How Should I Self-Study Math to Prepare for a Math Major?

[MJC684]

Hello all. I'm going to be entering my second year of college in the fall. I have a deep love for math and physics and I have decided to major in Mathematics. However I have a very limited background in mathematics. I will be starting Pre-Calc this upcoming semester and I am trying to hit the math books hard this summer. I am 100% committed to my goal of succeeding in mathematics and I am willing to do whatever it takes. However I could use some guidance as to what areas and textbooks I should be studying on my own.

Here is what I already have:

Precalc
Precalculus by Bittinger/Penna/Beecher 2nd Ed
Introductory Analysis by Dolciani
College Alg. & Trig A Functions Apprch. by Keedy/Bittinger
I also have a number of books from the 70's and 80's with titles like "Elementary Functions" or "Fundamentals of PreCalc Mathematical Analysis"

Trig
Trig by Larson/Hostetler 3rd Ed
Trig by Lial/Hornsby/Schneider

Calculus
Finney, Thomas 7th Ed
Swokowski, Olinick, Pence 6th Ed.
Larson,Hostetler, Edwards 2nd Ed
Stewart 2nd Ed.
Loomis 3rd Ed.
Courant Diff & Integ Vol 1
Spivak 3rd Ed

I also have "How to Prove it", "The Nuts and Bolts of Proofs", How to Read and Do Proofs, and the older book "The Principles of Mathematics" by Allendorfur and Oakley.

On top of these titles I have books which I figured I would need down the road like books on Mathematical Logic, Symbolic Logic and practically every Dover book a beggining math major could want.

I need someone with experience to guide me in what to study when, which book to use, which concepts to study the most etc. You know, set me on the right path? Any and all suggestions would be greatly appreciated.

 

[Howers]

Hello all. I'm going to be entering my second year of college in the fall. I have a deep love for math and physics and I have decided to major in Mathematics. However I have a very limited background in mathematics. I will be starting Pre-Calc this upcoming semester and I am trying to hit the math books hard this summer. I am 100% committed to my goal of succeeding in mathematics and I am willing to do whatever it takes. However I could use some guidance as to what areas and textbooks I should be studying on my own.

Here is what I already have:

Precalc
Precalculus by Bittinger/Penna/Beecher 2nd Ed
Introductory Analysis by Dolciani
College Alg. & Trig A Functions Apprch. by Keedy/Bittinger
I also have a number of books from the 70's and 80's with titles like "Elementary Functions" or "Fundamentals of PreCalc Mathematical Analysis"

Trig
Trig by Larson/Hostetler 3rd Ed
Trig by Lial/Hornsby/Schneider

Calculus
Finney, Thomas 7th Ed
Swokowski, Olinick, Pence 6th Ed.
Larson,Hostetler, Edwards 2nd Ed
Stewart 2nd Ed.
Loomis 3rd Ed.
Courant Diff & Integ Vol 1
Spivak 3rd Ed

I also have "How to Prove it", "The Nuts and Bolts of Proofs", How to Read and Do Proofs, and the older book "The Principles of Mathematics" by Allendorfur and Oakley.

On top of these titles I have books which I figured I would need down the road like books on Mathematical Logic, Symbolic Logic and practically every Dover book a beggining math major could want.

I need someone with experience to guide me in what to study when, which book to use, which concepts to study the most etc. You know, set me on the right path? Any and all suggestions would be greatly appreciated.

Thats a cute collection you got, but you will probably get through 1% with only summer break.

You are in second year and you are studying precalc? That is not a good sign at all. How did you decide you want to major in math without even knowing precalc? If all you have is high school algebra to go on, just note that real math is nothing like that. Neither is physics. It becomes all about proofs and highly abstract concepts, not plug and chug type problems.

The best thing you can do is avoid taking precalc, and get onto calc when you start next year. See, I don't know if you mean your taking precalc this summer (which would be good), or saving it for the fall semester next year (wich would be bad). Either way it means bust your balls (or muff) this summer learning precalc and trigonometry. Use one of the precalc books, and do a trig book, and do every single problem. That way you can start learning calc in college, and you would only be a year behind. If you are specializing in math (taking honors/theoretical courses), then you'd need to know proofs. So read the first 3 chapters of How to Prove it by Velleman, the rest is garbage. Also learn about planes, vectors, and matrices. I would suggest you do Euclidean Geometry for the proofs, but I don't think its needed - especially since your time is better wasted on learning precalc.

