Zero

by Charles Seife

  The lack of a year zero began to cause problems two centuries later. In 731 AD, about the time Dionysius’s Easter tables were set to run out, Bede, a soon-to-be-venerable monk from the northern part of England, extended them again. This is probably how he came to know of Dionysius’s work. When Bede wrote a history of the church in Britain, the Ecclesiastical History of the English People, he used the new calendar.

  The book was a huge success, but it had one significant flaw. Bede started his history with the year 60 BC—60 years before Dionysius’s reference year. Bede didn’t want to abandon the new dating system, so he extended Dionysius’s calendar backward. To Bede, also ignorant of the number zero, the year that came before 1 AD was 1 BC. There was no year zero. After all, to Bede, zero didn’t exist.

  At first glance this style of numbering might not seem so bad, but it guaranteed trouble. Think of the AD years as positive numbers and the BC years as negative ones. Bede’s style of counting went…, –3, –2, –1, 1, 2, 3,…. Zero, whose proper place is between –1 and 1, is nowhere to be seen. This throws everybody off. In 1996, an article about the calendar in the Washington Post told people “how to think” about the millennium controversy—and casually mentioned that since Jesus was born in 4 BC, the year 1996 was the 2,000th year since his birth. That makes perfect sense: 1996 – (–4) = 2000. But it is wrong. It was actually only 1,999 years.

  Imagine a child born on January 1 in the year 4 BC. In 3 BC he turns one year old. In 2 BC he turns two years old. In 1 BC he turns three years old. In 1 AD he turns four years old. In 2 AD he turns five years old. On January 1 in 2 AD, how many years has it been since he was born? Five years, obviously. But this isn’t what you get if you subtract the years: 2 – (–4) = 6 years old. You get the wrong answer because there is no year zero.

  By rights, the child should have turned four years old on January 1 in the year 0 AD, five in 1 AD, and six in 2 AD. Then all the numbers would come out right, and figuring out the child’s age would be a simple matter of subtracting - 4 from 2. But it isn’t so. You’ve got to subtract an additional year from the total to get the right answer. Hence, Jesus was not 2,000 years old in 1996; he was only 1,999. It’s very confusing, and it gets worse.

  Imagine a child born in the first second of the first day of the first year: January 1 in 1 AD. In the year 2, he would be one year old, in the year 3 he would be two, and so forth; in the year 99 he’d be 98 years old, and in the year 100 he’d be 99 years old. Now imagine that this child is named Century. The century is only 99 years old in the year 100, and only celebrates its hundredth birthday on January 1 in the year 101. Thus the second century begins in the year 101. Likewise, the third century begins in the year 201, and the twentieth century begins in the year 1901. This means that the twenty-first century—and the third millennium—begins in the year 2001. Not that you’d notice.

  Hotels and restaurants around the world were completely booked well in advance for December 31, 1999—not so for December 31, 2000. Everybody celebrated the turn of the millennium on the wrong date. Even the Royal Greenwich Observatory, the official keeper of the world’s time and arbiter of all things chronological, planned to be swamped by the revelers. While the atomic-precision clocks ticked away in the observatory on the hill, the masses down below awaited a state-sponsored Millennium Experience, complete with a “spectacular opening ceremony” that the organizers scheduled for—you guessed it—December 31, 1999. The exhibit’s close on December 31, 2000, is just when the astronomers on top of the hill crack open their champagne bottles to celebrate the turn of the millennium. That is, of course, assuming that astronomers care about the date at all.

  Astronomers can’t play with time as easily as everyone else can. After all, they are watching the clockwork of the heavens—a clockwork that does not hiccup on leap years or reset itself every time humans decide to change the calendar. Thus the astronomers decided to ignore human calendars altogether. They don’t measure time in years since the birth of Christ. They count days since January 1, 4713 BC, a pretty-much arbitrary date that the scholar Joseph Scaliger chose in 1583. His Julian Date (named after his father, Julius, rather than Julius Caesar) became the standard way to refer to astronomical events, because it avoided all the weirdness caused by calendars that were constantly under construction. (The system has since been modified slightly. Modified Julian Date is simply the Julian Date less 2,400,000 days and 12 hours, putting the zero hour at midnight on November 17,1858. Again, a more or less arbitrary date.) Perhaps astronomers will refuse to celebrate 51542 Modified Julian Date, and the Jews will ignore 23 Tevet, 5760 (anno Mundi), and the Muslims will forget about 23 Ramadan, 1420 (anno Hejirae). On second thought, probably not. They will all know that it is December 31, 1999 (anno Domini), and there is something very special about the year 2000.

  It’s hard to say just why, but we humans love nice, round numbers with lots of zeros. How many of us remember being a child and going for a ride in a car that was about to top the 20,000-mile mark? Everybody in the car waits, silently, as 19,999.9 slowly creeps forward…and then, with a click, 20,000! All the children cheer.

  December 31, 1999, is the evening when the great odometer in the sky clicks ahead.

  The Zeroth Number

  Waclaw Sierpinski, the great Polish mathematician…was worried that he’d lost one piece of his luggage. “No, dear!” said his wife. “All six pieces are here.” “That can’t be true,” said Sierpinski, “I’ve counted them several times: zero, one, two, three, four, five.”

  —JOHN CONWAY AND RICHARD GUY, THE BOOK OF NUMBERS

  It may seem bizarre to suggest that Dionysius and Bede made a mistake when they forgot to include zero in their calendar. After all, children count “one, two, three,” not “zero, one, two.” Except for the Mayans, nobody else had a year zero or started a month with day zero. It seems unnatural. On the other hand, when you count backward, it is second nature.

  Ten. Nine. Eight. Seven. Six. Five. Four. Three. Two. One. Liftoff.

  The space shuttle always waits for zero before it blasts into the air. An important event happens at the zero hour, not the one hour. When you drive toward the site where a bomb went off, you’re approaching ground zero.

  If you look carefully enough, you will see that people usually do start counting with zero. A stopwatch starts ticking from 0:00.00 and only reaches 0:01.00 after a second has elapsed. A car’s odometer comes from the factory set at 00000, though by the time the dealer’s done tooling around town, it’s probably got a few more miles on it. The military’s day officially begins at 0000 hours. But count aloud and you always start with “one,” unless you’re a mathematician or a computer programmer.* It has to do with order.

  When we are dealing with the counting numbers—1, 2, 3, and so on—it is easy to rank them in order. One is the first counting number, two is the second counting number, and three is the third. We don’t have to worry about mixing up the value of the number—its cardinality—with the order in which it arrives—its ordinality—since they are essentially the same thing. For years, this was the state of affairs, and everybody was happy. But as zero came into the fold, the neat relationship between a number’s cardinality and its ordinality was ruined. The numbers went 0, 1, 2, 3: zero came first, one was second in line, and two was in third place. No longer were cardinality and ordinality interchangable. This is the root of the calendar problem.

  The first hour of the day starts at zero seconds past midnight; the second hour starts at 1 AM, and the third hour starts at 2 AM. Though we count with the ordinals (first, second, third), we mark time with the cardinals (0, 1, 2). All of us have assimilated this way of thinking, whether we appreciate it or not. After a baby finishes his 12th month, we all say that the child is one year old; he has finished his first 12 months of life. If the baby turns one when she’s already lived a year, isn’t the only consistent choice to say that the baby is zero years old before that time? Of course, we say that the child is six weeks old or ni
ne months old instead—a clever way of getting around the fact that the baby is zero.

  Dionysius didn’t have a zero, so he started the calendar with year 1, just as the ancients before him had started theirs. People of those times thought in terms of the old-style equivalence of cardinality and ordinality. That was just fine…for them. If zero never entered their minds, it could hardly be a problem.

  The Gaping Void

  It was not absolute nothingness. It was a kind of formlessness without any definition…. True reasoning convinced me that I should wholly subtract all remnants of every kind of form if I wished to conceive the absolutely formless. I could not achieve this.

