Finding Zero (19 page)

“Incredible,” I said.

“OK, I need to wake up very early, you know, so I had better go.” We said goodbye and I tried to walk him to the hotel's entrance. “No, no, please,” he said. “I can find my way fine. Have a good stay, and anything you need, just let me know.” I thanked him warmly and promised to send him my write-up to accompany K-127 very soon.

I spent the entire next day in my hotel room working on the description for the display of K-127 to be placed in the museum.

I sent my work to Hab Touch by e-mail and waited for an answer.

Inscription K-127

Discovered in the nineteenth century at Trapang Prei, Site of Sambor On Mekong; seventh century, Pre-Angkor Period.

First Translated from Old Khmer, into French, by George Cœdès and published by him in 1931.

This inscription bears the earliest zero numeral ever discovered.

What is the importance of the zero? Zero is not only a concept of nothingness, which allows us to do arithmetic efficiently, but is also a place-holding device that enables our base-10 number system to work, so that the same 10 numerals can be used at different positions in a number, making our system extremely efficient. The Roman system, for example, which preceded our number system in Europe until the Late Middle Ages, employed Latin letters for quantities (I for 1, X for 10, L for 50, C for 100, M for 1,000); these letters had to be repeated, for example writing MMMCCCLXXIII for the number 3,373. We see that in our system the same numeral 3 is used in three different places, allowing for economy and ease of notation. None of the Latin letters could be repeated in different contexts. In our number system, it is the zero that enables the efficiency and power of the system: Thus, a 5 in the units location is a 5; but the same symbol in the tens location makes it a 50—if we can also use a zero as an empty place-holder for the units. Similarly, 505 can only be written in this highly efficient way because we use zero as a place-holder for the tens. The Babylonian system, for example, which predated the Greco-Roman letter-based number system by about two millennia, used base-60 with no place-holding zero. Hence, the difference between 62 and 3602 (where 3600 is the next-up power of 60) had to be guessed from the context. Our number system, using a much smaller base, and employing a special symbol for zero, derives its immense power and usefulness through the use of this place-holding zero, as compared with the Greco-Roman, Babylonian, and Egyptian systems. When we also consider the fact that arithmetic is so much more powerful with the use of zero, which helps define the whole realm of negative numbers, and the fact that everything we do with a computer (or cellular phone, GPS, or anything electronic) is controlled by strings of zeros and ones, it becomes clear that the invention of zero is perhaps the greatest intellectual achievement of the human mind.

So who invented it?

This inscription, bearing the earliest known zero ever discovered, is written in Old Khmer and begins with the words:

çaka parigraha 605 pankami roc . . .

Translated, it reads:

The çaka era has reached 605 on the fifth day of the waning moon . . .

The zero in the number 605 is the earliest zero of our system we have ever found.

These are Old Khmer numerals for “605,” and the dot in the center is a zero—the first zero ever made (as far as our present knowledge goes). The çaka era began in AD 78, so the year of this inscription in the Christian calendar is 605 + 78 = AD 683.

This inscription has a celebrated history. Until the 1930s, many scholars in the West believed that the zero—the key to the efficiency and versatility of our base-10 number system—was either a European or an Arab invention. The oldest known zero was in India, at the Chatur-bhuja temple in the city of Gwalior. That zero is dated to the mid-ninth century. Since that era coincided with extensive Arab trade, it could not be used to defeat the hypothesis that the zero was invented in Europe or in Arabia and from there moved east. The publication of George Cœdès's article in 1931 (see reference below) proved definitively that the zero is an Eastern, and perhaps Cambodian, invention since this inscription predates the Arab empire, as well as the Gwalior zero, by two centuries. It is of note that a zero that is one year younger, thus dating from AD 684, was found at around the same time near Palembang, Indonesia, and was also published by George Cœdès.

Inscription K-127 was kept for a time in this museum but was moved to Angkor Conservation in Siem Reap on November 22, 1969. During the Khmer Rouge regime of terror close to 10,000 artifacts were stolen or defaced and this inscription's whereabouts were uncertain. It was re-discovered in a shed at Angkor Conservation by Professor Amir D. Aczel of Boston University on January 2, 2013, and brought to the attention of His Excellency Hab Touch, whereupon it was reinstated at the museum.

References:

Cœdès, George, “A propos de l'origine des chiffres arabes,”
Bulletin of the School of Oriental Studies, University of London,
Vol. 6, No. 2, 1931, pp. 323–328.

