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Why is Pi² so close to 10?
- mathattack 9y agoI love math (and have recently gotten into the beauty of the Golden Ratio) but this seems remarkably non-profound. Squaring a number that’s a little higher than 3 will get you close to 10. If it were 9.999999 then it’s another story.
- NegativeLatency 9y agoEulers identity on the other hand...
- andrepd 9y agoIs almost a definition.
- gargarplex 9y agoOnce the identity was pointed out to me and I didn't immediately grok, I realized I would never be a first-class mathematician :p
- deleted 9y ago[deleted]
- wyager 9y agoThat’s not really “weird” though - it’s can be straightforwardly explained in any number of ways. You can think about it in terms of the circle group and its Lie algebra, in terms of Taylor expansions, etc. The OP is just not very interesting. Pi^2 isn’t close enough to 10 to trigger my not-a-coincidence detector. A much cooler problem of this nature is: why are musical notes generated by powers of the 12th root of 2? I remember seeing a good YouTube video about this on one of those math channels.
- deleted 9y ago[deleted]
- gowld 9y ago"explained in any number of ways", which involve complex numbers, geometry, trignometry, and calculus, is what makes it profound.
- jagthebeetle 9y agoIsn't this just by definition of equal temperament? If you want a pitch to double after 12 steps (multiplicatively), you choose 12-TET. Other musical systems exist (e.g. quarter-tone scales), and have existed. So the musical notes thing is by human fiat, which at least pi^2 isn't. Perhaps I'm missing your meaning though?
- thanatropism 9y agoBefore the current "equal temperament" system there was a "just temperament system" that basically harks back to Pythagoras. The main intervals in the C scale are very close to whole fractions of low numerator and denominator. The problem is that this doesn't work very well in every key. So the emergence of modern music came with a few tries at averaging these things out until equal temperament in the log scale arose. Some of the character of older music is actually lost because of this. But hey, now all keys work the same.
- wongarsu 9y ago>which at least pi^2 isn't pi^2 is 9.87. If that counts as close to 10, then half of all numbers are close to some integer. And the significance of 10 is entirely cultural. Nothing in math favours base 10, we use it because the right civilisations were dominant at the right time. Other civilisations counted using base 4, base 5, base 6 or base 12 (e.g. using the five fingers and a closed fist to count up to 6 for each hand).
- wyager 9y agoThe reason that powers of the 12th root of 2 are used is that these numbers are close to simple rational numbers, which the human auditory system can pick up.
- kerkeslager 9y agoI think what makes Euler's identity profound to me is that it is a tool for doing so many things. For example, if you want to calculate the log of a negative number: -1 = e^(i * pi) ln(-1) = i * pi ln(-x) = i * pi + ln(x) log_b(-x) = (i * pi + ln(x)) / ln(b) Now you have a formula for calculating the log of a negative number (-x) in any base.
- jerf 9y ago"seems remarkably non-profound." I'd point out the linked article doesn't particularly claim otherwise. It just provides a slightly deeper explanation but doesn't claim it's an explanation of anything particularly meaningful. In other news, if you'd like to see the Golden Ratio strut its stuff in a context not related to the supposed aesthetics of the ratio, this relatively recent (and relatively in-depth, for the channel) video and linked proof may be interesting to you: https://www.youtube.com/watch?v=FtNWzlfEQgY https://www.youtube.com/watch?v=FtNWzlfEQgY You'll seriously be asking "what was jerf talking about?" for like a third of the proof video before foom in comes phi, and it takes quite a bit of abuse before the end.
- deleted 9y ago[deleted]
- montalbano 9y agoThis one's a goodun though, the significance of the closeness of 196883 and 196884: https://en.wikipedia.org/wiki/Monstrous_moonshine https://en.wikipedia.org/wiki/Monstrous_moonshine https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/eli5_196884_196883_1/ https://www.reddit.com/r/explainlikeimfive/comments/2im1l1/e...
- mxwsn 9y agoWhy is e^2 so close to 7? Just to highlight how arbitrary pi and 10 are...
