7 ms·

The Mystery of 355/113 (2010)

198 points by LeoPanthera 4y ago

SergeAx 4y ago
For me it's 3.14 first, then it's 3.1415, because 15 is 14+1, then I just memorize 92 and 6. Better than 355/113, which I don't think are that easy to memorize.
- timbit42 4y ago
  Is 113355 easier to memorize?
  - SergeAx 4y ago
    Now that you meant it - yes, quite so :)
green_on_black 4y ago
It seems to me that an estimation of any computable number would scale with at least but not equal to O(n) of its denominator. Otherwise, it would be non-computable (please correct me, this is just my intuition). For 355/113 to be more than 16x closer to pi than 22/7 would be a matter of course. Furthermore, 1/791 = 1/7 * 1/113, and 355 is the floor of 22 * 7/113, as one might except of a slightly closer approximation. Not to undervalue 355/113. But (personally) intuitively, it's not particularly mysterious.
- sgtnoodle 4y ago
  For irrational numbers, you mean? A rational number is one that can be perfectly represented by a fraction. I'm not sure I follow your use of big-O notation here, but are you saying that the bound of the error of a rational approximation of an irrational number is inversely proportional to the denominator?
- eru 4y ago
  You might find a comparison to the continued fraction for the golden ration enlightening.
- alanbernstein 4y ago
  How does that explain the much bigger error in the approximation with an incremented denominator, 358/114?
jccalhoun 4y ago
I love the comments on the page trying to find correlation between 355/113 and the Quran and Bible.
- knaekhoved 4y ago
  I was reading Unrestricted Warfare, a book published by the PLA in the 90s about their thoughts and strategic considerations re 5th generation warfare. There was a lot of really interesting and insightful stuff in there, periodically interspersed with random Chinese numerology, e.g. how the Union won the American civil war due to an auspicious ratio of men to horses or something.
- mindcrime 4y ago
  I love the comments on the page trying to find correlation between 355/113 and the Quran and Bible. Hmm... well. jccalhoun 5 minutes ago 'jccalhoun' has 9 characters, and is succeeded by 2 words ('minutes ago'), and 9 times 2 is 18. And 'jccalhoun' is followed by three non-whitespace tokens ('5 minutes ago' and 18 divided by 3 is 6. And 6 repeated three times is 666, so clearly we can conclude ... that numerology is bullshit? Or 'jccalhouun' is actually SATAN?!?? Hmmm...
  - tzs 4y ago
    I don’t know if jccalhoun is Satan, but they are definitely evil. Proof [1]. Refresh that page to see other proofs of their evilness. [1] https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=jccalhoun https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=jccalhoun
    - eru 4y ago
      Luckily, we can trust Bob: https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=Bob https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=Bob
- JKCalhoun 4y ago
  Are we related?
zwkrt 4y ago
“As of yet undiscovered” Well by the author, maybe. [1] I had to do some more digging on the sequence stated by the author of that post. The idea is to construct a continued fraction, always choosing the next largest denominator you can that doesn’t make the resulting partial evaluation larger than pi. [1] https://www.quora.com/Why-is-355-113-so-close-to-pi/answer/Todd-Trimble-1?ch=15&oid=236476195&share=c7238daa&target_type=answer https://www.quora.com/Why-is-355-113-so-close-to-pi/answer/T...
- roywiggins 4y ago
  A comment on the original post brings that exact continued fraction up, the author responds: "the continued fraction sequence 3 7 15... is the continued fraction analogy to the decimal sequence 3 1 4 1 5 9 2... etc: they are both guaranteed-to-exist sequences of successively better approximations for pi that come from the real value of pi. Neither is a way to derive or compute pi if you don't already have the exact value." So it's not a sequence like the Wallis rational expression or Taylor series approximation, so it's not very helpful in saying "where 355/113 comes from." 355/113 is a good approximation of Pi, and if you engineer your continued fraction to include the best approximation without going over at each step, it will be in there. Doesn't tell you why there's a nice accurate Pi that has such a high "quality" measure. That is, the 292 showing up in the continued fraction is still (arguably) a surprise, or at least left unexplained.
  - civilized 4y ago
    "Why does pi have this awesome continued fraction" is exactly how a professional mathematician would ask the question.
    - roywiggins 4y ago
      Yeah, I changed my comment around to try and say that- the question just becomes "huh, why does the big integer 292 show up like that?" which is really the same question as before. Maybe the answer is "that's not actually that surprising", I don't know.
    - 4y ago
deleted 4y ago
[deleted]
russellbeattie 4y ago
If you were to count in base π, would the diameter be a transcendental number?
- roywiggins 4y ago
  If the circumference is rational (in "base pi"), then the diameter is also rational, I think. Suppose the circumference is π, or 10ₚ (base pi). Then the diameter is 1₁₀. What's that in base Pi? Well, how many Pis are in 1? 1/π, or π⁻¹. But in base Pi you write that 0.1ₚ. Ditto something like π²+π+1 (111ₚ). The diameter is π+1+π⁻¹, or 11.1ₚ.
  - russellbeattie 4y ago
    I'm both impressed with your answer, as well as your use of subscript/superscript in plain text! Thanks!
    - roywiggins 4y ago
      It's wrong after all that, 1 (base 10) is 1 base pi, of course. But the argument still works!
- kaashif 4y ago
  Transcendental means a number that is not a root of any polynomial with rational coefficients. The base you count in doesn't affect this definition, those are just different ways of writing down numbers, the numbers themselves don't change. The rational numbers and integers are still the same whatever base you use. 1 is still 1 in any base, there's only one multiplicative identity. In base pi, 10 (pi) is transcendental. I do get what you're trying to ask, the other reply has a more useful comment.
quickthrower2 4y ago
An interest question is given a random real number x in [0-4] range say, what is the probability there will be a fraction of quality > 3 (for approximating x) for denominators < n. (restricting n to make the problem easier to say 2048) Might be a fun programming project (or interview question ha ha) to monte carlo this.
- jakear 4y ago
  Little toy questions like this might make a good use case for the replit app. Might try this at the bar tonight.
  - jakear 4y ago
    Around 3% of Math.random’s numbers are similarly efficiently and accurately representable as small fractions. https://replit.com/@JacksonKearl1/Nodejs https://replit.com/@JacksonKearl1/Nodejs
- eru 4y ago
  To properly answer this, you'd need a healthy dose of math. Once you apply all the math, your resulting code is probably relatively simple. But justifying why the simple and short code is justified is a bit more complicated. A simple example: we have no way to uniformly sample a real number from a range in our computers. Our computers can not represent arbitrary real numbers. So we need to sample some kind of proxy, and justify that we are getting the same answer as if sampling properly. Or more formally: we need to estimate the error that our choice of proxy introduces, and justify that estimation.
dang 4y ago
Discussed at the time: The Mystery of 355/113 - https://news.ycombinator.com/item?id=4285531 https://news.ycombinator.com/item?id=4285531 - July 2012 (65 comments)
raydiatian 4y ago
I like to imagine this is the universe’s version of a stick with a piece of string on the end and we’re a cat maddeningly entranced by it flitting about.
handsclean 4y ago
22/7 has you memorize four digits (including the position of the /) to get pi to four digits (including the position of the .), 355/113 has you memorize seven digits to get pi to eight digits, 355/113-1/3748629 has you memorize 17 digits to get pi to 16 digits. I don’t get it. How is it easier than just memorizing pi? How is it surprising to get out about as many bits as you put in?
- a_t48 4y ago
  For 355/113 you get error correction - if the answer you get isn't around 3.14 you misremebered.
  - handsclean 4y ago
    Ok, now you’re memorizing 11 digits, including 3.14, to get 8 error-corrected digits. Edit: changed 7 to 8, my mistake
    - a_t48 4y ago
      Most people have the 3.14 down.
      - handsclean 4y ago
        Ok, then you’re memorizing seven additional digits to get four additional digits and error correction…
- sobriquet9 4y ago
  113\355 is easier to remember.
  - smilekzs 4y ago
    Exactly. Low https://en.wikipedia.org/wiki/Kolmogorov_complexity https://en.wikipedia.org/wiki/Kolmogorov_complexity
  - 4y ago
bardan 4y ago
Odd to see the religious numerology people in the comment section.
akomtu 4y ago
355/113 is the 4th continued fraction of pi, its 4th best approximation. Transcendent numbers have an interesting property: they are the best approximated by rational numbers, they are the closest to them in a sense; unlike some irrational numbers, such as phi, that are really hard to approximate. There is a curious hypothesis that the world arranges itself by the continued fraction powers of number e, to avoid destructive resonances that would occur if a rational number was used. In other words, the reason period of electron motion or a planet motion is so close to e^(p/q) because that number is very close to a rational fraction, but isn't rational.
- klodolph 4y ago
  Are you talking about Liouville numbers? Liouville numbers are numbers which can be "best approximated by rational numbers" in a certain sense. A number x is a Liouville number if for every integer n>0, there exist integers (p,q) such that 0<|x-p/q|<1/q^n No algebraic numbers have this property, therefore all of the Liouville numbers are transcendental. However! The reverse is not true; not all transcendental numbers are Liouville. Neither pi nor e are Liouville numbers.
  - akomtu 4y ago
    Liouville-ness is a very strict requirement. Instead, we can define "K-liouville numbers, that for all n < K have this property", and then pi is at least a 3-liouville number, which is pretty good.
    - klodolph 4y ago
      Is this the irrationality measure? https://mathworld.wolfram.com/IrrationalityMeasure.html https://mathworld.wolfram.com/IrrationalityMeasure.html I don't think the value for pi is known. The value for e is 2, which is the same value for e.g. sqrt 2.
- kbelder 4y ago
  >...to avoid destructive resonances that would occur if a rational number was used Hey, thank you. That is obvious upon hearing, and makes perfect intuitive sense, but I might never have thought of it.
klodolph 4y ago