  —SAINT AUGUSTINE, CONFESSIONS

  It’s hard to blame the monks for their ignorance. The world of Dionysius Exiguus, Boethius, and Bede was dark indeed. Rome had collapsed, and Western civilization seemed but a shadow of Rome’s past glory. The future seemed more horrid than the past. It is no wonder that in the search for wisdom medieval scholars didn’t look to their peers for ideas. Instead, they turned to the ancients like Aristotle and the Neoplatonists. As these medieval thinkers imported the philosophy and science of the ancients, they inherited the ancient prejudices: a fear of the infinite and a horror of the void.

  Medieval scholars branded void as evil—and evil as void. Satan was quite literally nothing. Boethius made the argument as follows: God is omnipotent. There is nothing God cannot do. But God, the ultimate goodness, cannot do evil. Therefore evil is nothing. It made perfect sense to the medieval mind.

  Lurking underneath the veil of medieval philosophy, however, was a conflict. The Aristotelian system was Greek, but the Judeo-Christian story of creation was Semitic—and Semites didn’t have such a fear of the void. The very act of creation was out of a chaotic void, and theologians like Saint Augustine, who lived in the fourth century, tried to explain it away by referring to the state before creation as “a nothing something” that is empty of form but yet “falls short of utter nothingness.” The fear of the void was so great that Christian scholars tried to fix the Bible to match Aristotle rather than vice versa.

  Luckily, not all civilizations were so afraid of zero.

  Chapter 3

  Nothing Ventured

  [ ZERO GOES EAST ]

  Where there is the Infinite there is joy. There is no joy in the finite.

  —THE CHANDOGYA UPANISHAD

  Though the West was afraid of the void, the East welcomed it. In Europe, zero was an outcast, but in India and later in the Arab lands, it flourished.

  When we last saw zero, it was simply a placeholder. It was a blank spot in the Babylonian system of numeration. Zero was useful but was not truly a number on its own—it had no value. It only took its meaning from the digits to its left; the symbol for zero literally meant nothing if it was by itself. In India, all this changed.

  In the fourth century BC, Alexander the Great marched with his Persian troops from Babylon to India. It was through this invasion that Indian mathematicians first learned about the Babylonian system of numbers—and about zero. When Alexander died in 323 BC, his squabbling generals split his empire into pieces. Rome rose to power in the second century BC and swallowed up Greece, but Rome’s power did not extend as far east as Alexander’s had. As a result, remote India was insulated from the rise of Christianity and the fall of Rome in the fourth and fifth centuries AD.

  India was also insulated from Aristotle’s philosophy. Though Alexander had been tutored by Aristotle, and no doubt introduced India to Aristotelian ideas, the Greek philosophy never took hold. Unlike Greece, India never had a fear of the infinite or of the void. Indeed, it embraced them.

  The void had an important place in the Hindu religion. Hinduism had started off as a polytheistic religion, a set of tales about warrior gods and battles similar in many ways to the Greek mythos. However, over centuries—centuries before Alexander arrived—the gods began to merge together. While Hinduism retained its popular rituals and devotion to its pantheon, at its core Hinduism became monotheistic and introspective. All the gods became aspects of an all-encompassing god, Brahma. At about the same time that the Greeks were rising in the Western world, Hinduism was becoming less like the Western myths; the individual gods became less distinct and the religion became more and more mystical. The mysticism was patently Eastern.

  Like many Eastern religions, Hinduism was steeped in the symbolism of duality. (Of course, this idea occasionally came up in the Western world, where it was promptly branded as heretical. One example is the Manichaean heresy, which saw the world as being under the influence of equal and opposite sources of good and evil.) As with the yin and yang of the Far East and Zoroaster’s dualism of good and evil in the Near East, creation and destruction were intermingled in Hinduism. The god Shiva was both creator and destroyer of the world and was depicted with the drum of creation in one hand and a flame of destruction in another (Figure 13). However, Shiva also represented nothingness. One aspect of the deity, Nishkala Shiva, was literally the Shiva “without parts.” He was the ultimate void, the supreme nothing—lifelessness incarnate. But out of the void, the universe was born, as was the infinite. Unlike the Western universe, the Hindu cosmos was infinite in extent; beyond our own universe were innumerable other universes.

