Zero: The Biography of a Dangerous Idea

by Charles Seife

Thus creatures could only inhabit the nutmeat in the center of the nutshell cosmos.

But how could Rome claim to be the seat of the one true Church if its authority could not extend to other solar systems? Were there other popes on other planets? It was a grim prospect for the Catholic Church, especially since it was beginning to have trouble with its subjects on even its own world.

The attack on the church began in earnest in 1517, when a constipated German monk nailed a list of complaints to the door of the church in Wittenberg. (Luther’s constipation was legendary. Some scholars believe that his great revelation about faith came to him when he was sitting on the privy. “Luther’s release from the constricting bondage of fear corresponded to the release of his bowels,” notes one text, commenting on this theory.)

I am in a sense something intermediate between God and nought. —RENÉ DESCARTES, DISCOURSE ON METHOD

What is man in nature? Nothing in relation to the infinite, everything in relation to nothing, a mean between nothing and everything. —BLAISE PASCAL, PENSÉES

When, after a thousand-year stupor, European thought shook off the effect of the sleeping powders so skilfully administered by the Christian Fathers, the problem of infinity was one of the first to be revived. —TOBIAS DANZIG, NUMBER: THE LANGUAGE OF SCIENCE

Consider the following series: 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – . . . . It’s not so hard to show that this series sums to zero. After all, (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + . . . is the same thing as 0 + 0 + 0 + 0 + 0 + 0 + . . . which clearly sums to zero. But beware! Group the series in a different way: 1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + 1) + (–1 + 1) + . . . is the same thing as 1 + 0 + 0 + 0 + 0 + 0 + . . . which clearly sums to one. The same infinite sum of zeros can equal 0 and 1 at the same time. An Italian priest, Father Guido Grandi, even used this series to prove that God could create the universe (1) out of nothing (0). In fact, the sequence can be set to equal anything at all. To make the sum equal 5, start with 5s and –5s instead of 1s and

if you know the way an object moves—whether it be a toy ball or a giant planet—the differential equation can tell you what kind of force is being applied. (Newton’s triumph was taking the equation that described the force of gravity and figuring out the shapes of the planets’ orbits. People had suspected that the force was proportional to 1/r2, and when ellipses popped right out of Newton’s differential equations, people began to believe that Newton was correct.) Despite the power of calculus, the key problem remained. Newton’s work was based on a very shaky foundation—dividing zero by itself. His rival’s work had the same flaw.

(In fact, faith was very much on Leibniz’s mind when he derived new mathematics, such as the binary numbers. Any number can be written as a string of zeros and ones; to Leibniz, this was the creation ex nihilo, the creation of the universe out of nothing more than God/1 and void/0. Leibniz even tried to get the Jesuits to use this knowledge to convert the Chinese to Christianity.)

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. For instance, consider the expression x/(sin x) when x=0; x=0, as does sin x, so the expression is equal to 0/0. Using L’Hopital’s rule, we see that the expression goes to 1/(cos x), as 1 is the derivative of x and cos x is the derivative of sin x. Cos x=1 when x=0, so the whole expression equals 1/1=1. Clever manipulations could also bring l’Hopital’s rule to resolve other odd expressions: ∞/∞, 00, 0∞, and ∞0. All of these expressions, but especially 0/0, could take on any value you desire them to have, depending on the functions you put in the numerator and denominator. This is why 0/0 is dubbed indeterminate. It was no longer a complete mystery; mathematicians could extract some information about 0/0 if they approached it very carefully. Zero was no longer an enemy to be avoided; it was an enigma to be studied.

In The Analyst, Berkeley pounced on Newton’s (and Leibniz’s) dirty tricks with zeros. Calling infinitesimals “ghosts of departed quantities,” Berkeley showed how making these infinitesimals disappear with impunity can lead to a contradiction. He concluded that “he who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in divinity.”

Nobody could explain how those infinitesimals disappeared when squared; they just accepted the fact because making them vanish at the right time gave the correct answer. Nobody worried about dividing by zero when conveniently ignoring the rules of mathematics explained everything from the fall of an apple to the orbits of the planets in the sky. Though it gave the right answer, using calculus was as much an act of faith as declaring a belief in God.

A quantity is something or nothing; if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera. —JEAN LE ROND D’ALEMBERT

Buddhists are laughing

Now, instead of thinking of the race as a sum of infinite parts, think of it as a limit of finite sub-races. For instance, in the first race Achilles runs to the one-foot mark. Achilles has run 1 1 foot in all. In the next race Achilles does the first two parts—first running 1 foot, and then a half foot. In total, Achilles has run 1 + 1/2 1.5 feet in all. The third race takes him as far as 1.75 feet, all told. Each of these sub-races is finite and well-defined; we never encounter an infinity.

