Break `abs` into two aspects

[Carter Schonwald]

One long running pain point is that the abs definition we have for complex
numbers is terrible.  Does anyone use it ?

On Tue, Jan 28, 2020 at 9:59 AM Carter Schonwald <carter.schonwald at gmail.com>
wrote:

> Well said Andrew!
>
> There’s a second twist : last I checked our abs for complex numbers isn’t
> the Euclidean norm or any Lp norm ..
>
> We could define the Pth power of the lpNorm for any complex a I think.
> Though that’s a weaker operation.
>
> On Tue, Jan 28, 2020 at 7:11 AM Andrew Lelechenko <
> andrew.lelechenko at gmail.com> wrote:
>
>> On Tue, 28 Jan 2020, Dannyu NDos wrote:
>>
>> > `abs` represents a norm, but its type is wrong
>>
>> There are two useful meanings of `abs`, which coincide for integers. One
>> is a norm. Another one is to define `abs` as a mapping from a ring R to a
>> factor ring R / U(R), where U(R) is a ring of units, and `signum` as a
>> mapping from R to U(R) such that `abs a * signum a = a`.
>>
>> > This enables us to implement rings (Num) and fields (Fractional) without
>> > concerning about norms. For example, Gaussian integers.
>>
>> For Gaussian integers I find convenient to define `signum z` with a
>> codomain {1, i, -1, -I} (basically, to which quadrant does z belong?) and
>> `abs z` with the first quadrant as a codomain.
>>
>> Best regards,
>> Andrew
>>
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>
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