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.

There was neither non-existence nor existence then; there was neither the realm of space nor the sky which is beyond. What stirred? Where? —THE RIG VEDA

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

In the very beginning of mathematics, it seems that people could only distinguish between one and many. A caveman owned one spearhead or many spearheads; he had eaten one crushed lizard or many crushed lizards. There was no way to express any quantities other than one and many. Over time, primitive languages evolved to distinguish between one, two, and many, and eventually one, two, three, many, but didn’t have terms for higher numbers. Some languages still have this shortcoming. The Siriona Indians of Bolivia and the Brazilian Yanoama people don’t have words for anything larger than three; instead, these two tribes use the words for “many” or “much.”

It was an accident of nature that gave humans five fingers on each hand, and because of this accident, five seemed to be a favorite base system across many cultures. The early Greeks, for instance, used the word “fiving” to describe the process of tallying.

Even in the South American binary counting schemes, linguists see the beginnings of a quinary system. A different phrase in Bororo for “two and two and one” is “this is my hand all together.” Apparently, ancient peoples liked to count with their body parts, and five (a hand), ten (both hands), and twenty (both hands and both feet) were the favorites. In English, eleven and twelve seem to be derived from “one over [ten]” and “two over [ten],” while thirteen, fourteen, fifteen, and so on are contractions of “three and ten,” “four and ten,” and “five and ten.”

In the Egyptian Book of the Dead, when a dead soul is challenged by Aqen, the ferryman who conveys departed spirits across a river in the netherworld, Aqen refuses to allow anyone aboard “who does not know the number of his fingers.” The soul must then recite a counting rhyme to tally his fingers, satisfying the ferryman. (The Greek ferryman, on the other hand, wanted money, which was stowed under the dead person’s tongue.)

A number of modern-day nations, like Israel and Saudi Arabia, still use a modified lunar calendar,

The Egyptian calendar had 12 months, like the lunar one, but each month was 30 days long. (Being base-10 sort of people, their week, the decade, was 10 days long.) At the end of the year, there were an extra five days, bringing the total up to 365. This calendar was the ancestor of our own calendar; the Egyptian system was adopted by Greece and then by Rome, where it was modified by adding leap years, and then became the standard calendar of the Western world.

(The Egyptians took property rights very seriously. 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. In Egypt, filching your neighbor’s land was considered as grave an offense as breaking an oath, murdering somebody, or masturbating in a temple.)

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

By around 300 BC the Babylonians had started using two slanted wedges, , to represent an empty space, an empty column on the abacus. This placeholder mark made it easy to tell which position a symbol was in. Before the advent of zero, could be interpreted as 61 or 3,601. But with zero, meant 61; 3,601 was written as (Figure 2). Zero was born out of the need to give any given sequence of Babylonian digits a unique, permanent meaning.

Because their system of counting was based on the number 20, the Mayans naturally divided their year into 18 months of 20 days each, totaling 360 days. A special period of five days at the end, called Uayeb, brought the count to 365. Unlike the Egyptians, though, the Mayans had a zero in their counting system, so they did the obvious thing: they started numbering days with the number zero. The first day of the month of Zip, for example, was usually called the “installation” or “seating” of Zip. The next day was 1 Zip, the following day was 2 Zip, and so forth, until they reached 19 Zip. The next day was the seating of Zotz’—0 Zotz’ followed by 1 Zotz’ and so forth. Each month had 20 days, numbered 0 through 19, not numbered 1 through 20 as we do today. (The Mayan calendar was wonderfully complicated. Along with this solar calendar, there was a ritual calendar that had 20 weeks, each of 13 days. Combined with the solar year, this created a calendar round

The Egyptians had an extremely cumbersome way of handling fractions. They didn’t think of 3/4 as a ratio of three to four as we do today; they saw it as the sum of 1/2 and 1/4. With the sole exception of 2/3, all Egyptian fractions were written as a sum of numbers in the form of 1/n (where n is a counting number)—the so-called unit fractions.

In the Babylonian system—with zero in it—it’s easy to write fractions. Just as we can write 0.5 for 1/2 and 0.75 for 3/4, the Babylonians used the numbers 0;30 for 1/2 and 0;45 for 3/4. (In fact, the Babylonian base-60 system is even better suited to writing down fractions than our modern-day base-10 system.) Unfortunately the Greeks and Romans hated zero so much that they clung to their own Egyptian-like notation rather than convert to the Babylonian system, even though the Babylonian system was easier to use. For intricate calculations, like those needed to create astronomical tables, the Greek system was so cumbersome that the mathematicians converted the unit fractions to the Babylonian sexagesimal system, did the calculations, and then translated the answers back into the Greek style. They could have saved many time-consuming steps. (We all know how fun it is to convert fractions back and forth!) However, the Greeks so despised zero that they refused to admit it into their writings, even though they saw how useful it was.

Dividing by zero once—just one time—allows you to prove, mathematically, anything at all in the universe. You can prove that 1 + 1=42, and from there you can prove that J. Edgar Hoover was a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof that Winston Churchill was a carrot.)

Zero clashed with one of the central tenets of Western philosophy, a dictum whose roots were in the number-philosophy of Pythagoras and whose importance came from the paradoxes of Zeno. The whole Greek universe rested upon this pillar: there is no void. 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. Because of this, the West could not accept zero for nearly two millennia. The consequences were dire. Zero’s absence would stunt the growth of mathematics, stifle innovation in science, and, incidentally, make a mess of the calendar. Before they could accept zero, philosophers

In ancient Greece, Pythagoras was remembered for a different invention: the musical scale. One day, according to legend, Pythagoras was toying with a monochord, a box with a string on it (Figure 7). By moving a sliding bridge up and down the monochord, Pythagoras changed the notes that the device played. He quickly discovered that strings have a peculiar, yet predictable, behavior. When you pluck the string without the bridge, you get a clear note, the tone known as the fundamental. Putting the bridge on the monochord so it touches the string changes the notes that are played. When you place the bridge exactly in the middle of the monochord, touching the center of the string, each half of the string plays the same note: a tone exactly one octave higher than the string’s fundamental. Shifting the bridge slightly might divide the string so that one side has three-fifths of the string and the other has two-fifths; in this case Pythagoras noticed that plucking the string segments creates two notes that form a perfect fifth, which is said to be the most powerful and evocative musical relationship. Different ratios gave different tones that could soothe or disturb. (The discordant tritone, for instance, was dubbed the “devil in music” and was rejected by medieval musicians.) Oddly enough, when Pythagoras put the bridge at a place that did not divide the string into a simple ratio, the plucked notes did not meld well. The sound was usually dissonant and sometimes even worse. Ofte...

Pythagoras argued that the earth sat at the center of the universe, and the sun, moon, planets, and stars revolved around the earth, each pinned inside a sphere (Figure 8). 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.”

the golden ratio. Achieving this blissful mean is a matter of dividing a line in a special way: divide it in two so that the ratio of the small part to the large part is the same as the ratio of the large part to the whole (see appendix B). In words, it doesn’t seem particularly special, but figures imbued with this golden ratio seem to be the most beautiful objects. 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. Some historians and mathematicians argue that

Some say that the Pythagoreans tossed Hippasus overboard, drowning him, a just punishment for ruining a beautiful theory with harsh facts. Ancient sources talk about his perishing at sea for his impiety, or alternatively, say that the brotherhood banished him and constructed a tomb for him, expelling him from the world of human beings. But whatever Hippasus’s true fate was, there is little doubt that he was reviled by his brothers. The secret he revealed shook the very foundations of the Pythagorean doctrine, but by considering the irrational an anomaly, the Pythagoreans could keep the irrationals from contaminating their view of the universe. Indeed, over time the Greeks reluctantly admitted the irrationals to the realm of numbers.

For astronomical purposes the Greeks adopted a sexagesimal number system and even divided hours into 60 minutes, and minutes into 60 seconds. 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.)

Pythagoras’s doctrine became the centerpiece of Western philosophy: all the universe was governed by ratios and shapes; the planets moved in heavenly spheres that made music as they turned. But what lay beyond these spheres? Were there more and more spheres, each larger than its neighbor? Or was the outermost sphere the end of the universe? Aristotle and later philosophers would insist that there could not be an infinite number of nested spheres. With the adoption of this philosophy, the West had no room for infinity or the infinite. They rejected it outright. For the infinite had already begun to gnaw at the roots of Western thought, thanks to Zeno of Elea, a philosopher reckoned by his contemporaries to be the most annoying man in the West.

