6 ms·
(pi^4+pi^5)^(1/6)
- ColinWright 12y agoEquivalently: https://www.google.com/search?q=%28pi^4%2Bpi^5%29^%28-1%2F6%29*e https://www.google.com/search?q=%28pi^4%2Bpi^5%29^%28-1%2F6%... Or: https://www.google.com/search?q=e^6%2F%28pi^4%2Bpi^5%29 https://www.google.com/search?q=e^6%2F%28pi^4%2Bpi^5%29 The point being, of course, that pi^4 + pi^5 is very, very nearly e^6. See also: https://news.ycombinator.com/item?id=7892430 https://news.ycombinator.com/item?id=7892430 The idea is that there are only so many small numbers, and there are lots of ways of combining things together. The Pigeonhole Principle then says that if you stuff too many things into a small enough space, some of them will be close together. Although apparently obvious, this is more widely applicable than people generally realize. It's used, for example, in one of the proofs that that every prime of the form 4k+1 is expressible as the sum of two squares (Examples: 29=4x7+1=5^2+2^2, 181=4x45+1=10^2+9^2, 193=4x48+1=12^2+7^2). Combine this with the birthday problem/paradox[0][1], and you end up with more coincidences than you might expect. [0] https://news.ycombinator.com/item?id=1312636 https://news.ycombinator.com/item?id=1312636 [1] https://news.ycombinator.com/item?id=4753014 https://news.ycombinator.com/item?id=4753014
- 3rd3 12y agoOn the other hand there are infinitely many small numbers. Maybe one could express it more precisely using Kolmogorov complexity? http://en.wikipedia.org/wiki/Kolmogorov_complexity http://en.wikipedia.org/wiki/Kolmogorov_complexity
- 3rd3 12y agoWhy the downvotes? Do you mean natural numbers?
- imaginenore 12y agopi^4 + pi^5 - e^6 = -0.00001767345...
- lucb1e 12y agoAm I missing something on mobile? I only see the result is 2.x and then google results that don't seem relevant.
- puzzlingcaptcha 12y agohttp://en.wikipedia.org/wiki/E_%28mathematical_constant%29 http://en.wikipedia.org/wiki/E_%28mathematical_constant%29 & http://en.wikipedia.org/wiki/Mathematical_coincidence http://en.wikipedia.org/wiki/Mathematical_coincidence
- ColinWright 12y agoYes, you're missing something. The "2.xxx..." is remarkably close to the mathematical constant e. See my comment[0] for more. [0] https://news.ycombinator.com/item?id=7892425 https://news.ycombinator.com/item?id=7892425
- deleted 12y ago[deleted]
- amphi 12y agoI was surprised about how relevant the google results were actually! http://en.wikipedia.org/wiki/Mathematical_coincidence http://en.wikipedia.org/wiki/Mathematical_coincidence (pi^4+pi^5)^(1/6) is a coincidental approximation to the mathematical constant e
- 6UwUFxLt 12y agohttp://mathworld.wolfram.com/AlmostInteger.html http://mathworld.wolfram.com/AlmostInteger.html
- thegeomaster 12y agoAh. Reminds me of http://xkcd.com/217 http://xkcd.com/217 :)
- ColinWright 12y agoRelated: http://xkcd.com/1047/ http://xkcd.com/1047/
- AliAdams 12y agoI'm feeling a bit ignorant. Any math guys want to elaborate on the relevance of the result?
- serf 12y agothat's (the/a) equation for finding an approximation of e, a constant in mathematics. http://en.wikipedia.org/wiki/E_(mathematical_constant) http://en.wikipedia.org/wiki/E_(mathematical_constant)
- mijoharas 12y agoHang on, I'd reword that as "that equation happens to result in an approximation of e". The way you've put it sounds like if mathematicians are trying to figure out e they are applying this equation (I'm of course giving assuming this is unintentional and the result of commenting quickly).
