Pi approximation games

Tim Wescott · 2012-05-01T23:16:25+00:00

Instead of doing productive work, I just spent a few enjoyable minutes with Scilab finding approximations to pi of the form m/n.Because I'm posting to a couple of nerd groups, I can be confident that most of you probably know 22/7 off the tops of your heads.What interested me is how spotty things are -- after 22/7, the error drops for a bit until you get down to 355/113 (which, if you're at an equal level of nerdiness to me will ring a bell, but not have been swimming around in your brain to be found).But what's _really_ interesting, is that the next better fit isn't found until you get up to 52163/16604. Then things get steadily better until you hit 104348/33215 -- at which point the next lowest ratio which improves anything is 208341/66317, then 312689/99532. At this point I decided that I would post my answers for your amusement, and get back to being productive.Discrete math is so fun. And these newfangled chips are just destroying the joy, by making floating point efficient and cheap enough that you don't need to know little tricks like pi = (almost) 355/113.-- My liberal friends think I'm a conservative kook.My conservative friends think I'm a liberal kook.Why am I not happy that they have found common ground?Tim Wescott, Communications, Control, Circuits & Software

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robert bristow-johnson 14 years ago

that's two different ways of defining the kilogram. but if you toss in other constants that are presently measured, you can fix both N_A and hbar. (not saying it all would be good.)

ir irridium. but i dunno how they would count the atoms.

nonetheless, it's because a Cavendish-like apparatus still measures G relatively poorly, that i don't ever in my lifetime expect units to get defined in terms of such a machine. but i *would* expect the watt-balance to become part of the unit standard (and replace the IPK).

if only we knew the mass of the Earth to a high degree of accuracy.

doesn't stop some folks from pushing "varying c" theories. of course they need to revert the definition of the meter to something pre-1960 to even speak of such.

L8r,

r b-j rbj@audioimagination.com "Imagination is more important than knowledge."

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Fred Abse 14 years ago

What did he/she teach?

History? drama? civics? languages? phys.ed?

"For a successful technology, reality must take precedence over public relations, for nature cannot be fooled." (Richard Feynman)

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Fred Abse 14 years ago

Neither have I.

Interestingly, the command line switch to start Octave for best Matlab compatibility is --braindead.

Octave script to calculate pi to the limit of its precision by integrating a circle:

1; output_precision (20) function y=f(x) y = sqrt (1 - x^2); endfunction [v, ier, nfun, err] = quad ("f", 0, 1); v*4

I first did that 30-something years ago on an HP41 programmable calculator. It took many more lines, including the numerical integrator, and took over an hour to run ;-)

"For a successful technology, reality must take precedence over public relations, for nature cannot be fooled." (Richard Feynman)

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John Devereux 14 years ago

Oh, everthing. Just a teacher for 9-10 year olds. She was actually a good teacher in general, but even at that age I knew it was not true.

John Devereux

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David Brown 14 years ago

True.

Absolutely.

As has been noted already, this discussion is pretty pedantic. All I am really saying is that the theoretical "true" values - that you can never measure, even if they can be said to exist at all - are irrational. In particular, I am saying that you can't call them "rational" nor "integer", unless of course your units are defined to make them rational.

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Jerry Avins 14 years ago

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You are almost right. The possibility exists that a pair of irrational numbers have a rational ratio. Sqrt(8)/sqtrt(2) is an example.

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Jerry

Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

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Steve Pope 14 years ago

Hence the qualifying term "unrelated". These two number are related.

Steve

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Jerry Avins 14 years ago

Numbers obtained by measurement rather than counting necessarily describe a range. There is an infinite count of numbers within that range. If you must describe measurements in terms of "rational" and "irrational", the correct description is "both".

Jerry

Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

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David Brown 14 years ago

If the numbers are unrelated, then the probability is /really/ small - it is infinitesimal.

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Mac Decman 14 years ago

Thanks Tim, (I've cross posted just for fun) for bombing out my usenet with this thread. I can't believe this was the one which was better than any troll post!

Mark DeArman

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josephkk 14 years ago

number

This is the first time i have seen virtual particles extended to complete nuclei with associated mass rather than the usual boson and lepton = virtual particles.

?-)

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Steve Pope 14 years ago

Cool. You heard it from me first. If the lump of silicon is large enough, this should happen.

Someone who is actually a physicist can tell you how large. :--)

Steve

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Przemek Klosowski 14 years ago

One of the candidates for kilogram artifact replacement is a perfect silicon sphere

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The idea is that we'll be able to count atoms in it, thus the accuracy. Common silicon isotopes are stable, so no problem there. Also, pure isotopic materials aren't that rare---isotopic separation not difficult, just tedious and requires prodigious amounts of electrical energy. Russians used to do it next to hydro power plants; my ad visor claims to have bartered a fancy electronic boxcar averager for a piece of isotopically pure cadmium that his Russian counterpart just pulled out of his desk and cut with scissors (probably would be worth a lot today).

By the way, both kilogram and the original meter were made out of Platinum-Iridium alloy.

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Ignacio G.T. 14 years ago

Wow. And given a whale big enough, do whole cells pop in and out of existence?

Saludos. Ignacio G.T.

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Pi approximation games

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