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I am a high school student who has finished the standard hs math curriculum. After working through an intro to proofs, logic and set theory (Velleman's How to Prove It), I am looking to study some proper pure maths. My motivations are to have fun, and to prepare myself for accelerated college classes such as Harvard's Math 55 and UChicago' Honors Analysis. I am considering learning either analysis or algebra.

If I pursue analysis, I will most likely use a combination of baby Rudin and Apostol, although I am open to suggestions (Tao and Abott?). If I learn algebra, I will use Artin, supplemented by Benedict Gross's youtube lectures.

I have not been greatly exposed to either field, but from the little I have read in the Princeton Companion of Mathematics, algebra seems more interesting. Is it more important to study what fascinates me more or what fits into my education better?

Any advice is appreciated.

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    $\begingroup$ It's most important to study what you like $\endgroup$ – Jakobian Jun 16 '18 at 15:36
  • $\begingroup$ My feeling is that if you have to prepare for Harvard's Math 55, then you shouldn't take it. It's kind of like a high school runner (of whose ability we have no information about) asking what kind of training he/she should do to get an individual invitation to the NCAA Division I Cross Country Championships the Fall of his/her's first year of college --- someone who doesn't already know what is needed for that level of competition is almost surely not going to be at that level his/her's first year of college (or any other year, for that matter). $\endgroup$ – Dave L. Renfro Jun 16 '18 at 16:33
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It sounds like you haven't read a rigorous calculus book based that focuses on proofs. If I'm correct in my understanding, you might benefit from reading a book like Spivak's Calculus before going further.

Apostol's Mathematical Analysis might work for you or it might not at this stage. Its level of difficulty is such that for some people with good mathematical ability and sufficient background in non-rigorous calculus, it could serve as a good first math book with complete proofs, while it might be too hard for others, even after reading Velleman's book. Alternatively, you could use Zorich's Mathematical Analysis I, II. I wouldn't recommend Rudin in your circumstances, as it starts out at a higher level of difficulty.

Artin is fine as an introduction to algebra, with similar reservations as for Apostol. Again, Spivak would provide sufficient background experience with rigorous math, but for some people such preparation wouldn't be necessary.

Anyone studying math seriously will naturally study both analysis and algebra, so it's really just a matter of the order you do it in. Either order is okay, or you can study them simultaneously.

Apart from university-level algebra and analysis, if you have time, you could also consider working through introductory textbooks or problem books on topics such as geometry, combinatorics and number theory. These will also build your problem-solving skills.

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