Topics in Algebra

I'm a big fan of Herstein's exposition. The text is plenty rigorous, without too getting bogged down in formalism, and he also manages to come across as personable and funny in a way that doesn't feel condescending. The people who update/glue stuff onto the 23rd edition of those 1200-page doorstop undergrad calculus texts would do well to revisit books of that era and learn the difference between "engaging" and "patronizing." That's not a dig at the books not being Spivak or Apostol - there's absolutely a place for that kind of undergrad textbook (though maybe not 1200 pages of it, updated every three years) - but in terms of attempts at being 'relevant' that transparently read as 'Hi, fellow kids!' I also dig Herstein's problem sets, there's a good breadth of difficulty, and they really elucidate key ideas, or point to other topics not covered in the text. Some of the problems are absolutely wicked, but in a good way. (IIRC, doesn't the introduction say that some are meant more to be attempted than actually solved?)

Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers! For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals. But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.…

Came here to say Rudin. His presentation is “odd” in that he introduces powerful concepts and whacks large numbers of problems on the head with them, whereas the usual pedagogical approach is to teach techniques that correspond to these concepts but never explain the concept. I also really liked Herstein on algebra but I believe it’s out of print now.

Great list. Another addition to the list would be 4. Topics in Algebra by I.N.Herstein. It has a slightly better (and tougher) set of problems than what you would find in the book by Artin.

"Can you please suggest on-line resource to learn great beginning and advanced probability concepts?" Yup! The subject at the level you ask is a major topic in applied math but is not very popular in US universities. So, you will not get many good answers. In particular, the 'computer science' community, with 'machine learning' and 'artificial' this, Bayesian that, likely won't have good answers. Your "on-line" part is asking a bit much; I can give you references to books but not all on-line. There may be some PDF files on-line, from TeX, that have such material; try some Google searching with the keywords used here. The intuitive foundations of probability went back to gambling. 'Probability' is a field of 'applied' math and as such is well defined: About 100 years ago, E. Borel student H. Lebesgue invented 'measure theory' which essentially 'rewrote' classic calculus, especially the part about integration. For the simple cases, what Lebesgue did gets the same numerical values as the classic Riemann integral. The difference is that in theoretical work Lebesgue's integral is much more general and much better 'behaved'.…