Foundations Recommended Math books as preparation for bachelor in math?

[christian0710]

For a long time I've been debating whether to study Physics or Math at university (Bachelor), but I'm going to jump into math this summer. I have this feeling that preparing for "logical thinking" and mathematical language/terminology would be a good idea, since all i have is my 3 year old high-school degree.

What books would you recommend to prepare me for a math bachelor at university?
So far I'm thinking about getting the following books based on other comments here.

1.The book of proof
2. calculus the infinitesimal approach (own this already)

 

[micromass]

I personally think doing proof books like Velleman or book of proof is pretty useless. They spend a lot on time and pages on stuff you could do way shorter.
Why do I think doing proof books is not good?
1) You learn a certain notation and a certain style to do proofs. This is not bad, but the style you learn from proof books tend to be really different from how the rest of mathematics approach their proofs.
2) You learn proofs in isolation. This is really bad, since proofs don't exist in isolation. They always have a certain context (like analysis, or discrete math). Eliminating the context is a really bad idea.

I think you would be better off getting a book on an actual mathematical topic. Keisler is good. I suggest perhaps some linear algebra book, or a discrete math book like Grimaldi. That would prepare you way better than a proof book.

Also, if possible, find somebody who is willing to guide you. This maybe some tutor, or even you posting solutions on physicsforums. But when learning basic proofs, you will NEED feedback. If you don't get feedback, you'll learn it all wrong.

 

[mathwonk]

I tend to agree that books merely about "proof" do not prepare one for real math books. Still there are books that try to do both, present rules of logic, then present actual math topics logically. I was going to suggest one I taught from, Introduction to Mathematical Thinking by Gilbert and Vanstone, until I searched and found it priced at over $100, for a slim paperback worth maybe $15. So I now suggest actually a better book, older and with more mathematical content, but also introducing logic, Principles of Mathematics, by Allendoerfer and Oakley. I liked it in high school and it helped get me ready to major in math in college. But nothing substitutes for just struggling with a real math book. Still this has some actual content. You might just go to a university library and sit in the stacks and browse until you find one that you like. There are some very negative reviews of Gilbert and Vanstone on amazon that make me shake my head and wonder what book would work for those reviewers, so everyone has different needs and different preparation.

here is a link to some used copies of Allendoerfer and Oakley's book:

http://www.abebooks.com/servlet/SearchResults?an=allendoerfer,+oakley&sts=t

 

[infinite.curve]

I recommend reading and understanding this:
http://www.mathdb.org/basic_calculus/BasicCalculus.pdf
Once you fully understand that, I say then delve into other books. Understanding the basic concepts of sets, subsets, elements, neighborhoods,..., are very crucial.

 

[micromass]

I tend to agree that books merely about "proof" do not prepare one for real math books. Still there are books that try to do both, present rules of logic, then present actual math topics logically. I was going to suggest one I taught from, Introduction to Mathematical Thinking by Gilbert and Vanstone, until I searched and found it priced at over $100, for a slim paperback worth maybe $15. So I now suggest actually a better book, older and with more mathematical content, but also introducing logic, Principles of Mathematics, by Allendoerfer and Oakley. I liked it in high school and it helped get me ready to major in math in college. But nothing substitutes for just struggling with a real math book. Still this has some actual content. You might just go to a university library and sit in the stacks and browse until you find one that you like. There are some very negative reviews of Gilbert and Vanstone on amazon that make me shake my head and wonder what book would work for those reviewers, so everyone has different needs and different preparation.

here is a link to some used copies of Allendoerfer and Oakley's book:

http://www.abebooks.com/servlet/SearchResults?an=allendoerfer,+oakley&sts=t

I second Allendoerfer and Oakley. Very good book that you could do now.

 

[christian0710]

Thank you very much guys,

So I'll skip the book of proof: Start with
Infinite.Curves recommendation http://www.mathdb.org/basic_calculus/BasicCalculus.pdf
Then try Allendoerfer and Oakley's book: after that.

 

[micromass]

Thank you very much guys,

So I'll skip the book of proof: Start with
Infinite.Curves recommendation http://www.mathdb.org/basic_calculus/BasicCalculus.pdf
Then try Allendoerfer and Oakley's book: after that.

