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How Many Decimals of Pi Do We Really Need?
- deleted 11y ago[deleted]
- joss82 11y agoTl;dr: 15
- mchahn 11y ago15 digits is about the precision hand-held calculators provide, right? Many early NASA missions took HP calculators along in missions with trajectory routines in case the computer failed.
- mchahn 11y agoBTW, the computers in the ship were also made by HP. There is a great story about an incident where a waste recycling problem caused a mission to be aborted. There was urine all over the inside of the capsule. NASA publicly reported that it was a computer failure. Unfortunately HP had just done an ad campaign about their computers in space. HP sued and NASA settled for an unknown amount.
- msl 11y agoThe flight computers on any manned NASA spacecraft have not been made by HP. The Gemini capsules and the Space Shuttle used IBM computers [1,2] and Apollo used computers made by MIT (primary computers) and TRW (Abort Guidance System) [3]. I could also not find any information on a mission that failed in such a way. [1] http://history.nasa.gov/computers/Ch1-2.html http://history.nasa.gov/computers/Ch1-2.html [2] http://history.nasa.gov/computers/Ch4-3.html http://history.nasa.gov/computers/Ch4-3.html [3] http://history.nasa.gov/computers/Ch2-8.html http://history.nasa.gov/computers/Ch2-8.html
- bitwize 11y agoHandheld calculators typically get between 8 and 12 digits of precision. A four-function calculator will top out at 8; a scientific calculator will offer more (plus scientific notation support). 15 digits is about what's offered in double precision floating point calculations.
- progers7 11y agoIf you're interested in common handheld calculators, you might enjoy this light-hearted series by Matt Parker titled "Calculator Unboxing": https://www.youtube.com/watch?v=8Nzi1h2m7pE&list=PLt5AfwLFPxWKAINNfxIdYmFVKuk_F_cQq https://www.youtube.com/watch?v=8Nzi1h2m7pE&list=PLt5AfwLFPx...
- todd8 11y agoHand-held calculators might have been carried on many NASA missions, but not the early ones. The first missions started in 1961 and hand-held HP calculators weren't invented until more than ten years later. By that time we had already been to the moon 6 or 7 times. The first actual hand-held calculator I every saw was a Bowman Brain (simple 4 function calculator) it was for sale in 1971 at the MIT COOP (the bookstore). I only knew one person that bought one; the rest of us continued carrying around our slide rules (they came in handy leather holsters with belt loops.) The HP that came out about a year later was a real scientific calculator. Years before that, sometime between 1965 and 1968, on an episode of Lost in Space (a TV program with a family of early space explorers lost in outer space) the son, Will Robinson, was carrying around a large device about 3 inches thick and a foot tall that looked like a calculator. I thought the idea quite marvelous and went to bed thinking about it and how much better it would be than my slide rule for playing around with calculations. (I was a weird kid.)
- mchahn 11y agoWe must be about the same age. I went through college with a leather holster for my slide rule. I saw the first TI calculator after graduating from college in '71. I lusted after it and later I went to work at HP and got an HP. I was a super-fan of reverse-polish at that time. HP stayed with RPN for some time.
- skykooler 11y agoDid the astronauts use slide rules? I know the E6B is pretty common for pilots still.
- mchahn 11y ago> I know the E6B is pretty common for pilots still. I've used that before. It is not a standard logarithm stick but a vector addition tool. Does one thing very quickly.
- yuubi 11y agoThe other side from the vector adder ("front" side at https://upload.wikimedia.org/wikipedia/commons/c/c4/StudentE6BFlightComputer.jpg https://upload.wikimedia.org/wikipedia/commons/c/c4/StudentE... ) includes a circular slide rule with perfectly normal log scales for fuel, time, distance calculations, an extra scale to help with hours/minutes conversions, and some marks for various conversion factors, including lb/gal fuel and lb/gal oil for use in weight/balance. The main difference between a straight and circular rule is that it has only one appearance of the index, so you don't have to move the slide around as much, and it's round so the equivalent of a 10" rule has around 3" diameter. It also has other scales for converting altimeter/airspeed (really pressure gauge) readings into other numbers more useful for certain purposes like true altitude (good for missing obstructions) and "density altitude" (for estimating takeoff performance, also helpful for missing obstructions).
- WalterBright 11y agoThis overlooks the issue that for repeated calculations, such as numerical integration, the trouble comes from accumulated roundoff errors. Even 16 digits of precision can become 0 digits pretty quickly if you're not very careful.
- colechristensen 11y agoRight but there are several other much less precise constants/measurements than 16 digits of pi. G, for example, is only known to 5 or 6 digits of precision. Nevermind the mass of the rocket/satellite/probe, positions, the mass of earth, orbital perturbations not accounted for ... etc. What NASA does do is know precisely the error bounds of any important number which is far more valuable than using arbitrary precision math for π.
