Is the Avogadro's constant equal to one? Question: Is the Avogadro's constant equal to one?
I was tasked with creating a presentation on Avogadro's work, and this is the first time I actually got introduced to the mole and to Avogadro's constant. And, to be honest, it doesn't make any mathematical sense to me.
1 mole = 6.022 * 10^23
Avogadro's constant = 6.022 * 10^23 * mole^(-1)

What?
This hole field seems very redundant. There are four names for the same thing! Since when is a number considered to be a measurement unit anyway?!
 A: Avogadro's number has some historical backgrund, and you can find that in Luboš Motl's answer.  I will explain here, why I think it is still useful today, even it is completely superfluous.
You want to calculate mass of the gas.
$$m = N m_1$$
Mass of gas is number of particles $N$ times mass of one particle $m_1$.  Nice, so what is the problem?  Problem is that first number is absurdly large and the second is absurdly small.  So you define some large number, Avogadro number and you do the following:
$$m = N m_1 = \frac{N}{N_A} (N_A m_1) = n M$$
Now you have product of two reasonable numbers, $n=\frac{N}{N_A}$ is number of moles and $M = N_A m_1$ is molar mass, that is mass of one mole of particles.  This funny unit mole is there just for accounting, so you won't forget that moles must in the end cancel out.
Actually Avogadro's number does not exist as a natural constant at all.  It is essentially 
$$N_A = \frac{10^{-3} \text{kg}}{m_P}$$
where ${m_P}$ is mass of proton, which is real natural constant.  Such definition is very handy, because it insures that all those numbers in periodic table of elements that represent molar mass of atoms are nice round numbers (e.g. Carbon 12, Oxygen 16, meaning that the mass of one carbon atom is 12 proton masses and the mass of one oxygen atom is 16 proton masses).
And thanks all who down-voted me without a comment.  Which only proves you have no arguments!
A: Yes, Avogadro's constant is a redundant artifact from the era in the history of chemistry in which people didn't know how many atoms there were in a macroscopic amount of a material and it is indeed legitimate to set Avogadro's constant equal to one and abandon the awkward obsolete unit "mole" along the way. This $N_A=1$ is equivalent to
$$ 1\,\,{\rm mole} = 6.023\times 10^{23} \text{molecules or atoms} $$
and the text "molecules or atoms" is usually omitted because they're formally dimensionless quantities and one doesn't earn much by considering "one molecule" to be a unit (because its number is integer and everyone may easily agree about the size of the unit). We may use the displayed formula above to replace "mole" (or its power) in any equation by the particular constant (or its power) in the same way as we may replace the word "dozen" by 12 everywhere (hat tip: Mark Eichenlaub). We can only do so today because we know how many atoms there are in macroscopic objects; people haven't had this knowledge from the beginning which made the usage of a special unit "mole" justified. But today, the particular magnitude of "one mole" is an obsolete artifact of social conventions that may be eliminated from science.
Setting $N_A=1$ is spiritually the same as the choice of natural units which have $c=1$ (helpful in relativity), $\hbar=1$ (helpful in any quantum theory), $G=1$ or $8\pi G=1$ (helpful in general relativity or quantum gravity), $k=1$ (helpful in discussions of thermodynamics and statistical physics: entropy may be converted to information and temperature may be converted to energy), $\mu_0=4\pi$ (vacuum permeability, a similar choice was done by Gauss in his CGSM units and with some powers of ten, it was inherited by the SI system as well: $4\pi$ is there because people didn't use the rationalized formulae yet) and others. See this article for the treatment of all these universal constants and the possible elimination of the independent units:

http://motls.blogspot.com/2012/04/lets-fix-value-of-plancks-constant.html

In every single case in this list, the right comment is that people used to use different units for quantities that were the same or convertible from a deeper physical viewpoint. (Heat and energy were another example that was unified before the 20th century began. Joule discovered the heat/energy equivalence which is why we usually don't use calories for heat anymore; we use joules both for heat and energy to celebrate him and the conversion factor that used to be a complicated number is one.) In particular, they were counting the number of molecules not in "units" but in "moles" where one mole turned out to be a very particular large number of molecules.
Setting the most universal constants to one requires one to use "coherent units" for previously independent physical quantities but it's worth doing so because the fundamental equations simplify: the universal constants may be dropped. It's still true that if you use a general unit such as "one mole" for the amount (which is useful e.g. because you often want the number of moles to be a reasonable number comparable to one, while the number of molecules is unreasonably large), you have to use a complicated numerical value of $N_A$. 
One additional terminological comment: the quantity that can be set to one and whose units are inverse moles is called the Avogadro constant, while the term "Avogadro's number" is obsolete and contains the numerical value of the Avogadro constant in the SI units. The Avogadro constant can be set to one; Avogadro's number, being dimensionless and different from one, obviously can't. Also, the inverse of Avogadro's number is the atomic mass unit in grams, with the units of grams removed. It's important to realize that the actual quantities, the atomic mass unit (with a unit of mass) and the Avogadro constant are not inverse to each other at all, having totally different units (when it comes both to grams and moles). Moreover, the basic unit of mass in the SI system is really 1 kilogram, not 1 gram, although multiples and fractions are constructed as if 1 gram were the basic unit.
A: Yes it's a little odd to have a unit of 'amount'. At least in English, it might make more sense in other languages
The second line is really "Avogadro's constant = 6.022 * 10^23 * items/mole"
A: Unfortunately, I can't post comments, so I have to write it this way. The statement 1 mole = 6.022 * 10^23 (which you use to show that N_A=1) simply doesn't hold - at least it didn't hold when they taught chemistry in my class. 
It's $1 \text{ mol} = 6.022\cdot10^{23}/N_A$, isn't it?
P. S. And what exactly did I do to get my -1?
