Math private study: what to focus on

[MWH]

I'm going to attend university and study mathematics in about 1 and a half years. As I want to devote myself and study the field for lifelong, I'd been thinking it wouldn't be a bad idea to start doing private study already. I'd like to ask for suggestions about what exactly would be the best to focus studying on, e.g. broadening my mathematics foundation necessary for admittance into university, focus on algebra/calculus or any other field or on subjects other than just plain math theory such as being able to provide a proof or the history of mathematics. Simply what would always be good to know already if I'm aiming to become a mathematician. Feedback and recommendations are more than welcome.

 

[chiro]

Its a little hard to give advice but most mathematicians at some point find problems in specific areas that they choose to give their attention to.

For you to get to that point, you need to know enough about a field and its problems and why they are still problems (ie unsolved).

There are many many different areas of mathematics and within these areas many many problems, whether you are looking in pure mathematics, appllied mathematics, or in statistics.

I guess to get closer to understanding problems you have to narrow down an area. Even if you don't have an undergraduate degree behind you, you can still get a taste of an area and its problems and maybe something might catch your attention.

What kind of things do you think you might be interested in? If you can't get out the specifics maybe you should browse the wikipedia website for math or areas of applied math like engineering, physics, computer science and so on.

 

[mathwonk]

You don't say what level you are at now, but if you are at high school or early undergrad level, you might enjoy a book like Courant and Robbins' What is mathematics? It is a survey of many basic classic areas by an expert mathematician.

 

[pergradus]

Geometry was the inspiration for the invention of much of modern mathematics. Maybe try reading some classic geometry? I've been meaning to read Euclid just for the historical perspective myself.

 

[Chaostamer]

Honestly, I think, in your position, I would've benefited most from extra proof experience. I would get a proof-oriented book and work through the basics of logic, proof methods, and set theory at the very least. Some experience with general functions and relations would be good too. That stuff is all very fundamental. Plus, if you have that foundation, you can focus more on taking as many rigorous classes as possible, as soon as possible.

 

[Sankaku]

Courant and Robbins' What is mathematics?

This is a great recommendation.

I would get a proof-oriented book and work through the basics of logic, proof methods, and set theory at the very least.

Yes! The Courant book has some of this but I liked Velleman for proofs:
https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

If you know what University you might go to, try to find out what their first proof-based course uses and track down a second-hand copy. It is often discrete math or linear algebra.

Our Discrete class used this:
https://www.amazon.com/dp/0131679953/?tag=pfamazon01-20

I found it quite good and it has the bonus of including a huge number of the solutions in the back (including proofs). That is very helpful when getting started on knowing what a proper proof looks like. The drawback is that it is hard to find a cheap copy of the book...

 

[MWH]

Thanks for your replies, I'll definitely check those out. Especially Courant and Robbins' What is mathematics? might just be what I was looking for.

Our Discrete class used this:
https://www.amazon.com/dp/0131679953/?tag=pfamazon01-20

I found it quite good and it has the bonus of including a huge number of the solutions in the back (including proofs). That is very helpful when getting started on knowing what a proper proof looks like. The drawback is that it is hard to find a cheap copy of the book...

Do most books include solutions? I've noticed some books/texts (sometimes) require a teacher or an already sufficient understanding to be able to fluently work through it, in which case it would be great to have the solutions at hand. At least that was the case with the Dutch math books I had at high school (I live in the Netherlands), as I preferred working on my own.

What kind of things do you think you might be interested in? If you can't get out the specifics maybe you should browse the wikipedia website for math or areas of applied math like engineering, physics, computer science and so on.

I've quite looked around the internet to see what specific areas are like. I'm interested in pure mathematics, but I'm not so interested in statistics or applied math. I do have a great interest in mathematical physics though. I would be really excited about studying the mathematics behind i.e. string theory. I've actually seriously considered studying mathematics AND physics. What are the opportunities for mathematics students, given they want to do research in a field of (theoretical) physics?

Geometry was the inspiration for the invention of much of modern mathematics. Maybe try reading some classic geometry? I've been meaning to read Euclid just for the historical perspective myself.

What would be a superb elementary geometry book?

 

[A. Neumaier]

Do most books include solutions?

Books from the Schaum Outline series are excellent in this respect.

I do have a great interest in mathematical physics though. I would be really excited about studying the mathematics behind i.e. string theory. I've actually seriously considered studying mathematics AND physics. What are the opportunities for mathematics students, given they want to do research in a field of (theoretical) physics?

In Vienna (a world center of mathematical physics0, you can study both simultaneously.

See also Chapter C4 ''How to do theoretical physics'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#C4

 

[MWH]

See also Chapter C4 ''How to do theoretical physics'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#C4

Very useful information via that link !

 

[Sankaku]

Do most books include solutions?

No. This is less of an issue as you get more comfortable with knowing what information should be fully written out or how a proof should look. However, as you say, it is very nice to have solutions when you are learning any challenging subject.

Some books have answers to computations in the back (with no detail on how it was generated). Some have fully worked solutions, but more commonly there is a separate solutions manual book. Usually these have only half the answers so teachers can assign marked work from the text.

It is a fact of life that some brilliant texts have no solutions and, surprisingly, some very average texts can have excellent solutions manuals. In that case you can learn as much from the solutions as you can from the text.

As A. Neumaier said above, the Schaum's series has many books with fully worked solutions. They are not usually enough to learn from alone, but a great source of extra material. Beware that, while they are very useful, they often have many errors. It keeps you on your toes!