- notjpl 11y agoI agree, that's a poor answer by NASA director and chief engineer. Here is a better answer: The precision used for calculations is dependent on the number of "steps" required to get to the final result. Roughly, for N repeated calculations you lose somewhere between sqrt(N) * eps to N * eps of precision (eps=2e-16 for IEEE64). Here are some actual examples: IEEE64 (~16 decimal digits) is OK for interplanetary navigation for few months, where relatively low accuracy is required. With the same precision, you start to lose phase accuracy above 24 hours if you're simulating GPS constellations. You need quad precision or above for simulations > 24 hours. For simulating planet trajectories and solar system stability (Lyapunov time of planets), IEEE64 is good for ~10 mya in the future (Neptune-Pluto Lyapunov time), IEEE128 for ~200-1000mya, above that it is recommended to use 256bit floats and above. This is assuming typically ~1000 steps per simulated orbit. Fun fact: we know from simulations that Pluto trajectory is stable for >10G years, but unpredictable above >10M years because of chaotic (but stable) interaction with Neptune. [1] https://en.wikipedia.org/wiki/Stability_of_the_Solar_System https://en.wikipedia.org/wiki/Stability_of_the_Solar_System
- Etheryte 11y agoDisagreed, the answer isn't aimed at engineers or actually at any science-oriented people, it's directed at the general public who don't even care what IEEE is.
- jacobolus 11y agoIn this particular case, they’re just using a standard double precision IEEE 754 floating point number. So I assume they do all of their arithmetic (“for JPL's highest accuracy calculations”) using double precision floats.
- sago 11y agoIn the 'Frontiers in Astrophysics' course on Open Yale, professor Bailyn says that, for the purpose of the course, pi = 3, and pi^2 = 10. Pi = 3, coincidentally, is the Hebrew Bible's approximation too.
- CydeWeys 11y agoWhen I took astronomy, anything within an order of magnitude (10) was considered to be the same number. Calculations are very easy when you're only worrying about the number in the exponent.
- tamana 11y agoIn astronomy, you only need to get the order of magnitude right to within an order of magnitude.
- archgoon 11y ago"Everything is linear if plotted on a log-log plot and with a fat enough magic marker." - Mar's Law
- sp332 11y agoFeynman was talking to some students, and he used some historical event as an illustration, but he got the date wrong by a few years and they called him on it. He laughed and said "Hey, three decimal places is pretty good for a theoretical physicist!"
- KMag 11y ago> Pi = 3, coincidentally, is the Hebrew Bible's approximation too. Certainly it's not explicitly spelled out. The example I've heard was the outer diameter and inner circumference of a vessel's circular rim were given. Pi comes out to 3 only if the thickness of the rim of the vessel is zero.
- lotharbot 11y agoeven if the measurements were both outer measurements, the actual value of pi is within the typically assumed error bounds (30/10 < pi but 30.5/9.5 > pi.)
- sneezeplease 11y agoDoesnt seem like you need that many digits of Pi when working with green screens and premiere.
- 13of40 11y agoAccording to Google NGrams, World Wars I and II both primarily used 3.1416 to represent pi. https://books.google.com/ngrams/graph?content=3.1416&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2C3.1416%3B%2Cc0 https://books.google.com/ngrams/graph?content=3.1416&year_st...
- brador 11y agoMiles and inches? Please learn and use standard international (SI) units. It's important.
- wtbob 11y agoHe's an American, writing for an American audience, and thus he's using the units Americans use. There's absolutely nothing more scientific about one set of units or another (although different sets of units may be more convenient in different situations).
- over 11y agoInch / foot / yard / mile is a really small range compared to what metric can handle (anything), so in practice most science is done with metric units.
- andrewflnr 11y agoWide ranges of values are handled exactly the same way in imperial units as in metric: multipliers. You write "0.00023 inches" or "5.48e4 miles", etc. It's just not as pretty.
- deleted 11y ago[deleted]
- duaneb 11y agoYea, but even in American high schools the metric system is taught for physics.
- brador 11y agoIt prevents mistakes when we all use the same units and the SI are agreed by an international committee of scientists and engineers. It's one less thing to go wrong.
- 11y ago
- dang 11y agoUrl changed from http://kottke.org/16/03/how-many-digits-of-pi-does-nasa-use http://kottke.org/16/03/how-many-digits-of-pi-does-nasa-use, which points to this.
- mabbo 11y agoI have heard, but never done the math the verify, that with 50-ish digits of pi, one's error on a circle the size of the Universe would be smaller than a plank length.
