Next semester I will be taking Abstract Alg 1, Adv Calc 1 (single variable) and Discrete structures, so I want to tune up my algebra skills a bit over the break. So, are there any books that go over some of the cheap tricks and/or techniques in algebra, some of the ones that people forget about, maybe an algebra reference book or something? General stuff that will be usefull in proofs and these classes, and for all of undergrad Pure Math. Thanks!
the only thing you need is an inquisitive and logical mind (you know that to show A imples B is the same as not B implies not A for instance), enjoy your holiday instead. not knowing what "algebra 1" means i wouldn;t be able to advise anyway. why do people think every one else knows what their syllabus will be?
Sorry, I don't even have the syllabus for the class. The book being used is A First Course in Abstract Algebra (3rd ed), by Rotman. The class is the very first Abstract Algebra class you can take, and its only prereq is Calc 3 (multivariable; vectors, planes, partials, double/tripple integration, etc). This is the description of it: Groups, rings, homomorphisms, permutation groups, quotient structure, ideal theory, fields.
How are you taking Calc 1 simultaneously with a course that has a Calc 3 prereq? I think you will find the skills required for this Abstract Algebra class a bit more then a few "techniques" of algebra! You may be surprised at how little of what you think of as Algebra, that you will be doing.
-Advanced Calc 1: Description: Rigorous treatment of calulus in one variable. Definition and topology of real numbers, sequences, limits, functions, continuity, differentiation and integration. Students will learn how to read, understand, and construct mathematical proofs. Prereq: Calc 3. Book: Analysis w/ Intro to Proof by Lay. I am thinking that knowing all the tricks to algebra, a reference book, would be useful for proofs and maybe some of the other undergrad math classes. Would a reference book be good for any math classes? does such a book even exist?
I guess that is why you have to be careful about posting Course titles..They vary wildly from institution to institution. You should adept at Algebra upon conclusion of your first calculus sequence. I am guessing you have done that so not sure what other tricks you need. The best teacher of Algebra is a Calculus course.
Yeah I should probably not abbreviate the course titles Well I guess I was expecting these classes to be even crazier when it comes to algebra, at least Advanced Calculus 1. If all I need are the algebra skills I got from the first calc sequence then I am good to go. Thanks.
it will probably be the first "real" math course, where you do mostly proofs & derivations. it might be really difficult & abstract compared to calculus but i don't think math gets any harder after a course like that.
there are no "general" tips and tricks that spring to mind for algebra specifically. for "real maths "though" 1. it is more a point of view that is required. the answer is no longer going to be 3.14, it is going to be an argument as to why something is true or false. 2. don't be surprised if you do not understand the arguments the first time you see them. 3. learn the rules of behaviour for the things you're manipulating: you wil deal with so called multiplicatively written groups where you will see things like xy=z, but that does not mean xy-z=0 since there is no addition defined on groups like that, and 0 makes no sense here. x=zy^[-1} does make sense. also remember and i cannot stress this enough that groups are NOT abelian, that is xy is not necessarily the same things as yx (think matrix multiplication) i have a set of group theory notes i'm writing that i can send you if i can access my work machine remotely. 4. when doing proofs for yourself it is more than likely that a proof you've already seeen can be adopted to prove it for you. 5. do you understand that iff (if and only if) is a two-direction implication? more later, though my website (at the bottom in the sig, or www.maths.bris.ac.uk/~maxmg and www.maths.bris.ac.uk/~maxmg/maths if youi''ve sigs turned off) has lots of garbage on it. as do lots of other places out there www.dpmms.cam.ac.uk/~wtg10 who has links to even more places
you might look at the logic and proofs chapters of the high school geometry book by harold jacobs, or the college math survey, principles of mathematics, by allendoerfer and oakley.
One way to expand your algebra skills is to do a couple of chapters of trigonometry from a Pre-Calculus course. Especially good are learning trig substitutions and simplifications so you can get more of a foundation of what is more general about algebraic manipulations and what is specific to special cases.