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Impearls: 20050821 Archive Earthdate 20050822
The Invention of Nothing
A commenter at the alwaysworthwhile Iraq the Model blog writes, among other things, that:
The invention of zero is indeed wonderful, and the commenter relates the conventional wisdom; however — not to criticize him too specifically (as I say this is the conventional wisdom) — the above is even more thoroughly wrong than much else that is conventionally held to be true in this world. Since one of my goals in life is to correct this particular misapprehension concerning the origin of zero wherever and whenever encountered, and encourage people to look into the fascinating real history of mathematics, I'll take the trouble to rebut it in detail. How is it wrong? Let me count the ways. 1. Our present decimal positional numbering system — the socalled “Arabic numerals” — was not invented by Arabs, nor Muslims more generally, but (decades before Mohammed) by Hindus in India. Mathematical historian Carl B. Boyer describes the advent of the decimal numerals thusly: ^{1}
Arabs later also picked up the system from the Hindus (along with algebra and much else) and eventually passed it along to the far West. 2. As the above quotation illustrates, the “base 10” system initially did not even have a zero, but for many centuries consisted of nine symbols only. When after the Dark Ages eased the system first came to the attention of the far West (during the late 10th century a.d.) this was still the case. ^{2} 3. Far more vital to the usefulness of a positional or placevalue numbering system (compared to nonpositional systems like Roman numerals) than the mere presence of a zero symbol is the capability of expressing fractions within the system (such as writing 3.14159… for π in base10), including the ability of multiplying and dividing by the system's base (i.e., 10) by simply shifting the “decimal” (radix) point right or left. For some reason, no one thought for many additional centuries of adding this capability to the decimal system. As historian Boyer put it: ^{3} “It is one of the ironies of history that the chief advantage of positional notation — its applicability to fractions — almost entirely escaped the users of the HinduArabic numerals for the first thousand years of their existence.” 4. Nobody knows when the zero was added to the “base 10” system, nor by whom it was done. It may have been Arabs who were responsible, but it just as easily could have been Greeks in Alexandria or elsewhere, or the Hindus in India who centuries earlier invented the system in the first place. All we really know regarding the latter possibility is that, as historian D. E. Smith pointed out, “the earliest undoubted occurrence of a zero in India is in an inscription of 876.” ^{4} 5. Even when the zero symbol was added to the other nine it cannot be said that that was the “invention” of zero — the invention of nothing! — as zero had long been known in the context of a positional numbering system. Zero was actually invented a thousand years before Mohammed by the Babylonians (the ancestors of modern Iraqis) — not long before, or perhaps a bit after, that venerable city (still going strong) was captured from the Persians by Alexander the Great (in 331 b.c.) and made his new capital — as part of the Babylonians' already ancient, base60, socalled “sexagesimal” placevalue numbering system. The Hellenistic Greeks held onto Mesopotamia for more than two hundred years following Alexander, picking up much lore (including, unfortunately, astrology), and in the process absorbed the Babylonians' mathematics and sexagesimal numbering system with its zero and spread it far and wide across the Western world as an ideal vehicle for trigonometric and astronomical calculations. Computations within the system were performed much as we do today using decimals. As Encyclopædia Britannica put it: “This zero symbol was in common use in all astronomical computations and reached Hellenistic mathematicians, along with the sexagesimal placevalue notation. Thus in all Greek trigonometric tables (for instance, in Ptolemy's Almagest of about 140 a.d.) a zero sign is used exactly as it is today. This holds for Islamic and Byzantine tables as well […].” ^{5} It's worth emphasizing this point: soon after the zero was introduced (during approximately the 4th century b.c.) Babylonian sexagesimal numerals — in a new noncuneiform graphical guise ^{6} — became pervasive across mathematical and astronomical circles of the West (extending as far east