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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?

#91

Post by Al-Khwarizmi »

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 appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc.

There is a list of formulae where π appears (some related to circles, some not) here: https://en.wikipedia.org/wiki/List_of_formulae_involving_π

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

#92

Post by cnvogel »

The 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.

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

#93

Post by [deleted] »

skykooler wrote: post #40
todd8 wrote: post #27

Earlier quoted context omitted.

Hand-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…

Did the astronauts use slide rules? I know the E6B is pretty common for pilots still.

[deleted]

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

#94

Post by ColinWright »

skykooler wrote: post #40
todd8 wrote: post #27

Earlier quoted context omitted.

Hand-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…

Did the astronauts use slide rules? I know the E6B is pretty common for pilots still.

I spoke at some length with Fred Haise Jr, Lunar Module pilot of Apollo 13, and when I showed him my slide rule, he enthusiastically took it and started to play with it. Yes, astronauts used to use slide rules.

He asked me what I used to lubricate it, and showed some simple calculations to our host. He said he'd mostly forgotten how to use it, but it was clear that it was very, very familiar to him.

He signed it. Later I got Jim Lovell to sign it as well. One of my most treasured possessions. I only wish I'd had the presence of mind to get TK Mattingly to have signed it when I met him a year earlier. There is yet time.

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

#95

Post by jonhohle »

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…

So don't use a lossy floating point representation and use arbitrary precision instead.

Just because π is estimated to 15 significant digits doesn't mean the entire calculation needs to be.

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

#96

Post by thoth »

julie1 wrote: post #22
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.

A branch of physics used to be taught a long time ago called "numerical analysis" to deal with this issue. We even used to be careful about the difference between 'precise and exact'. Pi = acos(0) is absolutely exact. But computer don't know about symbolic calculus. So to put the value in a register we used tricks. Pi as a the converging value at the infinite of the Taylor development is awesome. But computer don't k…

>A branch of physics used to be taught a long time ago called "numerical analysis" to deal with this issue.

Freshmen engineers had to take a year-long Numerical Analysis and FORTRAN programming class at my undergrad university. We would learn various iteration methods for solving equations and the homeworks would be more problems to solve with a program or two to write too.

This stuff is still taught, it just might be out of a different department than physics.

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

#97

Post by [deleted] »

Etheryte wrote: post #21
notjpl wrote: post #14

Earlier quoted context omitted.

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…

Disagreed, 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.

[deleted]

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

#99

Post by mc808 »

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.

It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the precision of numbers involved could also keep growing.

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

#100

Post by ikeboy »

Al-Khwarizmi wrote: post #91
api wrote: post #76

Earlier quoted context omitted.

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 appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc. There is a list of formulae where π appears (some related to circles, some not) here: https://en.wikipedia.org/wiki/List_…

But how many coins can you flip before the heat death of the universe, and does using 43 digits make your expectation off by more than say 1/1000000?

In other words, if you were trying to calculate pi by checking the actual probabilities, how many digits can you get?

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