https://mathworld.wolfram.com/PiContinuedFraction.html https://mathworld.wolfram.com/PiContinuedFraction.html > The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). > The very large term 292 means that the convergent > [3;7,15,1]=[3,7,16]=(355)/(113)=3.14159292... > is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). You can see how the terms in the sequence jump from 1 to 292. I'm not sure why simple continued fractions are not mentioned here when other continued fractions are not, seems like the author got within a hairs breadth of discovering this connection. I'm not going to say "explanation" because it doesn't explain things, it just connects it to something else. This isn't really a way that you would calculate pi, but you could compare it with, for example, the continued fraction for e: [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] and notice that there is no large numbers here... so there is no fraction for e that is suddenly "very good". Same with sqrt 2 (repeating 2s), same with golden ratio (repeating 1s). In a sense, it's the fact that pi DOESN'T have a neat continued fraction which gives it that nice fractional representation. In a sense, discovering a connection just gives you another question to ask. "Okay, why isn't there a neat little formula for calculating pi as a continued fraction?"
- eru 4y ago
  To go off on a tangent, the Golden Ratio's simple continued fraction makes it in some sense the most irrational number.
- svat 4y ago
  To actually quantify how "surprised" we should be, see e.g.: • Gauss-Kuzmin distribution: https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuzmin_distribution&oldid=1098003937 https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuz... • This post: https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.html https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.h... (Look at the tail of the distribution, i.e. sum(p(k)) for k ≥ 292 — that will give the probability that a specific term in the continued fraction of a "random" number is at least 292. This is about 31% I think, so not too surprising.)
- 4y ago
cadamsau 4y ago
I decided to try every integer denominator less than 1,000,000 to see how rare this is. A few fun facts: Constraining denominators to 4 digits (<10,000) there are NO better fractions than 355/113. Constraining denominators to 5 digits (<100,000) and excluding multiples (eg. 710/226), 1601 fractions exist which better approximate pi than 355/113. The best is 312689/99532 which is off by 2.914335439641036e-11 Considering denominators up to 6 digits <1,000,000 and excluding multiples, there are 162,141. The best is 3126535/995207 which is off by 1.1426415369442111e-12 This was done somewhat subptimally with a potentially buggy implementation in python. It's in a gist here: https://gist.github.com/thecadams/df3e12571294e3f7a9f5c07e91ebbfd0 https://gist.github.com/thecadams/df3e12571294e3f7a9f5c07e91...
- robin_reala 4y ago
  Out of interest, do you get the same sort of results for other numbers that look a little like pi (small irrational number)? Or is this something peculiar to pi?
- LeoPanthera 4y ago
  I did a similar thing (also with some ugly python) to generate the closest integer fraction approximations in increasing length: 3/1 = 3.0 22/7 = 3.142857142857143 311/99 = 3.1414141414141414 355/113 = 3.1415929203539825 99733/31746 = 3.1415926415926414 312689/99532 = 3.1415926536189365 The jump in length from 355/113 to the next one is quite noticeable, and it does not happen with other irrational numbers, such as e, which progress more smoothly: 8/3 = 2.6666666666666665 19/7 = 2.7142857142857144 87/32 = 2.71875 193/71 = 2.7183098591549295 878/323 = 2.718266253869969 1457/536 = 2.718283582089552 2721/1001 = 2.7182817182817183 25946/9545 = 2.7182818229439496 49171/18089 = 2.718281828735696 271801/99990 = 2.7182818281828185
  - Someone 4y ago
    You missed 13/4 = 3.25, 16/5 = 3.2, 19/6 = 3.16667, 179/57 = 3.14035, 201/64, 223/71, 245/78, 267/85, 289/92, 333/106, 52163/16604 https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction/ https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...
karakfa2 4y ago
If you already know 22/7 approximation, it's just a (0x10)^-1 correction in denominator. That is: 22/7 = 3 1/7 ==> 355/113 = 3 1/(7 + 1/16) This means just memorizing 3 numbers. {3,7,16} and continued fraction format.
1970-01-01 4y ago
Aren't these called "near misses" https://cs.uwaterloo.ca/~csk/other/nearmisses/ https://cs.uwaterloo.ca/~csk/other/nearmisses/
analog31 4y ago
I have a cheap circular slide rule with a bunch of integer fraction approximations for useful constants on the back of it. I guess that would have been just a bit quicker than eyeballing a decimal value.
Shorel 4y ago
I remember I was so proud of 'discovering' this ratio when I was a child. I used an electronic calculator, so I had some superhuman help, and I discovered it several centuries after it was truly discovered. But this is still a good memory for me, and for this reason I remember the ratio 355/113 better than I remember 22/7.
- sonovice 4y ago
  Same here. When I was a kid in school, we all had those TI-89 calculator that one could program using Basic. When first introduced to Pi and the concept of irrational numbers, I made a script to find close matches. 355/113 was the closest I got with the slow calculator and my (most probably) very slow code.
dkbrk 4y ago
