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All right, here's a fun one. It usually comes as part of a story. The story as told is mostly true, though a few details have been a little fudged by the winds of history. It goes like this: when the young Carl Gauss was a small child in school, his teacher wished to kill some time by making the students practice their addition by adding the integers from 1 to 100. Gauss worked for about 10 seconds and turned in the correct answer. His method was exceptionally clever. By doing the problem twice he made the calculation much easier.

Last time on Sunday Function we talked about two types of symmetries that a real function might have: odd and even symmetry under reflection about the y-axis. Much more than I expected even as an undergraduate student, these types of symmetries turn out to be of amazingly fundamental importance in fundamental physics. One of these fundamental symmetry results in the Pauli exclusion principle, which says no two electrons (or any Fermions for that matter) can be in the same quantum state.

If you look at an incandescent light in a spectroscope, you'll see a broad and continuous range of light emitted over a large portion of the visible spectrum. This combination of colors looks white to us. At the other extreme, laser light generally consists of just a tiny slice of the frequency spectrum, and so it appears highly monochromatic. The laser light is not literally mathematically all of the exact same frequency, instead it's distributed closely about the laser frequency according to some probability distribution that can vary depending on the specifics of the laser.

The vast majority of the functions we've talked about over all these Sundays have been ones that are expressible as a relationship between two numbers x and y. Sometimes the relationship is simple, sometimes it's fairly complicated. Mostly though, we just work according to the function as though it were a little machine. Take a number x, do to it what the function tells you to do, and the result is the number y. This is a function where the relationships between x and y are given in a very clear and explicit form.

Before her career took an unfortunate wrong turn, a young and talented Lindsay Lohan gave us a charming and popular comedy called Mean Girls. Time has been good to the careers of some of the others involved, Tiny Fey and Rachel McAdams perhaps most notably. But the film did something that very other movies have ever done - given us a climax explicitly involving a function of a real variable: