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Re: dimensionless units



Regarding Michael's comments about %:

If a student did a calculation and used a mass of 3.0 kg, then we suggested
to do the calculation again but multiply the mass by 50%, they would not
multiply 50% times 3.0 kg and come up with 150.0 %kg [alas, some would, but
most would not].

This result (i.e. 150 %kg) may look unconventional but it *is* 50% of 3 kg.
We can think of %kg as a weird metric unit which is equivalent to a Dg
(dekagram) once we interpret %kg to mean "percent of a kg". Thus,
150 %kg = 150 Dg = 1.5 kg which *is* 50% of 3 kg.

Or thinking in reverse, they know that 1.5 kg / 3.0 kg is
0.5 (dimensionless) but we multiply by 100 and tack on percent and call it
50%.

In essence we can think of the % to be a particular dimensionless unit
for numerical quantity like a dozen (or a maybe a mole; more on this
later) except that the percent unit is a fraction being less than 1
rather than an integer multiple unit like dozen, score, and gross are.

John gave a list of a few of these dimensionless units. Is that an
exhaustive list? Can others think of better or other examples?

No, it's not at all exhaustive. How about the examples here of %, doz,
gross, score, etc.? Also, don't forget all those units from elementary
particle physics, i.e. fermion number, baryon number, etc. and the units
of flavors like strangeness, charm, topness, bottomness, upness,
downness. (We aren't at all limited to the various dimensionless units
of fluid mechanics.) In information theory we have bits, bytes,
Gbytes, etc. not to mention the shannon as a unit of info theoretic
entropy (essentially a bit of entropy). In cosmology there is the OMEGA
quantity which measures the total mean mass/energy density of the
universe in units of the amount it would take to make the universe
spatially flat while expanding at the rate we observe it to expand today.
Often this OMEGA parameter is separated into a sum of 2 different
contributions--one from the actual matter and radiation present
(OMEGA_sub_m), and the other from the cosmological constant
(OMEGA_sub_lambda) which can sort of be thought of as the energy density
of the vacuum. There are scads of other dimensionless units extant. In
geometry we can generalize the concept of angle and solid angle to be a
generic "hypersolid" angle measure as relative measure's of a sector of a
hypersphere in a higher dimensional space. Actually, if we formulate our
theories of physics using the so-called Planck units, it ends up that
*all* the physical quantities of physics are dimensionless, since that
system of units has *no* base quantities at all. In this set of units
the fundamental constants of nature: c, G, h-bar, and k (Boltzmann's
constant) are all made to be equal to the pure dimensionless number 1.

I'm sure I could come up with many more examples of dimensionless units
if I tried to.

Regarding Herb's comment:

By the way, the number of atoms in an object is a dimensionless unit too.

True, a number count is usually thought to be dimensionless. But this
brings up another weird feature (I would call it a bug or defect) of the
SI system of units. This relates to the concept of a mole. In the
SI system the mole is taken as a *base unit* for the basic quantity
called "amount of substance". In doing dimensional analysis with the
base quantities given by those of the SI system we are to consider the
number of moles of a substance to have the dimension of 1st power of
"amount of substance" rather than to be just a dimensionless quantity
like a dozen or a gross or a percent. This is really wierd considering
that the "amount of substance" is just a number count of entities
present divided by Avogadro's number (being the number of C-12 atoms in
exactly 0.012 kg of C-12 in their ground state).

BTW, Avogadro's number N_A which defines the mole is a weirder numerical
quantity than other units of aggregate amount such as the dozen or the
gross. For these other familiar units of amount (dozen, gross, etc.) the
size of the unit is completely independent of any other choice of units
for our system. (The "size" of a dozen does not depend on how long a
meter is, for instance.) But not so for N_A which numerically depends on
the actual size of the standard artifact kilogram(s) which define the SI
unit of mass. So even though a mole just measures an amount of substance
as the (dimensionless) number of particles present divided by N_A, its
"size" depends on the "size" of the kilogram unit because we need to know
how large a kilogram is when we count the number of C-12 atoms in
0.012 kg of C-12.

I once asked a chemist collegue here at my college here why chemists are
so fond of moles & why they just don't measure the numerical quantities
of molecules by the actual number of molecules present, rather than
arbitrarily dividing that number by 6.02214199(47) x 10^23 and calling it
the number of moles present. It's not like chemists haven't heard of
scientific notation to handle the big number of particles present. (In
fact we could just give a nice Greek-named prefix to a factor of 10^24
(something better than tera tera) and append it to the number count if
they don't like the big exponent involved when quoting the numerical
amount of stuff present in a macroscopic sample.) I never got much of a
straight answer from him other than the preference being the inertia of
historical precedent.

All this N_A wierdness is, IMO, just another conceptual blot on the SI
system. The SI system is conceptually good in dealing with mechanical
quantities, but it is very atrocious in its treatment of electromagnetic
quantities, and it muddled treatment of "amount of substance" is not
much better. At least, as bad as the SI system is, it is a whole lot
better than the US customary system (which, BTW, for some reason, adopts
the *worst* part of the SI system by using the SI electromagnetic
quantities as its own).

David Bowman
David_Bowman@georgetowncollege.edu