7 ms·
Maybe powers of π don't have unexpectedly good approximations?
- rich_sasha 4y agoThis is stretching my long gone education, but pi is a transcendental number, meaning it us not a root of any integer-coefficient polynomial, and these tend to not have good rational approximations
- zeroonetwothree 4y agoActually the irrational number that is hardest to approximate is phi (the golden ratio) which is not transcendental.
- hprotagonist 4y ago“the most irrational” irrational:)
- ykonstant 4y agoOn the contrary, due to Roth's theorem [0], transcendental numbers are the only ones who have a chance to be atypically well approximable by rational numbers. [0] https://en.wikipedia.org/wiki/Roth%27s_theorem https://en.wikipedia.org/wiki/Roth%27s_theorem
- rich_sasha 4y agoAh so just off by 1 :) Now I remember, the "canonical" transcendental number is sum of reciprocals of n! Which is almost rational.
- ykonstant 4y agoPrecisely. In fact, that features in many proofs of transcendence: you find too good rational approximations, which contradicts things like Roth's theorem, so your number cannot be algebraic.
- madcaptenor 4y agoIIRC the first number proven to be transcendental was Liouville's number 0.110001000000000000000001... where the digits after the decimal place are 1 in the n!th place and 0 otherwise. This is explicitly constructed to be very close to the sequence of rational numbers 0.1, 0.11, 0.110001, ... (I expect there's nothing special about base 10 here; surely the proof works in binary as well.)
- henrydark 4y agoSo actually the number you're referring to the natural base of logarithm, usually written "e". It turns out that e has the same irrationality measure as irrational algebraic numbers (2), meaning it can be approximated similarly well as irrational algebraic numbers, and not as well as some other transcendstal numbers like Liouville's constant
- SilasX 4y agoHuh? The link says Roth's Theorem is about algebraic numbers, which are the opposite of transcendental numbers. And I'm pretty sure 100% of integers are approximable by rational numbers, which has got to be at least as good a figure as the transcendentals can claim.
- henrydark 4y agoActually integers are very poorly approximable by rational numbers, even though each integer is very well approximated by a single specific rational number The definition of "well approximated" is that there are infinitely many good approximations, not just one really good one, and this is what integers, and algebraic numbers in general, fail to have
- SilasX 4y agoEvery integer has an infinite number of combinations of integers that add up to it.
- henrydark 4y agoIndeed! Mathematics is a lot about defining something and then proving stuff (and sometimes going the other way around). Different combinations giving the same approximation are considered a single approximation, namely the result of the combination
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- madcaptenor 4y agoI wonder what happens for powers of e. This should be different than pi, since e has continued fraction coefficients that make a nice pattern [2, 1, 2, 1, 1, 4, 1, 1, ...]. e^2 does as well (https://oeis.org/A001204 https://oeis.org/A001204). (I know this fact but have never understood any of the proofs.) But e^3 (https://oeis.org/A058282 https://oeis.org/A058282) does not have a "nice pattern", and neither does e^4 (https://oeis.org/A058283 https://oeis.org/A058283)
- pyrolistical 4y agoOften a nice pattern can be found if one uses a generalized continued fraction
- westurner 4y ago/? 3blue1brown e^ipi https://m.youtube.com/results?sp=mAEA&search_query=3blue1brown+e%5Eipi https://m.youtube.com/results?sp=mAEA&search_query=3blue1bro... Given: e = limit((1 + 1/n)^n, +∞) # Euler's number i = √-1 # orthogonal; i_0^2 = -1 pi = (666/212 - 22/7)*π # circle circumference / diameter Euler's identity: e^iπ + 1 = 0 Euler's formula: e^ix = cos(x) + i*sin(x) Euler's formula: https://en.wikipedia.org/wiki/Euler's_formula https://en.wikipedia.org/wiki/Euler's_formula e (Euler's number) https://en.wikipedia.org/wiki/E_(mathematical_constant) https://en.wikipedia.org/wiki/E_(mathematical_constant)
- westurner 4y agoIs there something fundamental here - with e.g. radix base e - about countability and a continuum of reals, and maybe constructive interference?
- westurner 4y agoWhen EM waves that are in phase combine, the resultant amplitude of the combined waveform is the sum of the amplitudes; constructive interference is addition. And from addition, subtraction & multiplication and exponents and logarithms. And then this concept of phase and curl ( convergence and divergence ) in non-orthogonal, probably not conditionally-independent fluid fields that combine complexly and nonlinearly. Define distance between (fluid) field moments. A coherent multibody problem hopefully with unitarity and probably nonlocality. Can emergence of complex adaptive behavior in complex nonlinear systems of fields emerge from such observable phenomena as countability (perhaps just of application-domain-convenient field-combinatorial multiples in space Z)?
