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Authors: Charles Seife

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BOOK: Zero
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Calculating the date of Easter was no mean feat, thanks to a clash of calendars. The seat of the church was Rome, and Christians used the Roman solar calendar that was 365 days (and change) long. But Jesus was a Jew, and he used the Jewish lunar calendar that was only 354 days (and change) long. The big events in Jesus' life were marked with reference to the moon, while everyday life was ruled by the sun. The two calendars drifted with respect to each other, making it very difficult to predict when a holiday was due. Easter was just such a drifting holiday, so every few generations a monk was drafted to calculate the dates when Easter would fall for the next few hundred years.

Dionysius Exiguus was one of these monks. In the sixth century the pope, John I, asked him to extend the Easter tables. While translating and recalculating the tables, Dionysius did a little research on the side; he realized that he could figure out just when Jesus Christ was born. After chugging through a bit of math, he decided that the current year was the 525th year since the birth of Christ. Dionysius decided that the year of Christ's birth should, thenceforth, be the year 1
anno Domini,
or the first year of Our Lord. (Technically, Dionysius said that Christ's birth happened on December 25 the year before, and he started his calendar on January 1 to match the Roman year.) The next year after that was 2
AD
, and the next 3
AD
, and so forth, replacing the two dating systems then most commonly in use.
*
But there was a problem. Make that two.

For one thing, Dionysius got the date of Christ's birth wrong. The sources agree that Mary and Joseph fled the wrath of King Herod, since Herod had heard a prophecy about a newborn Messiah. But Herod died in 3
BC
, years before the supposed birth of Christ. Dionysius was clearly wrong; today most scholars believe that the birth of Christ was in 4
BC
. Dionysius was a few years off.

In truth, this mistake was not so terrible. When choosing the first year of a calendar, it really doesn't matter
which
year is chosen, so long as everything is consistent after that. A four-year error is inconsequential if everyone agrees to make the same mistake, as, indeed, we have. But there was a more serious problem with Dionysius's calendar: zero.

There was no year zero. Normally this would be no big deal; most calendars of that day started with the year one, not the year zero. Dionysius didn't even have a choice; he didn't know about zero. He was brought up after the decline of the Roman Empire. Even during the heyday of Rome, the Romans were not exactly math whizzes. In the year 525, at the start of the Dark Ages, all Westerners clung to the clunky Roman style of numbers, and there was no zero in that counting system. To Dionysius, the first year of Our Lord was, naturally, the year I. The next year was year II, and Dionysius came to this conclusion in the year DXXV. In most circumstances this would not have caused any trouble, especially since Dionysius's calendar did not catch on immediately. In 525 there was serious trouble for the intellectuals in the Roman court. Pope John died, and in the ensuing power shift all the philosophers and mathematicians like Dionysius were kicked out of office. They were lucky to escape with their lives. (Others were not so lucky. Anicius Boethius was a powerful courtier who was among the finest medieval Western mathematicians, which makes him worth noting. At about the same time that Dionysius was kicked out of office, Boethius, too, fell from power and was imprisoned. Boethius is not remembered for his math but for his
Consolation of Philosophy,
a tract in which he comforts himself with Aristotelian-style philosophy. He was clubbed to death soon afterward.) In any case, the new calendar languished for years.

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.”

—J
OHN
C
ONWAY AND
R
ICHARD
G
UY
,
T
HE
B
OOK OF
N
UMBERS

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 nine 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.

BOOK: Zero
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