---------- cut -------------

From ucgadkw@link-1.ts.bcc.ac.uk Sun May 3 15:28:53 1998

Date: Sun, 3 May 1998 15:28:52 +0100 (BST)

From: Dominik Wujastyk

To: letters@newscientist.com

Subject: Ian Stewart: Zero, Zilch and Zip

Dear Sir,

Ian Stewart writes engagingly about the origin of the mathematical zero and place-value notation ("Zero, zilch and zip", 25 April, p 41), but he suggests that these two concepts are connected, when they are in fact both logically and historically separate. You can count reasonably successfully with place-value notation but no explicit zero, and vice versa.

Ian Stewart is also quite wrong in saying that "place notation was born probably in India, maybe with Arab help, not too long after AD 200".

Three key elements -- a decimal base, place-value, and zero (I abbreviate this to "DPZ") -- occurred separately at earlier times both in India and in other parts of the ancient world. In particular, the Babylonians were using a place-value system, with a space for the null value, in the second millennium BC, but their base for counting was sixty, not ten. By the time of Alexander the Great, they were even using a special symbol for this null value. From perhaps as early as the third century AD the Mayans also used place-value and zero, but with the base twenty. But it does indeed seem to have been the Indians who first combined these key elements together to form the basis of the arithmetic system that has come down to the modern world.

The Arabs did not have anything to do with the invention, and indeed Arabs only arrived in India about five hundred years later than Stewart suggests. The Arabs (or rather, the Muslims of the Middle East) certainly did transmit knowledge about zero and the place-value notation to Europe, but they learned it all from the Indians. We call our numerals "Arabic", it is true, but Arabic writers called them "Hindu", meaning "Indian".

The Indian numerals are first mentioned outside India in the year 662, when the Syrian bishop Severus Sebokt, annoyed by the intellectual arrogance of immigrant Greek scholars, reminded them pointedly that other nations were also very learned, such as the Hindus with their admirable systems of astronomy and arithmetic, including calculating with nine symbols.

Indian works on arithmetic, translated from the original Sanskrit in to Arabic (perhaps through Persian or Syriac), began to reach the Islamic world in about the eighth century. The first book known to us from outside India that demonstrates Indian methods of calculating with nine digits and zero was composed in the ninth century, probably in Baghdad, by al-Khwarizmi (whose name, through the medieval Latin and Old French, gives us "algorism" and "algorithm"). From about AD 950 on, many Arabic works demonstrating these new Indian methods of working out arithmetical problems, including fractions, were circulated under the name "al-Hisab al-Hindi', or "Indian calculation". After 1100, Latin translations of Al-Khwarizmi's work spread throughout the centres of medieval learning in Europe, which is how the the Indian DZP system ultimately reached us today.

The earlier history of this number system in India is not perfectly clear, but the Indian astronomer Aryabhata, born in 473, was the first to describe the decimal place-value system explicitly, in a chapter of a work in which he also discusses algebra, geometry, and trigonometry. Before him, the third-century author Sphujidhvaja seems to be the first author to describe the use a symbol for zero in the decimal place-value system.

It is often claimed that the adoption of the DPZ system was a great epistemological change heralding the opening of vast new mathematical horizons, and a leap forward in knowledge generally. I do not see why a notational change of this type should be seen as so important, and there

is little actual historical evidence of such an effect. The counting system that has become second nature to us may seem consummate, but surely that is only a matter of what we are used to, what we have been taught from childhood. The Babylonians, using a non-DZP system, constructed vast tables of astronomical and arithmetical parameters which required extraordinary amounts of calculation, but we see no evidence that they were hampered by having a sexagesimal (60-based) and not a decimal system. Early Greek arithmetic was decimal, but was conducted without recourse to the use of zero. Sometimes a non-DZP notation can be positively helpful: to add ten and ten in Roman (X + X = XX) does not even require knowledge of another symbol, nor any notational manipulation beyond writing the symbols more closely together. What could be easier? That quintessence of modernity, the digital computer, abandons the internal use of decimals entirely, using only binary digits, or bits. The feeling that the DZP combination is in some way "better" than any other system is surely no

different in principle from any other chauvinistic belief, such as that that one's own mother language -- whatever it may be -- is the easiest, best, and most expressive language in the world.

In areas where sexagesimal (60-based) counting still lurks in our own mathematics, such as in the 360 degrees in a circle, we suffer no epistemological harm. That a right angle has ninety degrees has not held our civilization back in any obvious way, though measurement in radians is of course routine in higher maths.

If some future government, politically desperate for a "British Sausage" issue, decides to go for total, all-out decimalization and legislates that a circle shall have ten degrees, there may be a leap in learning, but it may in a direction of five degrees.

Yours faithfully,

Dominik Wujastyk

--

Dr Dominik Wujastyk, FAX/voice: +44 171 611 8545/8467

Wellcome Institute for URL: http://www.ucl.ac.uk/~ucgadkw/

the History of Medicine, Email: d.wujastyk@ucl.ac.uk

Wellcome Trust, 183 Euston Road, Trust URL: http://www.wellcome.ac.uk

London NW1 2BE, England.

First Rule of History:

History doesn't repeat itself -- historians merely repeat each other.

Good comments. (Was the letter published?)

ReplyDeleteBut I disagree with your final comments, that DZP was not a big deal. I say this not out of chauvinism (being both a user of the DZP system and Indian), but as a student of mathematics. Both Z and P *were* innovations of great consequences. You should not be dismissive of good notation; often the right notation, even (or especially) in higher mathematics immensely changes the way we think about things. (Not so much D, though if the base is too large we need too many different symbols and if it's too small numbers have too long representations). The concept of zero as a number *is*, of course, a great epistemological change — even Grothendieck thought so — comparable in magnitude, or even greater, to the idea of negative numbers. And P-with-Z was a great leap forward in practical ways.

We know this notation was better than the Roman system, because people using the Roman numerals did *not* get by conveniently. And throughout the Western world this new notation was adopted over the existing Roman numerals, despite being new and unfamiliar. Borwein quotes the advice given to a father in the 16th century: "given a wealthy German merchant in the 16th century: "If you only want him to be able to cope with addition and subtraction, then any French or German university will do. But if you are intent on your son going on to multiplication and division — assuming that he has sufficient gifts — then you will have to send him to Italy." (See here)

If you're really unconvinced that a positional number system (with 0, since we already have the concept) is easier than figuring with Roman numerals, you ought to try using Roman numerals for a year and see how it goes. :-) I do agree there's nothing special about base 10, of course.