# Fun and Rigorous math subjects

1. May 9, 2012

### jbmiller

Hey everyone!

Ok, this summer im self-studying Alg II so I can test out of it in August for my highschool. I was also planning on learning alot of pre-calc/calc from a friend, but I'm somewhat skeptical about it because he seems to not have that much time on his hands and I don't want to be a burden to him. The last thing I was going to study would be Alg II based Physics and some Calc based Physics (not to sure about the Calc based Physics, if anyone has a tremendousley good textbook PLEASE refer it too me).

If learning Pre-Calc/Calc from a friend doesn't work out then could anyone recommend a really fun and rigorous subject to me?

I'm looking along the lines of; Intro to Number Theory, Intermediate Counting & Probability, Abstract Algebra, Linear Algebra, Differential Geometry, and basically whatever else I would be capable of doing.

About 6 of the 12 weeks of summer I will be traveling, at camps, vacation, ect., so with the off time I have on my hands (which will be alot) I would really enjoy getting into a completely new and rigorous subject.

*Please keep in mind that I am in highschool, freshman to be exact, so please don't recommend a subject and textbook thats completely out of my league. Also, don't recommend Pre-Calc/Calc, I'm only interested in learning it this summer if my friend is teaching me. Otherwise I will just wait til my Sophomore year Calc course.

Thanks!

2. May 10, 2012

### Mépris

Coursera will have some courses that will start over the summer. Try Statistics One. Look into learning some combinatorics as well. (I think that falls under counting and probability?)

3. May 10, 2012

### jbmiller

I don't know if I really want to try Stats. It just doesn't seem to interest me that much and theres a possibility I will be taking AP stats next year.

I should probably study theories and proofs using Spivak, but I'm not too sure if I would be capable of doing that.

https://www.amazon.com/Calculus-4th...=sr_1_1?s=books&ie=UTF8&qid=1336628591&sr=1-1

Would I be able to handle learning from this book?

https://www.amazon.com/Calculus-Vol...=sr_1_5?s=books&ie=UTF8&qid=1336628591&sr=1-5

If not, this book also looks intruiging, but I don't know if I could handle it either.

4. May 10, 2012

### Mépris

Before deciding to take AP Statistics, try getting a feel of stats and see if you'll like it. It's a useful thing to know, apparently.

I don't think I can comment on the texts by Apostol or Spivak. I understand they're rigorous texts but I have not yet studied from either myself. Having said that, it is something I will do in the near future. Besides, I think you'll be the better judge of your abilities. Why don't you try and see for yourself?

Somebody suggested I look into this book. I have and I think this will keep you busy for a while. I can't wait to get started on it!

5. May 10, 2012

### Genericcoder

I self studied calculus this past 4 month it was really awesome. I took also calculus AB and it was easy not challenging like the problems I worked out in couple of books.
I didn't study calculus though rigoursly without going into some proofs I learned from couple of sources and all had really good methods of teaching.

Here are the sources that I learned from.

http://tutorial.math.lamar.edu

Solved many questions I found on the internet for preperation for AB calculus(altough the exams itself was pretty easy)
One of the site I solved from is this
http://www.math.ucdavis.edu/~kouba/ProblemsList.html

I also read Peterson Master the AP Calculus AB & BC it isn't rigorous but has some good problems list.

Anyway thats plan I did but I didn't learn everything rigorous because didn't have enough time I will though read Spivak and stewart book this summer and finish them since I got the knowledge to tackle now those books.

Remember also to do many problems especially in integration because its more tricky than derivatives you need to recognize patterns and it only comes from solving problems.

I hope this helps you :).

Last edited: May 10, 2012
6. May 10, 2012

### Gravitational

Wow, please don't attempt differential geometry, if you haven't even done calculus. In my opinion linear algebra and set theory are two very important subjects to learn, and can also get quite interesting. Highschools seem to put a lot of emphasis on calculus, but in actual fact, linear algebra and set theory are both equally useful. If you ask me, just go an self-study calculus. It's not an impossible thing to do

7. May 10, 2012

### jbmiller

Well, I'm not worried about Calculus being hard, it just doesn't seem to interest me as much. I really want to self-study subjects that my school doesn't offer, mainly for the fun of it and the knowledge.

Also, could you recommend me textbooks for linear algebra, set theory, and calculus. Who knows, I might end up trying calculus this summer and really enjoy it. If I do, my high school math classes should be super easy, even though they're already easy.

8. May 10, 2012

### jgens

You could try abstract algebra if you really wanted. I think Dummit and Foote is a pretty gentle introduction to the subject, but there are also people who disagree with that sentiment.

9. May 10, 2012

### jbmiller

Interesting, what would you say are the prerequisites to abstract algebra?

10. May 10, 2012

### jgens

I would say there are no formal prerequisites. You should have some acquaintance with the notation of set theory and you should also be familiar with some of the elementary divisibility results of the integers. But that is pretty much it.

11. May 10, 2012

### jbmiller

Should I learn set theory this summer, then procced to learn abstract algebra over the course of my sophomore year?

