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In short; money is really tight at the moment and need to save up for university. I'm a 17 year old student who have just finished basic multivariable calculus and differential equations.

I was wondering if you could recommend some free or very very cheap books, which have a reasonable standard. I'm mostly aiming for books covering either of these, but feel free to add others as well: Linear Algebra, Real/Complex analysis, Logic, (number theory, topology).

I am aware that free/gratis content might not be the best, and can be full of errors, but it's better than nothing, right?

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    $\begingroup$ Additionally to the ebooks, it might be helpful to consider Open Courseware (like MIT) as those typically also include notes and the like. On the book front, there are many on the web. Regards $\endgroup$ – Amzoti Feb 25 '13 at 22:26
  • $\begingroup$ @Amzoti, Actually I did the multivar. calc. on the MIT sites. It is very good, but I'd like some books to read as well. $\endgroup$ – Hans Groeffen Feb 25 '13 at 22:30
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    $\begingroup$ @MichaelGreinecker, Free as in legally free, and no piracy. $\endgroup$ – Hans Groeffen Feb 25 '13 at 22:32
  • $\begingroup$ @HansGroeffen I'm sure no mathematician would mind if you read their material without payment (rather, the publishers are the ones that care about the money, I guess). If you like it, and plan on using it, you can probably then find a used version for very few coins, or not. Don't let that slow you down... $\endgroup$ – Pedro Tamaroff Mar 28 '13 at 20:21
  • $\begingroup$ It might be a little difficult with regards your background, but Artin's Galois Theory is quite cheap (there is a legal free version online, and the printed version is ~$5 I think), and very good. And from what I remember, it introduces all of the concepts throughout the book. The first section covers some elements of linear algebra as well, before going into the field theory and main topic. $\endgroup$ – izœc Oct 10 '14 at 0:04
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Everything by Robert Ash, especially Real Variables with Basic Metric Space Topology. It covers a substantial part of Rudin's Principles with more intuitive explanations and even (gasp) illustrations. I haven't really read his books on other subjects (complex analysis, algebra, probability, etc), but they appear to be of reasonable quality, and definitely free.

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I have used many of the free math textbooks on this page and found that they are generally good.

If you are looking to buy some cheap books, dover publishes a lot of math textbooks for $<\$10$ but you will have to investigate yourself to find out which of these are worth buying (a few are dry and dated).

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I am a big fan of Kenneth Bogart's Combinatorics Through Guided Discovery, freely available here. Combinatorics is a beautiful subject and the problem oriented approach is particularly well-suited for this field.

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It is definitely worth looking into textbooks on Linear Algebra, as it is the stepping stone to many different areas of higher mathematics.

Matrix Analysis and Applied Linear Algebra by Meyer can be found here, subject to some conditions of use.

Linear Algebra by Hefferon is supposed to be good, though I've never tried it, and it can be found completely for free by the author here.

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Take a look at the books by The Trillia Group. The lecture notes by William Chen are nice too. Available in PDF for free. (But perhaps downloading and printing is more expensive than buying some of the Dover books...)

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See the free e-book Real Variables: With Basic Metric Space Topology by Robert B. Ash Published in 2007, 213 pages. For more free e-books see Free Mathematics Books. E-Books for online viewing and/or download.

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