Weird Constants in Math Problems | Hacker News Reader

Weird Constants in Math Problems

blog.plover.com

96 pointsweinzierl28 comments

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Seems the first constant is wrong

>>> 1/(2 * (1 - math.exp(-2)))

0.5782588213748329

Instead it should be

>>> 1/2 * (1 - math.exp(-2))

0.43233235838169365

Which handily appears in OEIS (a great resource for random constants and sequences) as http://oeis.org/A247847

Michael Lugo (author of the original post that Mark was responding to) here. I had written $1/2 (1-e^{-2})$ in my post, meaning the 0.4323 constant, but with the usual rules of precedence you do in fact get 0.5782; he just propagated my error.

I fixed it on my side also. Thanks to both of you for the correction.

This is nice. I think OEIS and in general libraries of constants will be of great help in tackling problems with no analytical solutions.

Actually I do a lot of integrals this way – symbolically but checking whether the transformation was legit evaluating definite integral on few random intervals. For me is very helpful and I believe it can be extended to problems beyond integrals.

> It's easy to use Monte Carlo methods and find that when n is large, the average cluster size is approximately 2.15013

It never occurred to me before, but running simulations and looking up the result in a dictionary of constants might be a helpful learning approach for people who are more adept at going from practice to theory.

The problem with this approach is that to get enough decimal places to be able to look things up in a dictionary of constants can require a very long simulation.

I solve a lot of problems this way, thanks to OEIS.

How many fromblaz of size N are there? lets count them for N=1 ... N=8 or whatever is tractable to count using naive methods then hope a useful result shows up in OEIS.

OEIS will also return equations for simple linear integer expressions without an OEIS entry.

Unfortunately, the tools for single constants are significantly less good than OEIS.

https://www.wolframalpha.com/input/?i=2.15013

It's there as one of "Possible closed forms".

it's there even for 0.4323 but further down, you need to request more closed forms.

One I found as a kid was 0.73908513321. It appears when you just press the Cos button on a calculator repeatedly (for any starting value). I can't find it used anywhere else.

I suppose that’s the limit value of repeatedly taking the cosine of any value and then the cosine of that iterating up to infinity.

Sounds easy enough to check, and easy enough to prove, if the hunch proves right.

Its name is the "Dottie Number":

[0] https://en.wikipedia.org/wiki/Dottie_number

[1] https://mathworld.wolfram.com/DottieNumber.html

You can see that your constant is the solution to x=cos(x) (if the equality is true, you can replace the x on the right side with cos(x), and repeat this process).

The solution apparently has a name, Dottie constant [1].

[1] https://mathworld.wolfram.com/DottieNumber.html

Beautifully, iterating any continuous function will yield a fixed point of the function if it converges.

https://en.wikipedia.org/wiki/Fixed-point_iteration

And in case it neither converges to a single value nor diverges to infinity, a whole new worldof wondrous math emerges [1][2][3]

[1] https://en.wikipedia.org/wiki/Logistic_map

[2] https://en.wikipedia.org/wiki/Mandelbrot_set

[3] https://en.wikipedia.org/wiki/Attractor#Strange_attractor

Interestingly, I've found this kinda sorta occasionally works on functionals of a differential equation too if you start with a good enough guess. Someone has probably made this concept and the necessary/sufficient conditions rigorous but its getting into territory that is a good bit too advanced for me to follow.

As I recall from my fiddling, you are probably most likely to end up with a series solution of sorts, so it's a good idea to guess with polynomials or exponentials so you end up with component functions that form a basis for analytic functions.

It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikip... article on fixed point iteration) gives lip service to everything working fine in arbitrary metric spaces. It's easy to define a useful metric between functions (e.g. via integrating their difference, ignoring edge cases like occasional pointwise differences since a more careful treatment can give a presentation where those don't matter for the problem at hand), so the same kinds of theorems that work with cos(x) on the reals also work with differential operators on function spaces.

I found that one too. I don't think it arises in any "real" mathematics but it's a nice example.

Also, if you alternate hitting the reciprocal key and adding 1 you get the golden ratio 1.618033... pretty quickly.

Yeah that’s the limit of the repeated map of the cosine function for any initial value; it can be shown to be equal to x such that cos(x) – x = 0

Too bad that Plouffe's inverter can't seem to stay reliably online, except in intermittently available cut down forms.

The database behind it grew huge, making it expensive to operate, and it seems any copy of it online goes down whenever someone graduates.

I find series even more fascinating than unexpected constants. Series also show up in the weirdest places and it's often very non-trivial to explain (or not yet explained at all) why the some series shows up in two different areas/applications. Check out the comments in OEIS[1] entries, it's fascinating.

[1] https://oeis.org

I recall doing a numerical approximation to something, and it seemed to converge to about 2.5

Enough for me to, in my overconfidence, assume the constant was 2.5. Later, when I found an exact derivation, it turned out the actual constant was pi^2/4 or 2.46740110027. Which was apparently close enough to fool me.

My PhD advisor wrote a paper titled "when is 0.999... equal to 1": https://www.math.upenn.edu/~pemantle/papers/nearly.pdf

(It wasn't, in the case he was interested in - he had a sum that he suspected might be 1, from numerical evidence, but that actually turned out to be a combination of zeta values equal to 0.999222...)

A while back I looked at a problem where Monte Carlo simulation said that the answer was probably 4.

But analysis showed that the actual answer was 9304682830147 ÷ 2329089562800 = 3.994987….

https://blog.plover.com/math/breaking-pills.html

Damn, I was expecting sqrt(2pi) which is 2.5066.

I think these kinds of constants are very normal (and very cool) in these kinds of problems. "e" shows up a lot. For example, the probability that a given bin is empty is the probability that every ball misses it, or (1 - 1/n)^n --> 1/e.

Another interesting way to formulate it is as a differential equation in the number of empty buckets as a function of time. Given k empty buckets, the probability of picking an empty bucket is k/n. For large values of n and k, we can pretend it's a continuous and not-particularly-stochastic differential equation dk/dt = -k/n, with k(0) = n.

That's classic exponential decay, the solution being k = ne^(-t/n). Thus k(n) = ne^(-1).

One bit of the analysis is left out (if you follow the link to the Math Stack Exchange post) -- the probability of a bin being empty is not independent of the probability of other bins being empty. From the perspective that the number of groupings Y(x) is contingent on the total number of bins filled, X, you might say they're assuming E[Y(X)] = E[Y(E[X])].

But as n -> infinity, the effect of this becomes negligible.

A probability problem has Euler's constant in its answer? And numeric approximations are close to that number?

Who cares? That's how probability works. Nothing weird.

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