![]() ![]() When you read a paper, try to stay a step or two ahead of the author. So how do you get a feel for your corner of mathematics? Well, you just do math. That said, yeah, there are often some technical theorems that are useful as black boxes (examples: Wagner's/Kuratowski's theorem in graph theory, the classification of finite simple groups, maybe structure theorems in closely related areas to the one in which you work), but if you use them often enough that it's an actual hassle to look up their statements, and they're reasonably simply stated, you'll probably memorize them through sheer force of habit. Of course, you also want to know the best tools in your area, but that often comes with understanding it on a deeper level. The best goal (IMO) is to get comfortable enough in your area and at your level that you can intuit whether or not some statement is true without necessarily knowing a proof (or an ad-hoc counterexample). I tend to agree that, nine times out of ten, memorizing theorems is a Bad Thing. So I try to get my students to remember that logs make hard things easier then they have some framework to put this identity inside. In this example, knowing a little history can help too logs were used heavily by all scientists until about 50 years ago, because they make arithmetic reasonable, by turning multiplication into addition and exponentiation into multiplication. And unless you understand log and its relation to exp, there's no reason to think one should hold but not the other. A trivial example is that my calculus students always have trouble remembering that log(ab) = log(a) + log(b), but log(a+b) isn't log(a)log(b). If you only have a shaky grasp on what the words in a theorem mean, it's hard to remember it. I found that time spent understanding definitions was more useful than time spent memorizing theorems. Those are both pretty generic pieces of advice, so here's some specific to math. Of course, you do risk being mistaken for a crazy mathematician that way. Also, if you really do need to memorize your theorems (for a test, for example), talking to yourself helps - I used to review math to myself out loud in the shower and while walking to class. The act of writing helps, and if I can keep my notes organized, then it's there next time I need it. I have a pretty mediocre memory, so when I need to remember something I write it down. Does the proof that a/b must repeat establish a reasonable bound on when? For instance, it seems unlikely that 1/29 requires a million digits to start to repeat. ![]() If every a/b must be a repeating decimal, then presumably you can find a bound for when it starts to repeat. So make a variation to have something more newsworthy to prove. This is a familiar fact and we all "know" it. Many variations of this equation certainly aren't true, so why is this one true? The goal is to learn the proof well enough to be able to persuade someone else who doesn't believe you.Įxample 2: Every fraction is a repeating decimal. If you haven't learned a proof, then the only evidence left is argument by example and argument by authority. ![]() ![]() It seems completely unreasonable to dismiss a theorem like this as "obvious". Would the same proof still work?Įxample 1: det(AB) = det(A)det(B) for square matrices. If the stated theorem just isn't news to you, ask yourself if you know whether it remains true if you change it slightly. My main method to care about a proof is to want to be convinced of the thing being proved, the way that a trial jury wants to be convinced. I have to care about something to remember it otherwise I am utterly incapable of memorizing anything. I'm teaching a course right now in which many students face these issues for the first time. ![]()
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