Is there a symbol for a quantity that we cannot count but is not infinite?

Question

So let's say I wanted to express mathematically what fraction of the sand on the earth is currently in my hand.

Obviously it would be like 1 over some very very huge number as there is tons of sand, but more importantly we know it is finite and is unmeasurable due to quantity and the constant creation and destruction of sand.

I know that there is a symbol for a very small quantity (epsilon), but do we have one for a very large quantity?

Answer 1

If $x$ is a very small positive number that you know is less than $1$, you could perhaps express this as $$x\ll 1$$ where $\ll$ (LaTeX code: \ll) signifies "much less than". Similarly, you could indicate a large positive number $$x\gg 0$$ where $\gg$ (LaTeX code: \gg) signifies "much greater than".

Answer 2

"Constant creation and destruction of sand" is describing something less like a number (finite, small, large, infinite, whatever) and more like something that isn't a number. Perhaps it describes a process, but other things, like the color blue and the feeling you get when you listen to Beethoven, also are not numbers, so I'm going to ignore that part of your question.

Things like $\epsilon$ are highly contextual conventions mathematicians use. Usually $\epsilon$ comes up in calculus proofs when the surrounding argument is, "no matter how small some number $w$ is, some other condition holds for some number $z$ anyway..." except we would usually say $\epsilon$ instead of $w$ because every mathematician in the world understands that better.

For large numbers, sometimes $N$ is used. $n$ and $N$ usually are integers, and an arbitrarily large number may as well be an integer, so $N$ is used.

If you want to follow the $\epsilon$ convention, you could invert the quantity, and say "the fraction of the world's sand I am holding is pretty much $\epsilon$." Obviously this isn't a very mathematical claim, but it makes sense enough.

Sometimes we use "much larger than" (>>) and "much smaller than" (<<) to a similar effect. What, precisely, "much larger than" means completely depends on the context, and it's up to the reader to figure that out. This comes up in physics in claims like, Newton's laws hold for velocities $v \ll c$, where of course they have bigger and bigger margins of error for velocities near the speed of light.