29 ms·
Stone tablet shows Babylonians able to approximate √2 with 99.9999% accuracy
- Buttons840 4y agoI don't understand the percentage. What, for example, would an approximation that was only 20% accurate look like?
- psychphysic 4y agoIt would mean their error would be less than it'd look like 1.2*sqrt(2). Read the thread he explains it very well. The percentage is 1 - (absolute error divided by the true value) * 100
- addaon 4y agoNote that the parent asked for 20% accurate, not 20% error (80% accurate).
- psychphysic 4y agoAccuracy is generally defined as (mean - true)/true But it's a common note that that's really the inaccuracy.
- deleted 4y ago[deleted]
- SomeHacker44 4y ago1.7 is about 20% off of accurate. 0.3 to 7.0 seem to be within the range of what 20% accurate would mean (1/5th to 5x the actual value). The thread shows what they mean a few tweets in.
- addaon 4y agoapprox_20pct * 0.2 <= sqrt(2) <= approx_20pct / 0.2; so it might look like 0.3, or 4.5.
- deleted 4y ago[deleted]
- olddustytrail 4y agoTo simplify your other responses, 20% accurate means you're 80% inaccurate. So +/- 80% of the correct value.
- Someone 4y agoI don’t think ‘Accuracy’ is used much in math. https://mathworld.wolfram.com/Accuracy.html https://mathworld.wolfram.com/Accuracy.html exists, but I think it would be better to report relative error (https://en.wikipedia.org/wiki/Approximation_error https://en.wikipedia.org/wiki/Approximation_error, https://mathworld.wolfram.com/RelativeError.html https://mathworld.wolfram.com/RelativeError.html), which is less than one in a million here. The math, in general, isn’t as good as it, IMO, should be. The claim “Thus, the interval [x₀, 2/x₀] envelopes √2. From this, it follows that the mid-point of the interval [x₀, 2/x₀] is a better approximation to √2” isn’t correct. The midpoint is a better approximation, but that doesn’t follow from “the interval [x₀, 2/x₀] envelopes √2.”. [1,999] envelopes √2, but 500 isn’t a better approximation of √2 than 1. (Aside: https://en.wikipedia.org/wiki/Accuracy_and_precision https://en.wikipedia.org/wiki/Accuracy_and_precision is useless for finding a definition of accuracy. The chapter “Common technical definition” doesn’t define anything, and the picture in that section seems to imply that, for both precision and accuracy, lower is better)
- moefh 4y agoThe math as explained in Twitter is correct, your example is not relevant because [1,999] is not in the form [x,2/x]. I'm sure there's a better way to see the conclusion stated in the tweet, but you can convince yourself with the brute-force way of graphing: - `abs(x-sqrt(2))` (the distance between x and sqrt(x)) - `abs(2/x-sqrt(2))` (the distance between 2/x and sqrt(x)) - `abs((x/2+1/x)-sqrt(2))` (the distance between the midpoint and sqrt(2)) and seeing that the last one is always smaller than the first two between 1 and sqrt(2), the values for x stated in a previous tweet.
- ummonk 4y agoThe details on the method are paywalled in his Substack, and it's not clear whether he has any evidence that the method he is proposing are actually how the Babylonians actually calculated square root 2.
- tiborsaas 4y agoI'm sure they didn't use floating point decimals :) I don't understand his confidence that this THE method.
- adrian_b 4y agoActually they were using floating point sexagesimals (base 60), where the exponent was usually not written, but understood from the context.
- Mathnerd314 4y agoThe substack loaded for me but all it says is that they used (a special case of) Newton-Raphson. No second tablet is mentioned showing that the Babylonians calculated that way. There is a link to https://www.jstor.org/stable/10.4169/j.ctt19b9k86 https://www.jstor.org/stable/10.4169/j.ctt19b9k86 which just says "this is an excellent approximation"
- ffssffss 4y agoI think this is the paper it's all coming from: https://www.sciencedirect.com/science/article/pii/S0315086098922091?via%3Dihub https://www.sciencedirect.com/science/article/pii/S031508609... They've found some tables of similar estimations, which apparently match the results you get when you apply the author's algorithm, suggesting that's what the Babylonians used as well. I don't think the method itself is actually attested anywhere in known texts.
