Suppose I want to implement some “numbers” (which should at least be a ring in the mathematical sense) in Julia. The documentation says that Number is the abstract supertype for all number types. Does Julia put any assumptions on elements of type Number that are needed for some functions accepting n…

This example suggest that Numbers need not have a commutative product. But it’s still unclear to me what the general assumptions are. Would (non-associative) octonions also be allowed? What about modular arithmetic? The integers mod n have positive characteristic and also zero-divisors unless n is p…

[image] matthias314: Will functions accepting Number arguments work with these kind of numbers? There are no guarantees. Some methods accepting Number might require only + and non-commutative * (and related functions like zero and one). Some methods might expect not only /, but even funct…

I agree that it’s nice to be able to extend or modify functions from other packages. But I also think it would be very helpful if it were clear what functionality I have to provide if I want to implement a subtype of a given abstract type (in this case, Number). I don’t want to start a general discu…

[image] stevengj: but even functions like exp or sqrt . Is it completely out of question having these functions implemented in pure Julia such that the extension of more basic operations made them work as well?

@matthias314 just make sure that you are not reinventing the wheel. I am positive that there are tons of high-level packages for abstract algebra out there where you can customize the operations and let the package do the heavy lifting for you.

exp is in pure Julia. The problem is that the method to compute exp (and similar functions) depends heavily on how the number is represented. (For example, the Float64 exp involves reinterpreting the memory as an unsigned integer at one point). Also, some objects like finite fields don’t have expone…

I’d say that one of the only things that one really expects from a number is that they behave immutably and usually are implemented with an immutable type. This is because mathematical objects in general are immutable and identified by value. You cannot change the value of an integer, you can only c…

What exactly is a `Number`?

General Usage

Mattriks March 3, 2022, 1:51am 2

A practical example: https://discourse.julialang.org/t/taking-quaternions-seriously/

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What exactly is a `Number`?

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