Have you never seen calculus before? The book you linked is very basic and easy calculus. If you have ever seen calculus, then the book you linked is useless. And Keisler covers way more material than the above book and it covers it better. If you're going into engineering or related, then by all means do this book. But you're going to be a math major, that means you should be doing very different kinds of books than the above (and then I'm not talking about the difficulty level).

 

[micromass]

Also, I did suggest skipping the book of proof, but that's under the very explicit assumption that you would study something else that is quite proofy. If you're just going to study an engineering calculus book, then I retract my suggestion. Proofs are important, you need to learn them well. The book of proof covers proofs nicely so that's good (my argument was not that it did proofs badly, but that there are better resources for learning proofs).

 

[christian0710]

Have you never seen calculus before? The book you linked is very basic and easy calculus. If you have ever seen calculus, then the book you linked is useless. And Keisler covers way more material than the above book and it covers it better. If you're going into engineering or related, then by all means do this book. But you're going to be a math major, that means you should be doing very different kinds of books than the above (and then I'm not talking about the difficulty level).

Sounds almost imtimidating. All i know is high school math and then the first 250 pages of Keisler (infinitesimal approach). I was hoping to become a high school teacher in math but I need to major in mat to do that where I'm from. I'm a very visual learner, what exactly will i be facing during such a course? And why is it so different from high school math? What books would get me prepared for this?

 

[MidgetDwarf]

Geometry by Edward M. Moise / Kisselev Geometry. Kisselev does not have any answers btw.

Friedberg, " Linear Algebra"

Axler, " Linear Algebra Done Right,". I haven't gotten around to doing Axler, but I did complete 2 Chapters and It was amazing.

First learn Calculus, however you play with the Geometry book

 

[kctong529]

If you are ambitious and want to do Calculus justice, try one of the following:

Spivak, Calculus
Apostol, Calculus Vol. 1 & 2
Courant & John, Introduction to Calculus and Analysis Vol. I & II/1, II/2​

Be forewarned anyone of them is enough to get you busy for at least a few months. Do not rush.
Do not turn to Stewart or Thomas unless you only care about exam.

Linear Algebra is also a possible choice, and I suggest (in order of difficulty):

Friedberg, Linear Algebra
Axler, Linear Algebra Done Right
Lang, Linear Algebra
Hoffman, Linear Algebra
Halmos, Finite-Dimensional Vector Spaces
Sergei Treil, Linear Algebra Done Wrong​

If you want a more computational taste you might turn to Strang or Lay for Matrix Algebra.
But the theory (i.e. Vector Space) is very important in that your next few years will be miserable if you ignore it.

For both Calculus and Linear Algebra you may take advantage of MIT OCW for their video lectures.
Bear in mind courses like 18.01, 18.02, 18.03, 18.06 (which have a lot of materials available) are intended for all majors,
as opposed to those favourable to math majors (18.014, 18.024, 18.034, 18.700).

All of the above can make you better in writing proofs.
Another very approachable choice is Number Theory, if you like it.

Davenport, The Higher Arithmetic
Silverman, A Friendly Introduction to Number Theory
Goldman, The Queen of Mathematics
Ireland, A Classical Introduction to Modern Number Theory
Hardy, An Introduction to the Theory of Numbers​

You may also step back to AoPS and start all over from Prealgebra to have a more solid background.

For a general picture of math major:

Lang, Basic Mathematics
Liebeck, A Concise Introduction to Pure Mathematics
Hardy, A Course of Pure Mathematics
Aleksandrov, Mathematics: Its Content, Methods and Meaning​

For more:
http://www.maths.cam.ac.uk/undergrad/course
https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
http://kiro.fraxy.net/books.php
http://www.cargalmathbooks.com/
http://www2.kenyon.edu/Depts/Math/schumacherc/public_html/Professional/CUPM/2015Guide/CUPMDraft.html
http://maths.mq.edu.au/~chris/notes/index.html

 

[micromass]

If you want a more computational taste you might turn to Strang or Lay for Matrix Algebra.

Actually, if you want a computational taste, you can really do no better than Meyer "Matrix analysis and applied linear algebra". It's one of the best linear algebra books I've ever read, even though it focuses a lot on computations and applications.