Zero: The Biography of a Dangerous Idea

by Charles Seife

The rubber band has broken. The whole number line has collapsed.

Divide by a number and you undo the multiplication:

Division by zero should be the opposite of multiplying by zero. It should undo the destruction of the number line. Unfortunately, this isn’t quite what happens.

Alas, this means that 0/0 equals 2, but it also equals 3, and it also equals 4. This just doesn’t make any sense.

Multiplication by zero should undo division by zero, so 1/0 × 0 should equal 1.

Multiplying by zero collapses the number line. But dividing by zero destroys

the entire framework of mathematics.

The whole Greek universe rested upon this pillar: there is no void.

To them, numbers and philosophy were inseparable,

The Greeks had inherited their numbers from the geometric Egyptians. As a result, in Greek mathematics there was no significant distinction between shapes and numbers.

The importance of the golden ratio comes from a Pythagorean discovery that is now barely remembered. In modern schools, children learn of Pythagoras

for his famed theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. However, this was in fact ancient news. It was known more than 1,000 years before Pythagoras’s time. In ancient

Greece, Pythagoras was remembered for a different invention:...

Understanding nature was as simple as understanding the mathematics of proportions.

To the Pythagorean mind, ratios controlled the universe, and what was true for the Pythagoreans soon became true for the entire West.

It precluded anyone from treating zero as a number. What shape, after all, could zero be? It is easy to visualize a square with width

two and height two, but what is a square with width zero and height zero? It’s hard to imagine something with no width and no height—with no substance at all—being a square.

Indeed, the Pythagoreans had tried to squelch another troublesome mathematical concept—the irrational. This concept was the first challenge to the Pythagorean point of view, and the brotherhood tried to keep it secret. When the secret leaked out, the cult turned to violence.

This meant trouble for the Pythagorean doctrine. How could nature be governed by ratios and proportions when something as simple as a square can confound the language of ratios?

Irrationality was dangerous to Pythagoras, as it threatened the basis of his ratio-universe. To add insult to injury, the Pythagoreans soon discovered that the golden ratio, the ultimate Pythagorean symbol of beauty and rationality, was an irrational number.

One day someone was going to let the secret out. This someone was Hippasus of Metapontum,

One day, according to a version of the legend, his house was set ablaze by his enemies (who were angry at not being considered worthy to be admitted into Pythagoras’s presence), and the brothers in the house scattered in all directions, running for their lives.

Zeno was born around 490 BC, at the beginning of the Persian wars—a great conflict between East and West. Greece would defeat the Persians;

for Zeno had a paradox, a logical puzzle that seemed intractable to the reasoning of Greek philosophers. It was the most troubling argument in Greece: Zeno had proved the impossible. According to Zeno, nothing in the universe could move. Of course, this is a silly statement; anyone can refute it by walking across the room.

The Greeks couldn’t do this neat little mathematical trick. They didn’t have the concept of a limit because they didn’t believe in zero.

This is the biggest failure in Greek mathematics,

There were other schools of thought. The atomists,

Of course, for these atoms to move, there has to be empty space for them to move into.

Thus, the atomic theory required that the universe be filled with emptiness—an

First, there could be an infinite amount of void—thus infinity exists. Second, there could be a finite amount of void, but since void is simply the lack of matter, there must be an infinite amount of matter to make sure that there is only a finite amount of void—thus

thus infinity exists.

both cases the existence of the void implies the existenc...

but if there is no such thing as infinity, there cannot be an infinite number of events. Thus, there must be a first event: creation. But what existed before creation? Void? That was unacceptable to Aristotle.

universe without both of them makes no sense.

Eventually, one ancient Greek surpassed Zeno in matters of the infinite: Archimedes,

so-called axiom of Archimedes,

As you may recall, this axiom says that any number added to itself over and over again can exceed any other number.

The infinite was not needed in the Greek universe.

Killing Archimedes was one of the biggest Roman contributions to mathematics. The Roman era lasted for about seven centuries.

Christianity swept through Europe, the Roman Empire fell, the Library at Alexandria burned, and the Dark Ages began. It would be another seven centuries before zero reappeared in the West. In the meantime two monks created a calendar without zero, damning us to eternal confusion.

This “silly, childish discussion”—whether the new century begins on the year 00 or the year 01—appears and reappears like clockwork every hundred years.

during the Middle Ages the only Westerners who studied math were the Christian monks.

Monks needed math for two things: prayer and money.

It was not a very demanding task, but by ancient standards it was the state of the art.

To pray, monks needed to know the time and the date.

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,

Dionysius Exiguus was one of these monks. In the sixth century the pope, John I, asked him to extend the Easter tables.

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,

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.”

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.