 

[MJC684]

I am aware of the fact that advanced mathematics is not "plug and chug" problems. As far as my math background..don't we all need to start somewhere? I got all the books becuase I thought a few of them might turn out useful. So I take it that proofs is what I shoud focus on? Will do.

 

[razored]

If you are 100% motivated, seriously dedicate 2-3 hours of learning precalculus on top of whatever math assignments or homework you have to do, DAILY until school ends. Also, obtain the syllabus for your scheduled precalculus class and switch to take Calculus instead. You can easily do the entire precalculus course within 1-2 month time frame (precalculus isn't challenging at all, at least to me). In addition, MASTER basic trigonometry (pythagoreans, reciprocals, inverses, double/half angle formulas, solving trig equations, look at the syllabus). In the summer, I would dedicate AT LEAST 5 hours DAILY of math reading and practice. Do not stop, it will become easier after a while sitting 5+ hours a day and thinking.

You are way behind and you should(I would if was in your situation) spend your entire summer starting and finishing precalculus( and trig) and touching SOME Calculus by the end of the summer.

Know that you are not alone, I am in a similar position as you. I will begin studying Calculus this summer; in fact, I have already started. I typically range 2 hours of Calculus math daily and so far, it is working very well. I am currently a junior in high school finishing precalculus (I think most high schools offer precalculus in sophomore year?).

Lastly, Enjoy the precalculus! The leap from Algebra 2 to precalc isn't great but it sure makes math more relevant and exciting.

Added :

Also, don't learn from all of those books at the same time. Pick 2-3 books for precalculus and 1 for trig to study from. I also have plenty of math books but I find it hard to learn from all of them at the same time. Two should suffice, three if you want extra information.

Also recommended that you connive your studying accordingly such as for instance, 5 chapters a week (hypothetical) or something.

 

[MJC684]

Thank you very much for the advice razored. I am definately spending every bit of my free time working through the precalc. Any good book recommendations?

 

[Howers]

Thank you very much for the advice razored. I am definately spending every bit of my free time working through the precalc. Any good book recommendations?

I think he meant 5 sections a week. At 5 chapters a week you'll learn nothing.

There are no good precalc books. So the ones you have should do. Getting through them should be your #1 priority. Once you completely understand precalc, you can begin learning the intuitive notions of calculus. Since you have Stewart, that is a good place to do that. Get through the first 3 chapters of it, or if possible through the first 5 and the appendix - that way, you'd have the same background as an AP high school student. "How to Prove" should give you a basic exposure to proofs, but you would need something more such as vectors and matrices which usually have harder proofs. Go to the library and borrow some high school algebra and geometry book. I think that book mathwonk made famous by Allendorfur and Oakley had some geometry and vectors/planes. I can't remember. I remember the first chapter is essentially the first 3 chapters of How to Prove it by Velleman, and the stuff afterwards is theorems of basic math. This could be read after Velleman for an exposure to theorem proof style math.

This can certainly be done, but you would need to work very hard this summer. And be sure to take a 3 week break before school. That will help the material sink in. You don't want to be doing a cotinous stream of precalc and then calc.

 

[ehrenfest]

I need someone with experience to guide me in what to study when, which book to use, which concepts to study the most etc. You know, set me on the right path? Any and all suggestions would be greatly appreciated.

I would recommend Stewart for calculus. You might want to get an edition higher than 2 though (they go up to 6 now). In terms of concepts, I would say focus on calculus since this is definitely the foundation of college-level physics and mathematics. You want to understand integrals and differentiation as well as possible. To get a working knowledge, you don't need to learn the proofs, just the basic integration and differentiation theorems and how to apply them. That will give you access to more advanced physics and is something you want to learn as soon as possible since it needs to become routine when you start doing more advanced stuff.

 

[Oberst Villa]

To get a working knowledge, you don't need to learn the proofs, just the basic integration and differentiation theorems and how to apply them. That will give you access to more advanced physics and is something you want to learn as soon as possible since it needs to become routine when you start doing more advanced stuff.

I guess this would be good advice for a physicist (or a would-be-lurking-hobby-physicist like me, I myself definitely don't need no stinkin proofs !). But the guy wants to major in mathematics, so I guess he has no chance at all to avoid doing mathematical proofs (doing them, not just learn them). So he might just as well start to practice them right now.

 

[MJC684]

Thanks for the feedback guys. Howers, I actually have the book by Allendorfur and Oakley that you are referring to - "Principles of Mathematics". I also have "Elementary Vector Geometry" and " A Vector Space Approach to Geometry". Would either of those be worth looking at for the vectors/planes material? Anyways, so let me get this straight, work the precalc books (5 sections a week) and read the first 3 chapters of 'How to Prove it? Right? Once again thanks for the input.