- ShorsHammer 9y agoThe real question should be why integers aren't factors of pi or e Everything else is quite arbitrary.
- thanatropism 9y agoBecause these are irrational numbers, they can't be expressed as a product of fractions. Integers are a special type of fractions.
- ShorsHammer 9y agoAhh I meant the base system we count in. Worded it badly.
- chasing 9y agoWhy is 3^2 so close to 9?
- singularity2001 9y agoin node.js with its bad math it's even exactly the same!
- SamReidHughes 9y agoBecause God carries a slide rule.
- scythe 9y agoIt's... not? It's 9.87. If you throw a dart at the number line, the average distance it will be from the nearest integer is 0.25. The distance from pi^2 to 10 is 0.13. That's below average, but in the second quintile. There's nothing special about 10, other than it happens to be the number of fingers on two human hands. However, deriving approximations for pi from the Basel series is sort of interesting. Except summing the Basel series requires at a bare minimum the theory of Taylor series, so it is not an accessible theorem for a primitive geometer.
- alphydan 9y agoBut it is accessible to the primitive geometer... https://www.youtube.com/watch?v=d-o3eB9sfls https://www.youtube.com/watch?v=d-o3eB9sfls Or for more details, http://www.math.chalmers.se/~wastlund/Cosmic.pdf http://www.math.chalmers.se/~wastlund/Cosmic.pdf
- deleted 9y ago[deleted]
- hazeii 9y agoIt's not 9.81, it's more like 9.87 (you got the distance correct at 0.13 so I guess a typo?).
- jandrese 9y agoOr as my dad used to say: Close only counts in horseshoes and hand grenades.
- 9y ago
- soVeryTired 9y agoI think this would get much better responses if the title was something like "You can use the zeta function to get an upper bound for Pi". Here's a more interesting result with the same flavour: https://math.stackexchange.com/questions/4544/why-is-e-pi-sqrt163-almost-an-integer https://math.stackexchange.com/questions/4544/why-is-e-pi-sq...
- pps43 9y agoWhy is Pi*10^7 seconds so close to one year?
- singularity2001 9y agobecause a year has ~ ½×1×2×3×4×5×τ days - 11 and π:=½τ
- lostmsu 9y agoWhy Pi^2 is so close to g, gravitational acceleration on the Earths surface?
- pleasecalllater 9y agoJust a coincidence. The g depends on the mass of the planet, while pi is rather universal for a perfect math circle. In reality the g constant is not a constant, the value depends on the place of the Earth.
- jovial_cavalier 9y agoThat's his point. It's actually closer to 9.81 than it is to 10.
- bunderbunder 9y agoYou must not be American. They're not that close in the USA.
- fenwick67 9y agoI'm an American but I have never used g in imperial units, I've only ever known it as -9.8m/s^2.
- bunderbunder 9y agoI love your high school so much better than my high school.
- evanb 9y ago...if you measure in a particular set of units. It's not so close to g = 21 mph/s, for example.
- davrosthedalek 9y agoWhat are you talking about, g is roughly 7e10 furlong/fortnight^2
- dmitrybrant 9y agoA much, much more interesting observation (attributed to Ramanujan) is that e^(pi * sqrt(163)) is extremely close to an integer. In fact it equals 262537412640768743.99999999999925... The most remarkable thing is that this is actually not a coincidence, but a consequence of complex multiplication: https://en.wikipedia.org/wiki/Complex_multiplication#Sample_consequence https://en.wikipedia.org/wiki/Complex_multiplication#Sample_...
- jandrese 9y agoIs the square root of 163 supposed to be a number I recognize? It looks awfully magical in that equation.
