Some other approximations: http://www.math.tamu.edu/~dallen/masters/alg_numtheory/pi.pd...
And: https://en.wikipedia.org/wiki/Approximations_of_%CF%80
Babylons and early Chinese just used pi = 3.
Romans used pi = 3.125.
How Many Decimals of Pi Do We Really Need?
111–120 of 134 posts
Re: How Many Decimals of Pi Do We Really Need?
#112In 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.
Another nice one is that there's pi*10^7 seconds in a year. Which is accurate to three decimal places.
Re: How Many Decimals of Pi Do We Really Need?
#113Earlier quoted context omitted.
The post i replied to started off with "If you know the diameter of the observable Universe" but if the result it depends on the size of the observable universe then you ought to say "given the size of the observable universe". Starting with 'if' would imply that the result only requires you 'knowing' the size of the obsv. unv. but does not depend on its actual size.
I think the meaning of "If you know," in this case was more like, "If you take as an input."
Yep. "If you know X" is commonly used as "given X".
Re: How Many Decimals of Pi Do We Really Need?
#114I 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.
> 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…
Pi 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.
Re: How Many Decimals of Pi Do We Really Need?
#115I 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.
:)
Physics is full of dirty shortcuts. I dread every time I see somebody using a natural units system.
Re: How Many Decimals of Pi Do We Really Need?
#116Earlier 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.
Never heard it stated that way. Can you elaborate on "the smallest possible distance is Plank's length"? Is that the smallest observable distance?
From what I understand the particles you need to probe distances on the order of the Planck length are so energetic that their own gravity would start interfering. Their Swarzschild radius would become bigger than the distance you're trying to measure.
Re: How Many Decimals of Pi Do We Really Need?
#117If 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.
I think this assumes the order of magnitude of the size of the observable universe. Clearly if the diameter of the universe is smaller than a proton then you don't need 43 digits of pi to calculate its circumference to smaller than the diameter of a proton. So if the diameter of the universe is 10^1000000^100000000 proton wide the precision you need for pi would be way higher than 43?
There's a difference between "the size of the universe" and "the size of the observable universe", see also "cosmic microwave background", "dark energy", and "inflation."
Re: How Many Decimals of Pi Do We Really Need?
#118Earlier quoted context omitted.
There were some exercises in our high school physics textbook which required you to simplify a fraction by assuming pi^2 = g.
Another nice one is that there's pi*10^7 seconds in a year. Which is accurate to three decimal places.
"pi seconds are a nanocentury" is attributed to Adm. Grace Murray Hopper and it's surprisingly useful to know.
Re: How Many Decimals of Pi Do We Really Need?
#119Earlier 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.
But pi is not an empirical constant. It is a mathematical constant, with well-defined formulas such as pi/4 = 1-1/3+1/5-1/7+...
Re: How Many Decimals of Pi Do We Really Need?
#12015 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.
BTW, 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.
The 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