Last week’s Occupy Math spoke a bit about Arabic numerals and declared that zero, a deep and subtle innovation, would be the topic of a future post. In this week’s post we look at zero, the empty set, and the ways these objects affect the way we do math. The starting point for appreciating zero is Roman numerals.

**Roman numerals have no zero. They also can’t be used on numbers bigger than a million.**

Roman numerals are still used all the time. The twenty-second Superbowl game is “Superbowl XXII”. A feature of Roman numerals is that they are not positional notation, but instead each numeral has a fixed value; you add all the numerals present to get the number represented — for instance, XXVIII is two tens, one five and three ones, for 28. Because it could be confusing to have too many of the same numeral in a sequence, they invented subtractive notation, putting the number to be subtracted first — IV is five minus one, for 4, instead of the hard-to-read IIII, for instance; IX is ten minus one, for 9, instead of VIIII; XL is fifty minus ten, for 40, instead of XXXX. As Roman civilization got more sophisticated, they started inventing really odd notational hacks. Fractions were a nightmare. So what do you do instead?