<p>So, obviously, from my thread title, I'm looking for a bit of advice. You see, next year, my senior year, I'll most likely be doing an independent study in math, but I have no idea what I should choose as a topic, or what an appropriate textbook would be.</p>
<p>I'll have finished Calc 1,2, and 3 by next year, and I kind of want to try something a little different. I was thinking, perhaps, Number Theory or something similar to Linear Algebra. However, I've had no exposure to either, so again, anyone's advice or personal experiences would much appreciated. My teacher also suggested non-euclidian geometry, which also sounded really neat.</p>
<p>So, again, any clarification on these subjects, and suggestions for textbooks would be greatly appreciated.</p>
<p>Also, I'm not interested in doing statistic-y stuff, or competition/test prep type stuff (unless you know it would be good even without participation in said competition)</p>
<p>Real Analysis is an interesting topic. It is more conceptual than say, Linear Algebra/Differential Equations…</p>
<p>Of course, there are plenty of topics that you could cover, such as Differential Equations/Partial Differentials, Linear Algebra/Matrix Theory, Fourier Transformations, Real Analysis I and II, Abstract Algebra etc. Or you could go further into calculus with Vector Calculus.</p>
<p>I am a senior in high school and this year I am taking abstract algebra, which is really interesting and is certainly quite different with the type of thinking it requires compared to calculus. Most of the first semester was an introduction to proofs and other things along those lines and towards the end of the semester we started doing some abstract algebra. The book for first semester that my teacher used was reading writing and proving, and the book I’m using know is a first course in abstract algebra.</p>
<p>Definitely superior to linear algebra, which is fairly straightforward and it is a course many people take. Analysis and differential equations have more similarities to calculus than abstract algebra and would be an extension to what you already know. If you have any other questions about the course I am taking I might be able to answer them.</p>
<p>Michael2, did you do anything with linear algebra beforehand, or jump right into Abstract Algebra? Also, so far, I’ve done nothing with proofs, aside from geometry 2 column proofs. Would that be a problem?</p>
<p>I haven’t done any linear algebra. I’ve only a done little bit of stuff with matrices in Calc 1 and dot product and cross product for physics, but I don’t know very much about it. I didn’t know about proving before I started the class and the first semester was mostly about proofs and set theory. So it is not a problem, but I would recommend starting with proofs and basic set theory, which I think is what the book reading writing and proving covers, but I’m not sure because I didn’t actually have the book; my teacher taught out of lecture packets he made based on the books, so I’m not sure exactly what he took from the book.</p>
<p>The book that I have now, a first course in abstract algebra, wouldn’t be good to start the year off before learning proofs and basic set theory. I could also ask my teacher if I could send you the pdf of the lecture packets he used if you think that would be helpful. If you look in the subsection ‘basic concepts’ in the Wikipedia page for set theory, you’ll see many of the topics I saw first semester and many proofs I had to do were related to those topics. The class also went over functions and proofs related to that as well as cardinality of sets. So when the class finished those topics a few weeks ago, we started with the book ‘a first course in abstract algebra’.</p>
<p>I’ve taken both courses in some capactiy, and number theory was by far more interesting/challenging/useful. </p>
<p>If you plan on being a math, compsci, physics, or engineering major, you will almost certainly end up taking a linear algebra course. However, you will most likely not end up receiving credit for any linear algebra course you take now, simply because linear algebra courses differ from college to college (of course, I’m certain there are exceptions!). You’ll probably end up repeating a lot of material if you take linear algebra.</p>
<p>Also, I would argue that number theory is significantly more abstract and, by extension, more fun. You probably will learn more re: logic, rigor, and elegance. My linear algebra proofs were more brute-forcey.</p>
<p>Re: abstract algebra: The mantra “Try some numerical examples” will not apply. If you haven’t taken a course like number theory, you might be frustrated with the complete paradigm shift from calculus to abstract algebra.</p>
<p>Non-euclidean geometry is very messy. I don’t have a lot of experience with the topic, so maybe another poster will be able to help you out more!</p>
<p>H. Davenport’s “The Higher Arithmetic, Eighth Edition” is a basic, intro-level read. It’s very clear and very easy to read.</p>
<p>G.H. Hardy and E.M Wright’s “An Introduction to the Theory of Numbers” is a more thorough approach to Number Theory. The book is encyclopedic in nature with very few exercises to complete. </p>
<p>Daniel Shanks’ “Solved and Unsolved Problems in Number Theory” strikes a pretty good balance of proofs and exercises - this is probably the closest to a “textbook” of the three I’ve listed.</p>
<p>Thanks, I was kind of leaning towards number theory because it seems like it would be very different from what I’m used to, though Algebra seems really interesting too.</p>
<p>I’d take number theory, but only because I think I’m not the biggest fan of linear algebra. I don’t know, my high school lumps everything into 3 compulsory years of math (the last one introducing us to calculus, amongst other things) and then gives you the option of taking calculus your senior year. It’s a very math-oriented school (oh how the athletes must suffer).</p>
<p>Maybe you can check out the books you’d use for each class and decide which problems would be more interesting to solve.</p>
<p>Do you know of any good textbooks, though, for any non-calculus topics? No matter what I do, I’ll most likely end up purchasing my own textbook. I think I’ll end up doing something somewhat number-theory-y, and perhaps a little proof stuff, since I haven’t had much exposure, but I am trying to get as many textbook and topic suggestions as possible from anyone with experience.</p>