4:03 AM, December 19, 2024
Shorter blog post tonight. A (generous) early Christmas gift from my parents that I’ve had the joy to start using has been the reMarkable 2, which I may start using in my blog posts as it’s SO much easier to write math notation on reMarkable than it is to type it in latex. I’ve still been trying to find the best marker/pencil setting to use, and it’s slightly tougher on my hand than ordinary handwriting, but it certainly feels way more like writing on paper than writing on an iPad does, and the writing looks way better than what I’ve had to do to submit written homework previously, which is to take pictures of my already poorly handwritten pages. And for those who might say ‘just use a pdf scanner for your handwriting’, that actually tends to come out worse than just taking a picture of the handwriting. I’ve only started using reMarkable a couple of days ago, but so far I’m really enjoying it and I plan to, for the first time in my life, actually start creating organized notes for self-study for the advanced exam (and for the TAing I’ll be doing next spring), as opposed to what I’ve done in the past, which is take awful hodgepodge notes in class (I really have a tough time both listening to the material and creating organized legible notes I can read later), only look at said notes if I’m doing a homework problem that seems similar to the notes, and then study for exams only with the slides provided by the Professor and the course textbook, also without taking notes.
The real beauty of organized notes, as I’ve realized to my chagrin, is that later if you see something that you remember learning but you can’t remember the details, you can go back into your notes and read an explanation that you can easily understand, because you’ve written it, and it can bring you back up to speed and remind you exactly what’s going on and how things are working as you understood them when you wrote the notes. For example, a few weeks ago Fokoué invited me to be a panelist on the RIT Statistics Majors alumni panel, so that current RIT Statistics Major undergrads could hear about what it’s like working in the stats field (or in my case going to grad school) after majoring in Statistics. I had a great time on the panel and went out to dinner at the Lovin’ Cup with the other panelists and some of my old professors after, which was a lot of fun. At one point during the panel though, one of the questions was for a panelist who worked in Quality Control, and she was asked what thing she learned that she used the most, and she said that it was Gage R&R analysis. And I remember learning about Gage R & R when I took Statistical Quality Control my Sophomore Year of college, and I remember thinking that even though I recognized the word and knew it had something to do with measurement accuracy had no knowledge now of how a Gage R & R works or how I would implement one, so I basically had remembered (almost) nothing from a whole semester of a class! And if I had happened to take notes on the material when I took the class (I don’t believe I did) I certainly wouldn’t know where to find them now. Meanwhile, my girlfriend has an organized set of notes for every single class that she’s taken since Freshman year of college saved onto her iCloud that she can access any time she wants. Anyway, I want to have organized notes of the major topics of the courses that I’ve taken so far in grad school, both for study for the Basic Exam and for my own reference, so that’ll be added to my list of personal projects. I think that I’m going to make these notes public, potentially on the professional page of this website, because if the notes help me, maybe they could be helpful for someone else.
Anyway, to show off this newfound digital note-taking technology, I decided to first learn about a proof that’s always been in the back of my mind but I’d yet to look into until tonight, and then write out that proof using the reMarkable. I think that it looks good, considering that my handwriting is just as legible on the reMarkable as it is on paper. I found the main part of this proof on Khan academy. And the proof, of course, is that the derivative of e^x = e^x, which is a very natural thing to wonder. Like, the cool thing about e is that the derivative of e^x is e^x, so where does it come from that for this to be true e had to be defined as the limit of (1 + 1/n)^n? I’m also super curious about why the ratio of a circle’s circumference to its radius comes out to be 3.14159265… and haven’t found an answer that’s as easy to understand as the one I’m about to give for why e is the limit of (1 + 1/n)^n, but that’s for another day. For our proof, note that we use ‘q’ to be the number that we’d like to find that has the property that the derivative of q^x = q^x, I use q so that its less confusing, e.g. we don’t know what q is yet and we want to find a q that makes it true that the derivative of q^x = q^x. Also, I didn’t include this but it’s an important note, the reason we can replace in the below proof the h approaches 0 with n approaches 0 is because the way n is defined, as h approaches 0 n also approaches 0. Without further ado, the proof, and goodnight to all: