When learning math, you don't want to skip anything, including theorems and proofs. Theorems tell you under what conditions the results can be applied and proofs show you how to use the math you already know to produce new uses and applications. Here are a few book suggestions if you want to go into more depth. But read through this page first to see if it helps you.

Let me say up front . . . most students just read theorem statements and skip proofs. If you work through each theorem statement and study the proof, you will begin to actually understand math and know when to use it. This will put you ahead in your class and in future classes. Some instructors will also give problems on homework and exams that the other students will miss because they didn't know when to apply which theorem. But you will get it right. This could potentially increase your grade. So, let's get to it.

How do you actually study proofs and theorems so that you can understand them? As you read on the page about how to read math books, you can't just read theorems and proofs like you would read a novel. Mathematicians write theorems and proofs as elegantly as they can and this will usually obscure what is going on. Even Ph.D.'s have to work through theorems and proofs, so you are in good company.

We will separate our discussion into two parts, the theorem statement and the proof. Start by getting out a pencil and several pieces of paper.

Understanding The Theorem Statement

Theorems are written in concise, very compact language which inevitably obscures what is being said. So you need to dig out the pieces in order to understand them. Theorem statements have two main parts, the conditions and the conclusions. The conditions tell you what has to be true in order to apply the theorem. The conclusions are what you can know for sure is true as long as all the conditions hold. The nice thing about theorem statements is every single condition that is required will be stated somewhere in the theorem. So you don't have to guess about a condition.

Your main task is to separate out the conditions and the conclusions. If the language of the theorem (like English) is a language you are very familiar with, then this will not be hard. Mathematicians are very concise, so every word is important. Do not skip any word, no matter how small.

At this point, it won't help you understand what to do if we just give you a bunch of generic rules to handle theorems. So we will use an example to show you how to do this, from which you should be able to extrapolate what you need to do. Here is a theorem from differential equations. You should know some math but don't worry if you don't understand all the advanced math, that's not the point. The point is to extract the information we need.

Theorem

If the functions \(p\) and \(g\) are continuous on an open interval \(I: \alpha < t < \beta\) containing the point \(t=t_0\), then there exists a unique function \(y=\phi(t)\) that satisfies the differential equation \(y'+p(t)y=g(t)\) for each \(t\) in \(I\), and that also satisfies the initial condition \(y(t_0)=y_0\) where \(y_0\) is an arbitrary prescribed initial value.

This is a theorem from differential equations that determines under what conditions a solution exists and is unique, i.e. it is the one and only solution.

Almost all theorems have an if-then format, where the if and/or then may not be explicitly stated but the structure of the sentences will tell you which section is which. In the theorem above, we do have if-then sections. So, first, we write down all the conditions covered by the if section.

conditions

\(p\) and \(g\) are functions which are continuous

\(p\) and \(g\) don't have to be continuous everywhere, just on the open interval \(I\)

\(I\) is an open interval \(\alpha < t < \beta\)

\(I\) must contain the point \(t=t_0\)

Although not stated up front, \(p\) and \(g\) are functions of t and can be written \(p(t)\) and \(g(t)\). You can see this from the differential equation \(y'+p(t)y=g(t)\).

conclusions

1. a solution is guaranteed to exist

2. the solution is unique, i.e. there is only one

They actually name the solution \(y=\phi(t)\). This name may be used in the proof or the subsequent discussion in the text but it doesn't come into play in the theorem statement.

Understanding The Proof

We are working on a detailed discussion on how to read proofs and we will post it here when it is finished. In the meantime, here are a couple of links to get you started and some book recommendations for more information.

Helping Undergraduates Learn to Read Mathematics

Reading Mathematics