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.

Instead of dividing by zero as Newton and Leibniz did, modern mathematicians divide by a number that they let approach zero. They do the division—perfectly legally, since there are no zeros—then they take the limit. The dirty tricks of making squared infinitesimals disappear and then dividing by zero to get a derivative were no longer necessary (see appendix C). This logic may seem like splitting hairs, like an argument as mystical as Newton’s “ghosts,” but in reality it’s not. It satisfies the mathematician’s strict requirement of logical rigor. There is a very firm, consistent basis for the concept of limits. Indeed, you can even dispense with the “I challenge you” argument entirely, as there are other ways of defining a limit, such as calling it the convergence of two numbers, the lim sup and lim inf. (I have a wonderful proof of this, but alas, this book is too small to contain it.) 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.

The Imaginary . . . a fine and wonderful refuge of the divine spirit—almost an amphibian between being and non-being. —GOTTFRIED WILHELM LEIBNIZ

Imaginary numbers never appeared in linear equations, but they began to crop up in quadratic ones. Consider the equation x2 + 1=0. No number seems to solve the equation; plugging in –1, 3, –750, 235.23, or any other positive or negative number you could think of doesn’t yield the correct answer. The expression simply will not split. Worse yet, when you try to apply the quadratic equation, you get two silly-sounding answers:

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. Algebraists loved i. Almost everyone else hated it. It was wonderful for solving polynomials—expressions like x3 + 3x + 1 that have x raised to various powers. In fact, once you allow i into the realm of numbers, every polynomial becomes solvable: x2 + 1 suddenly splits into (x – i)(x + i)—the roots of the equation are +i and –i. Cubic expressions like x3 – x2 + x – 1 split three ways, such as (x – 1)(x + i)(x – i). Quartic expressions—ones with a leading x4 term—always split into four terms, and quintics—ones with a leading x5 term—split five ways. All polynomials of degree n—those that have a leading term of xn—split into n distinct terms. This is the fundamental theorem of algebra. As early as the sixteenth century, mathematicians were using numbers with i included—the so-called complex numbers—to solve cubic and quartic polynomials. And while many mathematicians saw the complex numbers as a convenient fiction, others saw God. Leibniz thought that i was a bizarre mix between existence and nonexistence, something like a cross between 1 (God) and 0 (Void) in his binary scheme. Leibniz likened i to the Holy Spirit: both have an ethereal and barely substantial existence. But even Leibniz didn’t realize that i would finally reveal the rel...

Poncelet founded a new discipline: projective geometry. Poncelet’s mathematics was the culmination of the work begun by the artists and architects of the fifteenth century, like Filippo Brunelleschi and Leonardo da Vinci, who discovered how to draw realistically—in perspective. When “parallel” lines converge at the vanishing point in a painting, observers are tricked into believing that the lines never meet. Squares on the floor become trapezoids in a painting; everything gets gently distorted, but it looks perfectly natural to the viewer. This is the property of an infinitely distant point—a zero at infinity.

And all ellipses have the same property: if you had a mirror in the shape of an ellipse and you placed a lightbulb at one focus, all the beams of light would converge at the other focus, no matter how stretched-out the ellipses are (Figure 29). In his mind Kepler stretched an ellipse out more and more, dragging one focus farther and farther away. Then Kepler imagined that the second focus was infinitely far away: the second focus was a point at infinity. All of a sudden the ellipse becomes a parabola, and all of the lines that converged to a point become parallel lines. A parabola is simply an ellipse with one focus at infinity

Take, for example, the number i. The angle between i and the x-axis is 90 degrees (Figure 33). What happens when you square i? Well, by definition, i2=–1—a point whose angle is 180 degrees from the x-axis; the angle has doubled. The number i3 is equal to –i—270 degrees from the x-axis; the angle has tripled. The number i4=1; we have gone around 360 degrees—exactly four times the original angle (Figure 34). This is not a coincidence. Take any complex number and measure its angle. Raising a number to the nth power multiplies its angle by n. And as you keep raising the number to higher and higher powers, the number will spiral inward or outward, depending on whether the number is on the inside or on the outside of the unit circle, a circle centered at the origin with radius 1 (Figure 35). Multiplication and exponentiation in the complex plane became geometric ideas; you could actually see them happening.

