Pi approximation games

As a physicist I found the classic approximations

pi^2 = g pi x 10^7 seconds in a year

quite handy to within 1% slide rule accuracy

Dunno what it is like now, but arctan(1) was a risky choice on some old machines as the series convergence was at its worst for that argument value and the tradeoff between accuracy and speed could cause trouble. You were at the mercy of the trig library if you did it this way.

Wikipedia is often a great starting point for these sorts of things. It typically has enough information to give you some hints - but not so much that you can't have fun finding out more:

At university I remember a project that involved calculating all the digits of pi. It was written using a functional programming language (similar to Haskell) - the result was an unending list of the digits of pi. But since the language used lazy evaluation, it didn't bother calculating the entries until you tried to print them out. I used polynomial expansions of arctan() to do the sums.

So you are saying the inch is in fact metric?

David Brown schrieb:

Hello,

it is known since centuries that calculating all the digits of pi is not possible. Pi has an infinite number of digits.

Bye

Boasting again? ;)

Most imperial units are defined in terms of the metric units these days. Originally they were only rough definitions (I believe an inch was variously defined as the length from a thumb joint to the end of the thumb, or alternatively as the length of three grains of barley). Then they were a bit more standardised (such as the length of a particular metal rod). But now they use specific metric definitions - so an inch is precisely 25.4 mm - and will stay that way even if the definition of a millimetre varies!

The most relevant section, "working with infinite data structures", is missing - but I hope you get the point anyway.

mvh.,

David

Which is why a 1m pendulum has a half period of ~1 second using the classic formula two pi root ell over gee or:

l g / sqrt pi * 2 *

in RPN

Cheers

That is standard fare in continued fractions. Everybody interested in these kind of approximations should take a look at this fascinating subject.

It depends. Pi has been calculated to billions of decimals. Simple floating point doesn't get you there.

Groetjes Albert

--

There was a short PDP-8 assembly program that printed the digits of e forever.

In order ?

Mel.

ng

Grin... I always just let 2*pi =3D 10, so pi =3D 5!

(and then remember there's a 1.59 floating around)

George H.

ed text -

David Brown schrieb:

Hello,

even with infinite data structures, with finite time and finite RAM, it is not possible to compute all digits of pi.

Bye

That's where David Brown blew it -- when he wrote "it didn't bother calculating the entries until you tried to print them out". Without that, you can fall back on the idea that logical truths exist a priori, and argue that what's impossible is to *print* all digits of pi.

There's an argument about LISP that goes: it's easy to write a LISP interpreter in LISP such that the source code can be printed on a sheet of paper; it's also easy to quine that source code so that it's applied to itself; then you have a LISP interpreter executing on a sheet of paper. I/O bandwidth is the only problem. Maybe with e-paper ..

Mel.

Grin... I always just let 2*pi = 10, so pi = 5!

(and then remember there's a 1.59 floating around)

George H.

________________________________________

Shoulda been a congressman.

meter (m) "The metre is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second."

kilogram (kg)

"The kilogram is equal to the mass of the international prototype of the kilogram."

second (s)

"The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."

Note that the kilogram is now the only one defined by a physical object. ("International prototype").

Let's define pi as in Euler's identity: e^i*pi + 1 = 0

See:

"Meanwhile, only a few Americans know that the legal definitions of the English customary units are actually based on metric units. The U. S. and British governments have agreed that a yard equals exactly 0.9144 meter and an avoirdupois pound equals exactly 0.453 592 37 kilograms. In this way, all the units of measurement Americans use every day are based on the standards of the metric system. Since 1875, in fact, the United States has subscribed to the International System of Weights and Measures, the official version of the metric system."