- praptak 12y agoIt's a mathematical equivalent of Virgin Mary on a toast. Two expressions not related by any close theory yield almost the same result.
- ColinWright 12y agoWhat is there about my comment[0] that you feel is lacking? I'd be happy to elaborate further, but I'm not sure what you think I've missed out. [0] https://news.ycombinator.com/item?id=7892425 https://news.ycombinator.com/item?id=7892425
- deleted 12y ago[deleted]
- deleted 12y ago[deleted]
- scoot 12y agoThis is the missing link: http://mathworld.wolfram.com/e.html http://mathworld.wolfram.com/e.html (Compare the result of the expression to the value of the mathematical constant e. They coincide approximately [1]. [1] http://en.wikipedia.org/wiki/Mathematical_coincidence http://en.wikipedia.org/wiki/Mathematical_coincidence
- eps 12y agoPfft ... this is nothing compared to 1.49 ^ 2.87 yielding Pi, whereby 1.49 is a price of a 250cl bottle of vodka and 2.87 is the price of 0.5l of the same. Former is known as "chekushka" and latter - as "pol-litra", both are the staples of Soviet Union alcohol landscape with their prices remaining stable for decades and permanently etched into the heads of every Russian, drinking or not.
- Houshalter 12y ago3.141 takes up less space and is more accurate.
- raldi 12y agoI remember reading somewhere that in Russian culture the optimal number of people to drink together is customarily considered to be three, because they could each toss in a ruble and that would pay for a half-liter of vodka to share and a small snack.
- officialjunk 12y agoIf you're going to memorize 6 digits, I'd hope you would get more than 6 digits of accuracy at approximating pi. Try 355/113
- davyjones 12y agoi th root of i Not what you would expect.
- 3rd3 12y agoWhich is the same as i^(i^(i^i))).
- aylons 12y agoAnd 10! seconds is exactly 42 days. Not an approximation.
- dtech 12y agoThat's not really a fundamental coincidence, but a result of the mechanics of factorial and our time system. 10! = (10 * 9 * 8 * 5) * (4 3 2) * (7 * 6) = 3600 * 24 * 42 3600 = 1 hour in seconds 24 = 1 day in hours Waaay less a "coincidence" than strange (approximate) relationships between the fundamental constants of mathematics
- zhte415 12y ago42 is not a mathematical constance? It is the answer to the ultimate question of life, the universe, and everything. This is demonstrated by 42 being: Represented by 101010 in binary; The refraction angle of light off water in the forming of a rainbow; Light requires 10^-42 seconds to cross a proton; as well as being the result of 6*9. Or have I simply been misled?
- laxatives 12y agoThis is also highly dependent on how you define a day. A sidereal day is < 24 hours.
- ColinWright 12y agoMore interestingly, 10! seconds is exactly 6 weeks.
- RivieraKid 12y agoI wish people didn't upvote low quality submissions like this.
- ColinWright 12y agoWhile it's true that the submission itself is bare and unadorned, the topic is an interesting one. It would be nice if someone had actually taken the time to write a proper blog post and explained and expanded on the points, but that's happening here in this thread. And it's more relevant to computing, algorithms, statistics, and big data analysis than most people realize. People have a tendency to see too much in coincidences, people don't realize that there will be things that turn out to be equal, or nearly equal, more-or-less "by accident." So while you may think this is a low-quality submission, I think you are only half right, and I think it's worth having.
- okamiueru 12y agoI think the reason that it is of questionable interest and value, is the same for why there is no proper blog post explaining it. It's a coincidence found from playing with numbers. There are infinite "oh, neat, this simple expression is close to this other constant" coincidences, none of which provide any deeper understanding or appreciable use. Here is the wikipedia entry closes fits, and might actually be of interest: http://en.wikipedia.org/wiki/Mathematical_coincidence http://en.wikipedia.org/wiki/Mathematical_coincidence
- iopq 12y agoNot enough people have downvote powers and too many people have upvote powers.
