Help! Math Proofs!

Lurk

I have been working my butt off to understand practical math. The sort of math that just involves solving an equation or figuring out the derivative. For someone who hasn't done math in a long time, I feel this as an accomplishment. I am not normally a math person but I hoped I could tough it through university on willpower alone.

Now I think that my lack of a strong math background is catching up to me.

I have a hard time doing math proofs. When I would be given to the solution of something like "show that the square root of 2 is irrational", it feels like I am missing something important that can only be gotten from years of dedicated work studying math. For the example I gave, I understood that you need to solve it by contradiction but in no way it was apparent that "form" (forgot the proper word) of a rational number required that the numerator and denominator to be mutually prime. I doubt that I even heard the phrase "mutually prime" before seeing that solution.

What is the best way of tackling the process of learning how to do math proofs? Is there a system that I need to learn or do I need to grind it out through hard practice? Are there any resources that I could use (printed or on the web) to accomplish this?

musanman

Buy a geometry book modeled on Euclid's Elements and learn to prove all of the fundamental geometry theorems. Direct proofs, contradiction, and induction are probably the 3 you'll need to do all of the proofs you'd run into...it's not hard to come up with examples of those so just start digging.

There is a certain art to recognizing a good way to do a proof (one time I did a proof of the contrapositive by contradiction and felt like a god) and the biggest thing is just being exposed to different strategies that have worked for various types of problems.

Smug Duckling

The biggest thing about doing proofs is expanding definitions. Once you expand the definitions out enough, things usually because easy, or become a form that's easy to have intuition for.

For example, if someone gave me the problem "Prove that the square root of 2 is irrational", the first thing to do is expand the definition of "irrational". So you go look at some book where they've written down a definition of "irrational", and you see that it says something like, "a number is rational if it can be expressed as the ratio of two integers. A number is irrational if it is not rational."

So you'd rewrite the question as "Prove that the square root of 2 is not rational."

Which becomes "Prove that the square root of 2 cannot be expressed as a ratio of two integers."

At this point it's a bit more wide open what you could do, but it's never a bad idea to try proof by contradiction. So you can rewrite the question as "Show that there is a contradiction if the square root of 2 can be expressed as a ratio of two integers."

No real thinking has been done yet (just expanding definitions and rewriting the statement of the problem), but you're already at a point where you can do algebra to try to figure out an answer (you can now write sqrt(2) = a/b, and try to find a contradiction).

Math is really all about definitions (this is how it deals with high levels of abstraction), and the way to easily prove something, is to break down its abstraction to a level that you can understand. You can't try to prove something until you know what it is that you're trying to prove.

Contrapositives and proofs by contradictions are basically just fancy ways (with some formal logic) of rewriting the problem, and if you start my manipulating the problem statement, often these things just fall out, or at least give you a place to start.