Pi approximation games

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.

*although pi is *defined* as a ratio... I don't see how anything like that applies here.

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

Reply to
Les Cargill
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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

Reply to
glen herrmannsfeldt

No, it gets us to an arbitrary number, rather than a specific integer or other relationship. So all we know for sure is that it is a real number. Almost all real numbers are irrational - rational numbers only turn up in the real world if there is a particular reason (for example, orbital periods of planets' moons are often related by rational numbers).

Of course, it doesn't make much sense to talk about the irrationality of a measured number, since measurements are by definition limited in precision, and irrationality is a property of the pure number. It may also turn out that it is not constant - perhaps it varies gradually with the expansion of the universe or the strength of surrounding fields.

Without any explanation or definition otherwise, however, Avogadro's number is like any other arbitrary number - irrational.

That would work...

Nope.

Unfortunately, atomic masses don't work that way. A C12 atom weighs more than 12 times an H1 atom - but not an integral (or rational) amount more.

One of pi's many definitions is the ratio of a circle's circumference to its diameter. But that is completely unrelated to the definition of a "rational number", which is the ratio between two /integers/.

I agree here - it's not /quite/ a proof...

Reply to
David Brown

Thanks for this - I enjoyed reading the original paper.

--
Regards,
Martin Brown
Reply to
Martin Brown

Sure.

But it's *defined* as a number of atoms. That's the point. If it's an indeterminate quantity, it makes no sense to describe it as either real nor irrational.

Yes - the Andrew is specified by an natural number of C12 atoms. In a set-theoretic sense, it's really an integer, and They are going to rejigger the kilogram to make that happen... using... SILICON! :)

As they say, the natural numbers are God's work, and all the rest, man's. That's toungue in cheek, but ...

In that case, the thing is Avogadro's Vector, and the whole concept* is silly because it doesn't mean anything...

*it's only useful because there's a limit to the resolution of its measure - which is entertaining, but hardly the sort of thing to found a science on.

Meanwhile, the kids in Chem II are merrily mol-ing away...

-- Les Cargill

Reply to
Les Cargill

That would assume a standard (crystalline?) structure, no? Many atoms have several stable forms (carbon is a good example; graphite, diamond, etc.).

Reply to
krw

It is defined as the number of C12 atoms in 12 grams. But that does not make it an integer. If you define the "Foot Number" to be the number of centimetres in a foot, you will find it is not an integer. It's the same thing here.

There is a proposal to change the SI definition of a "mole" so that Avogadro's constant is /exactly/ 6.02214e23. Then it will be an integer. But N_A C12 atoms will no longer weigh exactly 12 grams.

You are inventing a problem that doesn't exist, by trying to think of Avogadro's number as an integer or a count of atoms. It is the ratio between 12 grams and the weight of a C12 atom - that's all.

You can happily take N_A atoms of Hydrogen (i.e., a mole of H) - but it weighs approximately 1.00794 grams, rather than exactly 1 gram. Even if you keep your sample as pure H1, it is still a little over a gram.

The only reason the numbers work out nicely for C12 is that the units of atomic mass are defined as one 12th of the mass of C12.

Reply to
David Brown

That is true by definition. The metre is defined as the distance travelled by light in vacuo in 1/299,792,458 second. Converting that back to light speed is simply taking the reciprocal of that number.

The current _definition_ of Avogadro's number does not such lead to such an exact integral relationship. As both you and I noted we could redefine _something_ - Avogadro's number itself, or the kilogram - to allow for such an exact relationship with no real-world impact, since it is so far beyond our current ability to measure. That doesn't alter the fundamental irrationality of the constant as it is _currently_ defined.

--
Andrew Smallshaw
andrews@sdf.lonestar.org
Reply to
Andrew Smallshaw

Then it's a *fraction*. It's not irrational... I thought I had adequately corrected "integer" to "rational"... in truth, there's a bijective map between them, so...

Right right right.

I am being modestly pedantic.

-- Les Cargill

Reply to
Les Cargill

The astronomer Edmund Halley used 6*ATAN(1/SQRT(3)). Writing in 1705 he gives the result to 12 decimal places. He says this took him half an hour. He had a table of square roots so he had the first term at once but then he must have been able to carry out division in less that a minute. I cannot get anywhere close to his time.

J.C.B.Sharp London

Reply to
J.C.B.Sharp

(snip)

It will if you redefine the gram, which is the whole reason for the project.

Note again what happened to the speed of light:

Previously there were standards for length and time, both with about the same relative error, and it was possible to measure the speed of light to about that same error.

At some point, it became possible to do optical frequency counting (slightly indirectly, but close enough). At that point, it became possible to measure time with a much much smaller relative error. (More or less, an integer number of cycles per second for visible light.)

Once that happens, and no corresponding change in distance the distance standard, the fix is to define the speed of light, and then redefine distance in terms of time and the speed of light.

Note that as a result the speed of light is now an integer,

299792458, in meters/second, by definition. The meter is now the distance light travels in 1/299792458 seconds.

The new definition has to be within the uncertainty of the previous definition, so it won't invalidate any previous measurement.

In the early days of spectroscopy, it became possible to measure wavelengths with less relative error than the measured speed of light. To keep the low error, reciprocal wavelength (wavenumber) was used instead of frequency, for example when plotting spectra.

