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How Many Decimals of Pi Do We Really Need?

jpl.nasa.gov

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Re: How Many Decimals of Pi Do We Really Need?

#81

Post by [deleted] »

julie1 wrote: post #56
ninguem2 wrote: post #48

Earlier quoted context omitted.

Physics? Numerical analysis is Mathematics. It's still active and very important, e.g., in solving PDEs using the finite element method.

Mathematics? The people trying to find solutions of an equation by solving them? Mathematicians is a weird education I hardly understand. The last PhD I met from Mc Gill university ignored the existence of non euclidean geometry. His excuse? He was on formal proof. I am sorry, I have a hard time with people that never were challenged to make equations spit their solutions in order to make something actually work in t…

[deleted]

Re: How Many Decimals of Pi Do We Really Need?

#82

Post by rzzzt »

sago wrote: post #5

In 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.

There were some exercises in our high school physics textbook which required you to simplify a fraction by assuming pi^2 = g.

Re: How Many Decimals of Pi Do We Really Need?

#83

Post by lostlogin »

yuubi wrote: post #72
mchahn wrote: post #43

Earlier quoted context omitted.

> 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.

The other side from the vector adder ("front" side at 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 s…

This is amazing. The idea of converting units of multiple types in a hurry when it matters with a slide or circular rule in imperial units is terrifying. Somehow doing that in metric seems less so, but the fact that the system got people to the moon relatively recently is still amazing.

Re: How Many Decimals of Pi Do We Really Need?

#84

Post by 21echoes »

api wrote: post #76
gmuslera wrote: post #44

Earlier quoted context omitted.

The smallest possible distance is the Plank lenght, 1,6 10^-35 (1 10^-15 is the diameter of a proton). And for that you only need around 60 digits of pi to calculate the circumference of the universe. Of course, that is just for the simple operation of calculate the circumference given the diameter, more complex operations with pi may require more precision.

Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nature of mathematics? Seems like you could argue that digits beyond what are needed to render measurement to within one Planck length are meaningless and therefore a kind of fiction... at least if you take that philosophical position.

Pi is far more than just the ratio of the diameter to the circumference, even as a constructivist: https://affinemess.quora.com/What-is-math-pi-math-and-while-...

Re: How Many Decimals of Pi Do We Really Need?

#85

Post by lostlogin »

niccaluim wrote: post #63
brador wrote: post #31

Earlier quoted context omitted.

It 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.

This isn't flight control software. It's a blog post. There aren't any "mistakes" to prevent. The website is not going to crash into Mars. Good writers write for their audience. His audience is accustomed to thinking in miles.

It would be interesting to see the analytics - I'm not disagreeing and the site is presumably funded by American taxes, but people like the site aren't all American.

Re: How Many Decimals of Pi Do We Really Need?

#86

Post by venomsnake »

api wrote: post #76
gmuslera wrote: post #44

Earlier quoted context omitted.

The smallest possible distance is the Plank lenght, 1,6 10^-35 (1 10^-15 is the diameter of a proton). And for that you only need around 60 digits of pi to calculate the circumference of the universe. Of course, that is just for the simple operation of calculate the circumference given the diameter, more complex operations with pi may require more precision.

Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nature of mathematics? Seems like you could argue that digits beyond what are needed to render measurement to within one Planck length are meaningless and therefore a kind of fiction... at least if you take that philosophical position.

Pi has other uses. The digits are real, but they just don't matter in practical engineering.

But lets assume that there really are 10 dimensions - in that case a volume of a 10 dimensional sphere will require (pi^5)*r^10.

If you want to measure the volume to 1 plank 10 dimensional cube, you will need more digits.

Re: How Many Decimals of Pi Do We Really Need?

#87

Post by brador »

niccaluim wrote: post #63
brador wrote: post #31

Earlier quoted context omitted.

It 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.

This isn't flight control software. It's a blog post. There aren't any "mistakes" to prevent. The website is not going to crash into Mars. Good writers write for their audience. His audience is accustomed to thinking in miles.

Great writers educate their audience.

Re: How Many Decimals of Pi Do We Really Need?

#88

Post by cperciva »

notjpl wrote: post #14
WalterBright wrote: post #3

This 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.

I 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 navigat…

Here are some actual examples:

Something to add to your list of examples: During the first Gulf war, 28 US soldiers died due to accumulated rounding errors in the Patriot Missile battery computers: https://www.ima.umn.edu/~arnold/disasters/patriot.html

(This was in fact a known issue, and operators had been instructed to reboot the computers every 8 hours. Unfortunately this instruction ignored the fact that, in the field, nobody wanted to be responsible for turning off their defensive systems for a minute.)

Re: How Many Decimals of Pi Do We Really Need?

#89

Post by lotharbot »

KMag wrote: post #69
sago wrote: post #5

In 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.

> 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.

even if the measurements were both outer measurements, the actual value of pi is within the typically assumed error bounds (30/10 pi.)
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