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I will be in the next month finishing up Spivak´s Calculus and I was wondering what would be a good continuation.

Background: I will probably start my university studies this fall and thought that I might be able to take on one more book before i start. I have been self studying mathematics for a about three years soon. I dropped out of high school at approximatly the same time as I started taking mathematics seriously and have since been very interested in mathematics as a whole. My studies can be summarized roughly as

  • The highschool material i did not complete while in school
  • How to prove it - Daniell Velleman
  • linear algebra and its applications(matrices, determinatns, vector spaces, euclidean space.)- Lay

  • Calculus - Spivak

  • A bit of introductory abstract algebra in the form of Fraleigh

Now i have been particulary pleased with the material of Spivak and I have found that the exercises in that book felt like they where proper exercises as compared to linear algebra and its applications , where , atleast for me, they seemed a bit too routine and not very interesting in general. I feel comfortable doing most proofs and have not struggled (too hard) on the material in spivak. I particulary enjoyed the later chapters where sums, uniform convergence and all that good stuff was introduced.

Should I continue my study of analysis? and if so what books do your reccomend?

or

Should I aim to take up some other branch (algebra, number theory, linear algebra, geometry)? and if so what books give a "simillar" approach as spivak to analysis?

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That's a lot of difficult independent study. Congratulations.

You might just have some fun with number theory before you get to school and start in on advanced courses at an appropriate level.

Here are my suggestions:

What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

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  • $\begingroup$ An adventurer´s guide to number theory looks pretty interesting but I can not see if it has any exercises, does it? If it does not have exercises, do you have any particular suggestion when it comes to books in elementary number theory with exercises? $\endgroup$ – André Armatowski May 13 at 17:48
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    $\begingroup$ @AndréArmatowski I can't seem to find my copy of Friedberg. My memory is that it does have exercises (at least informally). My number theory book (link follows) has many, some of which are challenging. Dover ebook, so inexpensive, and I get no royalties so there's no conflict recommending it. Used hard copies at Amazon. store.doverpublications.com/0486153096.html $\endgroup$ – Ethan Bolker May 13 at 20:38

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