-- glen

Reply to
glen herrmannsfeldt

You are /still/ missing the point - just because you divide two numbers, does not make the ratio a /rational/ number. It is only /rational/ if the two numbers are integers (or at least rationals). If the numbers are unrelated "random" numbers, such as different physical measurements, then their ratio will be irrational because there is no fundamental common unit of measurement.

It's simple probability. (Actually, it's quite hard probability to do this stuff rigorously - but the layman's "it works for finite cases, so we extend it to infinite cases" is good enough for now.) There are /many/ more irrational numbers than rational ones. In any given range, there are only countably infinite rational numbers. But there is 2 to the power that many /real/ numbers, so they vastly outweigh the rationals. Thus if you pick any two random numbers and divide them, the result will be irrational.

I am not sure about the "modestly", but I am certainly being pedantic.

mvh.,

David

Reply to
David Brown

I gather that is not the case (judging solely from the Wikipedia article):

To make N_A C12 atoms exactly 12 grams, you would need to redefine the gram (as you say) to make that the case. But that is not the plan (they are planing to redefine the gram, but to a different value - see earlier in the same article).

Reply to
David Brown

(snip, someone wrote)

In the case of measured quantities, there is no point in saying that one is irrational. That is, quantities that have an uncertainty.

Now, that doesn't mean that physical constants can't be irrational.

The SI mu-nought, by definition is 4e-7*pi H/m (or a few other combinations of units).

Since epsilon nought is 1/(mu nought c squared), it is also irrational.

Now, I could ask you to cut a stick that is pi inches long, but, given the finite diameter of the atoms that the stick is made from, there is zero probability that you could do it. (That is, limit as probability approaches zero.)

-- glen

Reply to
glen herrmannsfeldt

Ah, okay. Right. I was being thick. Thanks for your patience.

Right.

yeah, I got you now.

-- Les Cargill

Reply to
Les Cargill

real numbers are the set of numbers containing all the rational and irrational numbers, most of real numbers are irrational by a factor of some infinite amount more than the rational numbers.

it makes sense to describe any real number that is not known to be rational as irrational.

--
?? 100% natural

--- Posted via news://freenews.netfront.net/ - Complaints to news@netfront.net
Reply to
Jasen Betts

(snip, I wrote)

From

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it looks like the Planck constant based Watt balance is winning the race.

It seems that the goal is a relative uncertainty less than 2e-8, which they are getting close to, but haven't yet reached.

-- glen

Reply to
glen herrmannsfeldt

Right - I was ignoring the fact that it's arbitrary. DERP! :)

-- Les Cargill

Reply to
Les Cargill

What doesn't make sense is achieving that -- even by purely thermodynamic arguments, many crystals have necessary disorder, even at absolute zero; at the temperatures required to manufacture silicon, there is a well defined minimum number of vacancies.

Chemical purity becomes a somewhat bizarre thing. Semiconductors routinely operate with parts per billion impurities, but we're talking parts per ~10^-23. Seems to me, regardless of what method you use to synthesize the thing, you are going to trap, bind, dissolve or adsorb anything. Even the best vacuum in the world contains enough stray hydrogen atoms to adsorb onto the surface, and even if the surface is an atomically polished sphere, it has more than a few atoms worth of adsorption potential.

Very pure silicon is produced by distillation of SiCl4, which is thermally decomposed to yield polycrystalline silicon (or SiH4 is), which is in turn remelted and grown into single crystal boules. Evidently chlorine is volatile enough that it doesn't remain dissolved to an appreciable extent, but everything is appreciable in this extreme. Czochralsky can only be used once as the melt dissolves oxygen from the silica crucible (evidently, the few ppb is actually beneficial in its typical use, trapping impurities or something like that). I wonder just how much it is. Once crystallized, a boule could be zone refined, producing incrementally higher purity samples. Other methods of deposition and crystallization could be used, but I don't know what other processes are very selective. After zone refining, the physical shape would have to be polished, which I suppose would be easiest to define as a sphere, with diameter some integral number of wavelengths of Cs137 ground state transition, or a spectral line of Rb or Kr, and within as fine a tolerance as possible. Even so, what they got,

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was within "only" 300 picometers, which is maybe 5 atoms worth. As interferometry goes, that's not bad: visible light is in the 500nm range, so they're doing less than 1/1000th of a fringe band.

Tim

--
Deep Friar: a very philosophical monk.
Website: http://webpages.charter.net/dawill/tmoranwms
Reply to
Tim Williams

i thought they were redefining the molar mass of C-12 so that Avogadro's number is fixed and known. i am looking at

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and maybe that article is full of crap. i really hope they *don't* change the definition of mu_0 (let the electron charge remain a property of a particle or "thing" which is known as well as we can measure it, just like it is for the properties of mass or spin or radii of particles.

i think that the closer they get to Planck units (with some scaling tossed in) the better. with c and mu_0 fixed, and with hbar and Boltsmann k_B getting fixed, that would leave only G (which is known to sloppily nowadays, so they would still have to define the second in terms of Cs-133 rather than G, until they can get G down to 2e-8. if they did do all of that (redefine units so that all of c, mu_0, hbar, k_B, and G are fixed), then these units would be given multiples of Planck units. then we could stop hearing about this "variable c" or "varying G" crap. (why doesn't anyone pick on hbar? why no theories about a varying Planck's constant?)

--
r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."
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Reply to
robert bristow-johnson

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