Zero: The Biography of a Dangerous Idea

by Charles Seife

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. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero—and an infinity.

(Our word noon comes from the word nones, the midday prayer service of medieval clergy.)

Al-Khowarizmi wrote several important books, like Al-jabr wa’l muqabala, a treatise on how to solve elementary equations; the Al-jabr in the title (which means something like “completion”) gave us the term algebra. He also wrote a book about the Hindu numeral system, which allowed the new style of numbers to spread quickly through the Arab world—along with algorithms, the tricks for multiplying and dividing Hindu numerals quickly. In fact, the word algorithm was a corruption of al-Khowarizmi’s name.

Though the Arabs took the notation from India, the rest of the world would dub the new system Arabic numerals.

. These ratios approach a particularly interesting number: the golden ratio, which is 1.61803 . . . . Pythagoras had noticed that nature seemed to be governed by the golden ratio. Fibonacci discovered the sequence that is responsible. The size of the chambers of the nautilus and the number of clockwise grooves to counterclockwise grooves in the pineapple are governed by this sequence. This is why their ratios approach the golden ratio.

Before Arabic numerals came around, money counters had to make do with an abacus or a counting board. The Germans called the counting board a Rechenbank, which is why we call moneylenders banks.

At that time, banking methods were primitive. Not only did they use counting boards, they used tally sticks to record loans: a money value was written along the stick’s side, and it was split in two (Figure 16). The lender kept the biggest piece, the stock. After all, he was the stockholder.* Figure 16: A tally stick

But the advantages of zero and the other Arabic numerals were not so easily dispensed with; Italian merchants continued to use them, and even used them to send encrypted messages—which is how the word cipher came to mean “secret code.”

This is the definition of the infinite: it is something that can stay the same size even when you subtract from it.

A quantum-mechanical law called the Pauli exclusion principle keeps matter from squishing itself into a point. Discovered in the mid-1920s by German physicist Wolfgang Pauli, the exclusion principle states, roughly, that no two things can be in the same place at the same time. In particular, no two electrons of the same quantum state can be forced into the same spot. In 1933, the Indian physicist Subrahmanyan Chandrasekhar realized that the Pauli exclusion principle had only a limited ability to fight against the squeeze of gravity. As pressure in the star increases, the exclusion principle states that electrons inside must move faster and faster to avoid one another. But there’s a speed limit: electrons cannot move faster than the speed of light, so if you put enough pressure on a lump of matter, electrons cannot move fast enough to stop the matter from collapsing. Chandrasekhar showed that a collapsing star that has about 1.4 times the mass of our sun will have enough gravity to overwhelm the Pauli exclusion principle. Above this Chandrasekhar limit a star’s gravity will pull on itself so strongly that electrons can’t stop its collapse. The force of gravity is so great that the star’s electrons give up their struggle once and for all; the electrons smash into the star’s protons, creating neutrons. The massive star winds up being a gigantic ball of neutrons: a neutron star. Further calculations showed that when collapsing stars are a little more massive than the Chandrase...

The key to a black hole’s strange properties is the way it curves space-time. A black hole takes up no space at all, but it still has mass. Since the black hole has mass, it causes space-time to curve. Normally, this would not cause a problem. As you approach a heavy star, the curvature gets greater and greater, but once you have passed the outer edge of the star itself, the curvature decreases again, bottoming out at the center of the star. In contrast, a black hole is a point. It takes up zero space, so there is no outer edge, no place where space begins to flatten out again. The curvature of space gets greater and greater as you approach a black hole, and it never bottoms out. The curvature goes off to infinity because the black hole takes up zero space; the star has torn a hole in space-time (Figure 52). The zero of a black hole is a singularity, an open wound in the fabric of the universe.

The minimum speed you need to throw the rock to enable it to escape is called, naturally enough, the escape velocity. Black holes are so dense that if you get too close—past the so-called event horizon—the escape velocity is faster than the speed of light. Past the event horizon the pull of a black hole’s gravity is so strong—and space is so curved—that nothing can escape, not even light.

Even though a black hole is a star, none of the light it shines ever escapes past the event horizon; that’s why it’s black. The only way to view a black hole’s singularity is to go beyond the event horizon and see for yourself. However, even if you had an impossibly strong spacesuit that kept you from being stretched into a piece of astronaut spaghetti, you could never tell anybody about what you saw. Once you pass the event horizon, signals you broadcast can’t escape the black hole’s pull—neither can you. Traveling beyond the edge of the event horizon is like stepping off the edge of the universe. You will never return.

a singularity is a point of infinite curvature; it is a hole in the fabric of space and time. Under certain circumstances that hole can be stretched out. For instance, if a black hole is spinning or has an electric charge, mathematicians have calculated that the singularity is not a point—a pinpoint hole in space-time—but a ring. Physicists have speculated that two of these stretched-out singularities might be linked with a tunnel: a wormhole (Figure 53). A person who travels through a wormhole will emerge at another point in space—and perhaps in time. Just as wormholes can, in theory, send you halfway across the universe in the blink of an eye, they can send you backward and forward in time (see appendix E

All that scientists know is the cosmos was spawned from nothing, and will return to the nothing from whence it came. The universe begins and ends with zero.