Is Avogadro's constant closer to infinity than 10?

Question

Daniel Schroeder in his book, Introduction to Thermal Physics, mentions a quote,

It all works because Avogadro's number is closer to infinity than to 10.

—Ralph Baierlein, American Journal of Physics 46, 1045 (1978).

I don't understand how this is true. I know that Avogadro's constant is a huge number (23 orders of magnitude bigger than 10), but infinity is, well infinity. Infinity is more than than 23 order of magnitude away than Avogadro's constant. So, how can Avogadro's constant be closer to infinity than 10.

I also think that 10 is as close to infinity as the Avogadro's constant is. What am I missing in my process of thinking?

Answer 1

From a purely mathematical point of view you are correct ( though mathematically one could question what 'closer to infinity' would even mean given that infinity isn't even a point on the number line... and are we talking about some 'absolute difference' or 'relative difference' or ...)

But I doubt the author intended to make a precise mathematical claim here with a precise meaning. Rather, I suspect the author is making the claim for some conceptual or practical purpose. That is, in terms of practical real life scenario or situations, I can understand why treating the constant as 'closer to infinity' would make (some) sense, given that so many things in the real world, when quantified, are closer to something like $10$ than, say $10^{20}$. Hence, relative to the typical numbers we encounter in real life, you might as well treat numbers like the Avogadro's constant as a 'really big number', and maybe even see it as a kind of 'infinity' or at least something 'closer to infinity' than to $10$.

But obviously, you have to be really, really, really careful with doing such a thing!! Ultimately it all depends on how you are going to use this: in some situations it might be an ok move to think about things by treating some really big number as 'closer to infinity then to $10$' (not just conceptually, but even for some kind of physical explanation of prediction), but in other situations that can go really wrong!

Answer 2

Too much for a comment, so here goes:

Googling the quote gave me:

This statement is not necessarily true, but it may as well be, which is what Baierlein was hinting at. The reason we can accurately use entropy to describe spontaneity is that outliers are rendered obsolete due to the sheer quantity of the "sample size," or the number of molecules in this case, and, consequently, the "true" macroscopic quantities of a system are consistent from one experiment to the next to the real world.

So it is not "closer to infintiy", but just reeeeeeeeeally big...

I would compare it to thinking about Graham's Number. Some time ago I talked to first mathematic year students about "big numbers". And while everything is relative, some numbers are just "known" to be big and everyone has encountered such big numbers in his life. What qualifies as a big numbers changes with time, for a kid in the 2nd grade the biggest number to exist might be 1000, for someone in the 7th grade it could be 1000000, and while we all know that 100000 can still be small in some context, the number itself is big. Following that I tried to explain to my students, just how big Graham's Number is, so I introduced the arrow notation, gave some examples where I actually calculated $3\uparrow\uparrow\uparrow 3$ (or at least tried to calculate) and so on. In the end, we came to the conclusion that $g_{64}$ is an abnormal monster of a number (while not infinity yet).

One more point from a "theological" point of view: knowing that I might have to live forever after I die became much more frightening when I learned about Graham's number. Imagine doing one thing for $g_{64}$ minutes without ever getting bored of doing whatever it may be. Now imagine doing (any number of activities) for all eternity...