A Curious Integral
golem.ph.utexas.edu
golem.ph.utexas.edu
This author obviously knows this at a deeper level, especially the academic literature, where our group is more raw about it. So fun. Thanks to the author and the poster for a pleasant Friday afternoon read.
Heh heh! Glad am not the only one. I used to think this was purely an Asian habit, borne out of excessive focus on math problems during the 11-12 grades in order to pass the grueling entrance exams. I mostly share pdfs of math problems with former classmates thru linkedin.
Original post:
> three 6th graders I work with ended up acing the AMC 10, which is a contest that American kids take in the 10th grade. One of them even made the AIME cutoff,
This is 100% impossible. How could you "ace" a contest and not qualify for the lext level? No one did better than "acing" and there is no lottery.
Do you mean "100" on the AMC 10, not "acing" ? That's impressive and plausible, but "acing" is 150, which is achieved by only about 10 people in the world each year, and takes years of intense study, and all of them qualify for the AIME.
Also, AIME is currently top 2.5% of AMC 10, not top 1%.
Score reports at: http://amc-reg.maa.org/reports/GeneralReports.aspx
Sorry, I meant acing in the "scored way above what I'd expect 6th graders to do" sense. One of them made the AIME cutoff, two missed that cutoff but quite narrowly.
> "acing" is 150, which is achieved by only about 10 people in the world each year
Agreed, that is pretty impressive.
> Maybe host a math contest forum for working adults where we can work on interesting integrals & suchlike :)
I'm sure you must know about https://artofproblemsolving.com/community They have a huge collection of problems, and sometimes solutions.
I'm somewhat curious about the logistics of the courses. How many students are in your classes? Do you run one-on-ones online? And are they really one-on-one, i.e. only one student at a time? Or is it more like office-hours?
Also how much time does it take you weekly to prepare the course materials and to check the homeworks?
It would be great idea to see one such book if it doesn't exist.
What strikes me is the reminder that it is never possible to "prove" something by pointing out that it is true for all known cases. (See also Black Swan events). In Greg Egan's example, if you stopped testing at 10^43 iterations, you would be very tempted to conclude that the identity holds for all n, for example.
First prove that something is true for some n, usually n = 1.
Then prove that if it's true for n it's for n + 1, too.
Boom. It's true for all n.
But the second step is sometimes very hard or even perhaps impossible.
Or you might be able to prove that it is true for n + 2 but not be able to prove it is true for n + 1. Right?
Not exactly true. If you can prove that the cases you checked amount to all the cases in the theorem statement, you're done. The Four-color theorem is a well-known example that was originally proved in this way (accompanied by a controversy about whether that's a 'real' proof, which is arguably pretty much settled at this point). Induction would arguably be another way (that only requires checking a single case).
Somewhat similarly the ±23 chars of
"∫ 0 ∞ cos(2x)∏ n 1 ∞ cos(xn)dx"
giving 43 digits of another number is not _that_ surprising and in the realm of what you can expect with some likelihood.
I expect that one could theoretically find some double integral with less then 25 signs which approximates e^π or ANY number to 50 digits.
Now if you find some 20 char expression which approximates another 20 char expression up to 100000 digits and THEN suddenly takes a different turn, that would be really curious. Like those properties of natural numbers which are true for some orders of magnitude before someone found a counter example.
Jaded nonmathematicians told us it’s just a coincidence, so what is there to explain?
https://math.stackexchange.com/questions/1945026/an-amazing-...
lim n->\infty (1 + \frac{1}{n})^ n
is even easier to remember
(Nowadays, it’s easy to construct a transcendental number because it’s easy to construct a noncomputable one; constructing a transcendental computable number still requires additional ideas such as those bounds—I don’t really know of a simple way to do it.)
Not necessarily. For example:
0
10^-100000„However, to obtain an entirely closed form, it is necessary to do perform some analytic wizardry (see Watson 1939 for details). The fact that a closed form exists at all for this integral is therefore rather amazing.“
Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, by George Boros https://www.amazon.com/dp/0521796369/
Inside Interesting Integrals, by Paul J. Nahin https://www.amazon.com/Inside-Interesting-Integrals-Substitu...
Another possible lesson from these integrals: if someone has deliberately prepared the problem you're studying, and made it misleading on purpose, then solving the problem may be a different game from studying a problem that arose for some other reason.
Basically, the tldr version is that product under the integral can be considered a sum, of sorts, in the Fourier domain (convolution <-> product and products turn into sums under some transformation of exponentiation) and when the coefficients of that sum cross a constant, then the original integral becomes less than pi.
That is, when $\sum_{i=0}^n \frac{1}{2 i + 1} >= 1$, that's the transition point. 15 in the denominator is where that sum is greater than one.
Awesome stuff.
[1] http://www.paulgraham.com/nerds.html. Quote: "I know a lot of people who were nerds in school, and they all tell the same story: there is a strong correlation between being smart and being a nerd, and an even stronger inverse correlation between being a nerd and being popular. Being smart seems to make you unpopular."
And even if math/logic is granted its perfect world, it's never even self-complete.
Welcome back to the world of practical wisdom, where the rest of us live and work :)
It contained 41 letters and other miscellaneous math symbols.
Therefore, I find it unsurprising that it generates a specific constant having a magnitude of 10^-43.
It's a low complexity integral whose value matches pi/8 to 41 decimal places. Of course it's not a coincidence.