as India), maintaining its sway (with astronomers and mathematicians that is) throughout the Hellenistic, Roman, Byzantine, Islamic and Medieval periods, through the Renaissance and all the way into the modern era. Historian O. Neugebauer points out that “the Arabic form [in decimals] for the zero symbol (a little circle with a bar over it and related forms) is simply taken from Greek astronomical manuscripts” employing sexagesimals. ^{7} 6. Beyond their later addition of the zero, the Babylonians' base60 sexagesimal system had already, for basically forever (a millennium and a half before Alexander, or by around 1800 b.c.), incorporated the capability of expressing base60 radix fractions within the system. This usage is also well illustrated in Ptolemy's Almagest. 7. By whom and when did radix fractions finally get added to decimals, making the system for the first time halfway decent for technical use? First to do so was the mathematician Jamshid alKashi who, though writing in Arabic, actually lived and worked in the central Asian city of Samarkand (in what is now Uzbekistan) in the early 15th century a.d. AlKashi was master of both sexagesimals and decimals, developing the capability (perhaps under Chinese influence ^{8}) of expressing fractions in decimals as well as wielding the longestablished usage of fractions within sexagesimals. In addition to performing other sophisticated calculations (such as finding the sixth root of manydigit numbers), AlKashi computed π (actually 2π) to many places in both decimals and sexagesimals. For illustration purposes, here are the two equivalent results — decimal: 6.2831853071795865; sexagesimal: 6.16,59,28,01,34,51,46,15,50. ^{9} 8. Unfortunately, alKashi's superb results (he died around 1436 a.d.) did not at that time make it to the West, and the decimal system languished basically broken until the system was extended later in the 16th century by Western mathematicians (particularly Francois Viète ^{10} in France and Simon Stevin in Holland) into incorporating decimal fractions. Prior to that sexagesimals remained the only practical vehicle for scientific computations — thus Copernicus's worlddisplacing results, for example, were all derived using sexagesimals. Afterwards, however, decimals made steady progress visavis sexagesimals until today most people are hardly aware that there was a time when sexagesimals (for astronomical and trigonometric computations anyway) ruled! Moreover, lest folks get too haughty about the present prevalence of decimals over the ancient Babylonian sexagesimals — which, though a very capable positional notation, does require (naively at least) a times table sixty units on a side, with 3,600 (rather than 100 for decimals) elements within it — remember that even today virtually everyone in the civilized world still uses sexagesimals every day: in the degreesminutesseconds of angles, and especially the hoursminutesseconds of time. Thus we see that minutes and seconds aren't some weird “units” but rather are first and secondorder sexagesimal fractions (the equivalent of the tenth's and hundredth's places in decimals), what Ptolemy — in Latin translation — called partes minutae primae and partes minutae secundae. Not a bad legacy for a people who have vanished off the face of the earth, leaving seemingly nothing behind, to have invented nothing — zero, nullity, nil — i.e., the fundamental basis of our numbering system!
Given the reputation which still lingers today of the ancient Babylonians as mysterious purveyors of magic, that seems oddly appropriate.
UPDATE: 20050825 04:30 UT: Expanded the references, corrected Boyer's depiction of alKashi's sexagesimal 2π, along with minor other changes. UPDATE: 20050826 07:30 UT: The intriguing EBlogger blog linked to Impearls' article, in a piece called “A History of Nothing,” saying that “Michael McNeil has posted an interesting bit about the history of the number zero.” However, Eblogger incorrectly summarizes the main point of Impearls' piece, stating: “Michael traces the concept of zero not to the Arab world (a popular misconception, no doubt related to the etymology of the word ‘zero’, which is derived from the Arabic sifr), but to Indian mathematicians. Britannica suggests the concept goes back, in one form or another, at least to the 3rd century b.c. Babylonians.” On the contrary, Impearls does not suggest that Hindu mathematicians were responsible for the invention of the zero (though they did originate the forerunner of the base10, popularly called “Arabic,” numbering system that we use today). Rather, as Impearls wrote, “Zero was actually invented a thousand years before Mohammed by the Babylonians…,” thus putting it in full agreement with Britannica in this regard. UPDATE: 20050826 20:45 UT: In a followup comment to its piece, EBlogger has corrected the record with regard to the foregoing. UPDATE: 20050829 00:20 UT: Geitner Simmons of the perenniallyrewarding Regions of Mind blog, links to Impearls' piece (as well as to the Ibn Khaldun articles) in a posting entitled “The importance of zero.” Simmons wrote:
