# How to read mathematical definitions, theorems, etc. [closed]

I'm hoping someone can give me some advice on how to read and extrapolate information from Mathematical definitions, theorems, etc.

I understand that I need to read it till I understand what it means and to find examples to fit it. But it's not really doing anything for me. I'm spending more than 30 minutes on each definition/theorem but I don't know what the quickest method is for understanding these things.

here's one theorem that doesn't make sense to me

Theorem 2.4 (Cantor) No interval (a,b) of real numbers can be the range of some sequence

I'm currently reading Elementary Real Analysis and Abstract Algebra books and some of the definitions are pretty gnarly.

• I'm not sure why definitions are taking you so long -- generally those can be figured out fairly quickly. Theorems on the other hand should take a while to digest. When I really want to understand a theorem, I keep working at it until I can prove it myself (without just memorizing it). A good theorem might takes days (though thankfully, it's often less than that ;D).
– user137731
Dec 31, 2014 at 23:37
• Could you give an explicit example of a definition/theorem that is bothering you? Dec 31, 2014 at 23:38
• Lara Alcock's books, How to Study for a Mathematics Degree, (especially Chapter 3 on definitions and Chapter 4 on theorems), and How to Think About Analysis are worth consulting for advice. The first book is also available as How to Study as a Mathematics Major.
– J W
Dec 31, 2014 at 23:51
• @Bye_World: Some definitions are reasonably clear, but others can be fairly opaque initially. I remember struggling with the open sets definition of a topological space at first.
– J W
Jan 1, 2015 at 0:00
• The theorem that you quote is a theorem about two types of sets. One type is the range of a sequence, and the other is an interval. The theorem says that the two types are so different that no set can be both at the same time. Jan 1, 2015 at 0:08

## 1 Answer

When it comes to definitions, I'd follow the advice of Hilbert:

One must always begin with the simplest examples.


Math gets easier the more concrete it feels. Give yourself something simple to chew on before considering more complicated examples. If you're learning about, say vector spaces, spend some time just thinking about $\mathbb{R}^2$ for a night. I find that I understand most complicated examples by "projecting" them onto a simpler example where my intuition is stronger and then applying my result to the more complicated case.

There is no general formula for understanding theorems, or definitions for that matter. Besides reading slowly and making sure you understand the smallest details, I'd offer one further piece of advice, which I believe I read on MSE at some point. Once you understand the theorem, try to imagine how the theorem could not be true. Then, you'll find yourself mildly astonished by the time you hit the QED. I've always found this helps theorems "stick" with me more.

• This was helpful. Thanks. Jan 3, 2015 at 6:52