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"An introduction to the theory of numbers, G.H Hardy, E.M. Wright, revised by D.R. Heath-Brown, J.H. Silverman. Originally published 1938. Sixth edition 2008 with a foreword by Andrew Wiles" is AFAIK a highly praised book. - What seems odd to me is that there are no exercises in the book. This must be the first mathematics book I have ever seen that has no exercises in it. Exercises, with or without hints can't be missed IMO.

Question. Was this common for books written in that time, i.e. mathematics books without exercises? If not, it must have been noticed by the critics. What did they say about it? Or (shame ) have I perhaps missed an accompanying exercise book?

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"This must be the first mathematics book ... no exercises in the book." A random sample of 10 maths books I have handy shows that 4 doesn't have exercises. Their publication dates range from 1934 to 2000. –  Willie Wong Aug 16 '11 at 18:56
The book is one giant exercise. –  Mariano Suárez-Alvarez Aug 16 '11 at 18:57
Most (maybe all!) British math books, except for high school level ones, had no exercises. (High school level included the calculus). There were really no university level math textbooks. –  André Nicolas Aug 16 '11 at 19:16
It just demands a little bit more discipline from you. Try to prove the theorems yourself. –  Jonas Teuwen Aug 16 '11 at 20:22
I don't mind a research-level book with no exercises. What annoys me is a research-level book with no index!!!!!!!!!!!!!! –  GEdgar Aug 17 '11 at 0:25
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2 Answers

up vote 31 down vote accepted

Conspicuously, Lang's "Algebraic Number Theory" had no exercises in any of the 3 editions I've owned. I don't remember that Weil's "Basic Number Theory" did. Titchmarsh's "The theory of the Riemann zeta" does not. Artin-Tate's "Classfield Theory" does not. I had never thought about the fact that Hardy-Wright does not.

My own experience was contriving examples to illustrate the theorems, and trying to understand the (not always completely explicit) hypotheses or contexts of the theorems. I didn't think of looking for exercises-at-end-of-chapter.

When Lang's and Weil's books appeared, they were the first serious alg no th books in English. Lang's book is itself (as Mariano S-A says about Hardy-Wright, above) one large exercise. Ditto Weil's. What are "exercises" supposed to be, anyway? Derivative filler? Who'd want that?

As far as I know, the "textbook" concept, with exercises included, is a post-WWII invention of U.S. publishers, and/or publishers for the U.S. text-book market, which until recently may have been the dominant force in text-book publishing. Having published textbooks myself, I am well-acquainted with, and somewhat dismayed by, the fixation publishers have with "exercises".

The exercises in Rudin's "Real and Complex", and Lang's "Algebra" are notorious landmarks. One aspect that merits notoriety is that a certain small fraction of them is routine rewriting of definitions/theorems from the chapter, but many require a non-trivial idea. Really, is it true that exercises should test mathematical talent? Seems rather pointless, or even dishonest to make people pay tuition for "education" that is something else.

By this point, I am of the opinion that "the text" should give samples of all questions asked as "exercises", or it's a cheat. Trivial things oughtn't be given as exercises (busywork?), and non-trivial things need models.

Atiyah-MacDonald's "Commutative Algebra" is a crazy extreme case, in my opinion, despite its virtues: perhaps the hardest half (or 2/3) of the results in it are innocently-posed "exercises". Thus, the attractive slimness of the book. But it is a substantial dis-service to leave readers with dubious, un-enlightened, perhaps completely incorrect "solutions" to exercises... if the "exercises" are serious results worth executing well, and with emphasis on the critical features.

In many cases, in my experience, otherwise-conscientious students are too-often entrapped by pointless, make-work exercises, so find themselves feeling they've accomplished something by having spent much time, but, in fact, have not engaged with the central ideas.

The contrived necessity of homework, exams, and grades is very corrupting... Why is mathematics perceived as exclusively a "school subject", and its sense and habits defined by grading systems and publishers? It is bizarre.

In summary, I was at-first-surprised by the question, but soon recovered my equilibrium... and/but had the above reaction. :)

Edit/addendum: incorporating @Pete's comment and @Willie W's... : Indeed, it is not accurate to criticize the idea of "exercises" by noting that too many exercises are bad. It is possible, but very challenging, to give guided-exercises-with-hints in a non-combative, non-challenging fashion. One should try to do so! The false challenge of grading should/must be separated from explanation. To with-hold explanation as a "test" is a bad thing (despite its being standard).

It took me decades, but I only recently realized that the U.S. system has taught everyone to perceive "teachers" as antagonists, not (to say the least) unqualified supporters. Thus, everything said in lecture or notes or text is a potential challenge, a potential expression of doubt that the reader/student/audience understands. The UK system is a bit different, but I do not understand the current mind-set.

Hilariously, my attempts to rise above the corruption of adversarial game-playing grading by promising everyone an "A"... with required [sic] homework have not been as happy as I would have wished: I think the conditioning is too strong, and that combative, adversarial mindset does continue into grad-school. Sigh...

I do also think that discussion of the role of "exercises" deserves attention. Many students (not to mention old people...) misunderstand the "drill".

