Zero: The Biography of a Dangerous Idea

by Charles Seife

In fact, knowing about numbers at all was quite an ability in prehistoric times. Simply being able to count was considered a talent as mystical and arcane as casting spells and calling the gods by name. 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.

Yet despite the Egyptians’ brilliant geometric work, zero was nowhere to be found within Egypt. This was, in part, because the Egyptians were of a practical bent. They never progressed beyond measuring volumes and counting days and hours. Mathematics wasn’t used for anything impractical, except their system of astrology. As a result, their best mathematicians were unable to use the principles of geometry for anything unrelated to real world problems—they did not take their system of mathematics and turn it into an abstract system of logic. They were also not inclined to put math into their philosophy. The Greeks were different; they embraced the abstract and the philosophical, and brought mathematics to its highest point in ancient times. Yet it was not the Greeks who discovered zero. Zero came from the East, not the West.

Instead of using pictures to represent numbers as the Egyptians did, the Greeks used letters. H (eta) stood for hekaton: 100. M (mu) stood for myriori: 10,000—the myriad, the biggest grouping in the Greek system.

Unlike the Egyptians, the Greeks outgrew this primitive way of writing numbers and developed a more sophisticated system. Instead of using two strokes to represent 2, or three Hs to represent 300 as the Egyptian style of counting did, a newer Greek system of writing, appearing before 500 BC, had distinct letters for 2, 3, 300, and many other numbers (Figure 1). In this way the Greeks avoided repeated letters. For instance, writing the number 87 in the Egyptian system would require 15 symbols: eight heels and seven vertical marks. The new Greek system would need only two symbols: π for 80, and ζ for 7. (The Roman system, which supplanted Greek numbers, was a step backward toward the less sophisticated Egyptian system. The Roman 87, LXXXVII, requires seven symbols, with several repeats.)

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 abacus relies upon sliding stones to keep track of amounts. (The words calculate, calculus, and calcium all come from the Latin word for pebble: calculus.)

The Babylonian system of numbering was like an abacus inscribed symbolically onto a clay tablet. Each grouping of symbols represented a certain number of stones that had been moved on the abacus, and like each column of the abacus, each grouping had a different value, depending on its position. In this way the Babylonian system was not so different from the system we use today. Each 1 in the number 111 stands for a different value; from right to left, they stand for “one,” “ten,” and “one hundred,” respectively. Similarly, the symbol in stood for “one,” “sixty,” or “thirty-six hundred” in the three different positions. It was just like an abacus, except for one problem. How would a Babylonian write the number 60? The number 1 was easy to write: . Unfortunately, 60 was also written as ; the only difference was that was in the second position rather than the first. With the abacus it’s easy to tell which number is represented. A single stone in the first column is easy to distinguish from a single stone in the second column. The same isn’t true for writing. The Babylonians had no way to denote which column a written symbol was in; could represent 1, 60, or 3,600. It got worse when they mixed numbers. The symbol could mean 61; 3,601; 3,660; or even greater values. Zero was the solution to the problem. 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 w...

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

A number’s value comes from its place on the number line—from its position compared with other numbers.

In fact, Egyptian civilization was bad for math in more ways than one; it was not just the absence of a zero that caused future difficulties. 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. Long chains of these unit fractions made ratios extremely difficult to handle

digits in the Babylonian system, zero never was allowed to stand alone—for good reason.

Add a number to itself and it changes. One and one is not one—it’s two. Two and two is four. But zero and zero is zero. This violates a basic principle of numbers called the axiom of Archimedes, which says that if you add something to itself enough times, it will exceed any other number in magnitude. (The axiom of Archimedes was phrased in terms of areas; a number was viewed as the difference of two unequal areas.) Zero refuses to get bigger. It also refuses to make any other number bigger.

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 the West would have to destroy their universe.

The Egyptians, who had invented geometry, thought little about mathematics. For them it was a tool to reckon the passage of the days and to maintain plots of land. The Greeks had a very different attitude. To them, numbers and philosophy were inseparable, and they took both very seriously.

Hippasus of Metapontum stood on the deck, preparing to die. Around him stood the members of a cult, a secret brotherhood that he had betrayed. Hippasus had revealed a secret that was deadly to the Greek way of thinking, a secret that threatened to undermine the entire philosophy that the brotherhood had struggled to build. For revealing that secret, the great Pythagoras himself sentenced Hippasus to death by drowning. To protect their number-philosophy, the cult would kill. Yet as deadly as the secret that Hippasus revealed was, it was small compared to the dangers of zero. The leader of the cult was Pythagoras, an ancient radical. According to most accounts, he was born in the sixth century BC on Samos, a Greek island off the coast of Turkey famed for a temple to Hera and for really good wine. Even by the standards of the superstitious ancient Greeks, Pythagoras’s beliefs were eccentric. He was firmly convinced that he was the reincarnated soul of Euphorbus, a Trojan hero. This helped convince Pythagoras that all souls—including those of animals—transmigrated to other bodies after death. Because of this, he was a strict vegetarian. Beans, however, were taboo, as they generate flatulence and are like the genitalia.

But at the center of their philosophy was the most important tenet of the Pythagoreans: all is number.

To the Greek philosopher-mathematicians they were pretty much the same thing. (Even today, we have square numbers and triangular numbers thanks to their influence [Figure 5].)

And to Pythagoras the connection between shapes and numbers was deep and mystical. Every number-shape had a hidden meaning, and the most beautiful number-shapes were sacred.

The mystical symbol of the Pythagorean cult was, naturally, a number-shape: the pentagram, a five-pointed star. This simple figure is a glimpse at the infinite.

