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This function D on the integers might be interesting, but why call it a derivative? I can’t see much conceptual similarity with calculus derivatives.
by bjornsing 2y ago
This function D on the integers might be interesting, but why call it a derivative? I can’t see much conceptual similarity with calculus derivatives.
- keepamovin 2y agoI'm just reading about it now, but I think because the derivative power rule holds for it, and you can do partial derivatives based on primes. The linked wikipedia article covers it well in Elementary properties.
- tromp 2y agoAs Wikipedia puts it, the > number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.
- constantcrying 2y agoBecause it obeys a similar rule to derivatives on functions. The pattern of taking a concept and extending it to other things by looking only at what happens under one rule, is extremely common in mathematics. That way you can e.g. extend the definition of a derivative to non-continous functions.
- im3w1l 2y ago> The arithmetic derivative can also be extended to any unique factorization domain (UFD),[6] such as the Gaussian integers and the Eisenstein integers, and its associated field of fractions. If the UFD is a polynomial ring, then the arithmetic derivative is the same as the derivation over said polynomial ring I was just skimming the wikipedia article but this seems like a good argument.
- gizmo686 2y agoIt is pretty common to talk about derivatives in abstract algebra in contexts where the calculus definition makes no sense. Broadly, there is a class of functions refered to as "derivations" that can be viewed as a generalization of the derivative. In particular, a derivation satisfies 2 properties: 1) It is linear 2) It satisfies the product rule. Any function that satisfies these rules is often called a "derivative". Notably, the function discussed in the article fails the linearity test, which is a pretty big problem for calling it a derivative.
- codeflo 2y agoAlso, the abstract algebra definition is an actual generalization, because the (analysis) derivative acts as a linear operator in the vector space of (suitably smooth) funtions. It blew my mind when I first encountered this way of looking at derivatives. And from there, it makes complete sense to look at operators with similar properties in other vector spaces. The definition presented here is a loose analogy to derivatives rather than an actual generalization, which doesn't fully justify using the name IMO.
- deleted 2y ago[deleted]
- JadeNB 2y ago> And from there, it makes complete sense to look at operators with similar properties in other vector spaces. It does, but this one's on a rig (= ring - negatives), not a vector space over any field.
- GrantMoyer 2y agoD(n) is equal to f'(0) where f(x) = Π_(p∈P(n)) x + p and P(n) is the set of prime factors of n.
- rm445 2y agoIt seems pretty confusing given that operator D already exists as notation for the usual differential calculus. (Not the most common notation, but useful sometimes). It may not be very exciting that D{n} = 0 for all numbers, but that's the actual derivative of a number.
- Someone 2y ago> It may not be very exciting that D{n} = 0 for all numbers, but that's the actual derivative of a number. No, that’s the derivative of the function that return n whatever its argument, which also can be written as λx.n or in a zillion different ways. That can be written as n, but is different from the number n. Also, one man’s “pretty confusingly is another man’s “similar things should have similar names”. There’s a rich history in mathematics of overloading the meaning of terms and symbols as long as there’s some similarity between them, for example when using × for both the multiplication of numbers and of matrices (where the former is commutative, but the latter isn’t, barring some exceptions such as 1 × 1 matrices). (See also the comment elsewhere in this thread which says “Mathematicans like to call two things with the same "structure" by the same name, even if it's not obvious how they're otherwise related” (https://news.ycombinator.com/item?id=40327885 https://news.ycombinator.com/item?id=40327885)