You can use Euclid's algorithm [0] to get the continued fraction that produces 355/113. I did a quick search and found a worked example of it here [1]. [0]: https://en.wikipedia.org/wiki/Euclidean_algorithm#Continued_fractions https://en.wikipedia.org/wiki/Euclidean_algorithm#Continued_... [1]: https://nrich.maths.org/1397 https://nrich.maths.org/1397
NKosmatos 4y ago
This is interesting but has no real practical use to us, just something that mathematicians like to talk about and is related to fractions, approximations, convergence and other theorems. I knew about 22/7 but this fraction is more interesting! If NASA is using 15 digits of pi for their calculations ( https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-%20pi-do-we-really-need/ https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... ), I’m more than happy with the 11 digits I’ve memorized :-)
galaxyLogic 4y ago
Has anybody tried to apply the "Memory Palace" -technique to memorizing Pi? How might that work?
WiSaGaN 4y ago
It's amazing that the author never mentioned it was first discovered by Chinese mathematician Zu Chongzhi in 5 century AD: https://en.wikipedia.org/wiki/Mil%C3%BC https://en.wikipedia.org/wiki/Mil%C3%BC
pencilguin 4y ago
Speaking of coincidences, try 6phi²/5pi, or 20/(e^pi-pi).. Recall phi=(5^.5+1)/2, and phi²=phi+1.
lgxz 4y ago
But, why is pi 3.1415926 instead of 4.1415926?
- Sharlin 4y ago
  For one, if pi were > 4, a unit circle would have a greater area than a unit square.
Gordonjcp 4y ago
In the precision world of RF engineering, I approximate pi as "3, and then about a fifth more, and then trim to tune because you can cut it shorter but you can't cut it longer".
HelloNurse 4y ago
Averaging two rational approximations whose errors have opposite signs always yields an approximation with lower error and a bigger denominator than either "parent", making good approximations easy to find.
evanb 4y ago
A related puzzle is: why is sin(333) + sin(355) = sin(22) [angles in radians] good to 1e-9? It's easy to say why it's good to 1e-3, a bit more work to show it's 1e-6, and just a little bit more to say why it's 1e-9. If you know 22/7 and 355/113 then you just subtract them to get another continued fraction for π: (355 - 22)/(113 - 7) = 333/106 ~ pi; that's a bit of a hint. https://recursed.blogspot.com/2014/01/math-challenge-1.html https://recursed.blogspot.com/2014/01/math-challenge-1.html
drittich 4y ago
>The fraction 355/113 is incredibly close to pi, within a third of a millionth of the exact value. This level of accuracy is far beyond its rights as a fraction with such a small denominator, and it causes various oddities elsewhere in math. On the other hand, from a purely practical perspective, the answer is only accurate to six digits, which is the number of digits in the fraction you must remember. So, you're not really getting much of a free lunch.
- Quekid5 4y ago
  113355 seems pretty easy to remember tho
bikenaga 4y ago
Here's the specific result - it's a standard result in elementary number theory. Suppose x is irrational and p/q is a rational in lowest terms with q > 0, and |x - p/q| < 1/(2 q^2). Then p/q is a convergent in the continued fraction expansion for x. Using this, you can show that 355/113 is the best rational approximation to pi with denominator less than 1000. (Assume that there's a better approximation, and use the result to get a contradiction.) I wrote this up in my number theory notes (https://sites.millersville.edu/bikenaga/number-theory/approximation-by-rationals/approximation-by-rationals.html https://sites.millersville.edu/bikenaga/number-theory/approx... and https://sites.millersville.edu/bikenaga/number-theory/approximation-by-rationals/approximation-by-rationals.pdf https://sites.millersville.edu/bikenaga/number-theory/approx... [pdf]).
wazoox 4y ago
The easiest way to learn a quite large number of pi digits is with mnemonic poems like this one I've known since I was 10 or so (French): 3 1 4 1 5 9 2 6 5 3 5 Que j'aime à faire apprendre un nombre utile aux sages 8 9 7 9 Immortel Archimède, artiste, ingénieur, 3 2 3 8 4 6 2 6 Qui de ton jugement peut priser la valeur? 4 3 3 8 3 2 7 9 Pour moi ton problème eut de pareils avantages.
- dwringer 4y ago
  Whoa, thanks to your comment, which helpfully illustrates some of the "poetic" aspects of the digits' decimal arrangement, I became instantly convinced the digits of pi are the theme on which Elgar wrote his "Enigma" Variations. It's mentioned on Wikipedia but pretty far down as a recent theory (the source is here[0]). But now I feel almost foolish for not thinking it before. Quoting Wikipedia, "on the evening of 21 October 1898, after a tiring day's teaching, he sat down at the piano. A melody he played caught the attention of his wife, and he began to improvise variations on it in styles which reflected the character of some of his friends." It's pretty easy to imagine he was just plinking out scale degrees corresponding to the digits of pi in a pleasing rhythm and one thing led to another. [0]https://academiccommons.columbia.edu/doi/10.7916/D8377M82/download https://academiccommons.columbia.edu/doi/10.7916/D8377M82/do...
kloch 4y ago
Coincidences are everywhere and our monkey brains generally have a hard time accepting that they are just coincidences. One example that will make any physicist bang their head on their desk: 3 * l_P / alpha_em^2 = 9.10543E-31 m Which is within 4.3E-4 of the numerical value of the electron mass in KG except the formula has units of meters not kilograms so it's complete nonsense. l_P = Planck length alpha_em = "fine structure constant"
- tazjin 4y ago
  These are just eastereggs.