Edit-2: Thanks to @Gerry M. Indeed! Hardy's "Course..." I'd need to investigate years-of-publication and such, but I know the provenance of that thing needlessly well. My father (a high school math teacher in the U.S. required to obtain a higher degree [sic] in mathematics) endured a night-school course whose text was Hardy's. I was not very old, but old enough to be ... stunned... by the tendentiousness of that text. I had a fair understanding of analysis in the late 1960s, and/but Hardy's text effectively expressed doubt that its reader "had a brain in their body". And would not explain anything to them, either. Ack. I don't hold Hardy morally responsible for that text.

The Bourbaki impulse suggested things ... and publishers solicit. That is not the same as sincere intellectual expressions.

Edit... and about Zygmund, I am truly interested to look at my copy (in my campus office, tomorrow...) That would be an odder (counter-) example than many, to my mind. Relatedly, did Banach's monograph have "exercises"? Did Hausdorff's book? I do srsly think that "textbooks" were not what people were thinking about in those years. Advancing mathematics, monographs, yes. (What kind of "exercises" does EGA have? I'll also look tomorrow...)

Edit-edit: in response to @Bill D's query about my reaction to Hardy's "Course...": (First, certainly Hardy was a very good mathematician.) Surely it's a matter of taste, all the more so about the importance of "logical order" in mathematics, but/and the comparisons one has at hand, or sees as relevant (at least for oneself). Thus, for example, I would not think of comparing Hardy's "Course" to "calculus textbooks", almost all of which make much ado about nothing, not to mention that Hardy was actually a real mathematician, not a textbook writer. For that matter, I do think it's a pity his promotion of big-Oh and little-Oh didn't manage to displace the epsilon-delta version, especially for introductory texts. Nevertheless, in my opinion, it is too careful, and too long. It is (at some level) "logically complete", but I have never been a big fan of logical completeness per se, insofar as this tends to swamp highlights with an ocean of details, all too easily undifferentiated or undifferentiate-able.

Rather than "logical thinking", I think mathematics yields best to "critical thinking", which (to my understanding/experience) is often very different. My choice of charged language would be that critical thinking tries to discern which details matter, and allocate far fewer resources to the others. In contrast, too often a mere "logical order and logical completeness" is alleged to be what we want, and it's not what I want, anyway.

In other words, while I might agree that Hardy's "Course" is vastly better than almost all extant "calculus textbooks", that is very faint praise. I wouldn't recommend that people use those textbooks to actually learn calculus, in any case, since they make it too complicated, too fussy, too deus-ex-machina, too unpersuasive.

And while I'm editing: we aren't usually disappointed when a novel doesn't have attendant exercises, nor when a piece of music doesn't. Why should mathematical writing have exercises? :)

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Wow, this is a very interesting reaction. It gives me a lot to think about. (When I write lecture notes / expositions, I find myself both (i) putting in a fair number of exercises, mostly of the follow-your-nose variety and (ii) incorporating solutions to others' nontrivial exercises as part of the text.) –  Pete L. Clark Aug 16 '11 at 23:38
Good exercises can engage a student in central ideas. The problem is (a) a lot of exercises are not good (b) a lot of good exercises are given without any solution in sight, so if the student is truly stuck he will learn nothing from the experience. Perhaps (for self studying) a better idea would be to write complete examples with appropriately inserted "pause" marks encouraging the students to think about it on their own first. –  Willie Wong Aug 16 '11 at 23:46
I'm not sure the "post WWII invention of US publishers" is accurate. I have the 1944 Cambridge U Press edition of Hardy's A Course of Pure Mathematics, and it's chock-full of exercises (he calls them examples, but typically they call for proofs, which are not supplied). Hardy includes part of the preface to the 1908 edition, which says "There are plenty of hard examples...." Also, Zygmund's Trigonometrical Series, published in Wilno (Vilna) in 1935, has plenty of exercises. –  Gerry Myerson Aug 17 '11 at 0:15
Great answer! [No exercises in Weil's Basic Number Theory indeed. I'd have expected Serre's name to appear in such an answer...] –  Pierre-Yves Gaillard Aug 17 '11 at 2:20
Maybe this is getting too off-topic, but it would be wonderful if somebody could point me to a book that interpolates between the "attractive slimness" of Atiyah-Macdonald and the daunting bulkiness of Eisenbud. Both these books are completely unreadable for someone like me with only a superficial interest in the topic. Good exercises: they exist, definitely. Look at Pedersen's Analysis Now or Takesaki's Theory of operator algebra. From routine to rather challenging but always something to take away and excellent hints/outlines. Outstanding is Bogachev's approach in his measure theory. –  t.b. Aug 17 '11 at 2:23
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Here is a link to a commutative algebra course by Prof. Kleiman at MIT:


It has a link to his new text. It also includes a link to a pdf with problems and solutions.

Pertinent to this question, here is a quote from the syllabus page for the course:

"The solution set will also include solutions to the unassigned problems. Do try to solve each one before reading its solution, in order to better appreciate the issue. And do read the solution even if you think you already know it, just to make sure. Further, some problems have alternative solutions, which may enlighten you."

Perhaps this is a happy pedagogical medium.

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@t.b. Was motivated to post my answer based on your query. Regards. –  Andrew Aug 24 '12 at 14:38
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