4 ms·

Are you suggesting we're already good enough at measuring the mass of atoms and no further precision will ever be needed? Otherwise, your statement would have b

Are you suggesting we're already good enough at measuring the mass of atoms and no further precision will ever be needed? Otherwise, your statement would have been equally valid 100 years ago by just taking some significant figures off. By the way, 4!! ~= N_A was a funny surprise! If you were starting from scratch though, you could make it even simpler and define the unit of mass to be 1e27 times the mass of a hydrogen atom which is 1.7 kg. But again, you might find some other more accurate way and have to redefine it to include an ugly conversion factor just to get practical work done in some high-precision future world.

saalweachter 2y ago
Factorials are a base-independent way to make large numbers. You could go 10^27 or 10^25 or some other random power of ten, but why powers of ten over powers of two? And then once you choose a base, why 10^27 versus 10^25 versus 10^24? Which makes one the more natural choice than another? On the other hand, once you choose factorials over exponents, and furthermore, double-factorials, there's really only one option. 3!! is is 720, which is not really much of anything, in the grand scheme of things. 5!! is something like 7e198, which is probably more than there is of anything in the universe, or at least, in the known universe. 4!! is the only double-factorial which is a useful number of particles. The more annoying thing is that mass and electric charge are so far apart. If you were starting from scratch, it'd be really cool to have 1 number-unit of something be the mass unit, and 1 number-unit of electrons be the charge-unit. But 4!! electrons is like 100,000 coulombs, which is just a lot. 2^64 electrons is more like 3 coulombs, which is more workable, but 2^64 daltons is only around 30 migrograms, which is a helluva mass unit if roughly human-sized is the scale of most intelligent life forms. (Incidentally, powers of two have an even closer coincidence to Avogadro's number -- 2^79 is within half a percent of N_A. But 79 is such an ugly number -- there's nothing particularly elegant about 10^1001111.) Re: measurement, we're presumably going to get better at measuring the mass of a hydrogen atom over time. We might even eventually be able to calculate it from first principles (which is really just saying we might be able to establish an exact relationship between the mass of a proton, electron and various physical constants; according to a random search, a 2008 paper was able to calculate the mass of a nucleon within about 3% of experimental results, which isn't great accuracy wise but is still pretty interesting). (Aside #2: You could also pick a larger particle to try to bridge the gap between mass and charge. The Higgs Boson, for instance, masses around 130x the hydrogen atom.)
- EnigmaFlare 2y ago
  Good point about being natural rather than tied to base 10. Though natural conflicts with human convenience and ultimately is is for humans. Even more natural could be using the same unit for mass and energy, and make it dimensionless!
  - saalweachter 2y ago
    2^64 daltons (30 micrograms) is already about 2 gigajoules. 4!! photons -- specifically the 21cm photons corresponding to the hydrogen line [which, speaking of precision, we have the frequency of that down to around 1 part in ten trillion] -- is about 0.6 Joules. So 4!! hydrogen atoms gets you about 1 gram, 4!! H-I photons gets you about 0.6 Joules, and the wavelength of the H-I photon is around 21 cm.
    - EnigmaFlare 2y ago
      It all seems a bit marginal. Nobody actually uses the definitions to measure anything anyway, least of all by hand or memory, so it's not going to make anything more convenient having fewer digits in the still-arbitrary conversion factors.

saalweachter 2y ago

Factorials are a base-independent way to make large numbers. You could go 10^27 or 10^25 or some other random power of ten, but why powers of ten over powers of two? And then once you choose a base, why 10^27 versus 10^25 versus 10^24? Which makes one the more natural choice than another? On the other hand, once you choose factorials over exponents, and furthermore, double-factorials, there's really only one option. 3!! is is 720, which is not really much of anything, in the grand scheme of things. 5!! is something like 7e198, which is probably more than there is of anything in the universe, or at least, in the known universe. 4!! is the only double-factorial which is a useful number of particles. The more annoying thing is that mass and electric charge are so far apart. If you were starting from scratch, it'd be really cool to have 1 number-unit of something be the mass unit, and 1 number-unit of electrons be the charge-unit. But 4!! electrons is like 100,000 coulombs, which is just a lot. 2^64 electrons is more like 3 coulombs, which is more workable, but 2^64 daltons is only around 30 migrograms, which is a helluva mass unit if roughly human-sized is the scale of most intelligent life forms. (Incidentally, powers of two have an even closer coincidence to Avogadro's number -- 2^79 is within half a percent of N_A. But 79 is such an ugly number -- there's nothing particularly elegant about 10^1001111.) Re: measurement, we're presumably going to get better at measuring the mass of a hydrogen atom over time. We might even eventually be able to calculate it from first principles (which is really just saying we might be able to establish an exact relationship between the mass of a proton, electron and various physical constants; according to a random search, a 2008 paper was able to calculate the mass of a nucleon within about 3% of experimental results, which isn't great accuracy wise but is still pretty interesting). (Aside #2: You could also pick a larger particle to try to bridge the gap between mass and charge. The Higgs Boson, for instance, masses around 130x the hydrogen atom.)

EnigmaFlare 2y ago
Good point about being natural rather than tied to base 10. Though natural conflicts with human convenience and ultimately is is for humans. Even more natural could be using the same unit for mass and energy, and make it dimensionless!
- saalweachter 2y ago
  2^64 daltons (30 micrograms) is already about 2 gigajoules. 4!! photons -- specifically the 21cm photons corresponding to the hydrogen line [which, speaking of precision, we have the frequency of that down to around 1 part in ten trillion] -- is about 0.6 Joules. So 4!! hydrogen atoms gets you about 1 gram, 4!! H-I photons gets you about 0.6 Joules, and the wavelength of the H-I photon is around 21 cm.
  - EnigmaFlare 2y ago
    It all seems a bit marginal. Nobody actually uses the definitions to measure anything anyway, least of all by hand or memory, so it's not going to make anything more convenient having fewer digits in the still-arbitrary conversion factors.