Zero: The Biography of a Dangerous Idea: Chapters 0–1

Cultures girded themselves against zero, and philosophies crumbled under its influence, for zero is different from the other numbers.

It provides a glimpse of the ineffable and the infinite.

Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling.

And the most profound problems in physics—the dark core of a black hole and the brilliant flash of the big bang—are struggles to defeat zero.

An Eastern concept, born in the Fertile Crescent a few centuries before the birth of Christ, zero not only evoked images of a primal void, it also had dangerous mathematical properties.

Indeed, zero was so abhorrent to some cultures that they chose to live without it.

In the very beginning of mathematics, it seems that people could only distinguish between one and many.

But why five? Deep down, it’s an arbitrary decision. If Gog put his tallies in groups of four, and counted in groups of four and 16, his number system would have worked just as well, as would groups of six and 36.

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.”

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.”

When early civilizations started pressing reeds to clay tablets, carving figures in stone, and daubing ink on parchment and on papyrus, number systems had already been well-established.

Transcribing the oral number system into written form was a simple task: people just needed to figure out a coding method whereby scribes could set the numbers down in a more permanent form.

Some societies even found a way to do this before they discovered writing. The illiterate Incas, for one, used the quipu, a string of colored, knotted cords, to record calculations.

The first scribes wrote down numbers in a way that matched their base system, and predictably, did it in the most concise way they could think of.

Instead of having to write down 123 tick marks to denote the number "one hundred and twenty-three,” the scribe wrote six symbols: one snare, two heels, and three vertical marks.

Creating a stable calendar was a problem for most ancient peoples, because they generally started out with a lunar calendar: the length of a month was the time between successive full moons.

It was a natural choice; the waxing and waning of the moon in the heavens was hard to overlook, and it offered a convenient way of marking periodic cycles of time.

Thirteen lunar months yield roughly 19 days too many. Since it is the solar year, not the lunar year, that determines the time for harvest and planting, the seasons seem to drift when you reckon by an uncorrected lunar year.

Correcting the lunar calendar is a complicated undertaking.

The good news was that the flooding deposited rich, alluvial silt all over the fields, making the Nile delta the richest farmland in the ancient world.

The ancient pharaohs assigned surveyors to assess the damage and reset the boundary markers, and thus geometry was born.

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.

The Greeks were different; they embraced the abstract and the philosophical, and brought mathematics to its highest point in ancient times.

M (mu) stood for myriori: 10,000—the myriad, the biggest grouping in the Greek system.

The Roman system, which supplanted Greek numbers, was a step backward toward the less sophisticated Egyptian system.

At first glance the Babylonian system seems perverse. For one thing the system is sexagesimal—based on the number 60. This is an odd-looking choice, especially since most human societies chose 5, 10, or 20 as their base number.

The Babylonians, like many different cultures, had invented machines that helped them count. The most famous was the abacus. Known as the soroban in Japan, the suan-pan in China, the s’choty in Russia, the coulba in Turkey, the choreb in Armenia, and by a variety of other names in different cultures, 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.)

Each 1 in the number 111 stands for a different value; from right to left, they stand for “one,” "ten,” and “one hundred,” respectively.

Since the Western calendar was created at a time when there was no zero, we never see a day zero, or a year zero. This apparently insignificant omission caused a great deal of trouble; it kindled the controversy over the start of the millennium.

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.

Zero makes this cumbersome system obsolete.

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.

Yet zero was inexorably linked with the void—with nothing.

There was a primal fear of void and chaos.

Emptiness and disorder were the primeval, natural state of the cosmos, and there was always a nagging fear that at the end of time, disorder and void would reign once more.

To the ancients, zero’s mathematical properties were inexplicable, as shrouded in mystery as the birth of the universe.

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.

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

Worst of all, if you wantonly divide by zero, you can destroy the entire foundation of logic and mathematics.