Zero: The Biography of a Dangerous Idea

by Charles Seife

Warships are designed to withstand the strike of a torpedo or the blast of a mine. Though it was armored against weapons, nobody had thought to defend the Yorktown from zero. It was a grave mistake.

When the Yorktown’s computer system tried to divide by zero, 80,000 horsepower instantly became worthless. It took nearly three hours to attach emergency controls to the engines, and the Yorktown then limped into port. Engineers spent two days getting rid of the zero, repairing the engines, and putting the Yorktown back into fighting trim.

No other number can do such damage. Computer failures like the one that struck the Yorktown are just a faint shadow of the power of zero.

Cultures girded themselves against zero, and philosophies crumbled under its influence, for zero is different from the other numbers. It provides a glimpse of the ineffable and the infinite. Th...

And it is a history of the paradoxes posed by an innocent-looking number, rattling even this century’s brightest minds and threatening to unravel the whole framework of scientific thought.

Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity.

Yet through all its history, despite the rejection and the exile, zero has always defeated those who opposed it. Humanity could never force zero to fit its philosophies. Instead, zero shaped humanity’s view of the universe—and of God.

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought. —ALFRED NORTH WHITEHEAD

Even without a zero, the Egyptians had quickly become masters of mathematics. They had to, thanks to an angry river.

In the Egyptian Book of the Dead, a newly deceased person must swear to the gods that he hasn’t cheated his neighbor by stealing his land. It was a sin punishable by having his heart fed to a horrible beast called the devourer.

They were also not inclined to put math into their philosophy. The Greeks were different; they embraced the abstract and the philosophical, and brought mathematics to its highest point in ancient times.

In the history of culture the discovery of zero will always stand out as one of the greatest single achievements of the human race. —TOBIAS DANZIG, NUMBER: THE LANGUAGE OF SCIENCE

Zero was born out of the need to give any given sequence of Babylonian digits a unique, permanent meaning.

It is the number that separates the positive numbers from the negative numbers.

Yet zero sits at the end of the computer and at the bottom of the telephone because we always start counting with one.

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.

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.

This troublesome number crushes the number line into a point. But as annoying as this property was, the true power of zero becomes apparent with division, not multiplication.

Dividing by zero once—just one time—allows you to prove, mathematically, anything at all in the universe.

Multiplying by zero collapses the number line. But dividing by zero destroys the entire framework of mathematics.

There is a lot of power in this simple number. It was to become the most important tool in mathematics. But thanks to the odd mathematical and philosophical properties of zero, it would clash with the fundamental philosophy of the West.

The Greek universe, created by Pythagoras, Aristotle, and Ptolemy, survived long after the collapse of Greek civilization. In that universe there is no such thing as nothing. There is no zero.

Before they could accept zero, philosophers in the West would have to destroy their universe.

The Greeks had a very different attitude. To them, numbers and philosophy were inseparable, and they took both very seriously.

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: the musical scale.

Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions.

The ratios of the sizes of the spheres were nice and orderly, and as the spheres moved, they made music. The outermost planets, Jupiter and Saturn, moved the fastest and made the highest-pitched notes. The innermost ones, like the moon, made lower notes. Taken all together, the moving planets made a “harmony of the spheres,” and the heavens are a beautiful mathematical orchestra. This is what Pythagoras meant when he insisted, “All is number.”

Because ratios were the keys to understanding nature, the Pythagoreans and later Greek mathematicians spent much of their energy investigating their properties. Eventually, they categorized proportions into 10 different classes, with names like the harmonic mean.

Even today, artists and architects intuitively know that objects that have this ratio of length to width are the most aesthetically pleasing, and the ratio governs the proportions of many works of art and architecture.

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

Even if zero were a number in the Greek sense, the act of taking a ratio with zero in it would seem to defy nature.

In fact, no matter how tiny you make the bits, it is impossible to choose a common yardstick that will measure both the side and the diagonal perfectly: the diagonal is incommensurable with the side. However, without a common yardstick, it is impossible to express the two lines in a ratio.

In other words, the diagonal of that square is irrational—and nowadays we recognize that number as the square root of two.

One of the first mathematical proofs in history was about the incommensurability/irrationality of the square’s diagonal.

Zero conflicted with the fundamental philosophical beliefs of the West, for contained within zero are two ideas that were poisonous to Western doctrine. Indeed, these concepts would eventually destroy Aristotelian philosophy after its long reign. These dangerous ideas are the void and the infinite.

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; Greek philosophy would never quite defeat Zeno—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.

Void/zero destroys Aristotle’s neat argument, his refutation of Zeno, and his proof of God. So as Aristotle’s arguments were accepted, the Greeks were forced to reject zero, void, the infinite, and infinity.

The Aristotelian view of physics, as wrong as it was, was so influential that for more than a millennium it eclipsed all opposing views, including more realistic ones. Science would never progress until the world discarded Aristotle’s physics—along with Aristotle’s rejection of Zeno’s infinities.

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.

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.

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

Where there is the Infinite there is joy. There is no joy in the finite. —THE CHANDOGYA UPANISHAD

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.

This was the birth of what we now know as algebra. Though this mind-set prevented the Indians from contributing much to geometry, it had another, unexpected effect. It freed the Indians from the shortcomings of the Greek system of thought—and their rejection of zero.

After all, zero multiplied by anything is zero; it sucks everything into itself. And when you divide with it, all hell breaks loose.

The infinite and the void go hand in hand, and are both part of the divine creator. Better yet, the term ayin is an anagram of (and has the same numerical value as) the word aniy, the Hebrew “I.” It could scarcely be clearer: God was saying, in code, “I am nothing.” And at the same time, infinity.

God could make a vacuum if he wanted. All of a sudden the void was allowed, because an omnipotent deity doesn’t need to follow Aristotle’s rules if he doesn’t want to.

. . a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions. —PIERRE-SIMON LAPLACE

Christianity initially rejected zero, but trade would soon demand it. The man who reintroduced zero to the West was Leonardo of Pisa. The son of an Italian trader, he traveled to northern Africa. There the young man—better known as Fibonacci—learned mathematics from the Muslims and soon became a good mathematician in his own right.