Q&A for How to Do Math Proofs

My first tip is to realize that it is a difficult subject and that nobody is born knowing Math. We have to learn it over time and it's a sequential subject. Understand that there are a lot of steps that go into understanding more complicated math problems. It's okay to take time to learn, it's okay to fill in previous gaps in knowledge, and it's okay to relish the small achievement. Aiming for the small goal and realizing you are progressing as you go along is my main tip for how to tackle that.

Homework is just not fun, but we can give ourselves rewards, which can make it fun. For example, tell yourself I will get through this homework and once I'm done I will binge on my favorite show for two hours or I will go get an ice cream or whatever will propel you forward.

"Mathematical induction" begins with a statement (a conclusion) and proves that it is true in one case and then that it's true in other cases.

There is no especially easy way of doing it. All you can do is prove the corresponding sides equal in length and the corresponding angles of equal arc.

This is very difficult in general. It's very common in math for someone to find a clever way of looking at an old problem that results in a much shorter and simpler proof of a famous theorem. Anyway, here are two vague ideas: experience (ask what sort of tricks tend to be effective on problems similar to this one) and pattern recognition (what can you do to this problem to make it look like something you've seen before and have the tools to solve?) Both of these are easier (but still not necessarily easy) if you know the background material thoroughly, so study as many theorems as you can -- not just the result, but also how they are proven.

The reason that you are wrong is that 2 (and any other integer) is defined as a rational number. It's not possible to prove a proposition that runs contrary to a definition.

Yes. Theorems are formally written in a logically progressive manner where each step follows from and depends on the step(s) before it.

To convince yourself or others that a theorem or proposition is true. Sometimes something seems true but proves not to be upon close examination. Subjecting a proposition to a rigorous proof can eliminate all doubt.

This would be up to your teacher. In Chinese, you write a "proof:" , but it may not be necessary in English.