- kurthr 11y agoAlthough this really should have been posted 4 days ago, 2pi x 3x10^8 x 40x10^10 / 1.6x10^-35 is just over 10^54 so that's about all the digits you need to memorize. Unless you're looking for a really strong password, reciting 100,000 digits is probably more than necessary: http://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/ http://blogs.scientificamerican.com/observations/how-much-pi...
- drewolbrich 11y agoIf you know the diameter of the observable Universe and you want to calculate its circumference with the accuracy of the diameter of a proton, the number of digits of pi that you need is 43.
- deleted 11y ago[deleted]
- mabbo 11y agoI've always heard the number is around 50, but do you have any references to show that as being correct?
- delecti 11y agoThis [1] puts it at 44. [1] http://www.wolframalpha.com/input/?i=observable+universe+diameter+%2F+proton+diameter http://www.wolframalpha.com/input/?i=observable+universe+dia...
- deleted 11y ago[deleted]
- deleted 11y ago[deleted]
- jasonkostempski 11y agoWould have been so much better if that answer was 42.
- deleted 11y ago[deleted]
- ryanobjc 11y agoThe real answer: As many as it takes. Also, what about the quest for finding the largest prime? #keepthedreamalive
- deleted 11y ago[deleted]
- gunnihinn 11y agoI remember back in high school physics when we were calculating the volumes of a few stars and my teacher said "Just round out 4\pi/3 to 4". I completely understand why we'd do that -- the error terms in the radius of the star completely drown out that approximation -- but goddammit it still feels wrong. I guess I'm a mathematician and not a physicist for a reason.
- marcosdumay 11y ago:) Physics is full of dirty shortcuts. I dread every time I see somebody using a natural units system.
- gmuslera 11y agoMaybe for astronomy a few could be enough, but for computing all are needed for the perfect filesystem https://github.com/philipl/pifs https://github.com/philipl/pifs
- julie1 11y agobtw Pi = 4 (in taxicab geometry aka L1) http://math.stackexchange.com/questions/96835/are-there-any-geometries-spaces-where-pi-is-a-simple-or-at-least-rational-cons http://math.stackexchange.com/questions/96835/are-there-any-... Euclidean geometry is not the only one and some physical problems are solved using spaces in which pi is NOT 3.1459 ;)
- kordless 11y agoMaybe the new decimals are information from beyond this realm. Thanks, Sagan.
- justifier 11y agoi wonder if interest in measuring error of previous calculations is what encouraged this direction of computational rigor respecting accuracy encourages a self awareness with an almost conscious stead ignorant error i am always intrigued when it is discussed how a calculation began and the error of the initial values the first known attempt at measuring the speed of light(o) had an ignorant error of ~26% the first known attempt at measuring the circumference of the earth(i) had an ignorant error of ~15% > our planet Earth.. the circumference .. > .. would .. be if you used the limited version of pi above? > It would be off by the size of a molecule. our conscious error is the size of a molecule, but what will our ignorant error be? how will its significance manifest? the ignorant error is a result of the tools of measure, in this case observable measurements and numerical approximation for those who calculated using pi equal to 22/7, for the circumference, their error would only be ~.04% of the 15 digit rounded value >>>2*(22/7)*(7926/2)>>> 2*(22/7)*(7926/2) 24910.285714285714 >>> 2*(3.141592653589793)*(7926/2) #from the article 24900.2633723527 >>> 24910.285714285714/24900.2633723527 1.0004024994347707 >>> (1.0004024994347707-1)*100 0.04024994347706645 (o) https://en.wikipedia.org/wiki/Speed_of_light#First_measurement_attempts https://en.wikipedia.org/wiki/Speed_of_light#First_measureme... (i) https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_the_Earth.27s_circumference https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_th... .. edit, percentage error, left out the *100
- desdiv 11y ago>The primary purpose of the DATA statement is to give names to constants; instead of referring to pi as 3.141592653589793 at every appearance, the variable PI can be given that value with a DATA statement and used instead of the longer form of the constant. This also simplifies modifying the program, should the value of pi change. Xerox Basic FORTRAN and Basic FORTRAN IV Manual[0], attributed to David H. Owens. [0] https://www.textfiles.com/bitsavers/pdf/sds/sigma/lang/900967D_Sigma2_FORTRAN_Aug70.pdf https://www.textfiles.com/bitsavers/pdf/sds/sigma/lang/90096...
- brandmeyer 11y agoNot quite Pi, but something very closely related to Pi is retained to extremely high precision in computers. libm frequently contains 2/pi to very high precision. For example, Newlib's math library contains 476 decimal digits of 2/pi as part of its routines for calculating sine and cosine of numbers outside the range [-pi/4..pi/4]. See e_rem_pio2.c for more. Many of the open source math libraries are ultimately descended from the same root: the Sunpro fdlibm, archived at netlib: http://www.netlib.org/fdlibm/ http://www.netlib.org/fdlibm/
- x4m 11y agoHere is an article how precision of Pi could affect trigonometry https://randomascii.wordpress.com/2014/10/09/intel-underestimates-error-bounds-by-1-3-quintillion/ https://randomascii.wordpress.com/2014/10/09/intel-underesti...
