Self-study textbook help

[Herbert West]

I would really appreciate any help in regards to textbooks you could give that I may study undergraduate maths from. I have recently become fascinated by maths through recent exposure to Euclid's "Elements". I was already self-studying the maths A level and am just about finishing studying and learning everything contained in the revision book I learned A level mathematics from.

As I am in Year 12 I have over a year and a half to go before I can go to university and do not see any reason behind waiting that long to start studying undergraduate level work. What textbooks or other sources would you recommend so that I may increase my knowledge of mathematics? I am particularly interested in anything in any other fields like Euclid's "Elements" where everything is proven from a small foundation of axioms and such; that is if there are such comprehensive books in other areas. I do not really mind how complex they become as long as they begin with the knowledge that the average undergraduate would be in possession of and do not progress without explaining any new notation I would be unfamiliar with.

Thank you very much in advance to anyone who is able to help me.

 

[JasonRox]

I think Abstract Algebra might be what you're looking for. Lots of neat stuff in it, and it does start at a basic foundation as well.

I'd recommend Abstract Algebra by I.N. Herstein. You can find cheap used ones on Amazon.com.

Also, I'd recommend reading random books about the history of mathematics. It helps you find what you truly love about mathematics so you can make a path for yourself. Because right now, you're options are so broad it's hard to give you any directions at all. I can say find a Calculus textbook, but it doesn't seem fall in your category. They almost never start at the foundation. You'd have to go for an Analysis, which may be or may not be too advanced.

Anyways, Abstract Algebra in my opinion could be a great place to start or even Linear Algebra.

 

[ak416]

Spivak - Calculus. Starts from foundations and gets to some real advanced stuff. Exercises are difficult, but you know how it goes, whatever hurts you only makes you stronger...Just make sure youre strong in basic logic. Look for books in predicate logic, it will definitely help.

 

[Droner]

What do people think of somebody in the OP's position trying to self-study from baby Rudin?

 

[james5094]

What do people think of somebody in the OP's position trying to self-study from baby Rudin?

I'm studying baby Rudin before entering college as a math undergrad. I'm younger than the OP, but I find it to be stimulating. It's important that you try to prove theorems before going on to read Rudin's proof, and do the exercises at the end of each chapter so the concepts will cement themselves into your memory.