198 points by LeoPanthera 4y ago

For me it's 3.14 first, then it's 3.1415, because 15 is 14+1, then I just memorize 92 and 6. Better than 355/113, which I don't think are that easy to memorize.

timbit42 4y ago
Is 113355 easier to memorize?
- SergeAx 4y ago
  Now that you meant it - yes, quite so :)

green_on_black 4y ago

It seems to me that an estimation of any computable number would scale with at least but not equal to O(n) of its denominator. Otherwise, it would be non-computable (please correct me, this is just my intuition). For 355/113 to be more than 16x closer to pi than 22/7 would be a matter of course. Furthermore, 1/791 = 1/7 * 1/113, and 355 is the floor of 22 * 7/113, as one might except of a slightly closer approximation. Not to undervalue 355/113. But (personally) intuitively, it's not particularly mysterious.

sgtnoodle 4y ago
For irrational numbers, you mean? A rational number is one that can be perfectly represented by a fraction. I'm not sure I follow your use of big-O notation here, but are you saying that the bound of the error of a rational approximation of an irrational number is inversely proportional to the denominator?
eru 4y ago
You might find a comparison to the continued fraction for the golden ration enlightening.
alanbernstein 4y ago
How does that explain the much bigger error in the approximation with an incremented denominator, 358/114?

jccalhoun 4y ago

I love the comments on the page trying to find correlation between 355/113 and the Quran and Bible.

knaekhoved 4y ago
I was reading Unrestricted Warfare, a book published by the PLA in the 90s about their thoughts and strategic considerations re 5th generation warfare. There was a lot of really interesting and insightful stuff in there, periodically interspersed with random Chinese numerology, e.g. how the Union won the American civil war due to an auspicious ratio of men to horses or something.
mindcrime 4y ago
I love the comments on the page trying to find correlation between 355/113 and the Quran and Bible. Hmm... well. jccalhoun 5 minutes ago 'jccalhoun' has 9 characters, and is succeeded by 2 words ('minutes ago'), and 9 times 2 is 18. And 'jccalhoun' is followed by three non-whitespace tokens ('5 minutes ago' and 18 divided by 3 is 6. And 6 repeated three times is 666, so clearly we can conclude ... that numerology is bullshit? Or 'jccalhouun' is actually SATAN?!?? Hmmm...
- tzs 4y ago
  I don’t know if jccalhoun is Satan, but they are definitely evil. Proof [1]. Refresh that page to see other proofs of their evilness. [1] https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=jccalhoun https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=jccalhoun
  - eru 4y ago
    Luckily, we can trust Bob: https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=Bob https://lcamtuf.coredump.cx/evilfinder/ef.cgi?said=Bob
JKCalhoun 4y ago
Are we related?

zwkrt 4y ago

“As of yet undiscovered” Well by the author, maybe. [1] I had to do some more digging on the sequence stated by the author of that post. The idea is to construct a continued fraction, always choosing the next largest denominator you can that doesn’t make the resulting partial evaluation larger than pi. [1] https://www.quora.com/Why-is-355-113-so-close-to-pi/answer/Todd-Trimble-1?ch=15&oid=236476195&share=c7238daa&target_type=answer https://www.quora.com/Why-is-355-113-so-close-to-pi/answer/T...

roywiggins 4y ago
A comment on the original post brings that exact continued fraction up, the author responds: "the continued fraction sequence 3 7 15... is the continued fraction analogy to the decimal sequence 3 1 4 1 5 9 2... etc: they are both guaranteed-to-exist sequences of successively better approximations for pi that come from the real value of pi. Neither is a way to derive or compute pi if you don't already have the exact value." So it's not a sequence like the Wallis rational expression or Taylor series approximation, so it's not very helpful in saying "where 355/113 comes from." 355/113 is a good approximation of Pi, and if you engineer your continued fraction to include the best approximation without going over at each step, it will be in there. Doesn't tell you why there's a nice accurate Pi that has such a high "quality" measure. That is, the 292 showing up in the continued fraction is still (arguably) a surprise, or at least left unexplained.
- civilized 4y ago
  "Why does pi have this awesome continued fraction" is exactly how a professional mathematician would ask the question.
  - roywiggins 4y ago
    Yeah, I changed my comment around to try and say that- the question just becomes "huh, why does the big integer 292 show up like that?" which is really the same question as before. Maybe the answer is "that's not actually that surprising", I don't know.
  - 4y ago

deleted 4y ago

[deleted]

russellbeattie 4y ago

If you were to count in base π, would the diameter be a transcendental number?

roywiggins 4y ago
If the circumference is rational (in "base pi"), then the diameter is also rational, I think. Suppose the circumference is π, or 10ₚ (base pi). Then the diameter is 1₁₀. What's that in base Pi? Well, how many Pis are in 1? 1/π, or π⁻¹. But in base Pi you write that 0.1ₚ. Ditto something like π²+π+1 (111ₚ). The diameter is π+1+π⁻¹, or 11.1ₚ.
- russellbeattie 4y ago
  I'm both impressed with your answer, as well as your use of subscript/superscript in plain text! Thanks!
  - roywiggins 4y ago
    It's wrong after all that, 1 (base 10) is 1 base pi, of course. But the argument still works!
kaashif 4y ago
Transcendental means a number that is not a root of any polynomial with rational coefficients. The base you count in doesn't affect this definition, those are just different ways of writing down numbers, the numbers themselves don't change. The rational numbers and integers are still the same whatever base you use. 1 is still 1 in any base, there's only one multiplicative identity. In base pi, 10 (pi) is transcendental. I do get what you're trying to ask, the other reply has a more useful comment.

