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Wednesday, May 16, 2007
The answer to the question - "how did a pre-Petrine Russian express the notion of zero as a number, in writing and in speech?" - is supersimple: it didn't. The Greeks had no zero, the Romans had no zero, the Russians had no zero as long as they used their variant of Greek numeration.
Zero appears as placeholder in Hindu-Arabic numerals; it is first TREATED as number in using HA numerals for calculation (6-0=6); it is first RECOGNIZED as a number extremely late... the Flemish mathematician Simon Stevin at the end of the 16th century announced that the ONE is the same numerically as all other numbers, and called zero "the root of number." So it's not a number even here. Pascal thinks you're dumb if you don't understand that 0-4=0. One could do all sorts of incredible things in math before coming up with the notion of zero as a number.
Russians used the abacus to calculate - no one ever calculated in Slavic numerals because, like Greek and even Roman numerals, they are not intended for calculation but for setting down results. If your numeration or calculation has no place for zero, you won't understand why there should be a numeral that points to absence of number.
Zero entered Muscovy in the early- or mid-seventeenth century when introduction of HA numerals allowed for calculating on paper. Zero was called tsifra from the Arabic word for empty (sifr) and Medieval / Renaissance Latin for zero (cifra), which gave English both "zero" and "cipher." In Russian, already in the 17th century the word for zero started referring to all HA numerals (tsifry), also arithmetic (tsifir') and code (tsifir'). (Same denotative spread as in Latin, by the way.) I read in, I think, A. P. Iushkevich that in the 17th century the HA zero was also called "on" or "onik," like the letter O, which in Slavic numeration stands for 70.
Anyway, before the seventeenth century there really is nothing in any way like the concept of zero in Russia. And, of course, the concept of number at this point includes only natural numbers, nothing more.
I seem to remember an article on this subject by Simonov, the leading specialist on early Russian mathematical manuscripts, but I can't find it. For some discussion of this see ch. 2 of A. P. Iushkevich, Istoriia matematiki v Rossii, M. 1968. Also L.L. Kutina, Formirovanie iazyka russkoi nauki, M.,1964, pp. 14-20.
There are several coins, manuscripts and printed books of Russian origin in the seventeenth century which use or give lists of modern 'hindu-arabic' numerals, and Russian traders and officials in the Posolskii prikaz must certainly have been familiar with western numerals, so the concept must have been known and understood, at least by some.
Russian alphabetical numerals did not in fact need a zero since every number ending in zero in modern notation had a letter to designate it: e.g. k= 20. These could be modified by a preceding or subscript oblique line with two cross bars to produce thousands, and various kinds of circle round the letter for larger multiples of 1000. Arithmetical calculation was perfectly possible in this system as it had been for the Greeks from whom it was taken.
The earliest words for zero are recorded from the end of the 17th century and early eighteenth century - on, onik (i.e. the name of the latter o) in Peter I notebook (1688) and Kopievskii, later tsifra and nul'.
Emeritus Professor W. F. Ryan FBA, FSA
(School of Advanced Study, University of London)