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Splitting Cubic Formula into new page
Hi, I recently took steps to split information about the solutions to a cubic function to its own page, but my edit was undone to allow this to be discussed here. I would therefore like to formally propose that we split these two topics into distinct pages. This current page is around 80,000 characters, which WP:WHENSPLIT says "Probably should be divided (although the scope of a topic can sometimes justify the added reading time)". Beyond pure length, these two subjects deserve independent treatment - the cubic formula is interesting in and of itself, for reasons that are distinct from why cubic functions are interesting. We have separate articles for Quadratic formula and Quadratic function, and various other language Wikipedias treat Cubic Functions and the Cubic Formula independently, including German, Spanish, French, Chinese, and Russian, among others. As such, this split follows an established precedent, and I strongly recommend splitting these two subjects.
I would also like to ask for potential names for the new page. My original edit called the page "Cubic Formula", but I am open to alternative names. For the record, several other language wikis use some variation on "Cardano's Formula", but I do not feel this is appropriate, since the page treats other methods of finding cubic roots other than just Cardano's original formula.
- I agree to split the article, but, as I said in my edit summary, this is a major edit that require a consensus and a discussion on some details. Here are some of them
- What should be moved away? There are two possible choices: 1/ Moving all the content relative to the cubic equation. The drawback is that the remainder article would be very short (this was the motivation of a merge in 2007). 2/ Moving away only the cubic formula and its derivation. If this is chosen, many things that are in you article Cubic formula must be kept in Cubic function, such as the reduction to a depressed cubic (misplaced in the present version), the nature of the roots as a function of the discriminant, the Galois group, trigonometric solutions, etc.
- Which summary of the splitted article must be kept in Cubic function? The summary provided in your edit is not convenient, as being wrong in all cases where cubic roots are not well defined. In other words, it is correct only in the case of real coefficients and a single real root.
- My opinion is that it is worth to split away the cubic formula, but that, before splitting, one must reorganize the article for making a correct split simpler. This means having a single section "Cubic formula (for depressed cubics) and its derivations", and the other results regrouped in other sections. D.Lazard (talk) 09:04, 6 September 2019 (UTC)
As suggested before, this article is a mess. I have started to fix it. However, the number of issues makes the work harder than I thought previously. Here are the main issues:
- The order of presentation is not related to the importance of the the content nor to its technicality. For example, the section on depressed cubics was (before my edits) a subsection of an unrelated section, while reduction to a depressed cubic is fundamental for simplifying everything.
- Proof are presented with a lot of details (including very elementary computations, and proofs of side results that are better described in specific articles, generally not linked) that are confusing by making the exact results difficult to find, as well as the main ideas of the proofs.
- Some basic facts are lacking, such as that, up to translations and directional scalings along the axes, there is only three cubic functions, and thus three shapes for their graph.
So, restructuring the article will take some time, and the global reason of my edits needs to be explicited.
In a preceding thread is has been suggested to split the article.
In december 2007, Cubic equation were merged into Cubic function, with almost no discussion. The main reason was that the "function" part was very short. This is no more true, specially after my recent additions, and my restructuration for grouping together the "function" sections. On the other hand, the "equation" part is very long. It can certainly be dramatically shortened by giving less details in the proofs, per MOS:MATH#Proofs. Nevertheless, even after this, the article would remain very long. This makes obsolete the argument for the past merge.
Another argument in favor of splitting is that the coefficients are always real in the "function" part, while most of the "equation" part does not depend on the nature of the coefficients.
My suggestion is to split the article into Cubic function and Cubic equation. Another suggestion would be to split out the various proofs of the cubic formula into a subpage. I am not convinced by this. In any case, such a split must be decided after having simplified the proofs, as suggsted above.
- At first, I was confused as to which parts of the article would go to cubic function and which would go to cubic equation, since I would call "y = x^3 + px" a cubic equation (but not a cubic equation in one variable as defined in the lead). The suggestion seems reasonable otherwise though, and perhaps its implementation would be clearer than what I imagine right now. I'd say go for it. — MarkH21 (talk) 11:41, 24 September 2019 (UTC)
- To Aminabzz: The solution section has not been removed. It has been rewritten and renamed Cardano's formula for the depressed case, and General cubic formula for the general case. If you find the old version better, please, explain why. I'll try to solve your concerns, if any. By the way, this article has been splitted, and everything that concern equations is in Cubic equation except for a link in the lead of Cubic function. D.Lazard (talk) 20:09, 26 September 2019 (UTC)