quickthrower2 4y ago

An interest question is given a random real number x in [0-4] range say, what is the probability there will be a fraction of quality > 3 (for approximating x) for denominators < n. (restricting n to make the problem easier to say 2048) Might be a fun programming project (or interview question ha ha) to monte carlo this.

jakear 4y ago
Little toy questions like this might make a good use case for the replit app. Might try this at the bar tonight.
- jakear 4y ago
  Around 3% of Math.random’s numbers are similarly efficiently and accurately representable as small fractions. https://replit.com/@JacksonKearl1/Nodejs https://replit.com/@JacksonKearl1/Nodejs
eru 4y ago
To properly answer this, you'd need a healthy dose of math. Once you apply all the math, your resulting code is probably relatively simple. But justifying why the simple and short code is justified is a bit more complicated. A simple example: we have no way to uniformly sample a real number from a range in our computers. Our computers can not represent arbitrary real numbers. So we need to sample some kind of proxy, and justify that we are getting the same answer as if sampling properly. Or more formally: we need to estimate the error that our choice of proxy introduces, and justify that estimation.

dang 4y ago

Discussed at the time: The Mystery of 355/113 - https://news.ycombinator.com/item?id=4285531 https://news.ycombinator.com/item?id=4285531 - July 2012 (65 comments)

raydiatian 4y ago

I like to imagine this is the universe’s version of a stick with a piece of string on the end and we’re a cat maddeningly entranced by it flitting about.

handsclean 4y ago

22/7 has you memorize four digits (including the position of the /) to get pi to four digits (including the position of the .), 355/113 has you memorize seven digits to get pi to eight digits, 355/113-1/3748629 has you memorize 17 digits to get pi to 16 digits. I don’t get it. How is it easier than just memorizing pi? How is it surprising to get out about as many bits as you put in?

a_t48 4y ago
For 355/113 you get error correction - if the answer you get isn't around 3.14 you misremebered.
- handsclean 4y ago
  Ok, now you’re memorizing 11 digits, including 3.14, to get 8 error-corrected digits. Edit: changed 7 to 8, my mistake
  - a_t48 4y ago
    Most people have the 3.14 down.
    - handsclean 4y ago
      Ok, then you’re memorizing seven additional digits to get four additional digits and error correction…
sobriquet9 4y ago
113\355 is easier to remember.
- smilekzs 4y ago
  Exactly. Low https://en.wikipedia.org/wiki/Kolmogorov_complexity https://en.wikipedia.org/wiki/Kolmogorov_complexity
- 4y ago

bardan 4y ago

Odd to see the religious numerology people in the comment section.

akomtu 4y ago

355/113 is the 4th continued fraction of pi, its 4th best approximation. Transcendent numbers have an interesting property: they are the best approximated by rational numbers, they are the closest to them in a sense; unlike some irrational numbers, such as phi, that are really hard to approximate. There is a curious hypothesis that the world arranges itself by the continued fraction powers of number e, to avoid destructive resonances that would occur if a rational number was used. In other words, the reason period of electron motion or a planet motion is so close to e^(p/q) because that number is very close to a rational fraction, but isn't rational.

klodolph 4y ago
Are you talking about Liouville numbers? Liouville numbers are numbers which can be "best approximated by rational numbers" in a certain sense. A number x is a Liouville number if for every integer n>0, there exist integers (p,q) such that 0<|x-p/q|<1/q^n No algebraic numbers have this property, therefore all of the Liouville numbers are transcendental. However! The reverse is not true; not all transcendental numbers are Liouville. Neither pi nor e are Liouville numbers.
- akomtu 4y ago
  Liouville-ness is a very strict requirement. Instead, we can define "K-liouville numbers, that for all n < K have this property", and then pi is at least a 3-liouville number, which is pretty good.
  - klodolph 4y ago
    Is this the irrationality measure? https://mathworld.wolfram.com/IrrationalityMeasure.html https://mathworld.wolfram.com/IrrationalityMeasure.html I don't think the value for pi is known. The value for e is 2, which is the same value for e.g. sqrt 2.
kbelder 4y ago
>...to avoid destructive resonances that would occur if a rational number was used Hey, thank you. That is obvious upon hearing, and makes perfect intuitive sense, but I might never have thought of it.

