The Simpsons and continuous compounding

It appears the Simpsons have a mortgage that has 37% interest compounded every minute.

For the record, if the interest is compounded n times per year, then their interest wud be (1+0.37/n)ⁿ-1 compounded annually. If n = 12 (monthly), this is 43.97% per year; if n = 525,600 (every minute), this is 44.7734426% annually; compare 44.7734615% = exp(0.37)-1 for continuous compounding. In other words, compounding every minute might as well be continuous; the difference is one cent per $53,000 or so, per year.

The difference between every-minute and continuous compounding, at an interest rate of r, is the difference

exp(r) - 1 - (1+r/n)ⁿ

where n = 525,600; this is asymptotically r²/(2n). (This actually isn't a great approximation here; the next few terms of the series are reasonably large.)

End quintile disparity!

A fictional "presidential candidate" on The Simpsons said something about how the top fifth of Americans consume sixty percent of the resources and the bottom two fifths consume only one eighth of the resources, leading to his slogan "end quintile disparity".

You just don't see that many statistics jokes.

The math of Futurama

Dr. Sarah's Futurama Math, from Sarah Greenwald. Apparently a new Futurama DVD was just recently released, if you care about that sort of thing. (Personally, I like the show but not enough to go out of my way to watch it.) The DVD includes a lecture on the math of Futurama. I didn't know that a lot of the writers of the show had serious mathematical training, but it doesn't surprise me at all.

Also, simpsonsmath.com from Greenwald and Andrew Nestler. I like this one more, because I can get The Simpsons but not Futurama on my dirt-cheap cable package, so the Simpsons references are more current to me. I linked to this one a long time ago, but you probably weren't reading this blog then, because at the time I had maybe one percent of the readers I have now.

(In a not-all-that-strange coincidence, I'm reading William Poundstone's biography of Carl Sagan. Sagan was born in 1934, and often cited the "real" Futurama, a pavillion at the 1939 New York World's Fair, as one of the first things that pushed him towards being a scientist.)

The Simpsons use decimal numbers

In The Simpsons, people have four fingers on each hand. Eight fingers in total. Therefore, shouldn't they use numbers in base 8?

(The reason that they have four fingers is the same reason that most animated characters have four fingers -- it's easier to draw. In at least one episode, God appears; God has five fingers.)

This occurred to me while watching the episode The Canine Mutiny, in which "After using his credit card to buy another dog, Bart must choose between his new wonder-pooch and the bumbling but loyal Santa's Little Helper." Bart gets the credit card in the name of his old dog, Santa's Little Helper; to order the new dog from a catalog, he has to dial an 800 number. He says "I don't think our phone goes up to 800", which got me thinking about what kind of numbers they use in Simpsons-world.

simpsonsmath.com, by Sarah Greenwald and Andrew Nestler, has a list of mathematical references on the Simpsons. This is not one of them.

It's actually possible to prove that the Simpsons universe has numbers in base 10. The baseball attendance figures in Marge and Homer turn a Couple Play are 8191, 8128, and 8208. The use of 9 indicates that we're in base at least 10. If we assume this is supposed to be a mathematical joke, 8191 and 8128 are immediately recognizable as 2¹³-1 (Mersenne prime) and (2⁷-1)2⁶ (a perfect number.) 8208 is also 2¹³ + 2⁴, but more importantly it's the sum of the fourth powers of its digits. This would only be true in base 10. (Incidentally, most mathematicians regard properties of numbers that are based on their digits as not worthy of investigation, because they are basically accidents of the fact that we have ten fingers.)