One seems like the appropriate place to start counting, but doing so forces us to put zero in an unnatural place. To other cultures, like the Mayan people of Mexico and Central America, starting with one didn’t seem like the rational thing to do. In fact, the Mayans had a number system—and a calendar—that made more sense than ours does. Like the Babylonians, the Mayans had a place-value system of digits and places. The only real difference was that instead of basing their numbers on 60 as the Babylonians did, the Mayans had a vigesimal, base-20 system that had the remnants of an earlier base-10 system in it. And like the Babylonians, they needed a zero to keep track of what each digit meant. Just to make things interesting, the Mayans had two types of digits. The simple type was based on dots and lines, while the complicated type was based on glyphs—grotesque faces. To a modern eye, Mayan glyph writing is about as alien-looking as you can get (Figure 3).
Like the Egyptians, the Mayans also had an excellent solar calendar. Because their system of counting was based on the number 20, the Mayans naturally divided their year into 18 months of 20 days each, totaling 360 days. A special period of five days at the end, called Uayeb, brought the count to 365. Unlike the Egyptians, though, the Mayans had a zero in their counting system, so they did the obvious thing: they started numbering days with the number zero. The first day of the month of Zip, for example, was usually called the “installation” or “seating” of Zip. The next day was 1 Zip, the following day was 2 Zip, and so forth, until they reached 19 Zip. The next day was the seating of Zotz’—0 Zotz’ followed by 1 Zotz’ and so forth. Each month had 20 days, numbered 0 through 19, not numbered 1 through 20 as we do today. (The Mayan calendar was wonderfully complicated. Along with this solar calendar, there was a ritual calendar that had 20 weeks, each of 13 days. Combined with the solar year, this created a calendar round that had a different name for every day in a 52-year cycle.)
Figure 3: Mayan numbers
The Mayan system made more sense than the Western system does. Since the Western calendar was created at a time when there was no zero, we never see a day zero, or a year zero. This apparently insignificant omission caused a great deal of trouble; it kindled the controversy over the start of the millennium. The Mayans would never have argued about whether 2000 or 2001 was the first year in the twenty-first century. But it was not the Mayans who formed our calendar; it was the Egyptians and, later, the Romans. For this reason, we are stuck with a troublesome, zero-free calendar.
The Egyptian civilization’s lack of zero was bad for the calendar and bad for the future of Western mathematics. In fact, Egyptian civilization was bad for math in more ways than one; it was not just the absence of a zero that caused future difficulties. The Egyptians had an extremely cumbersome way of handling fractions. They didn’t think of ¾ as a ratio of three to four as we do today; they saw it as the sum of ½ and ¼. With the sole exception of 2/3, all Egyptian fractions were written as a sum of numbers in the form of 1/n (where n is a counting number)—the so-called unit fractions. Long chains of these unit fractions made ratios extremely difficult to handle in the Egyptian (and Greek) number systems.
Zero makes this cumbersome system obsolete. In the Babylonian system—with zero in it—it’s easy to write fractions. Just as we can write 0.5 for ½ and 0.75 for ¾, the Babylonians used the numbers 0;30 for ½ and 0;45 for ¾. (In fact, the Babylonian base-60 system is even better suited to writing down fractions than our modern-day base-10 system.)
Unfortunately, the Greeks and Romans hated zero so much that they clung to their own Egyptian-like notation rather than convert to the Babylonian system, even though the Babylonian system was easier to use. For intricate calculations, like those needed to create astronomical tables, the Greek system was so cumbersome that the mathematicians converted the unit fractions to the Babylonian sexagesimal system, did the calculations, and then translated the answers back into the Greek style. They could have saved many time-consuming steps. (We all know how fun it is to convert fractions back and forth!) However, the Greeks so despised zero that they refused to admit it into their writings, even though they saw how useful it was. The reason: zero was dangerous.
The Fearsome Properties of Nothing
In earliest times did Ymir live: was nor sea nor land nor salty waves, neither earth was there nor upper heaven, but a gaping nothing, and green things nowhere.
—THE ELDER EDDA
It is hard to imagine being afraid of a number. Yet zero was inexorably linked with the void—with nothing. There was a primal fear of void and chaos. There was also a fear of zero.
Most ancient peoples believed that only emptiness and chaos were present before the universe came to be. The Greeks claimed that at first Darkness was the mother of all things, and from Darkness sprang Chaos. Darkness and Chaos then spawned the rest of creation. The Hebrew creation myths say that the earth was chaotic and void before God showered it with light and formed its features. (The Hebrew phrase is tohu v’bohu. Robert Graves linked these tohu to Tehomot, a primal Semitic dragon that was present at the birth of the universe and whose body became the sky and earth. Bohu was linked to Behomot, the famed Behemoth monster of Hebrew legend.) The older Hindu tradition tells of a creator who churns the butter of chaos into the earth, and the Norse myth tells a tale of an open void that gets covered with ice, and from the chaos caused by the mingling of fire and ice was born the primal Giant. Emptiness and disorder were the primeval, natural state of the cosmos, and there was always a nagging fear that at the end of time, disorder and void would reign once more. Zero represented that void.
But the fear of zero went deeper than unease about the void. To the ancients, zero’s mathematical properties were inexplicable, as shrouded in mystery as the birth of the universe. This is because zero is different from the other numbers. Unlike the other digits in the Babylonian system, zero never was allowed to stand alone—for good reason. A lone zero always misbehaves. At the very least it does not behave the way other numbers do.
Add a number to itself and it changes. One and one is not one—it’s two. Two and two is four. But zero and zero is zero. This violates a basic principle of numbers called the axiom of Archimedes, which says that if you add something to itself enough times, it will exceed any other number in magnitude. (The axiom of Archimedes was phrased in terms of areas; a number was viewed as the difference of two unequal areas.) Zero refuses to get bigger. It also refuses to make any other number bigger. Add two and zero and you get two; it is as if you never bothered to add the numbers in the first place. The same thing happens with subtraction. Take zero away from two and you get two. Zero has no substance. Yet this substanceless number threatens to undermine the simplest operations in mathematics, like multiplication and division.
In the realm of numbers, multiplication is a stretch—literally. Imagine that the number line is a rubber band with tick marks on it (Figure 4). Multiplying by two can be thought of as stretching out the rubber band by a factor of two: the tick mark that was at one is now at two; the tick mark that was at three is now at six. Likewise, multiplying by one-half is like relaxing the rubber band a bit: the tick mark at two is now at one, and the tick mark at three winds up at one and a half. But what happens when you multiply by zero?