- hzhou321 11y agoSo it proves that the concept of irrational number is rather useless in practice ...
- gaur 11y agoNon-metric units... sigh...
- oniMaker 11y agoWe need all of them. Keep going until you reach the end.
- sharkjacobs 11y agoThis story is frustrating to me because it makes it sound like 15 digits of precision isn't a lot. Fifteen isn't a big number, but fifteen degrees of precision is almost incomprehensible. If you measured your height with fifteen degrees of precision, you would have a measurement in femtometres. A femtometre is roughly the diameter of a proton. That's really precise!
- rurban 11y agoSo they are using simple and fast double, not long double. Which makes sense.
- cnvogel 11y agoThe ratio of the observable universe's circumference to a proton diameter may be 10^-35, but that doesn't really say anything for the precision of Pi you'd need in practice for any calculation involving these scales. Because for everything involving real-world data, you'll have to measure quantities, and this is hardly ever done to more than just a few decimal digits. Whenever I want to state the circumfence of anything I know the diameter of down to single numbers of proton diameters, I first have to measure the diameter of to a precision of 1/3 proton diameter. Only when I reach such an absurdly nonsensical precision, I'd introduce errors by using an inadequately runded value for Pi. More practically: I might know that I could line up 2.611*10^25 protons (disregarding the fact that due to their charge they would repel each other) around the earth, but to calculate that I only need 5 decimal digits of the earth's diameter, and only 5 decimal places of Pi.
- jstoja 11y agoI really thought that the reason would have been for technical reasons, like a compromise between precision and how fast they can actually calculate with pi. The answer is simply awesome.
- tremguy 11y agoI think this is a bit of an oversimplification. You must consider compounding when talking about rounding errors. A single matrix operation with hundreds of rows and columns can easily have millions of multiplications. At every multiplication the previous error gets multiplied. That's why I don't feel the answer was exhaustive.
- carlob 11y ago> At every multiplication the previous error gets multiplied This is a bit of an oversimplification as well, it's not like you keep multiplying pi with itself over and over again and it's not like the error you introduce is random, if you've rounded pi once, you're gonna keep make a slight error in the same direction. If you were right there'd be no hope of ever getting sane results when multiplying largish matrices of doubles regardless of the presence of pi. I'm not saying that accumulation of error doesn't exist, I'm just saying that it's not to the extremes you're describing.
- marcosdumay 11y agoPi is kind of a worst case, because you round it only once, every operation will add errors on the same direction. Because of that you either use way to many decimal places, or make sure you don't keep multiplying pi with itself as you said. But both are optional, and must be designed into. A matrix of measurements, by its turn, normally has unbiased errors, what makes the resulting error grow much slower.
- albertzeyer 11y agoSome other approximations: http://www.math.tamu.edu/~dallen/masters/alg_numtheory/pi.pdf http://www.math.tamu.edu/~dallen/masters/alg_numtheory/pi.pd... And: https://en.wikipedia.org/wiki/Approximations_of_%CF%80 https://en.wikipedia.org/wiki/Approximations_of_%CF%80 Babylons and early Chinese just used pi = 3. Romans used pi = 3.125.
- sunstone 11y ago355/113 gets you more than you'll ever need.
- bbtn 11y agoUniversal constants [1] have about 6-9 significant digits today. I wouldn't use more than 10 digits of pi, if I am working on some physical calculations. [1] http://physics.nist.gov/cuu/Constants/index.html http://physics.nist.gov/cuu/Constants/index.html
- Houshalter 11y agoThe best way of looking at problems like this, is that it's an exponential process. The number of values you can represent with n digits increases exponentially. Each additional digit increases your precision by a factor of 10. If you have 15 digits, well imagine multiplying 10 over and over again 15 times, it's pretty big. The word "quadrillion" is rarely used in the English language. Because it's very rare you need numbers that large. And when you do, being off by a few digits doesn't matter. Calculators commonly only display up to 8-10 digits, for example. This applies to programming, since computers often only have a limited number of bits. Programmers often complain about floating point. One of the things about neural networks is that they don't actually need that many bits of precision, since they are by nature very "fuzzy". We can build computers that are bigger/cheaper by sacrificing a lot of bits. But one of the problems is, when adding a bunch of small numbers together, it rounds to the nearest whole number every time. And the inaccuracy builds up. So to really take advantage of less precision, we need to somehow build computers that can do stochastic rounding, where they sometimes round up, and sometimes round down, so the expected output is the same.