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I am looking for a book that contains mathematical proofs, not a book about proofs, what they are, how to produce them, etc. My focus is on Algebra and Calculus.

Searching the forum and Google I found the book "Thoughts Alpha: Basic Mathematics", and it is reasonably good, especially given that it's free, but there's too little algebra and also I imagine there must be excelent books on this.

Sigma
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  • It would help to know what level you are in order to give you appropriate books. – Thomas Andrews Apr 17 '17 at 17:11
  • I'm a Statistics major on my 2nd semester and I'm taking Calculus 1 (limits, derivatives and integrals at my university). Sometimes I struggle with algebra and have to go back to high school math. – Sigma Apr 17 '17 at 17:15

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It's pretty tough reading, but Landau's Foundations of Analysis starts with 1 is a number and works axiomatically through the basic proofs of algebra. I used to ask my Calculus instructors all sorts of questions about math after class, and one referred me to this book.

The pdf of a scan of this book can be found here. http://www.math.purdue.edu/~lipman/503/foundations-of-analysis.pdf

It's possible that I was referred to this book as a way of my instructor getting rid of a college kid that asked pesky questions.

If that isn't your speed, you might look into books on abstract algebra. This seems to be the first course for math majors where they really dig in and start going over proofs. (I don't really know, I was a physics major and didn't follow that path.)

I enjoyed working through this book:

http://abstract.ups.edu/download/aata-20130816.pdf

David Elm
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    Thank you so much. I'm one of those somewhat annoying students that has to understand everything deeply and see how things work "under the hood", so to speak. I'm annoyed when courses simply give you a tool and expect people to blindly accept them, as it is with my Calculus 1 course. The ridiculous part is that the teaching is shallow but the exercises often require deeep understanding for you to even know what to do. Again, thanks. – Sigma Apr 17 '17 at 18:17
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There are some proof-based calculus books, my favorite being Spivak: http://www.maa.org/press/maa-reviews/calculus-4 He starts simple, but will even show you the construction of numbers from set theory, and is very careful about defining functions and proving the essential theorems of calculus.

Trurl
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