Diller, Anthony, “New Zeros and Old Khmer,”
The Mon-Khmer Studies Journal,
Vol. 25, 1996, pp. 125–132.

Ifrah, Georges.
The Universal History of Numbers.
New York: Wiley, 2000.

26

I had been concentrating all along on the importance of zero as a place-holder within our number system, and on showing how our numbers work because we are able to insert a sign that says there are no tens or no hundreds or no thousands, and so on, in the representation of any number whatsoever using simply the ten numerals in our decimal number system. But what about the system as a whole?

Sitting at the departure lounge at the Phnom Penh airport waiting to board my flight back to Bangkok, I was pondering the rich history of zero the number—a concept I am convinced could only have arisen through a purely Eastern way of thinking (and, independently, by the Maya in the West).

Equally, I was thinking of the idea of infinity, also prevalent in Eastern thinking: the “endless sea,” Ananta the sea serpent, eternity, and innumerable other forms of extent that go beyond the simple numbers 1, 2, 3, and so on. But the development of our numbers in a purely mathematical setting, including both Eastern concepts of zero and infinity, was to take place in the West—or
rather, both in the East and in the West (the rational, irrational, and complex numbers were explored theoretically in Europe between the fifteenth and nineteenth centuries).

We've seen that it is possible to define the numbers starting simply from the void, the empty set, and proceeding through the use of set membership: the set containing the empty set for 1, the set containing the empty set and the set containing the empty set for 2, and so on. But of course this is a sophisticated way of defining the numbers, using sets and the idea of set membership. In reality, numbers developed in a very different way.

The ancient Babylonians and Egyptians, thousands of years ago, learned to assign numbers to objects and thus to abstract the concept of number from the magnitude of the sets of things they observed. Perhaps the greatest intellectual discovery of early antiquity—at the dawn of civilization, really—took place when someone, or very likely several individuals at different places and times, could look at three stones on the ground, three cows in a meadow, three people walking on a path, three grains of wheat, three pyramids, three goats, three children, and understand that all of them had one and only one aspect in common: the quality of “being three.” Similarly, four could be defined as that aspect of many different things that are four, and so on. The numbers could grow and grow, and the magic of this understanding—that things that are of the same discrete magnitude are in some sense the same—was, and is, overwhelmingly powerful.

Soon enough, people of antiquity added words to their languages to represent these numbers. In fact, in India especially and in several other Asian countries under its influence, there were
special words, nouns that everyone knew belonged to categories that were of universally accepted numbers, and these nouns became synonymous with the numbers. Here is an excellent example, from Cœdès's seminal 1931 article, commenting on the stele of Changal (my translation from the French): “The year of the king of the çaka expressed in numbers is: the flavors, the organs of sense, and the Veda.”

Cœdès explained that there were six known flavors of food, five senses, and four Veda (the ancient collections of Hindu holy writings). Thus, this is a way of expressing the number 654 in words. This method was widespread in Cambodia, India, and other countries in south and Southeast Asia.

Next, Cœdès gave the example of an inscription from a place called Dinaya, discovered in 1923, in which the date çaka 682 is given as “the flavors, the Vasu, and the eyes.” Again he noted that flavors stood for 6 and explained that Vasu (deities attending to Vishnu, of which there were eight) stood for 8, and we know that a person has two eyes.

But Cœdès also discussed the problems that arose here. Across geographical areas and through time, there was not always complete agreement on which number was represented by which noun; sometimes ambiguities existed.
1
This difficulty is similar to one we encounter today when we use the phonetic alphabet.

When asked to spell my name over the phone to someone whose English may not be perfect, or over a bad telephone connection, I often have to use this words-for-letters system. “Aczel,” I say, “A for apple, C for Charlie, Z for Zebra, E for Europe, L for Larry.”
I use these words because they are the first ones that pop into my head, and I usually have to repeat this a couple of times.

But of course I am mostly wrong, as the accepted NATO Phonetic Alphabet is: Alfa, Bravo, Charlie, Delta, Echo, Foxtrot, Golf, Hotel, India, Juliet, Kilo, Lima, Mike, November, Oscar, Papa, Quebec, Romeo, Sierra, Tango, Uniform, Victor, Whiskey, Xray, Yankee, Zulu. But does anyone remember these?

By analogy, a system of numbers, which was prevalent in the East for centuries—in which one says “Veda” for four, “flavors” for six, and so on—certainly could not have been uniformly well understood by everyone. This was one important reason why written signs for numbers had to be invented.