In an email to Impearls concerning his posting, Geitner also said: “Hello again from Omaha. I linked to your posts today. It was a pleasure to read your observations on all those topics. The piece on the history of mathematics was especially welldone — erudite, playful, rhetorical accessible. […] Congratulations on the high standard you continue to set at Impearls.” UPDATE: 20050829 04:50 UT: David Nishimura's fine historical blog Cronaca has linked to Impearls' article in a piece with the great title “Much ado about nothing,” commenting: “Who invented the zero? Michael McNeil reviews the literature here, and once again it appears that popular history isn't looking far enough back. A post well worth bookmarking….” UPDATE: 20050830 21:40 UT: A commenter, John Anderson, to the Cronaca piece wonders why sixty was selected as the base for the Babylonian numerals, writing:
According to Boyer ^{11}, base60 was apparently deliberately chosen by the Babylonians so as to minimize the sort of extended or even infinitely repeating fractions that one gets when taking a fractional quantity whose divisor (the number under the bar) is not an even divider or composed of factors of the base (in the case of decimals, ten is the base). Since only 2 and 5 evenly divide (“are factors of”) ten, only those particular fractions (1/2 and 1/5 and multiples thereof) can be singledigit fractions in decimal; thus 1/5 in decimal is 0.2. Beyond that, only numbers which incorporate the even dividers (factors) of the base as the factors of themselves (in decimal, values composed of factors 2 and/or 5; for example, 4) become simple, nonrepeating fractions (thus 1/4 is 0.25 in decimal). As a result of this effect, we see that 1/3 (because 3 isn't composed of 2 and/or 5 as factors) becomes a repeating fraction, or 0.3333… (infinitely extended), in decimal. Babylonians apparently didn't like those kind of fractions. Mathematicians sometimes suggest twelve as a more “rational” base than ten because more values will evenly divide 12, to wit: 2, 3, 4, and 6. Thus, 1/3 in base12 would simply be 0.4. However, with sixty as base the even divisors are: 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30! Therefore, many more common fractions become representable by a single fractional digit and far more are simple and nonrepeating when expressed in that system. UPDATE: 20050830 22:30 UT: A commenter, Sarah, to the Regions of Mind posting makes the comment:
The answer to the question I think is that positional, radix fractions did not elude mathematicians, as all along they had the Babylonians' sexagesimal (base60) placevalue notation for use in technical concerns. Indeed, the astonishing thing to me is how quickly sexagesimals picked up the capability (around 1800 b.c.). Once that facility was added to the system, mathematicians and scientists in my view probably avoided other systems for the most part, rather than trying to fix them, since they had a notation with the power they required. Sarah goes on:
After about the 4th century b.c. sexagesimals did have a zero, and thus mathematics after that point did not suffer from the problem suggested. (Merchants and traders, on the other hand, being unfamiliar with the sexagesimal system, would have experienced that difficulty, but not being professional mathematicians nor having expertise with a positional notation which surpassed the issue, lacked the technical horsepower to do anything about it.) Even during the millennium and a half that sexagesimals were in use before the system acquired its zero, recall that a base60 notation has only a one in sixty chance for any particular digit being zero, plus the number of digits needed to express any given number is fewer with base60; thus the sexagesimal system runs into the need for zero to hold the place in columns within numbers far less frequently than do decimals. One other point: though many examples of the use of zero within sexagesimal numbers have been found in ancient sources, none have been discovered where zero was placed at the end of numbers. ^{12} The conclusion therefore is that sexagesimals were inherently a form of floating point notation rather than being fixed point (as decimals are), and as a result were rather similar to modern engineering notation. UPDATE: 20050830 23:50 UT: Geitner Simmons has added an update to his Regions of Mind posting, noting the Olmecs/Mayas' invention and use of zero in the New World. Geitner wrote:
