Myp Mathematics 3 Answers

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Sourn Rose

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Aug 4, 2024, 5:27:34 PM8/4/24
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Forprinting a question together with all answers, the best option is still the same as two years ago: use Stack Printer. There is a handy bookmarklet on that page, which allows any question to be saved as PDF directly from the question page. The quality is pretty good.

Or, if you want the Q&A to be formatted just the way you want it, try Save Q&A as LaTeX, which converts the question with answers (and optionally, with comments) into a file that can be copy-pasted into, e.g., writeLaTeX or any other LaTeX compiler. Then get PDF from that. Some conversion errors may still occur; they require a minor manual adjustment of the source.


For graduate-level math books, the answer is typically not a value but a complete proof---typically of a related but relatively uninteresting topic. For example, one of the first exercises in the Neukirch book you reference is:


By the time that students are taking graduate-level mathematical courses, they are expected to have already mastered the general skills of constructing proofs. Seeing how somebody else has proved a point is thus not expected to be particularly educational, whereas struggling to prove something oneself forces a student to engage deeply with the material at hand.


Finally, examples of working with the concepts in the exercises are typically already given in the chapter, in the proofs of the main results, so adding extra examples by working proofs for the exercises would typically be of only incremental benefit, but undermine the value of students having to work through the proofs themselves.


Writing a graduate math textbook is a large effort that usually takes several years. The author typically has a vision of what material he/she wants to cover. After writing all the chapters and polishing everything, the exercises are probably the last part he/she works on. They are often meant as a pointer to additional more advanced topics in the literature that expand on the main content of the chapter, and adding solutions could require an effort comparable to writing an entirely new chapter (or several) to present that material in a polished, readable form. So, by that point the author feels that he/she is ready to move on to new projects and in any case the community is best served by releasing the book without exercise solutions. Solutions are sometimes added in later editions if the book is successful and the author is still passionate about the project.


Edit: Another thought that occurs to me is that adding exercise solutions can substantially increase the book's size. If the book is already of a good length (say 300 pages or more) then doing this could make the publisher very unhappy, and could potentially make the book less appealing to readers, who would start being intimidated/turned off by the book's length.


It is widely believed that there should be no solutions available, even privately, since this somehow ruins the game. This presumes that there should be "exercises" of the traditional sort in advanced mathematics courses, which is already partly dubious, since (as is often visible in commercially successful texts) it leads to make-work exercises often of questionable interest. I'd agree that there do exist significant, meaningful questions that may not fit into a small book... but would argue that then good write-ups of their solutions/resolutions should be available somewhere as models. Otherwise, all the students ever see is their peers' solutions... which in principle could be fine, but, observably, in practice, often overlook (through misunderstanding) ideas (from the text or otherwise) that make the resolution far more graceful and persuasive. That is, without good solutions, the only models anyone ever sees are "iffy".


(E.g., my abstract algebra text originally aimed to work a large fraction of the traditional significant questions as "examples", exactly to overcome the inertia of traditional-not-so-good alleged solutions of them, and have no "exercises" whatsoever. However, the publisher, who'd already made surprising concessions about intellectual property stuff, really-really wanted "exercises". So I made some near-clones of the worked exercises... And I've received several comments that I'm an anarchist for making those good solutions public!)


So, indeed, I think it's a bad idea to try somehow to suppress "good solutions". People will still grasp at bad solutions, and will be learning deficient versions of things to the extent they learn anything.


By the way, it is certainly not the case that the standard graduate mathematics texts provide means to resolve all their exercises. Often there is a considerable disjunction. Typically, the disjunction is that the theorems in the chapters do not at all suggest any quasi-algorithmic devices for doing computations in any particular case. E.g., abstract Galois theory usually disregards Lagrange resolvents, so does not hint at how to solve equations even when they can be proven solvable by radicals...


Nor is it the case that beginning math grad students are adepts at writing... so there is considerable feedback among them of marginal write-up style, marginal technical viewpoint, too much attention to secondary and tertiary details (often strictly demanded by in-my-opinion misguided texts or instructors), and needlessly distorted ambient language. Good writing models would help people "get over" this.


It's a commercial decision driven by the wishes of professors (who assign textbooks). Not having the answers makes them more important. I know of one author who wrote a very well regarded textbook with all the answers and had to remove them in second edition because his publisher said it wasn't selling as well.


I recommend people consult Schaum's, Kahn Academy, or look for books like Stroud or Granville that contain all the answers. It very much helps self study or even directed study (since the major learning comes NOT from the professor, but from working problems on your own).


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