<p>Does anyone know of any good books that teaches pre-calculus with emphasis on proofs and more difficult problems instead of the standard high school curriculum? </p>
<p>It doesn't necessarily have to say "Pre-calculus", but just algebra, geometry, trig, etc.</p>
<p>If you’re discontent with pecalc, just teach yourself calculus.</p>
<p>I am taking calculus</p>
<p>[How</a> to write Math Proofs](<a href=“How Can Beginners Improve Their Mathematical Proof Writing Skills?”>How Can Beginners Improve Their Mathematical Proof Writing Skills?)</p>
<p>This will tell you all you need to know about writing proofs. If you’re just trying to solidify your understanding in algebra/geo/trig, you ought to look into Art of Problem Solving Vol 1 and 2. Vol 1 is more Algebra I/Geometry stuff while Vol 2 is more Pre-Cal/NT kind of stuff. Both don’t have many “prove this” problems but definitely interesting problems.</p>
<p>If you tell us what your goal is, then perhaps we can give you better advice. Pre-Cal is very broad as most textbooks attempt to cover a little bit of everything and leave out too much depth. In my opinion, it’s better to take the superficial idea version but have a strong knowledge of trig/algebra, and then expand on those topics through calculus/linear alg/abstract alg via Spivak/Axler/Artin/Halmos/Rudin/Shiffrin/etc.</p>
<p>Thanks for the advice An0maly. I have looked into the art of problem solving books and they are definitely something I would like. I am indeed trying to solidify my understanding of the more elementary subjects. I just got Spivak’s calculus because honestly the material on the AP calculus test has not been very challenging. I’ve just glanced at the stuff in Spivak today and I found it has some non-calculus things at a more rigorous level. I’m mainly just looking for something with a similar style to Spivak. When I took pre-calc I found the material on certain topics like the binomial theorem, mathematical induction, conics, and on sequences and series to be lacking depth/difficult problems.</p>
<p>That’s right; those topics lack depth because:</p>
<ol>
<li>the average high school student struggles with those at a fundamental level</li>
<li>precal has to cover too much material to ever go in depth.</li>
</ol>
<p>For Number Theory, I would suggest Introduction to Theory of Numbers by Niven. It’s pretty solid but it doesn’t cover everything - no NT book can. In my view, I think it’s better to study Linear Algebra and just do NT problems instead of just straight theory because the concepts are much clearer through practice. The AoPS NT book is also great I’ve heard - actually they’re all great.</p>
<p>I don’t know what to suggest for conics, as that is a pretty specific subject. That topic is covered in differential geometry in college, and I’ve never read anything about it [only a HS senior]. For Seq.'s and Series, I’d stick with learning calculus. BC calculus introduces series, and later maths expand upon the ideas of power series [taylor, maclaurin mainly], and eventually develop to fourier and laplace series/transforms. </p>
<p>Don’t take this the wrong way, but I feel as if you just know several terms without any true sense of what you want to learn or how/why you want to learn it. For example, you mentioned that your pre-calc book lacked material on mathematical induction. For the most part, mathematical induction is a concept. I could teach it in three sentences and an example. The main way to develop that is by doing problems, and no book is devoted entirely to induction [or maybe there is, haha]. </p>
<p>I would suggest you stick with Spivak and work through the book. You said you glanced through it and found non-calc stuff at a rigorous level. This is a bad way to read it. I would suggest you literally read every single word in the book and understand where he’s going with it. Chapters 1-2 [and maybe 3, I forgot] work with non-calc stuff, but he goes into calc concepts pretty fast. </p>
<p>Nonetheless, being able to solve the problems in Spivak will give you the background you need to handle any math course, period. It’s good that you, as a high school student, have picked up a book most college students wouldn’t touch, but this isn’t Edwards & Penney or whatever they use to teach calc nowadays. That book is a read, recite, rewrite, and rediscover kind of book. You teach yourself with his guidance, but you have to do it diligently. Instead of reading on many topics, just work through this book, and you’ll be fine for everything else.</p>
<p>The topics I listed were mere examples I chose at random that I found to be lacking depth in my pre-calc class, meaning they are not topics that I would like an entire book on. I just want some books that are more rigorous and provide harder exercises than the standard public high school math curriculum. For example, I might find some difficult problems that involve the binomial theorem in the AoPS probability book. </p>
<p>And I will continue with Spivak; when I said I glanced at it I meant I just got the book today so I haven’t officially started reading it yet, I just skimmed over it because I was curious. So I think that Spivak and the AoPS books should be enough.</p>
<p>I guess a sub question would be: How does one get to a level of problem solving at, say, the USAMO? With the standard curriculum taught in many high schools, you cannot expect many high school students to have the math background for that. I understand that USAMO requires a lot of aptitude for math, but you cannot expect someone with great aptitude who has taken a standard pre-calculus with no knowledge of writing proofs to be able to compete and do well.</p>