If you want to get a flavor of the article without being too much of a math person, you can watch the 3Blue1Brown video linked in the article, which captures a similar surprise that for certain values a formula maps exactly to a "round" quantity, then starts to diverge after a while.
My ignorant intuition is since it's the area between sin(x)/x and the x axis, but as you get further into infinity the 1/x still keeps getting smaller slowly but sin(x) always is the same magnitude, the integral gets slightly further away the closer you get to infinity and that slight difference adds up?
I wondered if the Greg Egan he named provided a comment on the integral was THE Greg Egan (author of Permutation City) and following the comment indeed he was!
IIRC, THE Greg Egan has a (genuinely excellent, peer-reviewed) maths paper with John Baez, and a number of smaller or unpublished contributions to professional maths.
Yes, Greg is an active math guy. You can often find comments from Greg, John Baez, and Scott on each other's blogs. They are some of my favorite bloggers/writers!
I have been fascinated by these integrals for a long time and am happy to see them getting more attention. 3Blue1Brown recently made a video on the topic: https://www.youtube.com/watch?v=851U557j6HE
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.
> 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.
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).
It seems like an exercise in mathematical self-absorption, but it's really a fundamental question for science: assuming you have equations and even a model of the world that is really, really close, is that any kind of proof?
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's always heartwarming to see people passionately geek out, seeking to advance our civilization. It's also curious that a land of geeks like the US would have a culture of looking down upon geeks, to the point that Paul Graham would write essays like Why Nerds Are Unpopular[1]. I couldn't even understand that essay when reading it for the first time, as it was a novel concept in my country that people didn't appreciate hardworking and passion in hard subjects. It's even stranger that Americans thought it was a great virtual to toil in sports, like shooting hoops thousands of times a day, but it was a sin to toil in STEM, like solving maths problems for fun. But well, it's a topic for another day.
[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."
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.
It reminds me of those 'surprising' rational approximations of π where xyz/abc≈π with 7 digits precision and abcd/efgh≈π with 9 digits precision. It's not that surprising when you take into account that abcd/efgh does not save you many digits compared to just writing π out.
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.
This one isn't so interesting, as it's super easy to generate `e` in lots of different ways (such as the limit the expression is approximating). `e` is a very low-kolmogorov-complexity constant. The error term in the integral, on the other hand, has no apparent reason to be.
The kolmogorov complexity of the provided integral is vastly lower than the >140 bits needed to naively represent the error term. Something else is going on here.
Having good “small” rational approximations is actually a characterization of transcendental numbers: there are upper bounds for how good rational approximations to an algebraic number can be as the maximum allowed denominator grows, so by proving that this bound is violated for your number you can prove it’s not algebraic; that’s how people initially went about constructing transcendental numbers and—later—proving π and e were such.
(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.)
Proving it’s transcendental still requires the bound I was talking about, which to my admittedly very ignorant eyes looks like it doesn’t lead to anything interesting outside number theory. This is unlike the undecidability of halting needed to construct a noncomputable number, or Cantor’s theorem used to show they exist; both are also quite simple conceptually, containing a grand total of one idea each and basically no computations. (The Lindemann–Weierstrass theorem you’d usually use for proving e or π is transcendental at least looks like interesting algebra.)
> 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.
I have a social group of folks that work out hard integrals via social media -- friends and friends of friends over the years in quant jobs and graduate school where we work on fun things.
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.
> I have a social group of folks that work out hard integrals via social media
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.
Well, I live in the midwest, a small republican town in flyover country, with no fancypants private schools nearby :) The kids here go to public school. I thought I can help them out. So I volunteered my "math services" to the public school teachers & set up a side project website, weekly 1-1 math sessions. Long story short - this year 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, which is the top 1% i.e. top 3000 of the 300,000 contestants. So I'm thinking of increasing my efforts in that direction. Maybe host a math contest forum for working adults where we can work on interesting integrals & suchlike :) Lemme know what you think, perhaps we can collab.
EDIT: apparently the dictionary supports "acing" to mean "doing very well" even if there are clearly higher measures of performance. Still, "acing" generally means "the highest rank" which would at least be the Distinguished Honor Roll (approx 125-130) if you don't want to go all the way to exact 100% score.
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%.
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
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?
thanks! yeah aops is the mother ship, i am just plying a little ferry. my problem sets come from irodov, loney, hall and knight. These are 100+ year old classics, can get them off libgen. during covid i did zoom. nowadays the students just drop in on weekends. i don’t really prepare, most of this is high school math and thankfully my math muscles haven’t atrophied. honestly, i like the midwest. yes people are poor and the public school system is grossly underfunded. but the kids want to learn and i feel i am making a difference teaching them competition math. there’s no money in it, that’s forsure. but it makes me happy.
Do you people have any more open social channels? I used to be active a bit on CrossValidated (my favorite math problems are in probability and stats), but something more social could be nice.
If I may ask, do you have any book which has solutions of some of these hard integrals so that one can get a knack of ways to solve them than just start from first principles like by-parts ? Nntaleb and his twitter followers also often discuss solutions by methods which were not taught in Calculus 2.
It would be great idea to see one such book if it doesn't exist.
I only skimmed the article and skipped to the 3Brown1Blue [0] video which I found very enlightening.
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.
That video is excellent and I wish they existed when I studied mathematics years ago as I may have kept going! In particular he gives the broad overview first so you understand it at a squint level and then you feel motivated to learn more. Vs. turning up at “Convolutions 101” lecture and having a lemma written on the board. This sort of math is my favourite advanced mathematics. I never enjoyed stats and I never enjoyed anything more abstract I could not
visualize.
„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.“
If these two integrals differ, then you can subtract one from the other, and plot an error function. You would need a bit more storage than a difference engine, and a lot more accuracy than current computational methods.
64 comments
[ 2.1 ms ] story [ 2312 ms ] threadhttps://www.youtube.com/watch?v=851U557j6HE
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).
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 :)
[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."
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.
https://meta.wikimedia.org/wiki/Cunningham%27s_Law
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.
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:
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.
It's a low complexity integral whose value matches pi/8 to 41 decimal places. Of course it's not a coincidence.
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.
https://en.wikipedia.org/wiki/Kolmogorov_complexity
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.
[0] https://youtu.be/851U557j6HE
„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...