The Origins of the Zero

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The Origins of the Zero

Overview

The zero was invented three times in the history of the mathematics. The Babylonians, the Maya, and the Hindus all invented a symbol to represent nothing. However, only the Hindus came to understand the importance of what the zero represented. Today we use a descendant of the Hindu zero, which had a long journey and encountered much resistance until finally accepted in the West.

Background

Before any invention can be made and accepted in a society, there has to be a need. One of the reasons the zero was not developed along with other numbers is that many early number systems had no real need for a symbol to represent nothing. This may sound strange, but even today we rarely use the term zero in everyday speech. We say, "There are no apples," not "There are zero apples."

The Egyptian, Greek, and Roman number systems represented each number by a unique collection of symbols. For example, in the Roman system the number 23 is XXIII—two tens and three ones—while 203 is CCIII. There are two major faults with such a system. First, the number of symbols needed to represent some numbers can be very large. For example, in Roman numerals 338 is CCCXXXVIII. More importantly, however, such a number system makes complex calculations very difficult. It is hard enough to add and subtract Roman numbers, but just try multiplication and division.

There were other types of number systems that did, eventually, generate a need for a symbol to represent nothing. Place value number systems use a small set of numerals in different positions to represent numbers. The modern decimal system we use today is such a system, with a place for the ones, a place for the tens, a place for the hundreds, and so on. So the number six hundred and twenty-seven is written as 627, or 6 hundreds, 2 tens, and 7 ones.

Day-to-day calculation in the ancient world was done on a counting board or abacus, which allowed for quick and simple computation of addition and subtraction. Each column on an abacus represented a different place in the number system. In a decimal abacus the first column would be the ones, the next the tens, the third the hundreds, and so on. The amount of counters in each column represents the number in a visual manner.

However, place value systems have a problem when representing numbers that do not have any value in a particular place. For example, you may calculate a result on an abacus as three hundred and two, or three counters in the hundreds column, none in the tens, and two in the one column. In the Roman system the number is easily written as CCCII, but in a place value system, without a symbol for nothing, you run into trouble.

Impact

The zero was invented separately three times. In each case it was needed as a placeholder in a place-value number system. The Babylonians used numbers based on 60, a sexigesimal system. We still use their system for measuring the minutes in an hour, and the degrees in a circle (6 × 60 = 360°).

Without a zero symbol, Babylonian scribes had problems recording numbers that had no value in a certain place. To begin with they left a gap between numerals, which would be like writing two hundred and four as 2 4. However, not everyone followed this convention, and when copies were made the gaps were often left out. Even when this system was followed, it was difficult to tell a number like 204 from 2004, as they would both be written as 2 4.

Then, sometime in the third century b.c., an unknown scribe started to use a symbol to represent a place without a value, and so the first zero was invented. With a symbol for the zero there was no longer confusion over numbers like our 204 and 2004.

Yet, while the Babylonians can claim to be the first users of the zero, they did not understand it in the modern sense. Their zero did not represent a number in itself, it was just a place-holder. The idea of zero was still a little vague.

The Babylonians also refused to end numbers with zeroes. In our system that would be like writing 3,000 as 3. However, you would also write 30 as 3, and 3 as 3, which makes them hard to tell apart. Babylonian readers relied on context to determine the value of such numbers. We do this as well to some extent. If someone tells us an orange costs 15, we assume 15 cents, yet if we are told a new book costs 15 we assume they mean 1,500 cents, $15. Babylonian astronomers could not rely on context, and so they used the zero at the end of numbers just as we do, as it enabled then to note degrees and minutes of arc more accurately. Their innovation was not, however, accepted by wider society.

The Greek world encountered the Babylonian zero as part of the spoils of the conquests of Alexander the Great (356-323 b.c.). However, most Greeks had no use for it, as their number system was not a place value system. The concept of zero also raised some unsettling philosophical questions, and contradicted the teachings of Aristotle (384-322 b.c.). Again, like the Babylonians, only Greek astronomers used the zero, for the benefits it gave them outweighed the problems it caused.

Seven centuries after the Babylonian discovery, and half a world away, another culture using a place value number system also invented a zero of their own. The Maya of South America developed a sophisticated and complex time-keeping system. They used a number of different calendars for various purposes, one of which was their religious fear that time might one day come to an end if the calendars ran out.

The Maya used a place-value system using 20 as a base, although the second place only went up to 18. Sometime in about the fourth century the need to hold a place that had no value in their number system led them to develop the zero. They had a number of different symbols for the zero, from a bowl-like object, to a complex face. However, like the Babylonians, the Maya did not use the zero in any wider sense. It remained just a placeholder, and their complex number system limited calculations.

The third invention of the zero occurred in India, although some scholars still debate whether the Babylonian zero could have traveled to India. Again, it was created by the need for a placeholder.

In Hindu mathematics numbers were also written as symbolic words, which made mathematics a little like poetry, and had the added advantage of making copying very accurate. The first use of a Hindu mathematical word for zero dates from a 458 cosmology text, and the first surviving use of a numeral for zero in India dates from 628. In the intervening period, the idea of the zero appears to have become widely accepted in Hindu mathematics.

However, unlike the Maya and the Babylonians before them, the Hindus understood the zero as more than just a placeholder. Perhaps because of the practice of representing numbers with symbolic words, they realized that the zero represented the absence of a quantity. This was a big step, for it enabled mathematics to begin to use the zero in written calculations.

Previously, all calculations had been done on counting frames and abaci. With the zero used as a placeholder in any place value, written numbers became just as easy to understand as those on a counting frame. With the ability to write down all numbers accurately on a surface, it became possible to record the steps of a calculation, not just the result. This led to an interest in the rules of calculation, which marked the birth of algebra.

From India the use of the zero spread to Cambodia by the seventh century, and made it to China shortly after. By the eighth century the Hindu zero had traveled to Arab lands, and was adopted along with the entire Hindu numeral set. It is these numbers that we use today, and they are often called Arabic numbers, despite their origin in India. The Arab acceptance of the Hindu zero caused a revolution in the Arab sciences, and not just for the ease of calculation it offered. The Arabs had absorbed much of Greek learning, and had to deal with the same philosophical problems that had led the Greeks to reject the zero. In particular, the zero contradicted Aristotle's rejection of a vacuum, so to accept the zero meant denying a cornerstone of Greek philosophy.

The zero finally reached Europe in the twelfth century, though, again, it was not accepted with open arms. The Christian Church had united the Bible with the teachings of Aristotle, reinterpreting the pagan Greek's ideas so they became a proof of the existence of God. However, European merchants found the zero to be an invaluable tool in business, and eventually the zero became an accepted part of European mathematics.

The zero still causes arguments today. Asking someone what any number divided by zero is can often provoke debate. How many zeroes go into five, for example? The simple answer is that you cannot, under any circumstances, divide by zero, but why that should be the case is not always obvious. The late development of the zero, and its slow and difficult journey across the world is related to the surprisingly complex philosophical and intellectual questions the number raises.

DAVID TULLOCH