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 Babylonian style of counting. And thanks to this system, zero finally appeared in the East, in the Fertile Crescent of present-day Iraq.

(The words calculate, calculus, and calcium all come from the Latin word for pebble: calculus.)

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

Zero has no substance. Yet this substanceless number threatens to undermine the simplest operations in mathematics, like multiplication and division.

In the previous example we saw that 2 × 0 is 0. Thus to undo the multiplication, we have to assume that (2 × 0)/0 will get us back to 2. Likewise, (3 × 0)/0 should get us back to 3, and (4 × 0)/0 should equal 4. But 2 × 0 and 3 × 0 and 4 × 0 each equal zero, as we saw—so (2 × 0)/0 equals 0/0, as do (3 × 0)/0 and (4 × 0)/0. 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.

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.

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.

To the Pythagorean mind, ratios controlled the universe, and what was true for the Pythagoreans soon became true for the entire West. The supernatural link between aesthetics, ratios, and the universe became one of the central and long-lasting tenets of Western civilization.

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.

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.

Aristotle just wished infinity away by stating that it is simply a construct of the human mind.

When Christianity swept through the West, it became closely tied to the Aristotelian view of the universe and the proof of God’s existence. Atomism became associated with atheism. Questioning the Aristotelian doctrine was tantamount to questioning God’s existence.

After all, there were only two logical possibilities for the nature of the void, and both implied that the infinite exists. 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 infinity exists.

axiom of Archimedes, although Archimedes himself mentioned that earlier mathematicians deserved the credit. As you may recall, this axiom says that any number added to itself over and over again can exceed any other number. Zero, clearly, was not included.

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.

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.

God is omnipotent. There is nothing God cannot do. But God, the ultimate goodness, cannot do evil. Therefore evil is nothing.

Unlike the Western universe, the Hindu cosmos was infinite in extent; beyond our own universe were innumerable other universes.

To the Indians, negative numbers made perfect sense. Indeed, it was in India (and in China) that negative numbers first appeared. Brahmagupta, an Indian mathematician of the seventh century, gave rules for dividing numbers by each other, and he included the negatives.

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.

In fact, the word algorithm was a corruption of al-Khowarizmi’s name.

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.”

Descartes was a mathematician-philosopher; perhaps his most lasting legacy was a mathematical invention—what we now call Cartesian coordinates.

It is a subtle point: vacuums don’t suck; the atmosphere pushes.

Adding infinite things to each other can yield bizarre and contradictory results. Sometimes, when the terms go to zero, the sum is finite, a nice, normal number like 2 or 53. Other times the sum goes off to infinity. And an infinite sum of zeros can equal anything at all—and everything at the same time.

Calculus, on the other hand, gave scientists a way to express the laws that govern the motion of the celestial bodies—and laws that would eventually tell scientists how those moons and stars had formed.

Natural laws are described with equations, and equations, in a sense, are simply tools where you plug in numbers and get another number out.

For the truth is, nature doesn’t speak in ordinary equations. It speaks in differential equations, and calculus is the tool that you need to pose and solve these differential equations.

L’Hôpital’s rule took the first crack at the troubling 0/0 expressions that were popping up throughout calculus. The rule provided a way to figure out the true value of a mathematical function that goes to 0/0 at a point. L’Hôpital’s rule states that the value of the fraction was equal to the derivative of the top expression divided by the derivative of the bottom expression.

However, by putting this limit sign in front of a series, you separate the process from the goal. In this way you avoid manipulating infinities and zeros. Just as Achilles’ sub-races are each finite, each partial sum in a limit is finite. You can add them, divide them, square them; you can do whatever you want. The rules of mathematics still work, since everything is finite. Then, after all your manipulations are complete, you take the limit: you extrapolate and figure out where the expression is headed.

Since limits are logically airtight, by defining a derivative in terms of limits, it becomes airtight as well—and puts calculus on a solid foundation.

there is no number on the number line that gives you a negative number when you square it. The square root of a negative number seemed like a ridiculous concept. Descartes thought that these numbers were even worse than negative numbers; he came up with a scornful name for the square roots of negatives: imaginary numbers. The name stuck, and eventually, the symbol for the square root of –1 became i.

Riemann made it perfectly acceptable to go off to infinity; since infinity is just a point on the sphere like any other point, it was no longer something to be feared. In fact, mathematicians started analyzing and classifying the points where a function blows up: singularities.

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

If the volume of a gas is related to its temperature, a minimum volume means that there is a minimum temperature. A gas cannot keep getting colder and colder indefinitely; when you can’t shrink the balloon any further, you can’t lower the temperature any further. This is absolute zero. It is the lowest temperature possible, a little more than 273 degrees Celsius below the freezing point of water.

Every object is influenced by the environment it’s in, so it’s impossible to cool anything in the universe—a banana, an ice cube, a dollop of liquid helium—to absolute zero. It is an unbreakable barrier.

The concept of uncertainty pertains to scientists’ ability to describe the properties of a particle. For instance, if we want to find a particular particle, we need to determine the particle’s position and velocity—where it is and how fast it is going. Heisenberg’s uncertainty principle tells us that we can’t do even this simple act. No matter how hard we try, we cannot measure a particle’s position and its velocity with perfect accuracy at the same time. This is because the very act of measuring destroys some of the information we are trying to gather.

Casimir realized that the forbidden particle waves would affect the zero-point energy of the vacuum, since particles are everywhere winking in and out of existence. If you put two metal plates close together and some of those particles aren’t allowed between the plates, then there are more particles on the outside of the plates than on the inside. The undiminished zoo of particles presses on the outside of the plates, and without the full complement on the inside, the plates are crushed together, even in the deepest vacuum. This is the force of the vacuum, a force produced by nothing at all. This is the Casimir effect.

Zero in quantum mechanics invests the vacuum with infinite energy. A zero in the other great modern theory—relativity—creates another paradox: the infinite nothing of the black hole.

the flow of time changes, depending on an observer’s speed.

When a spaceship approaches the speed of light, time slows down more and more and more. If the ship were to travel at the speed of light, every tick of the clock on board would equal infinite seconds on the ground. In less than a fraction of a second, billions and billions of years would pass; the universe would have already met its ultimate fate and burned itself out. For an astronaut aboard the spaceship, time stops. The flow of time is multiplied by zero.

Thus, when you accelerate an object—when you subject it to any force, be it gravity or be it the push of a gigantic cosmic elephant—you change its motion through space and through time: through space-time.

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.

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.

There is a limit, though, to even the pressure that neutrons can bear. Some astrophysicists believe that a little more squeezing makes the neutrons break down into their component quarks, creating a quark star. But that is the last strong-hold. After that, all hell breaks loose.

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. This is a black hole, an object so paradoxical that some scientists believe that black holes can be used to travel faster than light—and backward in time.

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.

Back in the rubber-sheet analogy, 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). You might even be able to track down your mother and kill her before she meets your father, preventing you from being born and causing a terrible paradox.