klodolph 4y ago

https://mathworld.wolfram.com/PiContinuedFraction.html https://mathworld.wolfram.com/PiContinuedFraction.html > The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). > The very large term 292 means that the convergent > [3;7,15,1]=[3,7,16]=(355)/(113)=3.14159292... > is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). You can see how the terms in the sequence jump from 1 to 292. I'm not sure why simple continued fractions are not mentioned here when other continued fractions are not, seems like the author got within a hairs breadth of discovering this connection. I'm not going to say "explanation" because it doesn't explain things, it just connects it to something else. This isn't really a way that you would calculate pi, but you could compare it with, for example, the continued fraction for e: [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] and notice that there is no large numbers here... so there is no fraction for e that is suddenly "very good". Same with sqrt 2 (repeating 2s), same with golden ratio (repeating 1s). In a sense, it's the fact that pi DOESN'T have a neat continued fraction which gives it that nice fractional representation. In a sense, discovering a connection just gives you another question to ask. "Okay, why isn't there a neat little formula for calculating pi as a continued fraction?"

eru 4y ago
To go off on a tangent, the Golden Ratio's simple continued fraction makes it in some sense the most irrational number.
svat 4y ago
To actually quantify how "surprised" we should be, see e.g.: • Gauss-Kuzmin distribution: https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuzmin_distribution&oldid=1098003937 https://en.wikipedia.org/w/index.php?title=Gauss%E2%80%93Kuz... • This post: https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.html https://11011110.github.io/blog/2022/06/04/maybe-powers-pi.h... (Look at the tail of the distribution, i.e. sum(p(k)) for k ≥ 292 — that will give the probability that a specific term in the continued fraction of a "random" number is at least 292. This is about 31% I think, so not too surprising.)
4y ago

cadamsau 4y ago

I decided to try every integer denominator less than 1,000,000 to see how rare this is. A few fun facts: Constraining denominators to 4 digits (<10,000) there are NO better fractions than 355/113. Constraining denominators to 5 digits (<100,000) and excluding multiples (eg. 710/226), 1601 fractions exist which better approximate pi than 355/113. The best is 312689/99532 which is off by 2.914335439641036e-11 Considering denominators up to 6 digits <1,000,000 and excluding multiples, there are 162,141. The best is 3126535/995207 which is off by 1.1426415369442111e-12 This was done somewhat subptimally with a potentially buggy implementation in python. It's in a gist here: https://gist.github.com/thecadams/df3e12571294e3f7a9f5c07e91ebbfd0 https://gist.github.com/thecadams/df3e12571294e3f7a9f5c07e91...

robin_reala 4y ago
Out of interest, do you get the same sort of results for other numbers that look a little like pi (small irrational number)? Or is this something peculiar to pi?
LeoPanthera 4y ago
I did a similar thing (also with some ugly python) to generate the closest integer fraction approximations in increasing length: 3/1 = 3.0 22/7 = 3.142857142857143 311/99 = 3.1414141414141414 355/113 = 3.1415929203539825 99733/31746 = 3.1415926415926414 312689/99532 = 3.1415926536189365 The jump in length from 355/113 to the next one is quite noticeable, and it does not happen with other irrational numbers, such as e, which progress more smoothly: 8/3 = 2.6666666666666665 19/7 = 2.7142857142857144 87/32 = 2.71875 193/71 = 2.7183098591549295 878/323 = 2.718266253869969 1457/536 = 2.718283582089552 2721/1001 = 2.7182817182817183 25946/9545 = 2.7182818229439496 49171/18089 = 2.718281828735696 271801/99990 = 2.7182818281828185
- Someone 4y ago
  You missed 13/4 = 3.25, 16/5 = 3.2, 19/6 = 3.16667, 179/57 = 3.14035, 201/64, 223/71, 245/78, 267/85, 289/92, 333/106, 52163/16604 https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction/ https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...

karakfa2 4y ago

If you already know 22/7 approximation, it's just a (0x10)^-1 correction in denominator. That is: 22/7 = 3 1/7 ==> 355/113 = 3 1/(7 + 1/16) This means just memorizing 3 numbers. {3,7,16} and continued fraction format.

1970-01-01 4y ago

Aren't these called "near misses" https://cs.uwaterloo.ca/~csk/other/nearmisses/ https://cs.uwaterloo.ca/~csk/other/nearmisses/

analog31 4y ago

I have a cheap circular slide rule with a bunch of integer fraction approximations for useful constants on the back of it. I guess that would have been just a bit quicker than eyeballing a decimal value.

Shorel 4y ago

I remember I was so proud of 'discovering' this ratio when I was a child. I used an electronic calculator, so I had some superhuman help, and I discovered it several centuries after it was truly discovered. But this is still a good memory for me, and for this reason I remember the ratio 355/113 better than I remember 22/7.

sonovice 4y ago
Same here. When I was a kid in school, we all had those TI-89 calculator that one could program using Basic. When first introduced to Pi and the concept of irrational numbers, I made a script to find close matches. 355/113 was the closest I got with the slow calculator and my (most probably) very slow code.

dkbrk 4y ago

You can use Euclid's algorithm [0] to get the continued fraction that produces 355/113. I did a quick search and found a worked example of it here [1]. [0]: https://en.wikipedia.org/wiki/Euclidean_algorithm#Continued_fractions https://en.wikipedia.org/wiki/Euclidean_algorithm#Continued_... [1]: https://nrich.maths.org/1397 https://nrich.maths.org/1397

NKosmatos 4y ago

This is interesting but has no real practical use to us, just something that mathematicians like to talk about and is related to fractions, approximations, convergence and other theorems. I knew about 22/7 but this fraction is more interesting! If NASA is using 15 digits of pi for their calculations ( https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-%20pi-do-we-really-need/ https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... ), I’m more than happy with the 11 digits I’ve memorized :-)

galaxyLogic 4y ago

Has anybody tried to apply the "Memory Palace" -technique to memorizing Pi? How might that work?

