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I am currently going through the " Topics in Algebra" By I. N. Herstein. The problems are pretty good. But there are no answers on the back. Same is the case with "Mathematical Analysis" by Rudin.

I don't understand why is this so??

Even though some would argue that it robs the question of its charm, sometimes it is necessary to check out the answer just to get confirmation that what you have done is correct.

Moreover it becomes necessary in an environment where you study on your own. Even though I have solved exercises, there is always a doubt regarding the correctness of the process or answer.

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    $\begingroup$ Perhaps better suited for MESE. $\endgroup$ – Git Gud Apr 9 '14 at 15:33
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    $\begingroup$ I'd say this is in order to make you look at the definitions and theorems in the book instead of the solution when you are stuck, or to force you to communicate your question to other, which also forces you to recap. $\endgroup$ – Thomas Apr 9 '14 at 15:36
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    $\begingroup$ Perhaps the main reason is that including solutions in a textbook would add many many pages to a book that might already be very long. Including detailed solutions also takes some effort. $\endgroup$ – spin Apr 9 '14 at 15:37
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    $\begingroup$ And also, there is not always one solution. But the main thing is that it takes too much time to write a detailed solution and also that if there is a solution, then the reader will maybe not do the exercise. The best way is to compare with others. For example on this website? $\endgroup$ – Jérémy Blanc Apr 9 '14 at 15:40
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    $\begingroup$ Would be nice if for each book they also make a book with solutions. They would sell more and more students would be able to check their answers. $\endgroup$ – Integral Apr 9 '14 at 15:40
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Several answers to this question come to my mind, some of which are already covered by the comments. Actually, if I'd ever were to write a textbook I'd do the same.

Since you cite Rudin I'd like to note in advance that many consider his books on analysis among the best, and I don't think it is a coincidence that someone writing books which do get that attention and appreciation follows this practice.

Something everyone learning mathematics (not only there) has to learn is the reality (from which you are well protected at school) that you will usually (not sometimes, usually) face problems which you cannot solve. Maybe not at all, at least not with the first attempts or with little effort. You have to find your way to the solution one way or the other, either by thinking harder and smarter, by talking to people, whatever. But you don't get it for free, in real life. Almost never. It is an experience of people who teach (not just math), that this fact has to be taught as well, unless they want to produce many many people leaving university who will then have to learn it too late the hard way.

And, no insult intended, one word of advice: if you find you really need the solutions to make your way through such books, you may want to consider whether you've really chosen the right topic for you.

And one additional remark: why do you need the solution? By working seriously on a problem you may learn more -- even if you cannot solve it -- than by looking up the answer (and, very likely, missing most of the subtleties).

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    $\begingroup$ On the other hand, the deep psychological NEED to know is part of what drives all good mathematicians. We need it so badly that if the solution isn't in front of our noses then we work very hard to find it. $\endgroup$ – Lee Mosher Apr 9 '14 at 16:15
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    $\begingroup$ A lot of times, you learn from seeing somebody else's solution even if you can come up with your own. $\endgroup$ – jsmith Nov 23 '15 at 5:13
  • $\begingroup$ @jsmith This is not in contradiction to what I wrote. Actually, talking to someone else about the problem is among the things I explicitly mentioned. $\endgroup$ – Thomas Nov 23 '15 at 5:45
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    $\begingroup$ As a counterpoint to this I'll note that Don Knuth's Art of Computer Programming books (which despite their name are as much or more mathematics than computer science) do include answers for the huge majority of their exercises, to the point where at least a third of the book if not more is answers, and IMHO the work is immeasurably enriched by those answers. $\endgroup$ – Steven Stadnicki Oct 25 '16 at 16:02
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I think it is a commercial decision. Professors prefer it this way.

I personally think having the answers is beneficial. Even if you are confident in your methods, you may have made a dumb mistake (or have a flawed understanding). Usually if you are on top of the course, you can check the answer and if wrong, rework the problem correctly (if needed, checking the book). Also, working problems can be a chore--having immediate feedback is good psychology.

Note that many classic books have the answers in the back (Poly anything, Granville calculus, Hart College Algebra). This used to be much more the norm. I think the change is more of a commercial one and one that emphasizes the primacy of gatekeepers rather than of learning. [I'm sure the professors tell themselves they care about learning and are warding off lazy students, but how many of them really collect and grade daily homework in sufficient quantities to drive needed learning?]

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  • $\begingroup$ Having the answers in back is often a hindrance more than a help. In high school freshman year my answers were typically more exact than the textbook's, because the textbook used $3.14$ as an approximation for $\pi$. But much more to the point, in real life, you must personally confirm your answer. There is no "authority who knows best" in the field of math. Including answers always would set up a delusory situation wherein you can always check your answer against the "official" answer. That's not how math really works. $\endgroup$ – Wildcard Jan 14 '17 at 4:09
  • $\begingroup$ Well of course I disagree with you. FYI, I have a Ph.D. in the hard sciences. Have studied a variety of math, engineering, science, business, and military topics. I have had a full life (EIT, McKinsey, etc. etc.) Both with direct instruction and self study. $\endgroup$ – person Feb 1 '17 at 3:01
  • $\begingroup$ Having answers helps check for dumb mistakes or concept flaws. Often, just seeing the wrong answer is enough and I can redo the problem. If you don't have the answers, of course you do your best. But you are still turning away from a pedagogical aid. BTW, I'm fully capable of quickly realizing the difference in an answer that varies because of a small difference in the value of PI used. Also, having the answers is just psychologically beneficial. Like a video game. $\endgroup$ – person Feb 1 '17 at 3:08

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