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What would be a good textbook for Math 101, specifically to learn some advanced mathematical formalism without actually diving in applied science behind it?

Keeping with the free theme, Book Of Proof by Richard Hammack is a nice introduction to proofs and formalism. It's available free from the author as a PDF[1], and also as a physical book on Amazon[2].

An alternative if you're willing to spend a little is How to Prove It by Daniel J. Velleman, also available from Amazon[3] and probably many other retailers. Both books cover roughly the same topics.

[1]: http://www.people.vcu.edu/~rhammack/BookOfProof/

[2]: http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

[3]: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...

This stuff is usually rolled into courses called "abstract algebra". If a textbook is called "abstract algebra", it's usually designed for a first year undergraduate. If it's just called "algebra", it's usually aimed at a more mature audience. Herstein and Hungerford are the texts I learned from. This book seems more recent and popular:


The other big stream of basic undergraduate mathematics is analysis. For that I recommend Spivak's Calculus.

Coursera's Calculus One and Calculus Two courses.

I've found other branches of math to be much easier to understand if you have basic knowledge about calculus.

I really like Hubbard & Hubbard.

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