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I am a beginner. I have recently desired to teach myself the equivalent of a bachelor’s degree in mathematics. I looked up the course study requirements on the University of Michigan’s math department’s website and am in the process of ordering textbooks for each class (not necessarily the textbooks used by U of M, which are not listed on that website anyway). I have already delved into a calculus textbook. I have noticed that linear algebra is the next recommended course after calculus, so I ordered a Linear Algebra textbook on Amazon. My question is about the next recommended course after linear Algebra: Real Analysis. I have ordered, and have already received in the mail, the textbook Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbet. Is this textbook sufficient for mastering Real Analysis? I noticed on the U of M’s website that Real Analysis takes 2 semesters. Yet Introduction to Real Analysis is only about 400 pages long. That, combined with the fact that this textbook is only an “introduction”, makes me think that another textbook is required to fulfill the quota of a whole 2-semester course in real analysis. Is this true? And, if so, what textbook should I purchase?

Also: I noticed University of Michigan has a course in Advanced Calculus. Is advanced calculus different from analysis? If so, in what way?

OurBelovedWannaBe
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    It can take me anywhere from an hour to multiple days to understand a single page in a textbook; that is to say, don’t judge a book by its number of pages. –  Apr 04 '24 at 00:09
  • @WhileIAm Exactly. There are $400$ pages books that I can fully read (including solving the exercises) in a month, and there are books where it takes me many months to even read the first $100$ pages. – Mark Apr 04 '24 at 00:14
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    Usually real analysis is pretty self-contained. You won't need to have any special background except a general "mathematical maturity", ie, you should know your way around theorem proving to some extent, unless this text (which I'm not familiar with) is designed with beginners to proof in mind. It also helps to have some calculus background as motivation. – Jair Taylor Apr 04 '24 at 00:19
  • For beginners I would recommend Spivak's Calculus over other calculus textbooks, and you can even get an old version freely from the internet archive. – user21820 Apr 04 '24 at 02:24
  • For remarks about the distinction between advanced calculus and real analysis, see this MSE answer. – Dave L. Renfro Apr 04 '24 at 04:56

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You will have noticed on the Michigan website that a bachelor's degree in mathematics requires more than calculus, linear algebra and real analysis. Real analysis is often the last/hardest course in the major - you would not normally jump to it after those two courses without much more exposure to abstraction and proofs.

We cannot guide you through the full major here. I suggest that you look at the course prerequisite structure on that website and those of other universities (for comparison) and work your way through the courses in the suggested order.

I would recommend looking at more than one textbook for each course. You will understand more by comparing approaches.

Be sure to do lots of the exercises.

Ethan Bolker
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  • While it does seem like Michigan's math major can be completed with real analysis being the most advanced course taken, I would expect many (most?) of the students to also take some graduate courses. – ronno Apr 04 '24 at 10:00