Cœdès described the ancient Khmer number system, which was
not
decimal. Even today, despite the borrowing of numbers above 30, which are decimal, lower numbers in modern Khmer are not perfectly based on 10, but also on 5 and on 20. The Khmer, Cœdès noted, use many multiples of 20—as the French do only once, for 80 (quatre vingt, or four twenties) and numbers that go with it (for example, quatre vingt dix neuf, for 99). The Khmer use more of these multiples, clearly a vestige of a base-20 system due to our having ten fingers and ten toes, which persisted. This is reminiscent of the Maya number system, which was almost exclusively base 20 (with the exception of the partial calendrical use of base 18).

In antiquity, Cœdès explained, the Khmer possessed
only
the numbers: 1, 2, 3, 4, 5, 10, 20, and several multiples of 20. This was all they had as far as numbers they understood. At some point,
they borrowed the Sanskrit word
chata
for 100. With these numbers they expressed all numerical information.
2
All this was, of course, before the maturing of their numbers and the invention of zero (or its importation from India or some other place) as attested by inscription K-127.

What all this taught me was that fingers and toes are really important. If we had not had them, or had different quantities of them, maybe we would view numbers in a totally different way. If some day we were to meet aliens with only two fingers on each hand and two toes on each foot, their number system might well be binary, allowing them to communicate with the innards of their computers more directly than we do: Our numbers always have to be “translated” into a binary (0 and 1 only) code for a computer to understand them.

On the other hand, with two fingers per hand and two toes per foot, maybe their number system would be octal (based on 8). It was fun to speculate on such things, and it kept me entertained as I waited to hear about the fate of my precious find. In Bangkok, it helped me relieve the immense tension of waiting for news about the fate of K-127 and whether Hab Touch would follow through on his promise.

George Cœdès returned
to his native France some years after French colonial rule in Indochina ended, as these new nations grappled with questions of democracy, parliaments, monarchy, and Communism. In Paris, he had a prestigious academic position and continued to write papers and books about Southeast Asia. He was highly decorated, having been awarded the rank of commander
in Thailand's Order of the White Elephant, as well as France's prestigious Legion of Honor. He died in Paris in October 1969—a month before K-127 was brought to Angkor Conservation. Cœdès had several children, and one of them became the admiral of the Cambodian fleet. This fact made me feel a kind of nautical kinship with this great man.

27

On April 9, 2013, I finally got the e-mail message I had been waiting for:

Dear Professor Amir,

I apologize for having taken so long to write to you. It was a great pleasure to meeting with you in Phnom Penh and delighted to hear about the history of Zero. Thank you for your research article on Khmer Zero, which is now the earliest Zero in the world civilization. I have shared this exciting news with my colleagues and hope this inscription will be on display in the National Museum in Phnom Penh. I look forward to meeting with you again and please let me know if I can be of assistance to you with this important research.

With best wishes,

Touch

I was elated. I couldn't believe a successful conclusion was finally in sight. Could it be that my odyssey was now over? Following e-mails reassured me that what I had hoped for was going to take place at last. His Excellency Hab Touch would arrange for the priceless K-127 to be taken out of the hands of Lorella Pellegrino and placed in the Cambodian National Museum in Phnom Penh, where it once was, and where it belongs. From then on, scholars, mathematicians, historians of science, and the people of Cambodia and elsewhere would be able to see the very first zero of our numbers ever discovered—a find that changed our view of history, the one artifact in the history of science that proved definitively that the zero came from the East.

Debra met me again in Bangkok, and we flew to Paris together a week later, switching planes in Bahrain. At our small hotel on the Left Bank, I used the Internet and wrote a short article about the rediscovery of K-127 for the
Huffington Post.
It was published within hours. After I sent the link to Hab Touch, he responded that he was delighted that people would now be learning about “Khmer Zero,” as he called it. “Let the discussion begin!” he wrote me.

His country could benefit from displaying and explaining its antiquities, and I hoped he would succeed in his efforts to repatriate to Cambodia many statues that had been looted during the Khmer Rouge era and sold to museums around the world. The
Tribune
had an article about the New York Metropolitan Museum of Art agreeing to return two such statues, and other museums around the world were considering doing the same. I knew that this was the result of Hab Touch's negotiations with museums, and
I felt good that my own work had contributed in small measure to this larger effort.

Debra left to return to Boston and I stayed in France for a few more days. I had one last thing to do before this big adventure was over. After accompanying her to the counter for her transatlantic crossing, I walked over to another part of Terminal 2 at Charles de Gaulle Airport and boarded a domestic French flight to the south.