I like to think of zero not as a number but as a symbol that, when combined with other operators, becomes another operator. +/- 0 is an operator that leaves a number unchanged. *0 is an operator that produces zero when applied to any number. ^0 is an operator that produces 1 when appled to any number. There is no /0 operator; it is undefined.

One can, of course, play this game with any number, but in every other case, the /x "operator" is defined, which is why it is more convenient to treat numbers as numbers and conventional operators as operators. But because /0 is undefined, other operations involving zero are best understood as unique operators and not as operations with the "number" zero.