We have had multiple book-recommendation, but I have not found a question where the many free (yet legal) books are available under one umbrella. Therefore I propose a thread where all the best, free books can be found.
Please post the links with a slight explanation on what the reader may find (topic/ requirements).
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Project Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". It was founded in 1971 by American writer Michael S. Hart and is the oldest digital library. Most of the items in its collection are the full texts of public domain books. The project tries to make these as free as possible, in long-lasting, open formats that can be used on almost any computer. As of 23 June 2018, Project Gutenberg reached 57,000 items in its collection of free eBooks.
-Wikipedia
In a similar vein, https://github.com/rossant/awesome-math is another compilation of good math resources, including lecture notes.
http://builds.openlogicproject.org/ is an excellent resource on logic, beginner to advanced.
http://abstract.ups.edu/aata/index.html is something I have not read personally, but am told that it is a good resource for abstract algebra.
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I don't know if they strictly count as "books," but here's my contribution to this list: a set of eight manuscripts. I saw one linked elsewhere on MSE - I forget where though - and did some URL editing to find the entirety of the volume.
These texts (roughly 10-50 pages each) mostly are just good reference material for lots of combinatorial, series, summation, etc., identities. There's not a lot of proofs/elaboration so it's generally assumed you are familiar with the relevant material - you'd need to be very comfortable with calculus at minimum, and from there any gaps in knowledge ought to be resolved with a simple Google search of relevant terms.
They're not particularly good for casual reading or directly learning a ton, so much as they are like reference tables. Even so there's upwards of probably a thousand identities here, so they're definitely worth bookmarking or downloading.
The texts in question:
As far as attribution goes, they're based on notes by H.W. Gould taken between 1945 and 1990. The directory linking to these texts notes that this is not meant to be a replacement for a 1972 text that Gould wrote on (presumably) the same topic, which is still in print.
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For some introductory number theory topics, I would recommend Stein (2017), which covers areas from congruences to cryptography to continued fractions. Only some basic knowledge of properties of numbers is needed, and the book also drops in SAGE code for examples here and there.
Likewise, the textbook on complex analysis by Cain (1999) is elementary but shorter; major topics include calculus and series. Knowledge of these topics on the real numbers is needed, as some parts of the book are quite involved. Both these books have exercises at the end of each chapter which vary in difficulty.
For a large list of free textbooks (on many areas on mathematics) kindly offered by their authors, visit OpenCulture.
References:
Stein, W. (2017). Elementary Number Theory: Primes, Congruences, and Secrets. Available from: https://wstein.org/ent/ent.pdf.
Cain, G. (1999). Complex Analysis. Available from: http://people.math.gatech.edu/~cain/winter99/complex.html.
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During the Covid-19 lockdown Springer is making 408 expensive scientific books available for free (48 mathematics).
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I have created a website with precisely that.
Curated list of free (and legal!) math resources - textbooks, lecture notes and video lectures.
I am constantly updating it, both in content and features.
Recently we introduced a rating system. You can rate resources, so that others have an easier time finding the best ones for themselves.
Hope this helps!
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Openstax has Math (anywhere from pre-algebra to 3 volumes of calculus books), science, humanities, and business textbooks all for free.
-all you have to do is click the link, click the book you want to read, and click “view online” and start reading.
Ravi Vakil's The Rising Sea: Foundations of Algebraic Geometry is an excellent textbook on (you've guessed it) foundations of algebraic geometry, very nicely mixing rigorous proofs, intuitive explanations and illustrations. It is available for free on author's website, and the newest version (along with all other versions) is always available here:
http://math.stanford.edu/~vakil/216blog/index.html
There is also Fulton's book Algebraic Curves: An Introduction to Algebraic Geometry is another nice book whose latest edition has been published for free by the author. It gives an elementary introduction to the theory of algebraic curves, up to desingularization and Riemann-Roch theorem.
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
Both books require some familiarity with abstract algebra, but assume no prior contact with algebraic geometry.