The ancient pharaohs assigned surveyors to assess the damage and reset the boundary markers, and thus geometry was born. These surveyors, or rope stretchers (named for their measuring devices and knotted ropes designed to mark right angles), eventually learned to determine the areas of plots of land by dividing them into rectangles and triangles. The Egyptians also learned how to measure the volumes of objects—like pyramids. Egyptian mathematics was famed throughout the Mediterranean, and it is likely that the early Greek mathematicians, masters of geometry like Thales and Pythagoras, studied in Egypt. 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.
The Birth of Zero
In the history of culture the discovery of zero will always stand out as one of the greatest single achievements of the human race.
—TOBIAS DANZIG, NUMBER: THE LANGUAGE OF SCIENCE
The Greeks understood mathematics better than the Egyptians did; once they mastered the Egyptian art of geometry, Greek mathematicians quickly surpassed their teachers.
At first the Greek system of numbers was quite similar to the Egyptians’. Greeks also had a base-10 style of counting, and there was very little difference in the ways the two cultures wrote down their numbers. 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. They also had a symbol for five, indicating a mixed quinary-decimal system, but overall the Greek and Egyptian systems of writing numbers were almost identical—for a time. 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: ? for 80, and 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.)
Though the Greek number system was more sophisticated than the Egyptian system, it was not the most advanced way of writing numbers in the ancient world. That title was held by another Eastern invention: the Babylonian style of counting. And thanks to this system, zero finally appeared in the East, in the Fertile Crescent of present-day Iraq.
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. Also, the Babylonians used only two marks to represent their numbers: a wedge that represented 1 and a double wedge that represented 10. Groups of these marks, arranged in clumps that summed to 59 or less, were the basic symbols of the counting system, just as the Greek system was based on letters and the Egyptian system was based on pictures. But the really odd feature of the Babylonian system was that, instead of having a different symbol for each number like the Egyptian and Greek systems, each Babylonian symbol could represent a multitude of different numbers. A single wedge, for instance, could stand for 1; 60; 3,600; or countless others.
Figure 1: Numerals of different cultures
As strange as this system seems to modern eyes, it made perfect sense to ancient peoples. It was the Bronze Age equivalent of computer code. 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.)
Adding numbers on an abacus is as simple as moving the stones up and down. Stones in different columns have different values, and by manipulating them a skilled user can add large numbers with great speed. When a calculation is complete, all the user has to do is look at the final position of the stones and translate that into a number—a pretty straightforward operation.
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 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.
Though zero was useful, it was only a placeholder. It was merely a symbol for a blank place in the abacus, a column where all the stones were at the bottom. It did little more than make sure digits fell in the right places; it didn’t really have a numerical value of its own. After all, 000,002,148 means exactly the same thing as 2,148. A zero in a string of digits takes its meaning from some other digit to its left. On its own, it meant…nothing. Zero was a digit, not a number. It had no value.
Figure 2: Babylonian numbers
A number’s value comes from its place on the number line—from its position compared with other numbers. For instance, the number two comes before the number three and after the number one; nowhere else makes any sense. However, the 0 mark didn’t have a spot on the number line at first. It was just a symbol; it didn’t have a place in the hierarchy of numbers. Even today, we sometimes treat zero as a nonnumber even though we all know that zero has a numerical value of its own, using the digit 0 as a placeholder without connecting it to the number zero. Look at a telephone or the top of a computer keyboard. The 0 comes after the 9, not before the 1 where it belongs. It doesn’t matter where the placeholder 0 sits; it can be anywhere in the number sequence. But nowadays everybody knows that zero can’t really sit anywhere on the number line, because it has a definite numerical value of its own. It is the number that separates the positive numbers from the negative numbers. It is an even number, and it is the integer that precedes one. Zero must sit in its rightful place on the number line, before one and after negative one. Nowhere else makes any sense. Yet zero sits at the end of the computer and at the bottom of the telephone because we always start counting with one.