- Mathnerd314 4y agoWell if you read that, the one attested computation is said to be mathematically equivalent to Heron's method (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Heron's_method https://en.wikipedia.org/wiki/Methods_of_computing_square_ro... ). An iterative Newton's method would be pretty unlikely.
- version_five 4y agoThis is just a low quality ad for some guy's substack, and to like & follow him. It's too bad to see it ranking highly
- vesinisa 4y agoI don't think this is low-quality. I think he did a good job with the illustrations and explaining it. I guess that's why people are liking it. If someone publishes original content for free and in the end puts a link to their paid blog, I really don't mind it. I've seen far worse advertising.
- psychphysic 4y agoAgreed Twitter and gen z are mostly non-technical. It explains and illustrates a fact beautifully. You're not meant to be able to do the calculation yourself if you found yourself back in time
- tourist2d 4y ago[flagged]
- erredois 4y agoLike a lot of Twitter this days. Fact Threads or controversial opinions looking for engagement to sell something.
- version_five 4y ago
- wanderingstan 4y agoThe full post (non-twitter) is better and has the full explanation: https://thepalindrome.substack.com/p/how-did-the-babylonians-know-2-up https://thepalindrome.substack.com/p/how-did-the-babylonians... I didn’t realize Babylonians had a system for “decimals” in sexagesimal. Edit: another comment pointed out the post is an extrapolation of this paper: https://www.sciencedirect.com/science/article/pii/S0315086098922091?via%3Dihub https://www.sciencedirect.com/science/article/pii/S031508609...
- polygamous_bat 4y agoIt has the full explanation behind a paywall, which may as well be non-existent. No, thank you.
- imglorp 4y agoThis is the first substack I've seen with a full paywall, not just an engagement nag.
- Tams80 4y agoYou missed the whole Domonic Cummings UK thing then. Journalists were moaning about 'having to' subscribe to his SubStack. It was actually the first I heard of the place and I thought it was the norm to charge.
- sbierwagen 4y agoSubstack authors can choose which posts to paywall, and how far into the post to end the preview. It's interesting in that the paywall can be trivially circumvented... by asking a paid subscriber to forward the email of the full post. Costs nothing, but in practice, almost never happens.
- matheusmoreira 4y ago> Costs nothing, but in practice, almost never happens. Any information people consider important enough to share with the whole world will not be paywalled. Consequently, any paywalled information must not be important.
- sbaiddn 4y agoOnce they stopped knifing every fellow who figured out it's irrational, the Greeks had it to 100% ;)
- hgsgm 4y agoThat's not true, even as a joke about Pythagoras. √2 is just a name of a relationship, not a measurement, which implies finding its place on the number line. No one will ever know √2 to 100% accuracy, in the sense of answering every question of the the form "is x^2 < 2^n", for all exactly known x (rationals), computing each within a constant time limit.
- xigoi 4y agoBy this logic, no one will ever exactly know the value of 3. Can you compute in constant time for an arbitrary rational number x whether x < 3?
- sbaiddn 4y agoIm content with a Euclidean construction and a proof that its irrational. An infinite series definition. Actually evaluating the digits is for... technicians. (Besides, just choose an irrational number base and sqrt 2 is = 1)
- ArtWomb 4y agoEssential for astronomical calculations, like forecasting the paths of Jupiter and Venus. A daily brief of "omens" was submitted to the Assyrian King at breakfast. Has anything changed in modern day? I watch the weather and traffic at 6am and likewise predict how "long and glorious my reign shall live" ;)
- colechristensen 4y agoRomans relied on a few different omens for a lot of things which reduced to injecting randomness into a lot of decision making. I wonder if such things would be valuable to add back to modern life… roll dice to determine which tickets to add to the next sprint, legislatures have to flip a coin to trigger a floor vote for a piece of contentious legislation, that kind of thing.
- npongratz 4y agoIt could be sold as a meatspace chaos monkey.
- galaxyLogic 4y agoI think that makes sense. If you cannot be very sure of actual evidence why rely on it anyway? But randomness might mean that every time you use the result you learn something, at least you would learn whether your rule-based default value for the control variable is systematically wrong and tilted to one direction or another.
