Zero: The Biography of a Dangerous Idea

by Charles Seife

Though the Greek number system was more sophisticated than the Egyptian system, it was not the most advanced way of writing numbers in the ancient world. That title was held by another Eastern invention: the Babylonian style of counting. And thanks to this system, zero finally appeared in the East, in the Fertile Crescent of present-day Iraq.

A zero in a string of digits takes its meaning from some other digit to its left. On its own, it meant . . . nothing. Zero was a digit, not a number. It had no value.

Zero had no place within the Pythagorean framework. The equivalence of numbers and shapes made the ancient Greeks the masters of geometry, yet it had a serious drawback. 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. This meant that multiplication by zero didn’t make any sense either. Multiplying two numbers was equivalent to taking an area of a rectangle, but what could the area of a rectangle with zero height or zero width be?

The ratio of zero to anything—zero divided by a number—is always zero; the other number is completely consumed by the zero. And the ratio of anything to zero—a number divided by zero—can destroy logic.

The number-shape duality in Greek numbers made it easy; after all, zero didn’t have a shape and could thus not be a number.

Around 500 BC the placeholder zero began to appear in Babylonian writings; it naturally spread to the Greek astronomical community. During the peak of ancient astronomy, Greek astronomical tables regularly employed zero; its symbol was the lowercase omicron, o, which looks very much like our modern-day zero, though it’s probably a coincidence. (Perhaps the use of omicron came from the first letter of the Greek word for nothing, ouden.)

The Greeks didn’t like zero at all and used it as infrequently as possible. After doing their calculations with Babylonian notation, Greek astronomers usually converted the numbers back into clunky Greek-style numerals—without zero. Zero never worked its way into ancient Western numbers, so it is unlikely that the omicron is the mother of our 0. The Greeks saw the usefulness of zero in their calculations, yet they still rejected it.

So it was not ignorance that led the Greeks to reject zero, nor was it the restrictive Greek number-shape system. It was philosophy. 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 Aristote...

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. Though everybody knew that Zeno’s statement was false, nobody could find a flaw in Zeno’s argument. He had come up with a paradox. Zeno’s logical puzzle baffled Greek philosophers—as well as the philosophers who came after them. Zeno’s riddles plagued mathematicians for nearly two thousand years. In his most famous puzzle, “The Achilles,” Zeno proves that swift Achilles can never catch up with a lumbering tortoise that has a head start. To make things more concrete, let’s put some numbers on the problem. Imagine that Achilles runs at a foot a second, while the tortoise runs at half that speed. Imagine, too, that the tortoise starts off a foot ahead of Achilles. Achilles speeds ahead, and in a mere second he has caught up to where the tortoise was. But by the time he reaches that point, the tortoise, which is also running, has moved ahead by half a foot. No matter. Achilles is faster, so in half a second, he makes up the half foot. But again, the tortoise has moved ahead, this time by a quarter foot. In a flash—a quarter second—Achilles has made up the distance. But the tortoise lumbers ahead in that time by an eighth of a foot. Achilles runs and runs, but the tortoise scoots ahead each time; no matter how close Achilles gets to the tortoise, by the time he reaches the point where the tortoise was, the tortoise has moved. An eighth of...

Everybody knows that, in the real world, Achilles would quickly run past the tortoise, but Zeno’s argument seemed to prove that Achilles could never catch up. The philosophers of his day were unable to refute the paradox. Even though they knew that the conclusion was wrong, they could never find a mistake in Zeno’s mathematical proof. The philosophers’ main weapon was logic, but logical deduction seemed useless against Zeno’s argument. Each step along the way seemed airtight, and if all the steps are correct, how could the conclusion be wrong?

The Greeks were stumped by the problem, but they did find the source of the trouble: infinity. It is the infinite that lies at the heart of Zeno’s paradox: Zeno had taken continuous motion and divided it into an infinite number of tiny steps. Because there are an infinite number of steps, the Greeks assumed that the race would go on forever and ever, even though the steps get smaller and smaller. The race would never finish in finite time—or so they thought. The ancients didn’t have the equipment to deal with the infinite, but modern mathematicians have learned to handle it. The infinite must be approached very carefully, but it can be mastered, with the help of zero. Armed with 2,400 years’ worth of extra mathematics, it is not hard for us to go back and find Zeno’s Achilles’ heel. The Greeks did not have zero, but we do, and it is the key to solving Zeno’s puzzle. It is sometimes possible to add infinite terms together to get a finite result—but to do so, the terms being added together must approach zero.* This is the case with Achilles and the tortoise. When you add up the distance that Achilles runs, you start with the number 1, then add 1/2, then add 1/4, then add 1/8, and so on, with the terms getting smaller and smaller, getting closer and closer to zero; each term is like a step along a journey where the destination is zero. However, since the Greeks rejected the number zero, they couldn’t understand that this journey could ever have an end. To them, the numbers 1,...

doesn’t exist. Instead, the Greeks just saw the terms as simply getting smaller and smaller, meandering outside the realm of numbers. Modern mathematicians know that the terms have a limit; the numbers 1, 1/2, 1/4, 1/8, 1/16, and so forth are approaching zero as their limit. The journey has a destination. Once the journey has a destination, it is easy to ask how far away that destination is and how long it will take to get there. It is not that difficult to sum up the distances that Achilles runs: 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . + 1/2n + .... In the same way that the steps that Achilles takes get smaller and smaller, and closer and closer to zero, the sum of those steps gets closer and closer to 2. How do we know this? Well, let’s start off with 2, and subtract the terms of the sum, one by one. We begin with 2 – 1, which is, of course, 1. Next, we subtract 1/2, leaving 1/2. Then remove the next term: subtract 1/4, leaving 1/4 behind. Subtracting 1/8 leaves 1/8 behind. We’re back to our familiar sequence. We already know that 1, 1/2, 1/4, 1/8, and so forth has a limit of zero; thus, as we subtract the terms from 2, we have nothing left. The limit of the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . is 2 (Figure 11). Achilles runs 2 feet in catching up to the tortoise, even though he takes an infinite number of steps to do it. Better yet, look at the time it takes Achilles to overtake the tortoise: 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . — 2 seconds. Not only does Achilles take an in...