- cies 12y agoI think Eurler's identity is far more "interesting": e^(i * pi) + 1 = 0 It also connects "e" and "pi"; but on top of that also "i", "1" and "0". When I first read this equation it felt like a proof of God's existence :) http://en.wikipedia.org/wiki/Euler's_identity http://en.wikipedia.org/wiki/Euler's_identity
- waynecochran 12y agoIt's true that this may be one of most startling equations, but its far from being the concoction of the arbitrary approximation in the OP. This is just the special case of Euler's more general formula e^{ix} = \cos{x} + i\sin{x}, which can easily be derived from Taylor series for e^x, sine, and cosine.
- mijoharas 12y agoOne thing I've never understood is why people write it that way round, I always think of it as: e^(i*pi) = -1 I don't understand why people switch it around all the time, I don't think it loses any elegance going that way round, I also feel it loses clarity being shuffled around because it's just one more (admittedly tiny) operation you need to do to see why it happens.
- waynecochran 12y agoBecause it involves all of the most important a constants: 0, 1, e, pi, i.
- deleted 12y ago[deleted]
- okamiueru 12y agoOf course Euler's identity is far more interesting. That's an actual mathematical insight. This is just a coincidence from playing around with irrational constants. Two entirely different things.
- zatkin 12y agoAnother coincidence with pi and phi (golden ratio): https://www.google.com/search?q=4%2Fsqrt(phi) https://www.google.com/search?q=4%2Fsqrt(phi)
- jmount 12y agoNumerical near-miss coincidences are to be expected when working with transcendental numbers. First reason is the pigeon hole observation others have made. The other reason is: this is one of the properties of transcendental numbers. Roughly: one technique used in showing a number to be transcendental is you show it is not equal to any term in sequence of algebraic numbers and yet closer to them than (without equality) than any algebraic number could be. And this is something I never liked about a lot of "math puzzles." A lot of them point out some effect as surprising or esoteric when the effect is actually a specific example of something one should expect (given a developed mathematical intuition). In the end the observation is only hermetic or exotic to those who don't know math and ends up being a barrier to getting comfortable with known results and their consequences.
- jamesrom 12y agoBrief explanation for those who aren't sure what this is about. (pi^4+pi^5)^(1/6) is an approximation of the mathematical constant e. It's a noteworthy result, but nothing more than a curiosity. At first glance it may appear to show some relationship between these two separate constants, but it doesn't. Simply put, there's infinite ways to construct an approximation of e with pi (and vice versa) and some small subset of these ways are bound to look elegant or simple.
- GregBuchholz 12y agoIs there an theorem that says that you can't compute e from pi? Maybe this is just the second term/iteration of that algorithm. I'd be interested in seeing other formulas relating pi and e.
- knz42 12y agoe^(i * pi) + 1 = 0
- nhaehnle 12y agoA related mathematical concept is that of algebraic independence. Two numbers x and y are algebraically dependent (over the rationals) if there is a rational bivariate polynomial p such that p(x,y) = 0. As far as I know, pi and e are not known to be algebraically independent, though I guess most people would expect them to be. On the other hand, it is true that there must be some (actually infinitely many) infinite series in x with rational coefficients such that plugging x = pi gives e in the limit, but in itself, that's a trivial statement (just build the coefficients of the series in such a way that you force an approximation of e). It is also not a very interesting statement, because e is the limit of a series more or less by definition (what I mean here is that plugging 1 into the series that defines the exponential function). A much more interesting relationship is Euler's identity, e^(i pi) = -1. There are many more formulas in which e and pi appear together in more or less interesting ways once you start going deeper into mathematics, but they tend to be not of the form discussed in this submission (I'm partial to Stirling's approximation as the "next" step after Euler's identity).
- yammesicka 12y agoCool! Also: https://en.wikipedia.org/wiki/Mathematical_coincidence https://en.wikipedia.org/wiki/Mathematical_coincidence