[Jprogramming] Complex tolerant comparison

[Elijah Stone]

https://code.jsoftware.com/wiki/Essays/Tolerant_Comparison

A good read, explains the workings of the comparators.

  -E

On Wed, 23 Feb 2022, Michail L. Liarmakopoulos wrote:

> Thanks for the detailed explanation and for the counterexample to my
> argument.
>
> It seems that I haven't thought carefully how the "=" operates behind the
> scenes when comparing two complex numbers.
>
> Best regards,
>
> ---
> Michail L. Liarmakopoulos, MSc
>
> On Wed, Feb 23, 2022, 13:03 Elijah Stone <elronnd at elronnd.net> wrote:
>
>> Two arguments:
>>
>> 1. Comparison is tolerant anyway.  So:
>>
>>     3j1e100 = 5j1e100
>> 1
>>
>> Since _ is greater than 1e100, the (relative) difference between 3j_ and
>> 5j_ should be less than that between 3j1e100 and 5j1e100
>>
>> 2. Infinity is, of course, not a number.  So, when encountering an
>> expression involving infinity, it is necessary to ask how it should be
>> interpreted.  One way to interpret 3j_ is as follows: the limit as y
>> approaches _ of 3 j. y.  Well, that does not converge.  But its _angle_
>> does, as does the angle of 5 j. y, and they converge to the same result:
>> 0.5p1.  The j interpreter is kind enough to tell us as much:
>>
>>     0.5p1 =!.0 {:"1*.3j_ 5j_
>> 1 1
>>     3j_ -:!.0&*. 5j_
>> 1
>>
>> (The fact that these expressions evaluate to 1 even with a comparison
>> tolerance of 0--and would continue to do so, even given infinite
>> precision--is my argument for why 3j_=5j_ should be 1 even with a
>> comparison tolerance of 0.  I recognise, however, that there are other
>> tradeoffs involved, and this may not be the right choice; but I think
>> there is no excuse for a 0 result given nonzero comparison tolerance,
>> unless _=_ is changed.)
>>
>>   -E
>>
>> On Wed, 23 Feb 2022, Michail L. Liarmakopoulos wrote:
>>
>> > It seems correct to me, as they're two different complex numbers. If they
>> > were the same:
>> >
>> > 5j_=5j_
>> > 1
>> >
>> >
>> > They have the same modulus though:
>> >
>> >   %: 9 + _
>> >
>> > _
>> >
>> >   %: 25 + _
>> >
>> > _
>> >
>> >   (|3j_) = (|5j_)
>> >
>> > 1
>> >
>> > Best,
>> >
>> > ---
>> > Michail L. Liarmakopoulos, MSc
>> >
>> > On Wed, Feb 23, 2022, 10:15 Elijah Stone <elronnd at elronnd.net> wrote:
>> >
>> >>     3j_ = 5j_
>> >> 0
>> >>
>> >> I think it should be 1.  For nonzero comparison tolerance, at any rate.
>> >>
>> >>   -E
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