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I have a question that I have been wanting to ask for quite some time. I am currently a senior Computer Science student who will be graduating in a few months. I have taken all my required math courses (statistics, calculus 1-3, discrete mathematics, linear algebra, etc.) despite doing fairly well I feel like I lack any real understanding of mathematics. What I want to know is if there are any resources out there to learn math completely from the beginning starting at basic algebra and progressing into advanced topics. Right now I feel like I just know how to follow instructions and plug in formulas and numbers. I would like to eventually have a real understanding of exactly why I am doing what I am doing and how to use mathematics to solve future problems. Thanks for any advice!

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    $\begingroup$ Mathematics nowadays is quite vast. I think it would be best to try and limit the scope of your undertaking somewhat. For instance, when you write "advanced topics", how "advanced" are we going? Is it "advanced" as in mathemathics at an undergraduate university level, or is it "advanced" as in research level mathematics? And what branches of mathematics are you interested in learning more about? Calculus/analysis, geometry, discrete maths, linear/abstract algebra, graph theory, etc.? $\endgroup$ – Scounged May 8 '18 at 14:27
  • $\begingroup$ @Scounged You are correct my original question was a little broad. When saying advanced I more or less mean advanced at an undergraduate level with enough knowledge to purse further if the opportunity presents itself. The topics which interest me the most are without a doubt discrete mathematics, graph theory and linear/abstract algebra. Thank you for your response! $\endgroup$ – nick May 8 '18 at 14:33
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Here's a possible ordered list of undergraduate-level books that could help you in your quest, but I have to warn you it will take some time. You're wanting to go all the way from understanding the 'why' to problem-solving and applications. That's quite a broad sweep of mathematics.

  1. Language, Proof, & Logic, by Barwise and Etchemendy. Gotta have logic to understand advanced math. I prefer the natural deduction system to any other symbolic logic proof system.
  2. Foundations of Analysis, by Edmund Landau. You can find free electronic copies of this. This book actually proves a lot of the basic properties of the number systems used in advanced mathematics.
  3. Axiomatic Set Theory, by Patrick Suppes. Set theory and functions undergird all of advanced mathematics.
  4. Abstract Algebra, by Israel Herstein.
  5. Linear Algebra, by W. Keith Nicholson.
  6. Essential Topology, by Martin Crossley.
  7. Principles of Mathematical Analysis, by Walter Rudin.
  8. Numerical Analysis, by Burden and Faires.
  9. How to Solve It, by George Polya.
  10. How to Solve It: Modern Heuristics, by Michalewicz and Fogel.
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