This star, in turn, contains an even smaller pentagon, which contains a tinier star with its tiny pentagon, and so forth (Figure 6). As interesting as this was, to the Pythagoreans the most important property of the pentagram was not in this self-replication but was hidden within the lines of the star. They contained a number-shape that was the ultimate symbol of the Pythagorean view of the universe: the golden ratio.

Figure 5: Square and triangular numbers

Figure 6: The p...

To Pythagoras, playing music was a mathematical act. Like squares and triangles, lines were number-shapes, so dividing a string into two parts was the same as taking a ratio of two numbers. The harmony of the monochord was the harmony of mathematics—and the harmony of the universe.

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.

This philosophy—the interchangeability of music, math, and nature—led to the earliest Pythagorean model of the planets. 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.”

Figure 8: The Greek universe

The pentagram became the most sacred symbol of the Pythagorean brotherhood because the lines of the star are divided in this special way—the pentagram is chock-full of the golden ratio—and for Pythagoras, the golden ratio was the king of numbers. The golden ratio was favored by artists and nature alike and seemed to prove the Pythagorean assertion that music, beauty, architecture, nature, and the very construction of the cosmos were all intertwined and inseparable. 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. As late as Shakespeare’s time, scientists talked about the revolution of orbs of different proportions and discussed the heavenly music that reverberated throughout the cosmos.

Zero would punch a hole in the neat Pythagorean order of the universe, and for that reason it could not be tolerated.

Indeed, the Pythagoreans had tried to squelch another troublesome mathematical concept—the irrational. This concept was the first challenge to the Pythagorean point of view, and the brotherhood tried to keep it secret. When the secret leaked out, the cult turned to violence.

The brotherhood was being destroyed. Pythagoras himself fled for his life, and he might have gotten away had he not run smack into a bean field. There he stopped. He declared that he would rather be killed than cross the field of beans. His pursuers were more than happy to oblige. They cut his throat.

With Babylonian astronomy came Babylonian numbers. For astronomical purposes the Greeks adopted a sexagesimal number system and even divided hours into 60 minutes, and minutes into 60 seconds.

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.

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.

had a paradox, a logical puzzle that seemed intractable to the reasoning of Greek philosophers. It was the most troubling argument in Greece: Zeno had proved the impossible. According to Zeno, nothing in the universe could move.

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.

The world of Dionysius Exiguus, Boethius, and Bede was dark indeed. Rome had collapsed, and Western civilization seemed but a shadow of Rome’s past glory. The future seemed more horrid than the past. It is no wonder that in the search for wisdom medieval scholars didn’t look to their peers for ideas. Instead, they turned to the ancients like Aristotle and the Neoplatonists. As these medieval thinkers imported the philosophy and science of the ancients, they inherited the ancient prejudices: a fear of the infinite and a horror of the void.

Lurking underneath the veil of medieval philosophy, however, was a conflict. 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.

An Indian text written the same year that Rome fell—476 AD—shows the influence of Greek, Egyptian, and Babylonian mathematics, brought by Alexander as he penetrated Indian lands. Like the Egyptians, the Indians had rope stretchers to survey fields and lay out temples. They also had a sophisticated system of astronomy; like the Greeks, they tried to calculate the distance to the sun. That requires trigonometry; the Indian version was probably derived from the system that the Greeks had developed.

Sometime around the fifth century AD, Indian mathematicians changed their style of numbering; they moved from a Greek-like system to a Babylonian-style one. An important difference between the new Indian number system and the Babylonian style was that Indian numbers were base-10 instead of base-60. Our numbers evolved from the symbols that the Indians used; by rights they should be called Indian numerals rather than Arabic ones

Contests between the abacists and the so-called algorists who used Indian numerals were the medieval equivalents of the Kasparov versus Deep Blue chess match

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 Muslims seized upon the atomists’ ideas; after all, now that zero was around, the void was again a respectable idea. Aristotle hated the void; the atomists required it. The Bible told of the creation from the void, while the Greek doctrine rejected the possibility. The Christians, cowed by the power of Greek philosophy, chose Aristotle over their Bible. The Muslims, on the other hand, made the opposite choice.

In the thirteenth century a new doctrine spread: kabbalism, or Jewish mysticism. One centerpiece of kabbalistic thought is gematria—the search for coded messages within the text of the Bible. 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.

In 1277 the bishop of Paris, Étienne Tempier, called an assembly of scholars to discuss Aristotelianism, or rather, to attack it. Tempier abolished many Aristotelian doctrines that contradicted God’s omnipotence,

Tempier’s pronouncements were not the final blow to Aristotelian philosophy, but they certainly signaled that the foundations were crumbling. The church would cling to Aristotle for a few more centuries, but the fall of Aristotle and the rise of the void and the infinite were clearly beginning. It was a propitious time for zero to arrive in the West. In the mid-twelfth century the first adaptations of al-Khowarizmi’s Al-jabr were working their way through Spain, England, and the rest of Europe. Zero was on the way, and just as the church was breaking the shackles of Aristotelianism, it arrived.

Leonardo of Pisa. The son of an Italian trader, he traveled to northern Africa. There the young man—better known as Fibonacci—learned mathematics from the Muslims and soon became a good mathematician in his own right.

Fibonacci is best remembered for a silly little problem he posed in his book, Liber Abaci, which was published in 1202. Imagine that a farmer has a pair of baby rabbits. Babies take two months to reach maturity, and from then on they produce another pair of rabbits at the beginning of every month. As these rabbits mature and reproduce, and those rabbits mature and reproduce, and so on, how many pairs of rabbits do you have during any given month?

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

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

the end the governments had to relent in the face of commercial pressure. The Arabic notation was allowed into Italy and soon spread throughout Europe. Zero had arrived—as had the void. The Aristotelian wall was crumbling,