Thursday, March 02, 2023

Fun little oddity

I was doing a literature search and my eyes started glazing over. I took a break by skimming through a journal I'd never heard of before: Elemente der Mathematik "publishes survey articles and short notes about important developments in the field of mathematics; stimulating shorter communications that tackle more specialized questions; and papers that report on the latest advances in mathematics and applications in other disciplines. The journal does not focus on basic research." In other words, suited for both more applied and more random stuff. The discussion section is in German, unfortunately.

"The irrationality measure of $\pi$ as seen through the eyes of $\cos(n)$. A student asked: "What's the limit of $\cos(n)^n$?" Since it's always less than 1 in absolute value, as n becomes large, the result should go to zero. Except it doesn't. In fact, it "oscillates", because larger and larger fractions come closer and closer to approximating $\pi$. They go on to discuss qualitative irrationality, but that's more for specialists.

I suppose one conclusion to draw from this is: pay attention to student questions. Sometimes there's something weird hiding that nobody noticed before.

Drat. The article is too recent to qualify for the open access. I could read the paper in the library, but not from home. In order to cut down on the number of points, I didn't draw anything with abs() less than .01--pretend there's a line across the middle. Done with python, and I'm trusting that their $\cos$ function handles large numbers well.

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