- thomasahle 4y agoPretty cool that you can determine the exact probability distribution for the digits of fraction expansion of a random real number: lim_{n->inf} Pr[k_n = k] = -log_2(1 - 1/(k+1)^2) And this was determine already in 1929! I think fraction expansions was all the rage back then. https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribut...
- svat 4y agoYes! Not only that, here's something I found mind-blowing: for almost all real numbers, the nth root of the nth convergent's denominator has as limit the same value, and that value is e^(pi^2/12ln2). Am typing from phone so can't write it down here properly, but some details at an old blog post of mine: https://shreevatsa.wordpress.com/2010/04/30/some-incredibly-beautiful-things/ https://shreevatsa.wordpress.com/2010/04/30/some-incredibly-...
- madcaptenor 4y agothere's also Khinchin's constant - for almost all real numbers, the geometric mean of the first n continued fraction coefficients approaches a constant K as ngoes to infinity. K is about 2.68 (the geometric mean of the Gauss-Kuzmin distribution). I like to pair this with a more trivial fact - for almost all real numbers, the arithmetic mean of the first n digits of the decimal expansion, as n goes to infinity, approaches 4.5.
- tialaramex 4y ago> for almost all real numbers, the arithmetic mean of the first n digits of the decimal expansion, as n goes to infinity, approaches 4.5. Is this just because Almost All Reals are Normal, and so in every base they have an equal distribution of all digits ?
- madcaptenor 4y agoThat’s exactly it.
- bonzini 4y agoWouldn't the right question be "do the powers of pi have unusually big terms towards the beginning of their continued fractions?" Because even one such term is enough to have a surprisingly good approximation, but many small terms at the beginning would create an approximation with a larger numerator and denominator, which may seem less remarkable than 355/113.
- Dwedit 4y agoThe font I'm seeing this in makes the pi look like a lowercase N.
- colejohnson66 4y agoIt’s an uppercase pi; The lowercase one is what we’re all familiar with. What’s interesting is that the actual post has the lowercase one. Is it a bug in HN’s capitalization fixer?
- scapp 4y agoMaybe it's changed now, but the "π" in the title is unicode U+03C0 [0], which is a lowercase pi. In the font Verdana (the one that I see in the title, but not this comment as I type it[1]), U+03C0 looks like a small capital pi and looks a bit like "n").[2] [0] https://www.compart.com/en/unicode/U+03C0 https://www.compart.com/en/unicode/U+03C0 [1] EDIT: once the comment was submitted, the font changed to Verdana. [2] https://graphicdesign.stackexchange.com/questions/74608/why-is-verdanas-lower-case-pi-symbol-so-similar-to-an-n https://graphicdesign.stackexchange.com/questions/74608/why-...
- colejohnson66 4y agoAh. You’re right. I didn’t even think to check the actual codepoint. It still looks the same as it did when I commented hours ago, so it must just be my font then.
- ahazred8ta 4y agothere are also script-style 'mathematical pi symbol' codepoints π 𝜋 𝛑 𝝅 𝞹 ℼ
- charlieyu1 4y agoYou don’t want many terms of the same pi power, of course they will be random eventually. You should use the first few terms continued fractions of millions of pi powers for this
- ncmncm 4y agoPi, or rather pi/6=~0.523, has an odd coincidental relationship with a formula involving phi=(√5+1)/2=~1.618, to three places. And, it is just possible that whoever the hell did the Great Pyramid knew of this coincidence. There is also a strange relationship involving pi, phi, the cubit used there, and the modern meter (which is canonically taken from 1/40,000,000 of the Earth's circumference). The Egyptian cubit, by some mysterious, hermetical coincidence, exactly matches the French royal cubit. The ratio of that cubit to the modern meter is pi/6, to 3 places. https://www.willemwitteveen.com/the-royal-cubit/ https://www.willemwitteveen.com/the-royal-cubit/ This might all seem like reaching, but the Egyptians were very keen on numerology. They thought it had legitimately divine implications, and that coincidences were none. There are odd coincidences with the size of Earth, moon, and sun involving these values. You can go down the rabbit hole; https://www.sacredgeometry.blog/the-royal-cubit/ https://www.sacredgeometry.blog/the-royal-cubit/ Demanding only three digits of precision to match makes these coincidences a lot easier, but the ancients were probably satisfied by that precision.