Wow, I can't beleive it's already been two weeks! I've made it through my first 236 Problem Set, and I've gotta say, I'm really happy the way it turned out. I was initially scared after reading the questions, but then after I took a breath and calmed myself down, I realized that they were extremely similar to examples we did in class. In the first question I had to prove (using simple induction) that the right-most unit digits of 4^n (n being from the set of all natural numbers) were either a 1, 4 or 6. I used the example from class with 3^n to help me set up and structure the proof, but more importantly, I understood everything I was doing and why I was doing it. The second question on the other hand, took a little bit more thinking..
I'm not going to retype the entire question because I'm sure whoever is reading this knows what it was about haha, but I stumbled on this proof for a few unexpected reasons. I figured out the induction step pretty easily after doing some scratch work, but trying to explain the reasoning behind the induction step is where I had trouble. I went to Danny's office hours this week and overheard him talking to another student about how some proofs can consist of just writing, and that mathematical mumbo-jumbo isn't always neccessary. I think the way a person expresses themselves and conveys their thoughts into writing is super important, and it's actually one of the hardest parts of the proof. The math part is either right or wrong and is entirely based on whether you understand what you're doing -- but proving something using nothing but words? That's tough. I mean, what really makes a well-written, clear and concise proof?
What I realized is that it takes a lot of time and rough-drafts to get a proof exactly the way you want it. I swear I wrote up the answer to question two a million times before it met all of the TA's and Danny's requirements. What I thought were such little, trivial things actually constituted as major parts of the proof. I had everything written down, but in a somewhat unorganized and unclear manner. I didn't properly convey the connection between the claim and the Induction Step, and not because of what I had written down, but the way it was written down. However with some help and a little rephrasing I was able to write up a proof I couldn't have been happier with -- and now that I know the importance of "techincal writing", I think I'll be more prepared for future problem-sets, tests, quizzes and assignments.
Anyway, this is already turning out to be too long of a post so I'll try to wrap it up as quick as I can. The use of the tablet in lecture is awesome - I love it. Makes everything a lot easier to read and we have access to all the notes online, which is really cool. The 1st Assignment is coming along slowly but surely haha, I'm going to try to spend a lot of tommorow working on it. Other then that I'm just trying to grasp the concept of Complete Induction and figure out a strategy to know which one to use (Complete vs. Simple). Anyway I'm going to stop writing now because this is turning into a novel, but yeah, I'll be writing more next week.. so.. cya!
Subscribe to:
Post Comments (Atom)
2 comments:
I like the fact that the tablet notes are preserved on-line, but it makes me nervous. After all, any error I make on a traditional blackboard disappears quickly under layers of chalk dust. A mistake on a file from my tablet stays there winking at me forever.
You've keyed on the difficulty many people have with university level math. One TA in mathematics described it as "proof without words = answer without marks", admittedly that's an oversimplification but explaining why in words with a sufficient degree of rigour is often very challenging. It's also important to use a clean concise numerical explanation (sometimes those words can convert back into numbers). Learning to find the right balance is something that will serve you well with mathematical problems throughout your university life and potentially beyond.
Post a Comment