12. May 10, 2012

### jgens

Well you really only need some of the notation from set theory. For example, you should know what the symbols $\in,\subseteq,\cup,\cap,\dots$ mean. You should also know what things like $f:A \rightarrow B$ mean. But much more set theory than that is not particularly useful for a first course in abstract algebra.

13. May 10, 2012

### jbmiller

Alright, well, should I just spend my summer learning set theory?

And what about number theory, is it a good topic to learn?

I'm might just settle with learning Calculus from Spivak.

14. May 10, 2012

### jgens

I would say that is probably not the best idea. The notation of set theory is used everywhere in math, but a thorough study of set theory should probably wait until you have a little more mathematical maturity.

Number theory is always a good topic to learn. I do not know any good books on number theory at an elementary level, but I am sure there are members here who do.

That is also not a bad idea. Spivak is a really gentle introduction to formal mathematics, so it isn't a bad book to start learning proofs from. He also has some interesting exercises which is a plus. Calculus is a pretty fun subject to learn for the first time too.

15. May 10, 2012

### jbmiller

Alright, I will just have to do some poking around and see whats most useful/intruiging.

Thanks for the help!

16. May 10, 2012

### scurty

I took a course in Number Theory last semester at my university and we used a VERY user friendly text. I think it would be good for an aspiring high school student. For many of the major proofs in the book, the book has a little "preview section" to help guide you along in the proof and understand where you are going. Basically, it's a good self learning book because of this. It also is full of corny math jokes which makes it fun to read (it keeps the mathematical rigor though). I imagine a book like this was published by the authors to encourage reading the mathematical text instead of just sitting in the lecture hall and learning from just the board.

ISBN: 978-0470-42413-1

You can read some reviews here:
https://www.amazon.com/Number-Theor...=sr_1_1?s=books&ie=UTF8&qid=1336711780&sr=1-1

You can find it for cheaper though. Here's one site (may be cheaper other places):
http://www.abebooks.com/book-search...ggl-_-USA_ISBN_5-_-USA_ISBN_5_13-_-0470424133

Last edited by a moderator: May 6, 2017
17. May 11, 2012

### jbmiller

Would you advise me learning number theory before calc?

Last edited by a moderator: May 6, 2017
18. May 11, 2012

### scurty

There was very little Calculus in what I was taught from this book, so I wouldn't exactly call it a prerequisite. So in that sense I would say there is no harm done in learning Number Theory before Calculus.

This course was my easiest proof-based course I've taken by far. Mainly because proofs in other courses use unfamiliar (at the time) concepts and definitions that you have to master and be able to manipulate. Early on in this book you are proving very easy proofs and it is easy, visually, seeing why the proofs work. Integers should be a familiar concept to you. Hausdorff Spaces may not be!

My personal opinion is that you should look at Number Theory over the summer. I think I saw in another thread that you are taking pre-AP Calculus in the fall. Then you will likely take AP-Calc. Then possibly Cal II and III in college.. basically you'll be learning Calculus for a long time. You might not get to take Number Theory in college.

Do what you want to do though. Look at other subjects and see how they interest you. Most mathematics books are cold and dry books to read. It might be more enjoyable (especially for a high schooler) to read a book with a little bit more pip. You math title says fun and rigorous. Well, that book I linked to you has both of those in it. You're still in high school so I'm not exactly sure if you'd enjoy your summer plodding through a cold and dry math book. :)

19. May 11, 2012

### jbmiller

Well my senior year I would be taking Calc II at a local college, so I won't really have to worry about that. Depending on how it all works out I might even end up taking Calc III at a local college my senior year.

I will probably self-study number theory over the course of this summer, mainly because it sounds really interesting. Then, I will self-study Calculus over the course of my sophomore year when I'm taking pre-calc.

"You're still in high school so I'm not exactly sure if you'd enjoy your summer plodding through a cold and dry math book."

I'm traveling 6 out of the 12 weeks of summer, thus creating tons of extra time. I might as well spend it learning mathematics. (:

20. May 11, 2012

### lompocus

I'll jump in here and throw in my own 2 recommendations.

Regarding Calculus, you can try this book and slowly work through it over your summer and throughout highschool. I no longer have it, so I can't comment on the later parts covering what you would see at the end of your initial college calculus sequence (series, little bit of linear algebra, multivariable stuff, and all that), though I found it very useful
-> Calculus, Thomas (you'll want the 11th edition, not the newer ones)

I'm surprised no one has mentioned discrete math. It's my favorite course (I'm even procrastinating on my last problem right now! That's how much I love it...). I'm only being half sarcastic. The interests you've listed, and everything in general, can be understood more completely in limited time given some basis (heh.... :P) in discrete math. There are a few books out there, but I'll give you the more comprehensive: Discrete Math, Rosen. The problems will challenge you. The text will often force you to look elsewhere for supplementary information. However, that's my point in recommending this. You're a freshman. Let us put you through self-imposed bootcamp! (You may want to wait until after you've taken calculus, however.) Here's the last edition for twenty dollars-> https://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073229725

I'm too verbose. Oh well. Whatever you end up doing, please focus on only one thing at a time in these initial stages; it's easy to get distracted early on. Work steadily, consistently, and over a long period of time. Good luck!

Last edited by a moderator: May 6, 2017