- colechristensen 4y agoThat goes along with my thinking. Some things are obvious but when you move away from the top topic or two you’ll always have a long list of things where it’s not so clear which should be done next, and there’s plenty which is always important to do but never the most important thing. A bit of randomness injected into that process whether you’re dealing with software tickets, national legislation, or household chores might just lead to better outcomes than relying on the judgement of a few people without enough information to make nonarbitrary decisions and might have bad biases which lead to things getting overprioritized or neglected.
- hgsgm 4y agoFar less spammy and more readable presentation: https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/ https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-...
- yesenadam 4y agoThe comments on that page are wonderful - Baez, Greg Egan and others working out together the explanation/theory of everything mentioned in the article.
- danbruc 4y agoBinary search three 6 bit symbols, that is 18 multiplications. A bit less if you do not blindly pick the centre but estimate - or guess if you will - what the correct digit should approximately be. I would assume they could have done this to much larger precision if they had cared enough as I find it hard to imagine that they would have been limited by having to test every candidate. If I asked you to find the square root of two to eight fractional digits, you would probably come up with the idea of skipping ahead more than 0.00000001 each time pretty quickly. I guess my point is that it might be interesting that they knew about the idea of the square root of two but the precision does not really surprise me.
- deleted 4y ago[deleted]
- geysersam 4y agoIt does't sound like a lot of work. However, multiplying two six digit numbers by hand isn't that easy! They might have used Newtons algorithm (not using that name probably), but then they would have had to divide, which is even more horrible by hand. Consider that their notation was probably far worse than what we are used to today, and it all becomes quite a hassle.
- danbruc 4y agoI guess for them it would be more like a three - four with the leading one - digit number, would it not? Did they have a 60 x 60 multiplication table at hand? I am almost tempted to try it out, I would guess that it should be doable in two hours in modern notation.
- saalweachter 4y agoI assume it was in their head. I don't know exactly how large of a times table I have in my head, but it's larger than 10x10, more like 15x15. A 60x60 table is 16 times larger than that, but I also learned my times table in elementary school and being a scribe or whatever the job title was for a guy who multiplied was very likely a full time job for a grown Babylonian. Learning your times tables up to 60x60 seems totally plausible.
- prvc 4y agoThis is not a new discovery
- lindgrenj6 4y agoOnly four nines? Ridiculous. (/s)
- woopwoop 4y agoThe point is the update method they use is the same one you'd get from using Newton's method to find the roots of the function f(x) = x^2 - 2.
- spfzero 4y agoThat precision is fascinating. Other measurements of that age I'd guess were a lot less precise, such as time and distance. It seems like anything you used sqrt(2) for calculating, would be swamped with less-accurate factors and you'd get a much less accurate answer.
- PebblesRox 4y agoI wonder to what extent they did it for practicality and to what extent they did it because they could.
- mihaaly 4y agoWhy oh why write a study on twitter, oh why?! There are much more fitting mediums for that. I refuse to read it this way. Would you release the next Marvell movie on TikTok or a book in text messages? Grrr.
- rideontime 4y agoThat's why you read the comments :) https://news.ycombinator.com/item?id=34179449 https://news.ycombinator.com/item?id=34179449
- gleenn 4y agoIt's actually relatively complete but it's a lead for the full, pay-walled article.
- spacephysics 4y agoI hear ya, but think it’s the media medium targeted at that audience. A TikTok version would appeal to younger people, which I think would be a great thing!
- mihaaly 4y agoJesus!
- deleted 4y ago[deleted]
- pvg 4y agoPlease don't complain about tangential annoyances—e.g. article or website formats, name collisions, or back-button breakage. They're too common to be interesting. https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
- mihaaly 4y agoWhen it hinders the consumption of the topic then it is directly relevant. It was not about the font used or the colour.
- kxyvr 4y agoJoanna Papakonstantinou wrote a PhD thesis on the historical development of BFGS and the secant method back in 2009: https://scholarship.rice.edu/handle/1911/61898 https://scholarship.rice.edu/handle/1911/61898 Part of that thesis discusses the Babylonian use of the rule of the double false position and it's application to solving quadratics, which can be used to get the square root. Anyway, I'm not an expert in this history, but I believe she includes references to the related texts for those interested.
- IncRnd 4y agoThis was really interesting. PS The method described to find roots is quite similar to the method used in Quake by Carmack, Newton-Raphson. That's one of those pieces of code that's oft-copied by copilot.
