of their absolute values, or compare the angle the lines joining the point to the origin makes with the real(positive x) axis, comparing their phases.
Short answer is we don't. We don't write a + bi > c + di
In short: undefined
Any suitable vector-norm gives a preorder on ℂ.
You don't.
Iin my undergraduate course, we were told that they're not an ordered field so it makes no sense to try and order them. There was a neat proof of this:

If i>0, then multiplying both sides by i, we get -1>0, which is not true.

Whereas if i<0, then multiplying both sides (and flipping the inequality over) gives us -1>0 again.

So we can't order the complex numbers like we can the reals.
I've seen z_1 >> z_2 with the strict definition that Re(z_1) > Re(z_2) and Im(z_1) > Im(z_2) (WLOG for <<) but I'm not sure if it is just something that more than one person uses

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