# Re: Ordering in \bbold{C}

```Maybe it's just me, but I don't see the value of having some
*arbitrary* predefined order for complex numbers. If people really
want to order their complex numbers, let them do it themselves in
whatever way they want.```
```
Leon

On Mon, Mar 29, 2010 at 6:10 AM, Darren Duncan <dar...@darrenduncan.net> wrote:
> I was actually thinking, during the previous thread involving Complex
> numbers ...
>
> It may not have any practical use, but if one wanted to define an ordering
> for complex numbers that was deterministic and relatively unbiased, a way to
> do this would be based on what I'll call for now the "spiral distance".
>
> Conceptually, you take an infinite length spiral line that starts at and is
> centered on the origin, where for each turn the current spot on the spiral
> increases an infinitesimal radius from the origin, or a distance approaching
> zero, in the calculus sense.  Complex numbers closer to the origin on the
> spiral will be ordered earlier than those further from the spiral.
>
> Actually calculating this is a simple comparison of the radius and angle
> components of the two complex numbers in the polar coordinate system.  If
> the radius value is different, then the one with the smaller radius is
> ordered before the one with the larger; if the two radius values are the
> same, then the one with the smaller angle is ordered first; if both are the
> same, then the two complex numbers are equal.
>
> The math is just as simple as a naive comparison that just compares the real
> component and then imaginary component in a cartesian coordinate system, but
> the result is much more reasonable I think.
>
> This whole principle of "distance from origin" method of ordering does also,
> I suspect, scale to any number of dimensions; the one-dimensional version is
> simply comparing first the absolute value of the two numbers, and then
> saying that either the positive or negative version orders first.
>
> -- Darren Duncan
>
```