There is clear evidence that the Olmecs were using zero by 32 a.d., according to the book “1491: New Revelations of the Americas Before Columbus.” A timeline at that Web page has an item about the Olmecs and zero. There are problems with that timeline, especially its assertion concerning the origin of zero in the Old World. With regard to the New, it's true that the Mayas, and probably the Olmecs before them (Olmec civilization was largely over by about 800 b.c., whereas the Mayas were well on their way up by the turn of the Christian era), invented and knew about the use of zero in a positional notation. I didn't get into that in Impearls' foregoing piece because that's an entirely independent invention that had no influence on the development of our own numbering system. However, Impearls intends to publish a piece on that subject in December, as that will be approaching the preanniversary of the upcoming “Maya millennium bug” — when some say the Maya believed that the world will end (actually they didn't think that) — as a result of their base20 “vigesimal” positional notation rolling over to its next 400year “century” (number 13, which has special signficance in their system), due to occur on December 23, 2012 in our calendar. UPDATE: 20050831 00:25 UT: The interesting history of science oriented blog Copernicus Sashimi has linked to this piece. UPDATE: 20050831 09:00 UT: Fixed initially incorrect statement about the abovenoted upcoming Maya calendrical rollover in 2012. UPDATE: 20050831 16:40 UT: The blog G as in Good H as in Happy has linked to this article.
UPDATE:
20050903 00:45 UT:
ClioWeb
which is hosting the 15th installment of the
Carnival of History
has linked to Impearls' piece.
References
For more on the origin of zero and the Babylonian sexagesimal numbering system — and on the brilliance of ancient Babylonian mathematics, which goes far beyond their numerical system — see Carl B. Boyer's A History of Mathematics, 1991, John Wiley & Sons, New York (ISBN 0471097632, and [paperback] 0471543977), particularly Chapter 3: “Mesopotamia,” For much more on the Babylonian numerals and mathematics and its transition into Hellenistic and subsequent usage, see O. Neugebauer, The Exact Sciences in Antiquity, Second Edition, 1993, Barnes & Noble, New York (ISBN 1566192692), especially Chapter I: “Numbers” and Chapter II: “Babylonian Mathematics,”
^{1}
Boyer, op. cit.,
^{2}
Gerbert of Aurillac (ca. 9401003), who became Pope Sylvester II in 999, mentioned the nine decimal numerals.
Boyer, op. cit., ^{3} Boyer, op. cit., p. 255. ^{4} Quoted in Boyer, op. cit., p. 213. Smith, op. cit., Vol. II, p. 69. ^{5} “Mathematics, History of,” Encyclopædia Britannica, 15th Edition, 1974, Encyclopaedia Britannica, Inc., Chicago, Macropædia Vol. 11, p. 640.
^{6}
Neugebauer, op. cit., ^{7} Neugebauer notes that F. Woepcke first recognized this back in 1863. Neugebauer, op. cit., p. 26.
^{8}
The Chinese utilized two more or less decimal notations by at least a few hundred years b.c. (it's hard to tell prior to that due to the “burning of the books” in 213 b.c.).
One system was a decimal but non placevalue “multiplicative” scheme that specified pairs of symbols, a symbol indicating a decimal digit together with another indicating the power of ten, followed by other such pairings, in order to lay out a complete decimal number.
The other was a kind of “centesimal” or base100 positional notation involving socalled “rod numerals” set up so as to facilitate calculation using counting boards (a variation on the idea of the abacus).
As with the “multiplicative” scheme, rod numerals used pairings of symbols positionally specifying each “digit” in a centesimal, base100 number.
As with the Babylonian system, a zero symbol appeared relatively late, by around the 13th century a.d.
The Chinese early on (perhaps as early as the 14th century b.c., it is said) were creative in representing fractions within the system.
Boyer, op. cit.,
^{9}
Boyer, op. cit.,
^{10}
Boyer, op. cit., ^{11} Boyer, op. cit., p. 25. ^{12} Boyer, op. cit., p. 27; Neugebauer, op. cit., p. 27.

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