Advice on a book for expression manipulation?

[IanA23]

Books for self-teaching previously taken math courses, and other problems :)

What I want to focus on:
While I am a CS student, I am also interested in mathematics subjects as a whole and want to re-familiarize myself with some of the courses I took in the past. I received high grades, but to be honest, it was mostly a blur and I cannot recall any of it now.

The courses are:
Calculus I, II, III, Statistics, Combinatorics (w/ an intro to proofs), Abstract Algebra, and Diff. Equations.
Any suggestions on good books for this material? Books that have solutions I don't have to pay extra for. I don't plan to master the material, but in 2 years time I'd like to at least re-familiarize myself with it.

Also, I have another issue, which I think is a result of my psychology.

I took the GRE and received only a 700 verbal & 730 quantitative score. While this is not absolutely terrible, the quantitative score is not what is expected at the institutions I plan to attend when I go for my PhD. I am going for a masters first, and then going into a PhD, so I have some time.

The questions on the GRE were all quite simple; however, I ran out of time on the 22nd question and had to guess on all subsequent questions. My issue was a lack of speed. I went to a basic high school and never put a large amount of practice into math problems, so I do not recall any tricks that could speed up calculation. I also need to apply the tricks, so that they become habit.

I was wondering if anyone here happened to know some good material I could use to study various tricks (algebra). I have a good 2 years to study this material before I take the test again, so obviously time is not an issue :P. I can also dedicate 30~ minutes a day 4-5 days a week (of course there will be the odd vacation time). But I want to dedicate most of this time to other math subjects (look below).

Thanks for reading my long-winded post :)

-Ian

 

[Pengwuino]

2 years? At your level you should be able to pull a 800 within 2 weeks. The quantitative score can be boosted up far far easier and quicker than the verbal. The GRE practice books by Kaplan and the such should be fine to be honest...

 

[IanA23]

Well I'm not taking it for 2 years anyway, and I take it as a good opportunity to study a lot of math. I've always had great interest in math, but always just studied to get the As in the classes, not to truly grasp all the problems. I was more concerned with my GPA, but now that I am getting out of the period of my life where GPA will matter, I'd like to focus again on math and spend more time to appreciate it. The 800 on the gre quant is just a motivator for me.

I already prepared for the GRE for 3 weeks, and I cannot take it again for 1 month, which is useless because I will be applying for a masters in a few weeks (applications are due in december).

Even with all my 'experience' in various areas of math (I think more than most CS students, as I am theory oriented and also interested in Cryptography), I can say I am pretty unskilled. I don't know many tricks that I've seen a lot of hardcore math majors use. I don't remember any very useful formulas besides the fundamentals of calc, solving quadratic equations, all the trig stuff, etc. I remember the basic counting formulas for combinatorics. The basic idea of row reduction, echelon forms, etc for calculus II, the properties that define a vector space from abstract algebra, etc. I don't have a deep understanding of any of the courses I have taken because I took them to further my transcript, not my education.

I really need to improve my overall quantitative skill / problem solving skill, anyway. I plan to head into cryptography, and this area is extremely heavy in mathematics. I've read 'Applied Cryptography' and I had no issues with understanding the math behind it, so that is a good sign I suppose; however, it is a far cry from being able to do serious research in the field and write a thesis in the field. This is one of the main reasons I'm asking for advice on material. I want to build up my skills again over my masters career, so that I am better prepared during graduate school. I think I have fallen behind other students who are considering universities like Stanford, MIT, etc.

I do wish my memory was better.

 

[snipez90]

Are you conflating the general GRE quantitative section with the GRE math subject exam? I'm taking the general GRE in about 2 weeks, and while I haven't actually taken a quantitative practice section, the material looks about the same as the SAT math (other sources say it's about the same difficulty level, or even easier). For the general quantitative portion, there doesn't seem to be a whole lot of tricks you can really learn besides actually going back to the practice exams and figuring out the shortcuts on your own. In fact, doing this for SAT math problems was how I really understood algebra for the first time as a freshman in high school (I was really bad at algebra when I took it in 8th grade).

My advice is to practice with Barrons, since the questions tend to be harder than the actual exam, and this forces you to work quickly and find the "tricks" on your own.

 

[IanA23]

It's the same difficulty, yeah. I did better on the SAT math than I did on the GRE math, to be honest. Quite a lot better, a 790. I attribute this to being much more in practice. I also think that while I have become more knowledgeable over time, my IQ has decreased :). My last math class was abstract algebra, and that was 2 years ago. The classes at this college (Georgia Tech) either allow calculators (making many, though certainly not all, calculation tricks useless) or focus much more on the approach to solving the problem than having a final numerical value to place on the paper.

To further support my theory that my calculation speed and second-guessing is a problem, I can give evidence from the POWERPREP II gre practice test I took. In POWERPREP II, a calculator is made available to you. I'm sure I made a perfect 800 quantitative with 5 minutes to spare. I did not have to second/triple guess anything or worry, because I saw the exact answer I was looking for on the calculator. (http://img528.imageshack.us/img528/576/grelv.jpg - no, I don't know how I messed up so badly in the second verbal section! .

Time is a serious issue for me. In all classes, no matter the subject, I always take the entire test period to finish. I double check things, triple check, doubt myself, etc. I do great, and often avoid silly mistakes other people make because of this double checking & doubting; however, I also have 1 1/2 hours to complete a test that the professors usually design to be completed in 50-60 minutes. The GRE test is designed so that you complete it with a minute or less left, unless you are just awesome at math :). So, I think not doubting myself and not double checking / writing down the simpler problems, would help me a lot. Of course, if it was that simple, I'd do it. The problem is, if I don't double check things or doubt myself, then I make mistakes in reading a problem or in some equation in the first 5-10 questions of the math section, and as most people who have taken / are taking the gre know, the first 10~ questions of the test are extremely important (for CAT).

Anyway, as I said, practicing for gre is just a motivation to re-familiarize myself with some algebraic tricks, so that I can then have an easier time with working on harder material that I want to study on my own. I am far more interested in book suggestions for the courses I mentioned in my first post than in advice for studying the gre :). I think by the time I take the GRE again, I will have sorted out my timing issues. I will certainly make a conscious effort to.

 

[snipez90]

Ah okay, yeah it's important to make sure you have the basics down cold. I'll make two recommendations.

For basic calculus, see if you can find Paul Zeitz's Art and Craft of Problem Solving. The text is primarily intended for preparation for high school competitions that frown upon calculus usage, but there is a calculus section which is rather superb and contains a lot of good hard problems. Zeitz's text introduces the reader to epsilon-delta proofs, but he does so in a very intuitive manner, so you should be able to get something out of it. I don't think solutions are included, but most of the problems are well-known (some taken from competitions) enough that you can find the solutions if you really wanted to.

For abstract algebra, Herstein's Abstract Algebra is a good text. It is a somewhat more basic version of his other text called Topics in Algebra. There are LOTS of problems in this text, sorted by difficulty from the author's point of view. Many of them are nontrivial only for the fact that Herstein let's you develop the theory on your own, so that problems in an earlier section do show up as propositions in a later section. The exposition is good, and if you work hard enough it's pretty difficult to fail to get a sense of how to do algebra. There is a solutions manual somewhere online I think for Abstract Algebra.

Since you go to GT, you should have a decent chance of finding these texts in the library.