btw Pi = 4 (in taxicab geometry aka L1)
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
;)
How Many Decimals of Pi Do We Really Need?
51–60 of 134 posts
Re: How Many Decimals of Pi Do We Really Need?
#52Maybe the new decimals are information from beyond this realm. Thanks, Sagan.
Re: How Many Decimals of Pi Do We Really Need?
#53If 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.
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?
Re: How Many Decimals of Pi Do We Really Need?
#54Earlier quoted context omitted.
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?
It does depend on the size of the observable universe, which we do know because, well, it's observable.
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.
Re: How Many Decimals of Pi Do We Really Need?
#55i 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.
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
(i) https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_th...
.. edit, percentage error, left out the *100
Re: How Many Decimals of Pi Do We Really Need?
#56Earlier quoted context omitted.
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…
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 the real world with limited time and money.
It is the same difference I see between athletes and ergotherapists, or Einstein and Poincaré.
Re: How Many Decimals of Pi Do We Really Need?
#57Earlier 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?
If you have some time to spare, watch this talk "The astonishing simplicity of everything" by Neil Turok. He explains all this so beautifully.
Re: How Many Decimals of Pi Do We Really Need?
#58Earlier quoted context omitted.
I can't speak for other nations, but they still teach numeric analysis in Chinese universities as an undergraduate course. In my university it is a required subject. Many of us have countless dreadful memories of Runge-Kutta method, Euler's method, Newton's method, rate of convergence, numerical stability and error margins, just to name a few of the dreads...
And this ? https://zh.wikipedia.org/wiki/%E6%91%84%E5%8A%A8%E7%90%86%E8... (pt https://pt.wikipedia.org/wiki/Teoria_de_perturba%C3%A7%C3%B5... ) (fr https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_perturbations )
Yes, perturbation methods are still taught and are still recognized as important. The first math course I took during my graduate degree (Intro analytic methods) covered it, for example.
Re: How Many Decimals of Pi Do We Really Need?
#59Earlier 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?
> According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.
At least according to Wikipedia, it seems it is indeed the smallest observable distance. Although it has never been proven and follows from the theoretical generalized uncertainty principle. https://en.m.wikipedia.org/wiki/Planck_length#Theoretical_si...
Re: How Many Decimals of Pi Do We Really Need?
#60Earlier 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?
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