- soVeryTired 9y agoNope. It's one of a few very special numbers though. Every integer has a unique prime factorisation: that's something kids learn in primary school. The idea of factorisation goes a lot deeper though. Define the gaussian integers as integers of the form a + ib, where a and b are intgers. Unique factorisation holds in the Gaussian integers too, but some of the numbers we recognise as prime are no longer prime. For example, 2 = (1 + i)(1 – i). It turns out unique factorisation is a rare property of number systems. Consider the set of numbers whose ‘integers’ are of the form a + b sqrt(-5), with a and b standard integers. In this system, 6 = 2x3 = (1 + sqrt(-5)) x (1 - sqrt(-5)). But none of those factors can be factorised further, so 6 has two distinct factorisations. It turns out there are only nine positive numbers n so that unique factorisation holds in the system a + b*sqrt(-n). 163 is the largest! https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem
- ChuckMcM 9y agoNo, 163 is a number substituted into Euler's equation for the value of -1. When you use the square root of -1 in Euler's equation you get back -1. As far as I know there is no other square root of an integer that gives you an integer result. 163 comes close.
- soVeryTired 9y ago
- bdamm 9y agoWhile the mathematics here is completely inane, I really enjoyed the presentation whereby the side concepts that the author is building on top of become pop-up margin notes. Very nice.
- PokemonNoGo 9y agoThats a really neat inline pdf viewer! Rendered on the backend with pdf.js or something?
- jwilk 9y agoIt's pdf2htmlEX: https://coolwanglu.github.io/pdf2htmlEX/ https://coolwanglu.github.io/pdf2htmlEX/
- PokemonNoGo 9y agoSad to see that project isn't updated any more.
- deleted 9y ago[deleted]
- n4r9 9y agoProfound or not, these kinds of (near) identities are a lot of fun to explore. Here are some great lists: http://mathworld.wolfram.com/AlmostInteger.html http://mathworld.wolfram.com/AlmostInteger.html https://en.wikipedia.org/wiki/Mathematical_coincidence https://en.wikipedia.org/wiki/Mathematical_coincidence Some favourites: - 5 phi e / 7 pi = 1 (to 5dp) - pi^4 + pi^5 = e^6 (to 4dp) - e^pi - pi = 20 (to 3dp) Also, remember that many very profound discoveries come about from noticing fun little oddities.
- mnl 9y agoMore interestingly, why is 355/113 so close to Pi?
- dmurray 9y agoThere's a nice explanation for this one based on expressing Pi as an infinite continued fraction.
- gowld 9y agoObXkcd: https://xkcd.com/1047/ https://xkcd.com/1047/ and https://mrob.com/pub/ries/index.html https://mrob.com/pub/ries/index.html
- meuk 9y agoRelated, and slightly more interesting than the linked article (I think), is you can get similar 'almost equal to an integer'-results by using recursive relations. Take, for example, a Fibonacci-like sequence, defined by f(k + 2) = f(k) + f(k + 1) and f(0) = 2, f(1) = 1. Then you get the solution f(n) = a^n + b^n, where a = (1 - sqrt(5))/2 and b = (1 + sqrt(5))/2. So |a| < 1 and |b| > 1. If the recursive relation has integer coefficients, then a^n + b^n will be an integer. For large n, |a^n| will get very small, so b^n will be very close to an integer. For example ((1 + sqrt(5)) / 2) ^ 20 = 15126.9999 It should be possible to find a recursive relation where |a| is smaller, so that b^n is closer to an integer, but hey, I'm supposed to be working now. Edit: Coincidentally, this result is also presented in the math stackexchange post that soVeryTired linked to (actually, this is not a coincidence, since this is the most basic recursive relation, and has easily memorable forms for a and b).
- jovial_cavalier 9y agoIt isn't.
- madcaptenor 9y agoWhy is pi^3 so close to 31? Why is pi^2 + pi so close to 13?
- brador 9y agoWhat does this symbol ζ mean here? Like ζ(2)...what is that?
- detaro 9y agohttps://en.wikipedia.org/wiki/Riemann_zeta_function https://en.wikipedia.org/wiki/Riemann_zeta_function
- deleted 9y ago[deleted]
- Someone 9y ago”and the error is reasonably small because […], a sum whose first term is 1/60 and whose further terms are much smaller yet” That’s sloppy for a math paper. “Much smaller yet” doesn’t even guarantee that the series converges.
- IshKebab 9y agoErm, this isn't a maths paper.