# Math preparation

This page contains my recommendations for being well-prepared for ECN 606 from the point of view of math background. Success in ECN 605 is a necessary condition for success in ECN 606, but it is not sufficient. If you have previously succeeded in a rigorous proof-based math course such as real analysis, I suggest you simply review the topics (from a book of your choice) on the Math topics page and brush up on your proof-writing.

If you are not yet comfortable with mathematical proofs, the best thing you can do is to become more comfortable with *writing* proofs. I give my suggestions below, after a few caveats

- Very few people could accomplish this task in a short period of time, no matter how intensively they work. If you devote one hour per weekday for eight weeks, you will accomplish FAR more than you would in five eight-hour days.
- This kind of sophisticated thinking requires the brain to remodel itself, and that takes both struggle and recovery time between sessions and especially overnight, as the brain does a large part of consolidating new learning while you sleep. If you’re not getting enough sleep, you brain can’t properly consolidate the new learning (see these new findings on the effects of sleep deprivation).
- If you were to start this work on the first day of fall classes, you would be at least halfway through the semester before it really starts to bear fruit. But you need to use this kind of sophisticated thinking from day one. Start NOW.

## “How to Prove It” by Velleman

If you have little or no experience with writing mathematical proofs, start with the first five chapters of Velleman (except sections 3.7, 5.4 and 5.5, which I suggest you skip).

This will teach you step-by-step how to write proofs at the same time as it covers the majority of the topics under *Logic* and *Set Theory* on the Math topics list.

In order to make an impact, you **must** actually **work** the examples. As long as you can **work** the large majority of the examples for a given section, I suggest you skip the exercises for that section, at least for now.

What I mean by **work** is: Read the statement of the example. Cover up the answer. Do your best to generate the answer on your own. If you can’t figure it out, take a break and then try again. Perhaps ask a classmate for a hint. Think hard. Wrestle with it.

If you’ve done all this and still don’t know what to do, THEN:

- Go back, review, and do the exercises in the previous section.
- Try the example on which you were stuck again.
- If you can figure it out now, move forward in the book.
- If you’re still stuck, repeat Step 1, but go back an additional section. Repeat 2-3. If you’re still stuck, go back one more section. And so on.

## “Mathematical Analysis” by Binmore

Binmore is aimed more at giving a concise treatment of the essential concepts than teaching you how to actually prove the results. This should be fine once you’ve completed the Velleman chapters. I you work from start to back in Binmore, focusing on the sections below. Again, if you can do all the proofs in the examples yourself (without looking at the provided proofs), then you can probably skip the exercises.

- All of Chapter 1
- 2.2 – 2.10 (do the exercises in 2.10)
- 3.7 – 3.11 or Velleman 6.1 on induction proofs would be nice, but not top priority if you’re pressed for time
- 4.2 – 4.24
- I think you can skip Chapter 5 and 6 for the purposes of 606; I’d like to hear from people if this turns out not to be true, or if you need it for other classes (series are used a lot in econometrics)
- You’ve likely seen everything in Ch. 7; it’s a good idea to skim over it if you’re not sure
- All of Chapter 8
- All of Chapter 9, and it’s very important
- Chapter 10 is likely review. Make sure to look at 10.6.
- 11.1 – 11.7
- All of Chapter 12 except 12.11
- Chapter 13: We will use a little bit of integration, but I assume you will get more than we need in math camp / metrics
- Chapter 14 should be covered in undergraduate calculus; refresh if you need to
- For 606, I don’t think you need power series (Ch. 15), and I’m sure you don’t need trig (Ch. 16) or the gamma function (Ch. 17)
- 18.1 – 18.16, 18.21 – 18.23, 18.26 – 18.39
- Chapter 19 is perhaps useful as reference if you get caught in a proof about partial derivatives, but probably otherwise mostly a more complicated version of what you’ll go over in math camp.

# Other resources

See the Math references section of the Textbook page for further suggestions.