There is a belief that IMO style problems are what "extracurricular" maths in high school should look like. Others say that if you are serious about learning mathematics, you should try to do exactly that, learn mathematics as opposed to solving puzzles. Siding with the latter, here are some books that should be accessible to high school students (certainly I wish somebody mentioned them to me at the time!):
V.I. Arnold - Abel's theorem in problems and solutions- based on lectures read by Arnold to Moscow highschool students, discusses group theory, some complex analysis and ends with a geometric proof of Abel's theorem using complex valued functions. This was the first serious maths book for many Russian students. All problems have solutions at the back.
Abel's theorem in problems and ... - Google Books
M. Artin - Algebra - a great undergraduate abstract algebra textbook that assumes almost no background. Watch out for the typo in the matrix multiplication example in chapter 1!
Algebra - Google Books
F. Lawvere, S. Schanuel - Conceptual mathematics: a first introduction to categories - an elementary introduction to category theory that assumes no knowledge of advanced mathematics.
Conceptual mathematics: a first ... - Google Books
S. Axler - Linear algebra done right - a clear introduction to... linear algebra, avoiding determinants where possible.
Linear algebra done right - Google Books
O. Toeplitz - The Calculus - a genetic approach - traces the historical development of calculus from Archimedes to Newton, discussing the motivating problems and origins of ideas. Great for shedding some light on the awful mess that is the Ext 2. curriculum.
The calculus: a genetic approach - Google Books
The above books can be difficult to find in libraries, however, try searching for "_book title_ djvu" or "_book title_ pdf" in google.
Some advice when reading maths books:
I will add some more later if people are interested.
V.I. Arnold - Abel's theorem in problems and solutions- based on lectures read by Arnold to Moscow highschool students, discusses group theory, some complex analysis and ends with a geometric proof of Abel's theorem using complex valued functions. This was the first serious maths book for many Russian students. All problems have solutions at the back.
Abel's theorem in problems and ... - Google Books
M. Artin - Algebra - a great undergraduate abstract algebra textbook that assumes almost no background. Watch out for the typo in the matrix multiplication example in chapter 1!
Algebra - Google Books
F. Lawvere, S. Schanuel - Conceptual mathematics: a first introduction to categories - an elementary introduction to category theory that assumes no knowledge of advanced mathematics.
Conceptual mathematics: a first ... - Google Books
S. Axler - Linear algebra done right - a clear introduction to... linear algebra, avoiding determinants where possible.
Linear algebra done right - Google Books
O. Toeplitz - The Calculus - a genetic approach - traces the historical development of calculus from Archimedes to Newton, discussing the motivating problems and origins of ideas. Great for shedding some light on the awful mess that is the Ext 2. curriculum.
The calculus: a genetic approach - Google Books
The above books can be difficult to find in libraries, however, try searching for "_book title_ djvu" or "_book title_ pdf" in google.
Some advice when reading maths books:
- If you get stuck on something, do not spend too long - move on to the next section and come back later
- More advanced textbooks often contain typos - if you think something doesn't make sense, there is a good chance that it doesn't
- Try reading several books on the same topic at once
I will add some more later if people are interested.
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