Good Proofs for Math Prep: High School Senior

[lj18]

I am a high school senior who is planning to major in math in college. I am currently in a break until the second semester of calculus at a local college starts at the end of January. I took the first half as an AP class at my school last year. I have been going back and reviewing topics from before calculus and the calculus that I have already learned to make sure that I am ready for this semester.
I know that proofs are very important to math, but I feel like the topic has really been neglected in former classes. I would really like to read through some really interesting ones that I would be able to follow with my background. Any suggestions? Also, any advise in general on things I should do now to prepare for for my future math education/career?

 

[dcpo]

There are some classic, simple proofs that nevertheless use important ideas. An example is the reductio ad absurdum proof of the irrationality of the square root of 2 (see here, "proof by infinite descent"). This is a proof by contradiction, you begin by supposing something is true, and proceed to show that a contradiction follows from this assumption, concluding that the original assumption must be false. Proofs of this kind appear everywhere in maths, so getting an idea of how it works at this stage will be very useful.

Another classic proof by contradiction is a proof that there are an infinite number of primes.

Both these proofs are used as examples in the famous "a mathematician's apology" by G.H. Hardy, which is well worth reading. It was written in 1940 and in some respects is a little dated (at the time of writing Hardy was firmly convinced of the practical uselessness of his field, number theory, which is now hugely important due to its role in cryptography), but it is a very readable peek into the thought processes of a pure mathematician.

 

[jedishrfu]

Another good resource is the book: Proofs from the Book by Aigner written in honor of Paul Erdos who always said upon seeing a truly elegant proof that's one from the Book.

The book is referenced at wikipedia:

http://en.wikipedia.org/wiki/Proofs_from_the_book

and on amazon:

https://www.amazon.com/dp/3642008550/?tag=pfamazon01-20

 

[lj18]

Thank you. I looked at those and I'll look into getting those books.

 

[Edgardo]

Have a look at our thread How to write math proofs. It contains a collection of free pdfs that teach you proof writing. They are geared towards beginners.

 

[ahsanxr]

If you have a good mathematical intuition, your transition to proof writing shouldn't be hard at all. Just read a few chapters of the book "How to Prove It" by Velleman to get acquainted with the techniques, methods and formalisms used in proofs and you should be set.