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.

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.

Around 435 BC, he conspired to overthrow the tyrant of Elea, Nearchus. He was smuggling arms to support the cause. Unfortunately for Zeno, Nearchus found out about the plot, and Zeno was arrested. Hoping to discover who the coconspirators were, Nearchus had Zeno tortured. Soon Zeno begged the torturers to stop and promised that he would name his colleagues. When Nearchus drew near, Zeno insisted that the tyrant come closer, since it was best to keep the names a secret. Nearchus leaned over, tilting his head toward Zeno. All of a sudden Zeno sank his teeth into Nearchus’s ear. Nearchus screamed, but Zeno refused to let go. The torturers could only force Zeno to let go by stabbing him to death. Thus died the master of the infinite.

Archimedes first glimpsed the infinite in the polish of his war mirrors. For centuries the Greeks had been fascinated with conic sections. Take a cone and cut it up; you get circles, ellipses, parabolas, and hyperbolas, depending on how you slice it. The parabola has a special property: it takes the rays of light from the sun, or any distant source, and focuses them to a point, concentrating all the light’s energy on a very small area. Any mirror that could set ships afire must be in the shape of a parabola. Archimedes studied the properties of the parabola, and it is here that he first started toying with the infinite.

Figure 12: Archimedes’ parabola

1051 is a really, really big number. Take 1051 molecules of water, for instance. It would take every man, woman, and child now on Earth, each drinking a ton of water a second, over 150,000 years to drink it all.)

It is a silly, childish discussion and only exposes the want of brains of those who maintain a contrary opinion to that we have stated. —THE TIMES (LONDON), DECEMBER 26, 1799 This “silly, childish discussion”—whether the new century begins on the year 00 or the year 01—appears and reappears like clockwork every hundred years. If medieval monks had only known of zero, our calendar would not be in such a muddle.

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

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.

When we are dealing with the counting numbers—1, 2, 3, and so on—it is easy to rank them in order. One is the first counting number, two is the second counting number, and three is the third. We don’t have to worry about mixing up the value of the number—its cardinality—with the order in which it arrives—its ordinality—since they are essentially the same thing. For years, this was the state of affairs, and everybody was happy. 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. This is the root of the calendar problem.

Though we count with the ordinals (first, second, third), we mark time with the cardinals (0, 1, 2). All of us have assimilated this way of thinking, whether we appreciate it or not. After a baby finishes his 12th month, we all say that the child is one year old; he has finished his first 12 months of life. If the baby turns one when she’s already lived a year, isn’t the only consistent choice to say that the baby is zero years old before that time? Of course, we say that the child is six weeks old or nine months old instead—a clever way of getting around the fact that the baby is zero.

It was not absolute nothingness. It was a kind of formlessness without any definition. . . . True reasoning convinced me that I should wholly subtract all remnants of every kind of form if I wished to conceive the absolutely formless. I could not achieve this. —SAINT AUGUSTINE, CONFESSIONS

The Aristotelian system was Greek, but the Judeo-Christian story of creation was Semitic—and Semites didn’t have such a fear of the void. The very act of creation was out of a chaotic void, and theologians like Saint Augustine, who lived in the fourth century, tried to explain it away by referring to the state before creation as “a nothing something” that is empty of form but yet “falls short of utter nothingness.” The fear of the void was so great that Christian scholars tried to fix the Bible to match Aristotle rather than vice versa.

Hinduism was steeped in the symbolism of duality. (Of course, this idea occasionally came up in the Western world, where it was promptly branded as heretical. One example is the Manichaean heresy, which saw the world as being under the influence of equal and opposite sources of good and evil.) As with the yin and yang of the Far East and Zoroaster’s dualism of good and evil in the Near East, creation and destruction were intermingled in Hinduism. The god Shiva was both creator and destroyer of the world and was depicted with the drum of creation in one hand and a flame of destruction in another (Figure 13). However, Shiva also represented nothingness. One aspect of the deity, Nishkala Shiva, was literally the Shiva “without parts.” He was the ultimate void, the supreme nothing—lifelessness incarnate. But out of the void, the universe was born, as was the infinite. Unlike the Western universe, the Hindu cosmos was infinite in extent; beyond our own universe were innumerable other universes.

