Hardest level of college math?

Question

Ive been looking to buy a college math book for a friend of mine who is a mathematical genius. Beyond calc 3, whats the most difficult level of math that i can find used books for?

thanks

Answer 1

I had a hard time with real analysis. That's where you get into the theorietical underpinnings of calculus, and have to rigorously prove all the stuff that seemed so nice and intuitively obvious in calculus. But what is the most difficult math is a matter of opinion. Some people sailed thru real analsysis and have a hard time with abstract algebra or linear algebra. I did ok with those.

Instead of the hardest math, why not go after something interesting and pertinent, like fractal geometry, which a lot of people talk about but not many people really understand.

Answer 2

What does he like? If he is looking for something for fun I would avoid college math, most of it is not so interesting, and is building for future stuff. If he wants something to prepare, algebra is best (algebra has to do with sets, it is not the same as the "algebra" taught in high school). It is conceptually different from math that comes before, I found it best to have a little more time to get accustomed to the concepts.

I would recommend something like Euclid's Elements. It is a historically interesting book, the math in it is not trivial, and not taught well in school today. It also has inspired mathematicians for thousands of years, including many today. (and you should be able to find it used)

There are lots of other things that are not taught in school anymore but are fascinating mathematically, and accessible to a bright and interested high schooler. Checking my bookshelf, there are a couple called "A Friendly Introduction to Number Theory" (Joseph Silverman) and "Concrete Mathematics" by Ronald Graham et al. These have thorough expositions of things people might have been exposed to but never really mastered. I am not sure what can be found in used book stores, but this type is good.

How about a gift certificate?

Answer 3

It depends on what "College Level Math" is at your school. At my university there are a bunch of different math classes. We don't have a college level math class, but we have "College Algebra" which is just a combination of Algebra 1 and Algebra 2/Trigonometry. I never took that class because I got placed into Calculus but my boyfriend is taking the class and it is super easy. But again, that depends on a person's math background. It will probably be a lot of work but if you study hard, the material is definitely manageable.

Answer 4

Personally I would go to Dover books, and look at what math books they have.

FYI Dover reprints classic text books that other publishers will not print.

A personal favorite is 1,2,3, infinity by George Gamow. This is a book of the pursuit of math and science in the first half of the 20th century. It is a bit outdated but the author makes it a very interesting and educational read.

Answer 5

the undergraduate level Real Analysis (on metric spaces) is generally considered to be the hardest under graduate class (quite a bit beyond calc 3) elements of analysis by Rudin is a classic

Answer 6

Sheaf cohomology. In reality, you may want to get him a book on category theory, as its currently the most used/abused language in modern math circles.

Answer 7

Topology

Ana