Inconsistent definition of "algebraic operation"
The lead section of the dot product article describes the dot product as an algebraic operation, but the algebraic operation article limits the definition of dot product to addition, subtraction, multiplication, and taking roots. Is there a factual error in either of these two articles? Jarble (talk) 02:42, 6 January 2015 (UTC)
I'm going to go out on a limb here since I can't reference everything I'm going to say. The operations listed in this article have been called the "Fundamental Algebraic Operations" in at least one old source that I have and I am sure that I have seen the expression "basic algebraic operations" used for the same collection. This implies that there exist algebraic operations which are not basic (or perhaps we should say not atomic). Certainly a candidate for this more general type of algebraic operation would be a finite algorithm each of whose steps was an application of a basic algebraic operation. Again borrowing from chemistry, we could call such an algorithm a "compound algebraic operation". In this sense, both finding a dot product, completing the square, and a host of other computations (calculating a factorial, factoring a polynomial by grouping terms appropriately, come to mind) are algebraic operations. I don't think that I have ever seen this formally defined, but I am pretty sure that this is what most authors, who use this term loosely, would have in mind. A more modern approach would be to say that an algebraic operation is the result of applying an (algebraic) operator in some appropriate domain where the descriptor indicates that there exists some representation of the operator which only involves the basic algebraic operations. This is a little too fancy for most of the uses of this expression. Historically I think that the expression arose when a distinction was needed in taking operations on (real) numbers and expressing their corresponding action in the realm of algebra. This is a very limited use and I can see that there would be pressure to expand the envelope somewhat. I am not making a suggestion as to what to do with the articles in question, but I don't think that what is wrong with them can be called a factual error. Bill Cherowitzo (talk) 05:07, 6 January 2015 (UTC)