A math odyssey
Hello again, faithful readers. I spent today’s study time reading the next section of my calculus book and investigating some of the ideas it presented. In particular, I was interested in what pairs of functions, $f(x)$ and $g(x)$, have the property that $f(g(x))=g(f(x))$.1 I made some minor discoveries, but I am too tired this evening to write them up. Perhaps tomorrow.
I had a difficult day and spent my study time watching videos, including a nice documentary-style one about the math of relativity. I admit that I’m looking forward to my day of rest tomorrow.
I am actually contemplating taking all of next week off study and blogging. I will be traveling again, settling in as a live-in sitter for two cats, visiting with family, and attending Pride events. It will be a lot for me.
I’m feeling much better, but struggling to get my various plates spinning again. Today I read more of Infinite Powers, finishing the section focused on Galileo as well as the succeeding one on Kepler. I’m developing an affection for Kepler, for whom math and religious devotion were definitely intertwined.
Today I read more of Infinite Powers. I’ve finished the section on Archimedes and moved on to the next protagonist of Strogatz’s tale, Galileo. What I read about him today proved interesting. I don’t think I ever knew that he discovered the properties of falling bodies, not by dropping balls vertically, but by rolling them down ramps. Perhaps if I had ever taken a traditional physic course, I would have learned that. On the other hand, perhaps this is an example of the tendency in science education to present results without explaining how they were discovered. My scientist father has lamented the disservice that approach does to the subject.
I suppose the analog in math education is the fact that nearly all pre-college math courses are focused on calculation rather than proof. (Traditionally, the exception was geometry, but I think that is often calculation-focused now, as well. My course was, for the most part.) I don’t know whether that is a disservice or not. For me, proof is the most exciting thing. I enjoyed math in middle school and high school, but didn’t truly love it until I was aske to discover and prove facts for myself. I think I’m probably in the minority there, though. Most people, to the extent that they are interested in math at all, are interested in its applications.
Today I felt unwell again. I also had a discussion with my father about approximating pi, however, that inspired me to do some research on the subject. I ended up reading the historical section of the Wikipedia article on pi. When I was in college, the consensus was that you should never read about math on Wikipedia. My understanding is that the mathematical articles have improved since then, however, and since I’m sure there are many eyes on this particular article, I decided it would serve my purpose.
As I told my father, ancient mathematicians (such as Archimedes, but also including mathematicians in India and China) calculated pi by approximating the circumference of a circle using the perimeters of polygons with increasingly many sides. According to the article, after 1500, mathematicians in both Europe and India began to use infinite series that could be shown to converge to pi, instead. In the 20th century, they continued to use infinite series, but also developed very fast iterative algorithms in which each step involved applying the same calculation to the results of the previous step. Computers following these algorithms were able to generate millions of digits of pi. Currently, the limiting factor in finding new digits of pi seems to be, not processor speed, but availability of storage for the huge numbers needed for each calculation.
Today was another bad day, though of a different flavor than the bad days last week. I read more from my calc book, but didn’t manage to do any exercises. I find my life very frustrating sometimes.
So far, this week is going better than last week. I did feel unwell for part of today, but I also finished the set of calculus exercises I had been working on and read part of the next section in my textbook. I’ve been hoping for a while to experiment with two study periods, one for calculus review and one more open-ended. That did not happen today but perhaps conditions will be favorable tomorrow or Wednesday.