You add a bunch of stuff and you get a total. You add nothing and 0 is the total because it's the identity and starting point for addition. You multiply a bunch

You add a bunch of stuff and you get a total. You add nothing and 0 is the total because it's the identity and starting point for addition. You multiply a bunch of stuff and you get a product. You multiply nothing and you get 1 because that's the identity for multiplication. (That's the same as saying "the rules of algebra work out" but there's maybe something intuitive about multiplying nothing and getting back the thing that doesn't change the result of multiplication?)