should complex, non-real, objects be comparable?

[Chris Smith]

Python and SymPy both raise an error for something like I < 2*I -- is there a good reason to disallow comparison of numbers that have the same arg even though it is not 0 (as in the comparison of real numbers)? Allowing this would allow 1 + 2*I < 2 + 4*I to be true, too.

[Richard Fateman]

What would the result  from   a>b   if a and b are not comparable by your rule?  false? error?  abs(a)>abs(b)?

[Aaron Meurer]

Aside from consistency with Python, the reason is that you can't
really define a so-called algebraic ordering on complex numbers.
Algebraic means that you can add both sides of an inequality (if a > b
then a + c > b + c), and multiply them by something positive (if a > b
and c > 0, then a*c > b*c).

For example, if you allow 1 + 2*I < 2 + 4*I, then rearranging by the
first rule, you get 2*I + 1> 0. By the second rule, this is positive,
so we can multiply it by both sides of the first inequality, giving -3
+ 4*I < -6 + 8*I. Rearranging gives 4*I > 3. Since we still would have
normal positive, numbers, 3 > 0, so by transitivity, 4*I > 0. But then
multiplying that equation by 4*I by the first rule gives -8 > 0, which
is not true.

Basically, no matter how you try to define it, you can't make complex
numbers into an algebraic ordering, because the way that I**2 gives a
negative number screws up the second rule, and the first rule always
lets you get to I > 0 or I < 0 (i.e., -I > 0), and since I**2 =
(-I)**2 = -1, you get a contradiction either way.

We *definitely* want > to mean an algebraic ordering in SymPy, since
it implicitly lets us perform operations on inequalities like moving
things to one side (saying a > b is equivalent to a - b being
positive) and multiplying by positive things, which is really how you
combine inequalities to do useful algebra with them. I say implicitly
because things like, "solve (x - 1)*(x + 4) > 0 by finding where x - 1
and x + 4 are both positive or both negative" inherently rely on this
rule.

Aaron Meurer

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[Chris Smith]

Very informative. Thanks for taking the time to give this answer, Aaron.