In one story, Death tells a disciple about the soul: “Concealed in the heart of all beings is the Atman, the Spirit, the Self,” he says. “Smaller than the smallest atom, greater than the vast spaces.” This Atman, which resides in every thing, is part of the essence of the universe, and is immortal. When a person dies, the Atman is released from the body and soon enters another being; the soul transmigrates and the person is reincarnated. The goal of the Hindu is to free the Atman entirely from the cycle of rebirth, to stop wandering from death to death. The way to achieve the ultimate liberation through lifelessness is to cease paying heed to the illusion of reality. “The body, the house of the spirit, is under the power of pleasure and pain,” explains a god. “And if a man is ruled by his body then this man can never be free.” But once you are able to separate yourself from the whims of the flesh and embrace the silence and nothingness of your soul, you will be liberated. Your Atman will fly from the web of human desire and join the collective consciousness—the infinite soul that suffuses the universe, at once everywhere and nowhere at the same time. It is infinity, and it is nothing.

Does man forget that We created him out of the void? —THE KORAN

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.

The Indian name for zero was sunya, meaning “empty,” which the Arabs turned into sifr. When some Western scholars described the new number to their colleagues, they turned sifr into a Latin-sounding word, yielding zephirus, which is the root of our word zero. Other Western mathematicians didn’t change the word so heavily and called zero cifra, which became cipher. Zero was so important to the new set of numbers that people started calling all numbers ciphers, which gave the French their term chiffre, digit.

Like the Greeks, the Hebrews used letters from their alphabet to represent numbers, so every word had a numerical value. This could be used to interpret the hidden meaning of words. For instance, Gulf War participants might have noticed that Saddam has the following value: samech (60) + aleph (1) + daled (4) + aleph (1) + mem (600)=666—a number that Christians associate with the evil Beast that appears during the Apocalypse. (Whether “Saddam” has two daleds or one would make no difference to the kabbalists, who often used alternate spellings of words to make sums come out right.) Kabbalists thought that words and phrases with the same numerical value were mystically linked.

Aristotle still had a firm grip on the church, and its finest thinkers still rejected the infinitely large, the infinitely small, and the void. Even as the Crusades drew to a close in the thirteenth century, Saint Thomas Aquinas declared that God could not make something that was infinite any more than he could make a scholarly horse. But that implied that God was not omnipotent—a forbidden thought in Christian theology.

The number of rabbits goes as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .; the number of rabbits you have in any given month is the sum of the rabbits that you had in each of the two previous months. Mathematicians instantly realized the importance of this series. Take any term and divide it by its previous term. For instance, 8/5=1.6; 13/8=1.625; 21/13=1.61538 . . . . 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.

And new philosophy calls all in doubt, The element of fire is quite put out; The sun is lost, and th’ earth, and no man’s wit Can well direct him where to look for it. . . . ’Tis all in pieces, all coherence gone; All just supply, and all relation: Prince, subject, Father; Son, are things forgot. —JOHN DONNE, “AN ANATOMY OF THE WORLD

As objects recede into the distance in the painting, they get closer and closer to the vanishing point, getting more compressed as they get farther away from the viewer. Everything sufficiently distant—people, trees, buildings—is squashed into a zero-dimensional point and disappears. The zero in the center of the painting contains an infinity of space.

The vanishing point turned a two-dimensional drawing into a perfect simulation of a three-dimensional building. Figure 18: The vanishing point It is no coincidence that zero and infinity are linked in the vanishing point. Just as multiplying by zero causes the number line to collapse into a point, the vanishing point has caused most of the universe to sit in a tiny dot. This is a singularity,

How could a vanishing point be used to simulate four dimensionl space? Smeared sculptures? Tralfamadorian vision?

A contemporary of Brunelleschi, a German cardinal named Nicholas of Cusa, looked at infinity and promptly declared, “Terra non est centra mundi”: the earth is not the center of the universe.