Does it make sense to compare complex numbers in certain circumstances?

Question

I know that $\mathbb{C}$ is an unordered field and that (strictly non-real) complex numbers cannot be 'compared' in the sense that one is less than/greater than another.

However, we can compare real numbers; geometrically, this is because they lie on a straight line through the origin, $0+0i$, and one real is greater than another if it's 'further' along the real line than the other.

I'm wondering whether, given that two (or more) strictly complex (i.e. ones with nonzero real and imaginary parts) numbers can be compared given that they lie on a straight line through the origin. e.g. to me, it makes sense to say that $1+i<2+2i$, for instance, and that $i<10i,$ as one is a real multiple of the other.

Essentially, what I'm asking, is: can we compare complex numbers in any way other than comparing their moduli?

Thanks

Answer 1

Yes, you can put a lexicographical ordering on the complex numbers, which makes it a totally ordered set. However, we cannot introduce a totally ordered relation on the complex numbers (as a field such that the field operations are compatible with the defined order) since, every ordered field is a formally real field.

Answer 2

Yes, it totally makes sense.

In order to use $\gt$, $\lt$ and $=$ the same way they are used on real numbers, a many-to-one relation that maps complex numbers into real numbers will suffice.

More generally, an order is nothing but a relation. A total order is a relation $R$ that is transitive, asymmetrical and connected.

Transitive: if $xRy$ and $yRz$ then $xRz$, e.g. if 3>2 and 2>1 then 3>1;

Asymmetrical: if $xRy$ then $yRx$ must be false, e.g. if 3 > 1 then 1 > 3 must be false;

Connected: Given any two terms $x$ and $y$, either $xRy$ or $yRx$.

Source: Russell, Bertrand. Introduction to Mathematical Philosophy, "Chapter IV The Definition of Order"

Example1: Who gets off the Ferris Wheel first

Given any two complex numbers $a=r_1e^{i\phi}$ and $b=r_2e^{i\psi}$,

we define $a>b$ as

$(\phi > \psi) $ or $(\phi=\psi$ and $r_1>r_2 )$

This means that the person who has travelled a greater degree from the starting point or, if the travelled degrees are the same, the person who is on the further end of the spoke gets off first.

Thus, $3e^{i\pi}> 2e^{i\pi}> 3e^{i\dfrac{\pi}{2}}$

Example2: Sailing in crosswinds

Suppose you are sailing towards Northeast$(\dfrac{π}{4})$. You can produce various lifts by adjusting your sail, but some lifts are better than others because they produce more forward thrusts. Given a lift $fe^{i\theta}$, the forward thrust it produces is $fcos(θ−\dfrac{π}{4})$. This is the value you use to compare various lifts.

Edit: Depending on the field of terms, this comparison does not necessarily give rise to a total order. Different lifts can produce the same forward thrusts, thus the asymmetrical requirement is violated. Nevertheless, you can still compare lifts.

If you know Ruby programming language, you can customize the spaceship operator to sort complex numbers.