Books with a lot of problems but all with solutions

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Discussion Overview

The discussion revolves around recommendations for books that contain numerous problems with complete solutions in various fields of mathematics, including number theory, abstract algebra, geometry and topology, and probability. Participants also touch on the broader topic of studying mathematics, particularly for those who may not have a formal background in the subject.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • The original poster (OP) seeks books with many problems and complete solutions, citing examples like "Calculus Problem Solver" and "3000 Solved Problems in Calculus."
  • One participant suggests "Schaum's Outlines" as a resource for problem-solving.
  • Another participant advises that the OP should focus less on solutions manuals, suggesting that many classic books have solutions available online and emphasizing the importance of understanding the material through exercises.
  • A participant recommends "Problems In Algebraic Number Theory," noting it has many problems with solutions and includes a crash course in elementary number theory.
  • Another participant expresses agreement with the previous advice, cautioning that the OP should first learn algebra and analysis before tackling more advanced texts like algebraic number theory.
  • Recommendations for foundational texts include Dummit and Foote for algebra and Rudin for analysis, with an additional suggestion for Herstein as an algebra text.

Areas of Agreement / Disagreement

Participants generally agree on the importance of foundational knowledge in algebra and analysis before approaching more advanced topics. However, there is some disagreement regarding the necessity of solutions manuals, with differing views on their role in learning mathematics.

Contextual Notes

Some participants emphasize the need for a solid understanding of fundamental concepts before engaging with more complex material, suggesting that the OP may need to adjust their study approach based on their current level of knowledge.

Muhammad Ali
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Hi guys,
I am eagerly looking for books which contain a lot of problems with complete solutions. Please tell me the name of such books/resources for the following fields.
Number theory
Abstract Algebra
Geometry & Topology
Probability
I mean books like Calculus Problem Solver or Advanced Calculus Problem Solver published by REA or 3000 Solved Problems in Calculus from Schaum’s Series.
Please also advise me about studying mathematics because mathematics is not my major subject. I have BSc. degree in Electronic Engineering and recently I have got tremendous interest in mathematics (particularly pure mathematics) that now I want to study mathematics.
 
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schaum's outlines
 
Good to know you have a newfound interest in mathematics.

I advise, however, that you not worry so much about solutions manuals. The classic books have solution to many problems floating around on the web anyway, and you should be able to use those as reference. Additionally, once you learn how to work with mathematics, you do exercises mainly to check whether you have a functioning understanding of the material - so you shouldn't need solutions, because these are proofs - you know it's true, but want to see if you can check why, so the steps you take help you exercise your understanding.

If you are truly new to mathematics, I would first get a good understanding of algebra and analysis. Books by Dummit + Foote in algebra and Rudin in basic analysis should introduce the fundamentals very well, plus provide more advanced reading.
 
I used a book called "Problems In Algebraic Number Theory", which is a graduate text. It has a lot of problems, and the second half of the book has all the solutions. Also, if you are not familiar with number theory it has a crash course in elementary number theory at the beginning.
 
I mostly agree with deRham. I don't think the OP has any business touching an algebraic number theory text without first learning analysis and algebra (in particular algebra). The tools you learn in these two courses are fundamental, and there are many good concrete problems to work on. You can save the really abstract stuff for later, but I would recommend finding a good analysis text (deRham already mentioned Rudin, but there are many threads about good intro analysis texts), or an algebra text (I recommend Herstein, though our school uses Dummit Foote).
 

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