It's handy that the product of zero numbers is 1 because it means nothing special has to be done to make it so that, given a fraction x / (xy)
, you can cancel out the x
es by simply removing them as factors from their respective products, yielding 1 / y
.
I'm not saying that really why the empty sum is 1. But it's a case where you may have benefited from it.
In discrete arithmetic, which admits to a combinatoric interpretation, the sum a + b is the number of ways to select an item taken either from a pile of a things or a separate pile of b things. More generally, a sum of arbitrarily many (or few) addends is the number of ways to select an item taken from one of separate piles with those addends as their sizes. If there are no piles, there are no ways to select an item from one of them, so the empty sum is 0.
Also in discrete arithmetic and with the combinatoric interpretation, the product a × b is the number of ways to pick from a things and then, independently, from b things. More generally, a product of arbitrarily many (or few) factors is the number of ways to pick one item independently from each of separate piles with those factors as their sizes.
If there are no piles, there is one way to pick one item from each of them (in that you're immediately done, having already made all zero choices).
In case that's not clear or is not sufficiently compelling, consider that a × b × c is the number of lists of length 3 where there are a possibilities for the first item, b possibilities for the second item, and c possibilities for the third item;
that a × b is the number of lists of length 2 where there are a possibilities for the first item and b possibilities for the second item; that a is the number of lists of length 1 where there are a possibilities for the first (and only) item. There is exactly one length 0 list. If you write the length 3 lists like ⟨u,v,w⟩ and the length 2 lists like ⟨u,v⟩ and the length 1 lists like ⟨u⟩, then the length 0 list is written like ⟨⟩.
So the empty sum being 0 and the empty product being 1 are not just conventions for convenience. They are meaningful. Similarly, this is why 0! = 1. The factorial of n is the number of permutations of n distinct items. There's exactly one permutation of 0 items.
(It also explains the behavior of 0 exponents, but the combinatoric meaning of exponentiation is hard to elucidate without first being very familiar with functions, i.e., with the sort of mathematical object that is called a function. IIRC you'd said you were shaky on that, so I'll wait to give that example.)