<p>So I definitely want to major in English because that's what I'm really passionate about, far and away over all other subjects, but the higher I get in math the more interested in it I become. I particularly loved taking calculus this year, because the pure math emphasis as opposed to an engineering emphasis was a lot more interesting than what I had in high school. I'm considering a math minor or possibly even a double major (yes, it's possible to do that if I take a lot of summer classes). I have two questions:</p>
<p>1) Is it a really awful idea to take some upper-division classes like Theoretical Concepts of Calculus and Abstract Algebra 1 before taking linear algebra and differential equations? It's just that it would work with my schedule better if I could take those in the summer, since they're some of the only math classes offered in the summer. (They don't have prerequisites except Calc 1 and Calc 2, and I think I would be most interested in Theoretical Concepts of Calculus). </p>
<p>2) I'm making a 99 in Calculus 1 right now, and I got an 800 on the math SAT and math SAT II and a 5 on the Calculus BC test, but I know that my HS scores don't really matter much for upper-level math anyway. And I wasn't in math club or anything like that. One of my high school math teachers wanted me to be on some math UIL team, but I felt so intimidated because I'd never seen anything like the problems they were working and I had no idea how to even start. What are some books I could use to get more experience working in math and understanding the upper-division concepts that would be appropriate to my level? I have heard that Spivak's Calculus is a really good book to use if you want challenging problems and a focus on proofs. Would I be able to handle that, and what are some introductory books I could use to understand what sorts of things I would be learning about if I took upper-division math classes? I'd like an overview of concepts like topology, abstract algebra, different orders of infinity, why specific indeterminate forms are indeterminate, and more detailed explanations of where all the trigonometric identities come from as well as a deeper understanding of calculus--I get the binomial theorem on some level, but I'd like that to be better connected in my head to Pascal's triangle and combinatorics. I'd also like to understand how to work those problems from high school math competitions, even though I don't remember what they were. If any of you are in math honors classes (not offered at my school) and have any texts that you think would challenge me while still being feasible for me, do you think you tell me what books you're using and how good they are? Thanks for your help.</p>
<p>One of my friends did this double major at Yale back in the day (he and I are both middle-aged)–graduating with highest honors in both subjects. He later attended law school, and is now one of the nation’s experts on the implications of pension law. Most lawyers don’t have the math background to begin to understand the math that goes into setting policies on pensions and the associated taxes.</p>
<p>If you have a 5 on the calculus BC exam, you should be able to start with multivariable calculus when you get to college, and you will have plenty of time to do both majors. A basic English major does not usually have many requirements, though you will need a more rigorous background if you’re intending to apply to grad school.</p>
<p>Thanks, it’s encouraging that it’s possible to do a math/English double major! That’s really what I want to do, and from planning out my schedule it looks like it could be feasible. I already took Calculus 1 at the university (my school was hesitant to accept AP scores, and I heard that it was a good idea to retake it) but I still think I can make it work.</p>
<p>You don’t need linear algebra or diff eq’s before real analysis (which is what the theoretical calculus class probably is like). You might want to have experience with linear algebra before taking abstract algebra.</p>
<p>That’s great that you want to double-major in math! To me, I think that it is kind of sad that that most humanities majors are almost proud of their lack of math knowledge.</p>
<p>Regarding books: I’d reccommend working through a ‘intro to higher math/proof writing’ book. It’ll teach you stuff about the basic objects in math (logic, sets, functions, etc) and it’ll give you practice reasoning about them in the more systematic way that you’d be expected to in later math classes. </p>
<p>The good thing about these books is that they don’t assume much prior math knowledge, so they are an excellent place to start from. The bad thing about these books is that the math itself may not be as interesting as what you’d learn about in an algebra or analysis class, but it is math that you need to know for future classes.</p>
<p>I don’t have any good reccomendations for books like this. Here’s a link to a ~$30 book on amazon. </p>
<p>In doing a google on what a Theoretical Concepts of Calculus class might entail, if youre comfortable with Integral Calculus and sequences and series - I doubt you’d have much trouble in this course.</p>
<p>Abstract Algebra is another story. I’m not sure how this course is taught, however the course description makes it sound like any other abstract algebra course at any other campus, and the assumption is made that you have had some introduction to logic, proofs and how to construct them, as well as some theory: set, functions, equivalance classes, and binary operations. I can not imagine your instructor will spend more than a week or so on this stuff, before jumping into the meat of a first semester abstract algebra course. </p>
<p>Or, since an introductory course isn’t offered - perhaps that is somehow merged into the course. Really, only you know the course offerings at your campus. If there isn’t some sort of transitionary course from lower division computational math (calc, diff eq) to upper division work (algebra, analysis) - then its more than likely built into the course structure. </p>
<p>As for a book - I would pick up A Transition to Advanced Mathematics; any edition would do. Your other interests are far too broad to label down a few books (I mean really they all cover their own texts), but I would start with the above book if only because if you can get through that - they you can get through (in theory) what they have planned for you in upper division mathematics. The book linked in the previous post would be fine as well, but looks like it might lack a bridge between equivalence classes and functions.</p>