Pi approximation games

Hmm.. true, but I think I ned pi more often than the number of my fingers, so I think it would be a win.

-jm

I thought that the PC term for "Murricans" is "USAnians".

Stephen

Yes, it is pretty easy to write a program to calculate e. e has the value 2.11111111... in factorial radix. The first digit after the radix point is in the 1/2 place, the next is in the 1/6 place, and so on. All you have to do to print out the value of e is convert to base 10.

Scott

(snip)

Not necessarily more efficient, but you can compute the Nth hexadecimal digit of pi without computing the N-1 digits in between.

One use is for testing a computation using a different algorithm.

There is a web page by someone who has computed trillions of decimal digits, doing the computation in binary and then converting to decimal. At various points, the result is compared against hex digits computed using the other algorithm.

-- glen

As far as I know, it is that no-one has yet found the algorithm that expands to decimal digits. It is convenient for users of binary computers that there is one for base 16. (One could have been less lucky, and had it come out in base 13.)

Pi is a little unusual, in that there are many different ways to compute it. It might be that square roots could have a similar algorithm, but then, as I remember it, they are just irrational but not transcendental.

There are some irrational numbers that have nice simple definitions, such as ln(2). Roots of Bessel functions might be another interesting case to look at.

-- glen

Actually, the US has been officially metric since 1866. At that time, the inch was defined as 1/39.37 meters. (The standard meter was then the distance between two marks on an iron -- later platinum-iridium -- bar in Paris.) That made the inch 25.40000508001 millimeters long.

In 1960 the meter was defined as 1,650,763.73 wavelengths of the orange-red line of krypton 86, and the inch was redefined to be 25.4 mm exactly about a year earlier. (I saw no need to replace my micrometer.) In order to maintain the integrity of old geodetic records, The old foot is now named a survey foot. A survey foot is 1.000002 modern foot. Life is messy.

Jerry

Eh? Suppose a bag of shot with each piece weighing an ounce. An integer number of pieces will surely not fill any arbitrary volume.

Jerry

We're not so far apart. I often use 25 (~ circumference of earth in kilomiles) / 8 (~ approximate diameter).

Jerry

A lump of pure silicon must, by definition, contain an integral number of silicon atoms.

Not quite... Certainly there should be an integral number of atoms in any physical object. However, it is very likely that the present SI kilogram (which is platinum, n'est pas?) may not equal an integral number of Silicon atoms.

It's surely not completely platinum. The purest platinum contains a ppm or so of similar metals (rhodium, palladium etc.). One ppm is ~8x10^15 atoms, rather a lot

Also Pt has five stable isotopes.

If we ever gain the technology to make perfect stacks of absolutely pure atoms, something like Beryllium might be a better choice.

Best regards, Spehro Pefhany

I have a 30's-era CRC handbook which confirms that the Amer-inch and Brit-inch were then different, and bracketed the value of the Modern Inch.

Steve

I haven't hit a case with Octave where it left me needing MATLAB... but I don't use it very heavily.

-- Les Cargill

No. It stands to reason that Avogadro's number is not only not integral but irrational at truly arbitrary levels of precision. To make the arithmetic slightly easier to grasp consider for a moment an alternative reality where atoms are rather more massive and a C-12 atom has a mass of exactly 5g. A mole of C-12 MUST have a mass of 12g and so a mole works out at 2.4 atoms.

In reality the mass of an atom is almost infinitely less, but there is still the problem that n C-12 atoms have a mass of 11.9999...9998712g but n+1 atoms have a mass of 12.0000...0003425g. There's no reason at all to suppose that an exact integral relationship must exist. In practice of course you could round off to the nearest atom or even tens of billions of atoms with no real-world impact at all, but that is not how Avogadro's number is _currently_ defined.

Too bad the binding energy amounts to more than several atoms' mass, and likely isn't an integral number.

The mass of a silicon atom is around 27GeV/c^2, so at an average binding energy of 2eV or so, it only takes 1.3e9 atoms in a pile (a grain of about fractional um size) to accrue an extra atom's worth of mass-energy.

They could define it in terms of mass at infinity (unbound) per atom, but that doesn't make much sense when you have a polished sphere or whatever containing a few faraday volts of 'excess' binding energy.

Tim

-- Deep Friar: a very philos>>

I assume they include the mass associated with binding energy when they do the calculation.

Steve

Ooh, good point. but that still only gets us to fractions, not irrational quantities*.

So let us define a unit of mass - the Andrew - to be exactly one integral-Avogardo's-number of C12 atoms - we'll pick a value for Advogadro's Number and round down. Doesn't matter which, really.

For any given object, there must be a rational relationship between the mass of that object expressed in kilograms and expressed in Andrews.

That's not quite proof by contradiction, but it's the schematic of one...

I cheated and looked - Avogadro's Number is good to 6 decimal places. :) We are so deep in the noise... still, the processes that produce Advogadro's number are inherently about counting, so I maintain that it is at least rational.

-- Les Cargill

OK, but using that argument you would have a hard time explaining why the speed of light is an integer number of m/s.

Until you can measure it more accurately than +/- 1, which is one part in 6.02e23, you might as well round to an integer.

But more practically, you only have to do it more accurately than the current standard, which is way worse than 1 in 6.02e23.

No, but the idea is to redefine the mass standard in terms of Avogadro's number. Similar to the way length is defined in terms of time and the defined speed of light.

-- glen