 

[ice109]

you're insane if you bought all these books.

nm most of what razored said. don't worry about being behind, it doesn't matter.

study the precalc book by bittinger. i don't remember a single thing from precalc which means i haven't had to use any of it and I'm a physics/pure math major. from trig you should know the definitions and the graphs of sin cos tan and their recipricals. that's about it. knowing trig identities is utterly useless in real life cause you can look them up. yes some teachers will be pricks and expect you to know them by heart but they'll warn you.

i would read through polya's "how to solve it" and "how to read and do proofs" by david or daniel or whoever he is. post problems on these boards to be checked. once you're comfortable with writing proofs then move on a rigorous calc book like apostol's or spivak's.

again there's no rush, there's no need to "bust your balls". you'll make the decision on your own when you're ready to commit yourself to studying but if you do it before then you'll hate your life and you'll begin to hate math.

edit

actually i realize now you never stated whether you want to do pure math or applied. my suggestions are for a pure math major. if you're interested in applied math i can give suggestions on that too.

edit 2

you also don't need mathematical logic or symbolic logic. those are books for logicians not mathematicians.

 

[Howers]

Thanks for the feedback guys. Howers, I actually have the book by Allendorfur and Oakley that you are referring to - "Principles of Mathematics". I also have "Elementary Vector Geometry" and " A Vector Space Approach to Geometry". Would either of those be worth looking at for the vectors/planes material? Anyways, so let me get this straight, work the precalc books (5 sections a week) and read the first 3 chapters of 'How to Prove it? Right? Once again thanks for the input.

Yes, read the first 3 chapters of how to prove it. You might also try the induction chapter (skip the stuff between), which I think is CH6. The reason I don't recommend CH4 and 5 is because it is set theory that you will learn anyway in upper years, and Velleman doesn't do a fantastic job with it. And its not needed at your level. His CH6 on induction isn't what I would call the best approach either, but it has some good stuff. You should know induction before starting though, so its your choice if you use his book for it. You don't really need Oakley if you have Velleman, but it might be a useful refrence when you get stuck. Once you get all of this, you might try some Euclidean geometry. But I wouldn't worry too much about it now.

Your problem is you have too many books. I would not recommend those vector books, because they sound like you need some linear algebra. You don't need to know a whole lot about vectors at this point, just know the basics. In fact, here is a list:

Planes, lines, and intersections.
Vectors, components, unit vectors, algebraic vectors, geometric vectors
Vector addition, vector representation of a plane
Vector producs: dot product and cross product
Matrices, matrix inverses, how to find a determinant of a 3x3

Any book that has this stuff is good. So if those vector books you have start with it, then you know what to do.

You don't really need to follow 5 sections a week strictly. I would say no more than 2 a day, so you can do up to 7/14. Some might be harder, and may require more time. The most important thing is that you feel comfortable with a section before moving on. At the same time don't obsess about it as you don't want to fall behind.

So here's some kind of plan you might consider:
June:
day- Precalc(numbers, exponents, algebra, inequalities, graphing, functions, transforming functions).
afternoon/evening- Proofs(CH 1-2/3)

July:
day- Precalc (inverses, exponential functions, trigonometric functions, sigma/pi notation, sequences, series) + Stewart's Appendix + Basic Calc if time permits
evening- More on Proofs(Chaters 3, skip and do CH6 after you've done sequences and series)

August:
day - Basic Calculus - Stewart (Try to get to CH5. Rush through CH1, which is hopefull review at this point. Do one section a day, even if your not entirely comfortable - this stuff will be covered in college. Skip precise definition of limits, and any fancy sections. You just need a feel for calc)
evening - Vectors and planes, see topics above. Basic Euclidean geometry if time permits.

MOST IMPORTANT: 3 weeks rest before school. Unless you still need to do precalc.



You do this and you're set. Somehow, like most pupils, I doubt you will :-)

 

[MJC684]

Good stuff. This is exactly the type of information I was looking for. I really appreciate the help. And ice109, yes Pure Math would definately be my choice. For some reason I thought mathematical logic was needed to be able to write proofs.

By the way, does anyone know if Saxon Precalc is any good?

 

[Indeed]

Good luck. Here a good advice for math majors. Get the fundamentals down as solid as possible. Always get the instructor's solution manual if you're studying on your own, they help so much. And you have to be as scrupulous a teacher as any if you want to avoid bad mistakes/misconceptions.