- mproud 4y agohttps://thepalindrome.substack.com/p/how-did-the-babylonians-know-2-up https://thepalindrome.substack.com/p/how-did-the-babylonians...
- civilized 4y agoThe author never provides any evidence or references showing that the Babylonians used any of the methods he describes.
- moloch-hai 4y agoFun fact, 2phi²-5pi/3 is 0.0000802. The ancients certainly knew about phi, because it defines the ratios between segments of a pentagram. They loved pentagrams. They probably would have considered the match above exact, having no independent method to estimate pi. (Phi is (sqrt(5)+1)/2. Phi²=phi+1, 1/phi=phi-1.) A measurement system used by builders in medieval times had units that were related by powers of phi. Nobody knows how far back the convention goes, passed from master to apprentice.
- moloch-hai 4y agoAnother form is 6phi²/5 - pi = 0.000048.
- deleted 4y ago[deleted]
- szundi 4y agoI would call it calculation, not approximation.
- Maursault 4y agoI find it fascinating that we have names for base 60 and base 12. Is there a name for every base? What is base-11? base-59?
- adastra22 4y agoThose names are literally just “60-per-digit” and “12-per-digit” in a combination of Greek and Latin.
- cyco130 4y agoThat's because base 12 (dozens) and base 60 (minutes) are commonly used across cultures. So is 20 (vigesimal). Base 11 is called undecimal[1] and Wikipedia has a comprehensive list[2]. It doesn't list 59 but it would be enneaquinquagesimal if you follow the rules. [1] https://en.wikipedia.org/wiki/Undecimal https://en.wikipedia.org/wiki/Undecimal [2] https://en.wikipedia.org/wiki/List_of_numeral_systems#Standard_positional_numeral_systems https://en.wikipedia.org/wiki/List_of_numeral_systems#Standa...
- cyco130 4y agoAnd the rules are: - Use Latin for 1-10 - Use Latin 10n, 10n+1 and 10n+2 - Use Greek for the units + Latin for the tens for 10n+3..10n+9 The Greek-Latin mixup was probably caused by whoever invented the word hexadecimal (16). The older "sexadecimal" was maybe avoided due to the word "sex". "sedecimal" would be a bit more correct Latin but it doesn't seem to be attested.
- peoplefromibiza 4y ago> The Greek-Latin mixup was probably caused by whoever invented the word hexadecimal it was simply very common to use Greek prefixes in Latin to indicate measures. Micro, mega, nano, etto, etc. are all Greek words. micro -> mikrós (small) mega -> mégas (big) nano -> νάνος (small in its own kind) etto -> ἑκατόν (100) hexadecimal comes from the Greek prefix έξι (pronounced exi in English it sounds exactly like sexy without the s) there were absolutely no issues with the word sex, sexagesimal (from the Latin sexagesimus which meant the 60th) is still a thing.
- adictator 4y ago[dead]
- aaws11 4y agohttps://threadreaderapp.com/thread/1608419325706391554.html https://threadreaderapp.com/thread/1608419325706391554.html
- deleted 4y ago[deleted]
- deafpolygon 4y agoIt’s things like this that make me wonder how much knowledge has been lost and rediscovered over the ages.
- aaws11 4y agoauthor's detailed blog post on this same topic: https://thepalindrome.substack.com/p/how-did-the-babylonians-know-2-up https://thepalindrome.substack.com/p/how-did-the-babylonians...
- piuvas 4y agoI don't understand how it's 99.9999% accurate if it's infinitely long
- EmilioMartinez 4y agoAccuracy is best thought as relative, though the usage of percentages here is not a good fit imho. You compare your result to the wanted value by dividing the error by the value. In this case |x - √2|/√2 < 0.000001 (or 0.0001% if you insist) For comparison, the initial value of the iteration (1.2) would be 85% accurate, which without context might sounds like a lot but is pretty abysmal
- piuvas 4y agoAh, thank you very much for the explanation!
- zestyping 4y agoHe explains the first row of symbols: 1, 24, 51, 10. This represents 1 + 24/60 + 51/60^2 + 10/60^3 ~= 1.41421296, very close to sqrt(2) ~= 1.41421356. But what about the second row? The numbers on the second row are 42, 25, 35. So I tried working out 42/60 + 25/60^2 + 35/60^3 and it works out to 0.70710648, which is very close to 1/sqrt(2) ~= 0.70710678. So cool!