Riemann saw that the complex plane (with a point at infinity) was the same thing as a sphere,

Most interesting of all, if you take a number x and replace it with its reciprocal 1/x, that is equivalent to flipping the sphere upside down and reflecting it in a mirror. The north pole becomes the south pole and the south pole becomes the north pole: zero becomes infinity and infinity becomes zero. It’s all built into the geometry of the sphere; 1/0=∞ and 1/∞=0. Infinity and zero are simply opposite poles on the Riemann sphere, and they can switch places in a blink. And they have equal and opposite powers. Figure 37: Lines and circles are the same.

In fact, we can remove an infinite number of elements from the set of whole numbers—we can delete the odd numbers, for example—and the size of the set remains unchanged. Everybody still has a seat, and every seat is filled: This is the definition of the infinite: it is something that can stay the same size even when you subtract from it.

The even numbers, the odd numbers, the whole numbers, the integers—all of these sets were the same size, a size that Cantor soon dubbed (aleph nought, named after the first letter of the Hebrew alphabet). Since these numbers are the same size as the counting numbers, any set of size is called countable.

both the rationals and irrationals make up the so-called real numbers. Cantor discovered that the set of real numbers is much, much bigger than the rationals.

The real numbers are a bigger infinity than the rational numbers. The term for this type of infinity was , the first uncountable infinity.

In Cantor’s mind there were an infinite number of infinities—the transfinite numbers—each nested in the other. is smaller than , which is smaller than , which is smaller than , and so forth. At the top of the chain sits the ultimate infinity that engulfs all other infinities: God, the infinity that defies all comprehension.

When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science. —WILLIAM THOMSON, LORD KELVIN

Statistical mechanics—the theory of wiggles—explained

Indeed, one of the most important thermodynamic laws—the so-called Stefan-Boltzmann equation—seems to tie the wiggles of molecules to the wiggles of light. It relates the temperature of an object to the total amount of light energy it radiates. This was the biggest victory for statistical mechanics and the wave theory of light. (The equation states that the radiated energy is proportional to the temperature raised to the fourth power. It not only tells how much radiation an object gives off, but also how hot an object gets when irradiated with a given amount of energy. This is the law that physicists used—along with a passage in the book of Isaiah—to determine that heaven is more than 500 degrees Kelvin.)

According to the Rayleigh-Jeans equation, every object is constantly radiating an infinite amount of energy, no matter what its temperature is; even an ice cube would be radiating enough ultraviolet rays, x rays, and gamma rays to vaporize everything around. This was the “ultraviolet catastrophe.” Zero wavelength equals infinite energy; zero and infinity conspired to break a nice, neat system of laws. Solving this paradox quickly became the leading puzzle in physics. Rayleigh and Jeans had done nothing wrong. They used equations that physicists thought were valid, manipulated them in an accepted way, and came out with an answer that didn’t reflect the way the world works. Ice cubes don’t wipe out civilizations with bursts of gamma rays, though following the then-accepted rules of physics led inexorably to that conclusion.

To physicists, vacuum has all particles and forces latent in it. It’s a far richer substance than the philosopher’s nothing. —SIR MARTIN REES

E=mc2. This simple formula relates mass and energy: the mass of an object is equivalent to a certain amount of energy. (In fact, particle physicists don’t measure the mass of the electron, say, in kilograms or pounds or any of the usual units of mass or weight. They say that the electron’s rest mass is 0.511 MeV [million electron volts]—a lump of energy.) The fluctuation in the energy in the vacuum is the same thing as a fluctuation in the amount of mass. Particles are constantly winking in and out of existence, like tiny Cheshire cats. The vacuum is never truly empty. Instead, it is seething with these virtual particles; at every point in space, an infinite number are happily popping up and disappearing. This is the zero-point energy, an infinity in the formulas of quantum theory. Interpreted strictly, the zero-point energy is limitless. According to the equations of quantum mechanics, more power than is stored in all the coal mines, oil fields, and nuclear weapons in the world is sitting in the space inside your toaster.

They don’t go all the way to zero distance from the electron when they calculate the electron’s true mass and charge; they stop short of zero at an arbitrary distance. Once a scientist chooses a suitably close distance, all the calculations using the “true” mass and charge agree with one another. This is a process called renormalization. “It is what I would call a dippy process,” wrote physicist Richard Feynman, even though Feynman won his Nobel Prize for perfecting the art of renormalization.

Tally sticks caused no end of trouble. The English Exchequer used to keep accounts on a variant of the tally stick until 1826. Charles Dickens told of the outcome of that long-outdated practice: “In 1834, it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur to any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighborhood. However, they never had been useful, and official routine required that they should never be, and so the order went out that they were to be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, over-gorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost thereof.”

When Newton was three, his mother remarried and moved. Newton didn’t accompany his mother and stepfather. As a result, he had little contact with his parents after that, unless you count the time he threatened to come over and burn their house down with them inside.