SciFri 4.22.05: Zero tolerance

the Medieval Version

By Deane Morrison

Published on April 22, 2005

This is the Jerry Seinfeld column: a column about nothing at all. Zero.

Of course, zero is not nothing at all; our counting system depends on it. Yet, when this little oval slipped into the Western world, it caused no dancing in the streets and sparked no renaissance of mathematics. It was an interloper in what was regarded as a perfectly good system of counting, using an abacus or counting board. (On a counting board, objects like beads were moved between carved or painted lines; on an abacus, beads were slid up and down rods.) With no symbols to represent zero, spaces were left to indicate nothing in the 10s, 100s, etc. column.

"The first person in history we know of who pondered zero as a concept was the

Hindu mathematician Brahmagupta," says astronomy professor Terry Jones. "Mathematicians did not have the symbolic system we use today (in algebra, for example), so everything was written out in words." Brahmagupta, who lived from about A.D. 598 to 660, put forth rules of arithmetic, including:

A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
The product of zero multiplied by a debt or fortune is zero.
The product or quotient of a fortune and a debt is a debt.

Here, "debt" means a negative number.

Some who resisted trading in their counting boards for the new numerals regarded them as the work of the devil; while others had a field day ridiculing them.

The zero as a figure used in calculations was invented several times, including by the Mayans in the New World and the ancient Greeks, with Babylonian influence. It arrived in Europe in more or less modern form in the Middle Ages, having originated in India, where it was called sunya ("empty") in Sanskirt. It appeared in the Arabic language in the 9th century as as-sifr, a literal translation of "empty." It passed to Spain and then the rest of Europe.

Zero was called two names in Latin: cifra and cephirum. Cifra became the French chiffre around the 14th century and the German Ziffer in the 15th century. Cephirum also was rendered as zefirum, then, in Italian, zefiro or zevero or zero, hence our word. The French also adopted the Italian zero, and so had two names for the digit. Chiffre came to mean digit and also evolved into cipher. So chiffre meant not only zero, but any other digit.

This meant that there were two names for zero, and one of those names also referred to digits that have numerical value. "There could be no better symptom of the confusion and insecurity which the zero produced in the minds of people in the Middle Ages," wrote Karl Menninger in his book, Number Words and Number Symbols: A Cultural History of Numbers, about the fact that there were two names for zero, and one of those names also referred to digits with numerical value.

But zero wasn't the only difficulty. The whole system of Hindu-Arabic numerals--which we use today--threw Medieval Europe for a loop. People calculated on a counting board, then wrote down the answer in Roman numerals. It worked fine, and the counters could see and feel what they were doing. But the new numerals were symbolic--more purely intellectual and hard to visualize, as well as a whole new system that had to be learned. And the zero only added to the burden.

The new numerals popped in and out of European writings throughout the Middle Ages, often in monastic manuscripts, and were notably adopted by the Salem Monastery in Germany. But, according to Menninger, they took a long time to filter down to ordinary people, who continued to use their counting boards for calculations. The boards were visual, and they got the job done. Consequently, the "abacists," who used visual counting systems, and the "algorithmicists," who favored numerals, formed two opposing camps for a long time.

One thing in the numerals' favor is that mistakes could be traced, says Jones.

"[Counting boards] were just as fast as positional arithmetic, but it had the problem that the details of mistakes made with the counting board were lost," he says. "All you had was the answer in Roman numerals. With our modern arithmetic, you could see all the intermediate steps on the page of paper--until mechanical calculators, that is."

We who are used to zero find the Medievalists' difficulty hard to comprehend, but zero really is a strange bird. It has no value of its own. Put it in front of a number, and the number doesn't change: 3 and 03 mean the same. Add it or subtract it from a number, and the number doesn't change; put it after a number, and you multiply it by 10: 3 becomes 30. But if other numbers precede the zero, the rules change. A zero in front of a number now means something: 103 is not the same as 13. And if the number includes a decimal, a final zero no longer multiplies by 10: 1.03 and 1.030 are the same.

No wonder zero was confusing to people who had never seen it before. Was zero a digit or not? It had no value, like the numerals 1 through 9, but it had the power to hold a place and to multiply or negate the value of other numerals. Some of the time. While Europe was struggling with this phenomenon, its mathematics floundered.

"There was essentially nothing of note accomplished in mathematics in Europe from A.D. 500 to 1400," says Alfred Aeppli, professor of mathematics at the U of M.

A Web story credited to Bent E. Peterson, professor of mathematics at Oregon State University, offers this bit of lore: "In Medieval Europe some communities banned the positional number system. The bankers of Florence, for example, were forbidden, in 1299, to use Hindu-Arabic numerals. Instead they had to use Roman numerals. Thus the more convenient Hindu-Arabic numbers had to be used secretly. As a result "ciphra" came to mean a secret code, a usage that continues in English. The resolution of a code is of course "deciphering"--a very popular word in modern English."

Some who resisted trading in their counting boards for the new numerals regarded them as the work of the devil, while others had a field day ridiculing them. Menninger lists several examples of how the zero turned up in literature:

• "Just as the rag doll wanted to be an eagle, the donkey a lion, and the monkey a queen, the 'cifra' put on airs and pretended to be a digit," wrote a 15th-century French man.
• Another French source said that calling somebody a person who used the "ciphering" system (rather than counting boards) was equivalent to calling that person a blockhead. Even as late as the mid-20th century, French had a term, "faire par algorisme [the modern system of numerals]," meaning "to miscalculate."
• Shakespeare had King Lear's Fool point out the king's abandonment thusly: "Now thou art an 0 without a figure. I am better than thou art now. I am a fool, thou art nothing."

Did anyone but a few monastic eggheads love the unlovely zero when it first appeared in Europe? Yes, says Menninger: the astrologers. They gladly adopted the new numerals, including the dark and mysterious zero, because "like every form of secret writing, [the numerals] helped to raise their status."