  Figure 13: Shiva’s dance

  At the same time, though, the cosmos never truly abandoned its original emptiness. Nothingness was what the world came from, and to achieve nothingness again became the ultimate goal of mankind. In one story, Death tells a disciple about the soul: “Concealed in the heart of all beings is the Atman, the Spirit, the Self,” he says. “Smaller than the smallest atom, greater than the vast spaces.” This Atman, which resides in every thing, is part of the essence of the universe, and is immortal. When a person dies, the Atman is released from the body and soon enters another being; the soul transmigrates and the person is reincarnated. The goal of the Hindu is to free the Atman entirely from the cycle of rebirth, to stop wandering from death to death. The way to achieve the ultimate liberation through lifelessness is to cease paying heed to the illusion of reality. “The body, the house of the spirit, is under the power of pleasure and pain,” explains a god. “And if a man is ruled by his body then this man can never be free.” But once you are able to separate yourself from the whims of the flesh and embrace the silence and nothingness of your soul, you will be liberated. Your Atman will fly from the web of human desire and join the collective consciousness—the infinite soul that suffuses the universe, at once everywhere and nowhere at the same time. It is infinity, and it is nothing.

  So India, as a society that actively explored the void and the infinite, accepted zero.

  Zero’s Reincarnation

  In the earliest age of the gods, existence was born from non-existence.

  —THE RIG VEDA

  Indian mathematicians did more than simply accept zero. They transformed it, changing its role from mere placeholder to number. This reincarnation was what gave zero its power.

  The roots of Indian mathematics are hidden by time. An Indian text written the same year that Rome fell—476 AD—shows the influence of Greek, Egyptian, and Babylonian mathematics, brought by Alexander as he penetrated Indian lands. Like the Egyptians, the Indians had rope stretchers to survey fields and lay out temples. They also had a sophisticated system of astronomy; like the Greeks, they tried to calculate the distance to the sun. That requires trigonometry; the Indian version was probably derived from the system that the Greeks had developed.

  Sometime around the fifth century AD, Indian mathematicians changed their style of numbering; they moved from a Greek-like system to a Babylonian-style one. An important difference between the new Indian number system and the Babylonian style was that Indian numbers were base-10 instead of base-60. Our numbers evolved from the symbols that the Indians used; by rights they should be called Indian numerals rather than Arabic ones (Figure 14).

  Nobody knows when the Indians made the swit
ch to a Babylonian-style place-value number system. The earliest reference to the Hindu numerals comes from a Syrian bishop who wrote, in 662, of how the Indians did calculations “by means of nine signs.” Nine—not ten. Zero was evidently not among them. But it’s hard to tell for sure. It is fairly clear that the Hindu numerals were around before the bishop wrote about them; there is evidence that zero appeared in some variants of the Indian system by that time, though the bishop hadn’t heard about it. In any case, a symbol for zero—the placeholder in the base-10 numbering system—was certainly in use by the ninth century. By then Indian mathematicians had already made a giant leap.

  The Indians had borrowed little of Greek geometry. They apparently didn’t have a deep interest in the plane figures that the Greeks loved so much. They never worried about whether the diagonal of a square was rational or irrational, nor did they investigate the conic sections as Archimedes had. But they did learn how to play with numbers.

  The Indian system of numbering allowed them to use fancy tricks to add, subtract, multiply, and divide numbers without using an abacus to help them. Thanks to their place-number system, they could add and subtract large numbers in roughly the same way we do today. With training, a person could multiply with Indian numerals faster than an abacist could tally. Contests between the abacists and the so-called algorists who used Indian numerals were the medieval equivalents of the Kasparov versus Deep Blue chess match (Figure 15). Like Deep Blue, the algorists would win in the end.

  Figure 14: The evolution of our numerals