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I dislike modern textbooks; their cookie-cutter approach and appearance, over reliance on breaking things down into little boxes, the general spoon-feeding they engender and most of all the poor exposition (in my opinion). I feel that reading a mathematics text is a skill in itself, a skill that becomes lost if one subsists on these "modern" books and the bite-sized morsels of knowledge they impart.

I'm in need of a thorough and dry pre-calculus text that will prove worthwhile to work through. I'm not looking for a laundry of list of definitions and theorems. I want as much rigour as a textbook at this level can allow, however not at the expense of clarity.

I understand that the better textbooks were released in 1950-1960s, or perhaps even before then, so I do not mind textbooks that aren't "modern" – the earlier the better.

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    $\begingroup$ I'm glad I'm not the only one who dislikes modern textbooks...I have to teach out of them! $\endgroup$ – icurays1 Aug 18 '14 at 14:14
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    $\begingroup$ "dry" is relative, someone's "dry" is another-one's "juicy", Higher Algebra by Hall & Knight, Geometry by Loney, Trigonometery by Loney $\endgroup$ – Vikram Aug 18 '14 at 16:01
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    $\begingroup$ Fairly rigorous is the very widely used (in the U.S., from the late 1960s through the 1970s) Dolciani text Modern Introductory Analysis. Also worth looking at is Olmsted's Prelude to Calculus and Linear Algebra (1968). $\endgroup$ – Dave L. Renfro Aug 18 '14 at 16:03
  • $\begingroup$ @seeker Did you ever get your hands on any of the books mentioned in Ishfaaq's answer? I'm curious if either Advanced Level Pure Mathematics by S.L. Green or Pure Mathematics 1 or Pure Mathematics 2 by Bostock and Chandler contain material on sets, logic, and proofs. $\endgroup$ – user185744 Mar 15 '16 at 23:23
  • $\begingroup$ @K.Hotz I'm pretty confident that geometry by Hall and trigonometry by Loney don't contain set theory at all, I'll check the others when I get home. $\endgroup$ – seeker Mar 17 '16 at 10:33
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Yes I hear your point. Most books released these days look to spoon-feed. But that does not apply to all new books. I mean Spivak's books, Chapman Pugh's text on Analysis are examples.

Now these are the books I perused during A Levels. I only got my hands on them because the government sells them for dirt cheap prices (I mean for less than 20 cents US).

  • Advanced Level Pure Mathematics by SL Green is an excellent book. Very precise and written in the classical style. Summarised but thorough.
  • Pure Mathematics by Bostock and Chandler is another text which is less rigorous but still fits into your category. I read the translation so things might be a little different.
  • Trigonometry by SL Loney is a must read during your A Levels. It prepares you for mathematical reading. And very precise and rigorous. Solid collection of exercises too. This is where I probably learnt the concept of rigour.
  • Geometry by Hall is apparently something my father read and recommended highly to me. But my syllabus contained very little plane geometry so I never got to reading it.

Judging by the last one you can probably guess that all these books were written well before our time. So if age is what you are looking for these should serve you well.

But the book I still treasure and hold close is one named "Elements of Pure Mathematics by S Nadarasar". Written in the 50's. It's a Sri Lankan book and very rare even here. They don't print the original English version anymore - only the translation. So this is of no use to you since you can't get a hold of it but I owe a lot to this text and not mentioning it on this thread would be a crime.

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  • $\begingroup$ I'm intrigued by the Nadarasar book, could you explain why it was such a special book? $\endgroup$ – seeker Aug 21 '14 at 12:16
  • $\begingroup$ Well its special for exactly the reasons you specified above. Think about Rudin written for A Level students. That's what it is. $\endgroup$ – Ishfaaq Aug 21 '14 at 14:57
  • $\begingroup$ that sounds great, I've tried to google it to no avail, is it not possible to source this book anywhere? $\endgroup$ – seeker Aug 21 '14 at 14:58
  • $\begingroup$ Don't think so. Sorry. Like I said hard enough to find it here. $\endgroup$ – Ishfaaq Aug 21 '14 at 14:59
  • $\begingroup$ Can I borrow yours? $\endgroup$ – seeker Aug 21 '14 at 15:12
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The first few chapters of GH Hardy's "A Course of Pure Mathematics" may be worth a read.

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  • $\begingroup$ Too scary for a pre-cal text isn't it? I mean it almost qualifies for an Analysis book. $\endgroup$ – Ishfaaq Aug 21 '14 at 9:37
  • $\begingroup$ True, though I think the first few chapters covers what could be considered "pre-calculus" topics in a rather dry and rigorous manner $\endgroup$ – yepikhodov Aug 21 '14 at 16:15
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As dry, old and rigorous as it gets "Advanced Mathematics Precalculus with discrete mathematics and data analysis." It's what I had in High School, although I had a modern textbook as a suppliment. There might be newer versions out, but I assume you want the older ones.

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    $\begingroup$ FYI, I taught 6 precalculus courses in the 1990s using an earlier version of this book: Brown/Robbins, Advanced Mathematics: A Precalculus Course, revised edition, 1987. $\endgroup$ – Dave L. Renfro Aug 18 '14 at 15:56
  • $\begingroup$ So it's not as old as I though. Nice, I remembering it having some great problems. $\endgroup$ – dylan7 Aug 18 '14 at 16:00
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I think you will probably like any of the introductory books by Rey Pastor. The issue there is that he was Spanish, so you won't probably be able to find a book by him in English. I have the three volumes of his Calculus course, and it's the most comprehensive book I've ever seen on the subject.

A book I like that has a small introduction including some pre-calculus concepts is Calculus by Tom Apostol. I'm not sure if that's what you're looking for, I suppose you're looking for a complete book on the subject.

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You might be interested in the books Algebra and Trigonometry by Gelfand.

Also, it's not dry or old, but the Precalculus textbook from artofproblemsolving.com won't spoonfeed, at least.

From the book description:

It includes nearly 1000 problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics competitions such as the American Invitational Mathematics Exam and the USA Mathematical Olympiad. Almost half of the problems have full, detailed solutions in the text, and the rest have full solutions in the accompanying Solutions Manual.

As with all of the books in Art of Problem Solving's Introduction and Intermediate series, Precalculus is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which new techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text.

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