WiSaGaN 4y ago

It's amazing that the author never mentioned it was first discovered by Chinese mathematician Zu Chongzhi in 5 century AD: https://en.wikipedia.org/wiki/Mil%C3%BC https://en.wikipedia.org/wiki/Mil%C3%BC

pencilguin 4y ago

Speaking of coincidences, try 6phi²/5pi, or 20/(e^pi-pi).. Recall phi=(5^.5+1)/2, and phi²=phi+1.

lgxz 4y ago

But, why is pi 3.1415926 instead of 4.1415926?

Sharlin 4y ago
For one, if pi were > 4, a unit circle would have a greater area than a unit square.

Gordonjcp 4y ago

In the precision world of RF engineering, I approximate pi as "3, and then about a fifth more, and then trim to tune because you can cut it shorter but you can't cut it longer".

HelloNurse 4y ago

Averaging two rational approximations whose errors have opposite signs always yields an approximation with lower error and a bigger denominator than either "parent", making good approximations easy to find.

evanb 4y ago

A related puzzle is: why is sin(333) + sin(355) = sin(22) [angles in radians] good to 1e-9? It's easy to say why it's good to 1e-3, a bit more work to show it's 1e-6, and just a little bit more to say why it's 1e-9. If you know 22/7 and 355/113 then you just subtract them to get another continued fraction for π: (355 - 22)/(113 - 7) = 333/106 ~ pi; that's a bit of a hint. https://recursed.blogspot.com/2014/01/math-challenge-1.html https://recursed.blogspot.com/2014/01/math-challenge-1.html

drittich 4y ago

>The fraction 355/113 is incredibly close to pi, within a third of a millionth of the exact value. This level of accuracy is far beyond its rights as a fraction with such a small denominator, and it causes various oddities elsewhere in math. On the other hand, from a purely practical perspective, the answer is only accurate to six digits, which is the number of digits in the fraction you must remember. So, you're not really getting much of a free lunch.

Quekid5 4y ago
113355 seems pretty easy to remember tho

bikenaga 4y ago

Here's the specific result - it's a standard result in elementary number theory. Suppose x is irrational and p/q is a rational in lowest terms with q > 0, and |x - p/q| < 1/(2 q^2). Then p/q is a convergent in the continued fraction expansion for x. Using this, you can show that 355/113 is the best rational approximation to pi with denominator less than 1000. (Assume that there's a better approximation, and use the result to get a contradiction.) I wrote this up in my number theory notes (https://sites.millersville.edu/bikenaga/number-theory/approximation-by-rationals/approximation-by-rationals.html https://sites.millersville.edu/bikenaga/number-theory/approx... and https://sites.millersville.edu/bikenaga/number-theory/approximation-by-rationals/approximation-by-rationals.pdf https://sites.millersville.edu/bikenaga/number-theory/approx... [pdf]).

wazoox 4y ago

The easiest way to learn a quite large number of pi digits is with mnemonic poems like this one I've known since I was 10 or so (French): 3 1 4 1 5 9 2 6 5 3 5 Que j'aime à faire apprendre un nombre utile aux sages 8 9 7 9 Immortel Archimède, artiste, ingénieur, 3 2 3 8 4 6 2 6 Qui de ton jugement peut priser la valeur? 4 3 3 8 3 2 7 9 Pour moi ton problème eut de pareils avantages.

dwringer 4y ago
Whoa, thanks to your comment, which helpfully illustrates some of the "poetic" aspects of the digits' decimal arrangement, I became instantly convinced the digits of pi are the theme on which Elgar wrote his "Enigma" Variations. It's mentioned on Wikipedia but pretty far down as a recent theory (the source is here[0]). But now I feel almost foolish for not thinking it before. Quoting Wikipedia, "on the evening of 21 October 1898, after a tiring day's teaching, he sat down at the piano. A melody he played caught the attention of his wife, and he began to improvise variations on it in styles which reflected the character of some of his friends." It's pretty easy to imagine he was just plinking out scale degrees corresponding to the digits of pi in a pleasing rhythm and one thing led to another. [0]https://academiccommons.columbia.edu/doi/10.7916/D8377M82/download https://academiccommons.columbia.edu/doi/10.7916/D8377M82/do...

kloch 4y ago

Coincidences are everywhere and our monkey brains generally have a hard time accepting that they are just coincidences. One example that will make any physicist bang their head on their desk: 3 * l_P / alpha_em^2 = 9.10543E-31 m Which is within 4.3E-4 of the numerical value of the electron mass in KG except the formula has units of meters not kilograms so it's complete nonsense. l_P = Planck length alpha_em = "fine structure constant"

tazjin 4y ago
These are just eastereggs.