If you do a google search for how to derive the facts that and , most of the derivations you’ll find rely on knowing something like the double angle formula and go through a direct, non-trivial evaluation of . It’s actually possible to compute these derivatives this without really remembering any specific facts about trigonometry […]
You can often get the gist of a mathematical subject via an informal explanation involving infinitesimals. But I often find that questions arise from that informal explanation that I’d like resolved, but I don’t want to jump all the way to the full definitions. Without a rigorous basis for reasoning about infinitesimals, it can be […]
Until this morning, I didn’t really understand how it was possible for a sailboat to sail into the wind: popular descriptions like Wikipedia’s talk about keels and lift and the Bernoulli effect and so forth, but this feels like a leap beyond my understanding: I wanted an account of how it is even possible in […]
A while ago, I wrote a blog post called Making Money Disappear Through Infinite Iteration, and I just put out a video version of this post on youtube. Note: it’s very rough and unpolished, but I hope to make more videos and get better at their production as time goes on. I’d been interested in […]
CGP Grey is a youtuber with a variety of interesting videos, often about the quirks of geography and political boundaries. In this video, he asks the question “How many continents are there?”, discusses a variety of subtleties in the notion of “continent”, and concludes that it is not well-defined enough to provide an answer. Let’s […]
Actually, thermodynamics is hard and I don’t understand it. But even without totally understanding thermodynamics, it turns out its possible to do a surprising number of useful calculations with just a couple of simple rules about entropy. The setup is as follows: Imagine that there is some set of states of the world, called the […]
I’m not a physicist, and I’d always supposed that, while the Earth has a significant gravitational pull because it’s so massive, the gravitational pull between everyday objects must be completely undetectable, or maybe only detectable with modern laboratory equipment. But I only thought that because I never bothered to actually plug in any numbers. Using […]
