So today’s calculus training went pretty well! By pretty, I mean very.
Anyway, let’s talk about some cool af stuff: namely, what we learned, enjoyed, and waded through. We started by taking some practice derivatives, and then migrated into anti-differentiation. We drew some curves and connected the lines (hah!) and realized that areas underneath curves could be found with anti-derivatives, and then we went into some useful differentiation techniques.
During a bit of an afterparty, we listened to “New Math” by Tom Lehrer, “A finite simple group of order two” (we turned this one off after realizing most didn’t get the jokes), and finally explained fractions using two numbers in a non-zero commutative ring.
Anyway… onto some cool stuff that is related to integrals.
So there’s a guy named Richard Feynman (you might’ve heard of him!), who was a famous physicist during the 20th century. Renowned for his lockpicking skills and ability to do mental math really, really quickly, Feynman also made some groundbreaking contributions to quantum mechanics.
There’s another guy, whose corpse (I’m sad to say), is much, much older than Feynman’s. Pierre de Fermat was a Mr. Stark-like lawyer (he did math!) during the 17th century. Back when calculus was still a (very) big deal, he wrote in the margins of a book the fact that for whole numbers , the equation: does not have a solution where a, b, and c are counting numbers. However, the proof was too long to fit into the margins (or so he said), and so he didn’t include one. Said proof, after a many-century obssession with the problem, was finally found in 1995. Unfortunately, it was 110 pages long, and it was far beyond any math that Fermat could’ve done, margins or no, so it looks like he never did have a proof for his theorem.
Anyway, so at some point Feynman decided to play around with Fermat’s last theorem. (From this point on I’m basically summarizing this blogpost. He didn’t come up with a perfect proof, but he did come up with a pretty cool way to check if it was true or not. Here’s what he did:
NOTE: I use “perfect power” to mean a number x^n where both x, n, and the result are positive integers (counting numbers).
- Found the probability that a given number N is a perfect Nth power. This turned out to be roughly: . (Again, the exact derivation of this is in the paper. It involves some pretty basic probability and some taylor series, but shouldn’t be too hard to follow.)
- Plugged in for N. This gave him the odds that the sum of two given “perfect powers” (power of an integer) has an Nth root that is an integer. His new formula:
- Summed up all the probabilities from some two starting numbers, to infinity, to get the odds that a given “nth” root has two counting numbers x and y that add up to a perfect power. To do this, he approximated said sum with two integrals, and did some substitution inside of said integrals. NOTE: To do a “2d integral”, you basically first treat one variable as a constant (u), and integrate with respect to the other one (v), and then do the opposite (integrate with respect to u and treat v like a constant).
- After some mathemagic, he gets the following fact: The odds of there being integer solutions to is: , where is the starting value after which you want to know the solution.
- You’re done! Evaluate the integral for some and some , and you get a really nifty graph
Basically, what this graph proves is that Fermat’s theorem is probably true – I mean, come on, we’re talking about a probability that’s less than a percent for a given
The moral of the story: Just because probability, Fermat’s theorem, and integrals seem completely unrelated, doesn’t mean they aren’t. Math isn’t a loose collection of randomly scattered subjects, it’s a giant amalgamous katamari ball of ideas. Things are connected in the most unexpected ways - and that’s what’s so amazing about it.