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. The terms in the infinite series didn’t have a limit or a destination; they seemed to get smaller and smaller without any particular end in sight. As a result, the Greeks couldn’t handle the infinite. They pondered the concept of the void but rejected zero as a number, and they toyed with the concept of the infinite but refused to allow infinity—numbers that are infinitely small and infinitely large—anywhere near the realm of numbers. This is the biggest failure in Greek mathematics, and it is the only thing that kept them from discovering calculus.

There were other schools of thought. The atomists, for example, believed that the universe is made up of little particles called atoms, which are indivisible and eternal. Motion, according to the atomists, was the movement of these little particles. Of course, for these atoms to move, there has to be empty space for them to move into. After all, these little atoms had to move around somehow; if there were no such thing as a vacuum, the atoms would be constantly pressed against one another. Everything would be stuck in one position for eternity, unable to move. Thus, the atomic theory required that the universe be filled with emptiness—an infinite void. The atomists embraced the concept of the infinite vacuum—infinity and zero wrapped into one. This was a shocking conclusion, but the indivisible kernels of matter in atomic theory got around the problem of Zeno’s paradoxes. Because atoms are indivisible, there is a point beyond which things could not be divided. Zeno’s hair-splitting doesn’t go on ad infinitum. After a number of strides, Achilles would be taking tiny steps that can’t get any smaller; eventually he would have to hurdle an atom that the tortoise doesn’t. Achilles would finally catch up to the elusive turtle. Another philosophy vied with the atomic theory, and instead of posing such bizarre concepts as the infinite vacuum, it turned the universe into a cozy nutshell. There was no infinity, no void—just beautiful spheres that surrounded the earth, which was natu...

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.

Brahmagupta tried to figure out what 0 ÷ 0 and 1 ÷ 0 were, and failed. “Cipher divided by cipher is naught,” he wrote. “Positive or negative divided by cipher is a fraction with that for a denominator.” In other words, he thought 0 ÷ 0 was 0 (he was wrong, as we will see), and he thought that 1 ÷ 0 was, well, we don’t really know, because he doesn’t make a whole lot of sense. Basically, he was waving his hands and hoping that the problem would go away. Brahmagupta’s mistake did not last for very long. In time the Indians realized that 1 ÷ 0 was infinite. “This fraction of which the denominator is a cipher, is termed an infinite quantity,” writes Bhaskara, a twelfth-century Indian mathematician, who tells of what happens when you add a number to 1 ÷ 0. “There is no alteration, though many be inserted or extracted; as no change takes place in the infinite and immutable God.” God was found in infinity—and in zero.

For the Jews, the years after Maimonides’ death became the era of nothing. In the thirteenth century a new doctrine spread: kabbalism, or Jewish mysticism.

The kabbalah was much more than number crunching; it was a tradition so mystical that some scholars say that it bears a striking resemblance to Hinduism. For instance, the kabbalah seized upon the idea of the dual nature of God. The Hebrew term ein sof, which meant “infinite,” represented the creator aspect of God, the part of the deity that made the universe and that permeates every corner of the cosmos. But at the same time it had a different name: ayin, or “nothing.” 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.

At the very center of the coordinate system—where the two axes cross—sits a zero.

It was d’Alembert who realized that it was important to consider the journey as well as the destination. He was the one who hatched the idea of limit and solved calculus’s problems with zeros.

A black hole begins, like all stars, as a big ball of hot gas—mostly hydrogen. If left to its own devices, a sufficiently large ball of gas would collapse under the weight of its own gravity; it would crush itself into a tiny lump. Luckily for us, stars don’t collapse because there is another force at work: nuclear fusion. As a cloud of gas collapses, it gets hotter and denser, and hydrogen atoms slam into one another with increasing force. Eventually, the star gets so hot and dense that the hydrogen atoms stick to one another and fuse, creating helium and releasing large quantities of energy. This energy shoots out from the center of the star, causing it to expand a little bit. During most of its life, a star is in an uneasy equilibrium: the propensity to collapse under its own gravity is balanced by the energy that comes from the fusing hydrogen in its center.

When an extremely massive star collapses, it disappears. The gravitational attraction is so great that physicists know of no force in the universe that can stop its collapse—not the repulsion of its electrons, not the pressure of neutron against neutron or quark against quark—nothing. The dying star gets smaller and smaller and smaller. Then . . . zero. The star crams itself into zero space.

Modern physics is a struggle of two titans. General relativity holds sway in the realm of the very, very big: the most massive objects in the universe, such as stars, solar systems, and galaxies. Quantum mechanics rules the domain of the very, very small: atoms and electrons and subatomic particles. It would seem that these two theories could live in harmony together, each dictating the rules of physics for different aspects of the universe.

Zero is behind all of the big puzzles in physics. The infinite density of the black hole is a division by zero. The big bang creation from the void is a division by zero. The infinite energy of the vacuum is a division by zero. Yet dividing by zero destroys the fabric of mathematics and

the framework of logic—and threatens to undermine the very basis of science.

For Newton to explain the laws of the universe, he had to ignore the illogic within his calculus—an illogic caused by a division by zero.