[Jprogramming] Complex tolerant comparison

[Elijah Stone]

So you do not find it inconsistent that 3j1e100=5j1e100 but 3j_~:5j_; nor 
that 3j_ and 5j_ seem to represent the same thing (as evinced by *.)?

It does not seem at all inconsistent or problematic to define = such that 
(x j. _) = (y j. _) (for finite, real x,y), and it does not follow from 
such a definition that _ must compare equal with finite values.

  -E

On Wed, 23 Feb 2022, Henry Rich wrote:

> As the essay points out, according to the rules _ is tolerantly equal to 
> everything.  To avoid that, comparisons to _ are intolerant.
>
> Henry Rich
>
> On 2/23/2022 7:09 AM, Elijah Stone wrote:
>> 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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