Books Best for Mathematics & Algebra Self-Study with Proofs?

  • #1
Rin
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Hello,

I've been trying to improve my algebra since I've never been particularly good with math. I'm going through Serge Lang's Basic Mathematics textbook and while I have been learning a lot his proof-based exercises are a pain to get through and the back of the book only provides answers for a few questions in each section. This makes it quite difficult for me to verify that my proofs are right, or even where to start.

With that in mind, are there any other basic mathematics / algebra & geometry / precalc & trigonometry books with the same level of rigor but also provide comprehensive explanations and answers to exercises?

Thank you!
 

Answers and Replies

  • #2
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I used "Matrix analysis and applied linear algebra" by Meyer as a textbook, and i felt satisfied with that. Besides, you can have a look at "Linear Algebra Done Right" by Axler or "Introduction to Linear Algebra" by Gilbert Strang. Here is the series of lectures by Prof. Gilbert Strang, one of my most favorite teacher of all time https://www.youtube.com/playlist?list=PL49CF3715CB9EF31D
Good luck :wink:
 
  • #3
840
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Unfortunately, you can count on one hand the books that cover basic mathematics rigorously, and the trend for the past few decades has been to not give answers or solutions in textbooks, so I'm afraid you're stuck in that regard. My suggestion would be to learn proofs and basic maths (algebra, trig, etc.) seperately.

For proofs there are many books, such as Hammack's Book of Proof, which the author has made available for free (direct link to pdf). The best way to learn how to do proofs is to study many of them. I don't know of many books that show you how to do the proofs in the solutions, but I have been told that Halmos' Linear Algebra Problem Book is like that.

For studying algebra, trig, etc. I suppose the Schaum's Outline series would be good for practice, not so much theory. Unfortunately I don't hink there are many, if any, books that meet all your requirements.
 
  • #4
mathwonk
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I don't know if this is helpful, but there should ideally be no need to have an answer book to tell you whether your proof is or is not correct. A proof is a sequence of statements that begin from the given information and follow logically, each one from the previous ones, and end with the desired final statement. If your proof has these features then it is correct, whether or not it resembles the author's proof. Indeed it is preferable to create a proof of your own that differs from the author's. since it shows originality. So knowing whether your proof is or is not correct depends on whether or not you know whata proof is, and when an implication is logically valid. This can be gained by studying a good proof book, i.e. a book that explains clearly the rules of (informal) mathematical logic. I got mine from the book Principles of Mathematics, by Allendoerfer and Oakley, some 50 years or so ago. Another good source for beginners is the first edition of Harold Jacobs Geometry, but not the later editions from which most of the logic chapter was removed. There are many other more recent books on proof, but I am not too well acquainted with them. The original book of Euclid's Elements is quite wonderful in my opinion.

Another rule for making proofs is not to give up too soon. I am a professional mathematician but I still have trouble filling in details of proofs in books. My method now, and I recommend trying it as soon as possible in your career, is to read the statement, close the book, and try to prove it myself. I usually feel frustrated as to where to begin, but if I am stubborn enough to keep thinking about it, often for days, and nights, eventually I get some idea that works at least in part. I.e. if I give up and look at the proof in the book after trying long enough, I usually see that I have correctly identified some piece of the proof. Another trick is to make the problem easier, by adding more special hypotheses, or taking a very special case, where you can get started. The ideas you find for the easier case usually resemble the ones that do the harder cases, and this helps you see the underlying idea. In a book the author frequently goes straight to the hardest most general case to save time, since going through several illustrative and easier cases takes too many pages he needs.

Anyway, good luck, it is quite pleasing to actually come up with a proof yourself, and it helps if you are not in a hurry. Proofs are less painful if you give them enough thought to become familiar with them.
 
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  • #5
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I have been searching for a rigorous basic mathematics book for myself for quite some time and have had no luck. I just went out and got an intro to proofs text by Chartrand and it has served me well. The first few chapters of such a book would probably suit you well.

I have Basic Mathematics by Lang. I used it mostly for exercises. It's not the worst text, but it definitely feels like it was written over a weekend.
 
  • #6
mathwonk
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all lang's books are famous for having been written quickly. I do not recommend most of them. On the other hand since he was so smart and knew so much, occasionally he makes a very succinct and useful remark, but as books go, his are among the least carefully written, which does not always mean least well written. Still I did like his book Analysis I, at least in places, as stated above, and his Complex analysis, but since complex analysis is such a beautiful well understood subject, almost all books on that topic tend to be excellent.

as i have said many times before, basic mathematics, i.e. algebra and geometry, are best treated in the classics by Euclid and Euler, i.e. Euclid's Elements, and Euler's Elements of Algebra. These are both available free online, but I recommend the Green Lion press version of Euclid. I own the Cambridge library collection paperback of Euler. but it's a little pricy for a student. I was a college professor and taught graduate students about theory of equations, including cubic formulas, and galois theory, etc, and even wroote my own graduate algebra book, but after i read euler's lucid explanation of the cubic formula i understood it so well i could teach it to (very bright) 10 year olds. It is really embarrassing to spend decades studying advanced math, think you understand it, and then find out after reading a real master that you did not understand diddly. So always try to read someone who actually understands what you are trying to learn. You can't go wrong with Euclid and Euler.

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20

https://www.amazon.com/dp/150890118X/?tag=pfamazon01-20

https://www.amazon.com/dp/110800296X/?tag=pfamazon01-20


here is the book I was helped by as as a senior in high school, already possessing a good grasp of high school algebra: I had always heard about "converses" and "negations" in geometry, class but never knew ezactly what that meant until reading the excellent and clear explanation of propositional calculus, i.e. informal logic, in A&O. It stood me in good stead the rest of my life in math courses involving proofs.

https://www.amazon.com/Principles-Mathematics-C-Oakley-Allendoerfer/dp/B001CD9834/ref=sr_1_1?s=books&ie=UTF8&qid=1520046478&sr=1-1&keywords=principles+of+mathematics,+allendoerfer#customerReviews
 
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