With my ability to announce my posts blocked by the Chinese government, now’s my chance to indulge in a selfish post without destroying my modesty, since fewer people will find out about it.
A few months ago I had the most amazing dream. It even impressed me. Somehow I dreamt about this:
It’s called a continued fraction. I know I’d seen the fraction before. I don’t remember if anyone showed me how to solve it. I know I hadn’t seen anything about it in years, maybe over a decade.
The mildly rock star thing in my dream was that I realized how to solve it. I’m impressed I remembered it at all. I have no idea how my mind came to be thinking about it.
Realizing the solution is a cool trick. If the whole thing equals x, then the part under the large horizontal line is also x. Once you see that, you get x = 1 + 1/x, which you can solve for x.
The above was only mildly rock star in comparison to the next part, which was to recognize that the trick only works if you prove existence and uniqueness. That part was totally rock star and way more impressive. Mathematicians think that way, not physicists.
Anyway, that’s as far as I got in the dream — to recognize the fraction could be solved. I didn’t completely solve it or prove uniqueness and existence. I mean, you can tell by looking at it it would be unique and exist, so I was pretty sure of it in the dream.
Things kept getting cooler when I solved x = 1 + 1/x. You get a quadratic equation. One root is (1 + sqrt(5))/2. The number looked familiar, like the golden ratio. Wikipedia not only showed me it was, but also showed that the continued fraction gave it.
I appreciate your indulging my indulgence.
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