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
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@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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