Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero
Berenice Rojo-Garibaldi, Costanza Rangoni, Diego L. González, Julyan H. E. Cartwright
NNon-Power Positional Number RepresentationSystems, Bijective Numeration, and theMesoamerican Discovery of Zero
Berenice Rojo-Garibaldi, , Costanza Rangoni, ,Diego L. Gonz´alez , , and Julyan H. E. Cartwright , Posgrado en Ciencias del Mar y Limnolog´ıa,Universidad Nacional Aut´onoma de M´exico,Av. Universidad 3000, Col. Copilco,Del. Coyoac´an, Cd.Mx. 04510, M´exico Dipartimento di Scienze Statistiche “Paolo Fortunati”,Universit`a di Bologna, 40126 Bologna, Italy Istituto per la Microelettronica e i Microsistemi,Area della Ricerca CNR di Bologna, 40129 Bologna, Italy Instituto Andaluz de Ciencias de la Tierra,CSIC–Universidad de Granada, 18100 Armilla, Granada, Spain Instituto Carlos I de F´ısica Te´orica y Computacional,Universidad de Granada, 18071 Granada, SpainMay 21, 2020
Abstract
Pre-Columbian Mesoamerica was a fertile crescent for the developmentof number systems. A form of vigesimal system seems to have been presentfrom the first Olmec civilization onwards, to which succeeding peoplesmade contributions. We discuss the Maya use of the representationalredundancy present in their Long Count calendar, a non-power positionalnumber representation system with multipliers 1, 20, 18 × . . . , 18 × n . We demonstrate that the Mesoamericans did not need to inventpositional notation and discover zero at the same time because they werenot afraid of using a number system in which the same number can bewritten in different ways. A Long Count number system with digits from0 to 20 is seen later to pass to one using digits 0 to 19, which leads us topropose that even earlier there may have been an initial zeroless bijectivenumeration system whose digits ran from 1 to 20. Mesoamerica was ableto make this conceptual leap to the concept of a cardinal zero to performarithmetic owing to a familiarity with multiple and redundant numberrepresentation systems. a r X i v : . [ m a t h . HO ] M a y igure 1: The Maya Long Count calendar. The stela on the left contains thedates shown enlarged on the right, including the date of the last completion ofa full cycle of the Maya calendar, which occurred on 13.0.0.0.0, 23rd December2012, many centuries in the future when this stela was inscribed. (Museo Mayade Canc´un, Instituto Nacional de Antropolog´ıa e Historia, Mexico.) Alongside the decimal positional number system, Fibonacci popularized a newnumber in Europe: zero. Partially owing to this historical link, it has almostbeen, as it were, a truth universally acknowledged that a civilization in pos-session of a good number system must be in want of a zero. But this is notnecessarily so. It is perfectly possible to have a positional number system with-out a zero (see the Appendix for an introduction to number representationsystems). This is called bijective numeration, and we argue that Mesoamericamay well have invented the positional number system first as a bijective sys-tem without a zero. Only some time later do we see zeros beginning to appearin the Maya Long Count, depicted in Fig. 1. Because the Maya were used toa redundant number system they were not afraid of writing the same numberin various ways, and they found that the zero they had discovered and initiallyused in a non-positional system could be introduced into their positional systemwith minimal problems. Thus they were able to make the conceptual leap to acardinal zero — a zero used in arithmetic — in stages aided by their familiarity2ith multiple number representation systems. “No people in history has shown such interest in time as the Maya.Records of its passage were inscribed on practically every stela, onlintels of wood and stone, on stairways, cornices, friezes and panels”J. E. S. Thompson [1].The Maya understood well what we now call deep time. The Long Count isa positional notation system that, as its name indicates, enables complicatedarithmetical calculations over arbitrarily long periods of time. By the classicalperiod as it reached its apogee under the Maya, scribes were writing of timeperiods of millions of years into the past and thousands of years into the future[2]. The Maya developed a very sophisticated astronomical culture [3] in theircivilization centred around the Yucat´an peninsula in what is today Mexico,Guatemala and Belize, whose classic period of greatest splendour ran fromaround 3rd century to 10th century CE before falling into decline [4]. A numer-ical calendar is a revolutionary idea: to enumerate the passage of time, ratherthan merely giving it a descriptive label; the year the big tree in the village blewdown ; the the day the sun rises over that mountain , etc. Enumerating ratherthan just labelling time permits one to know how long ago in the past some-thing occurred, or how far into the future it will occur. The Maya used theircalendar to record astronomical events for astrological purposes [1, 5] and therecontinues to be much interest in understanding the Maya concepts of time andon what astronomical observations it may have been based [6]. It has been as-serted that the Maya numeration system would be superior to today’s, at leastfor the ease of recognizing small divisors of large numbers [7]. They certainlyinherited parts of their number system, such as base 20, which was commonacross Mesoamerica, from earlier civilizations such as the Olmecs, and sharedthese aspects with succeeding peoples of Mesoamerica such as the Aztecs [8, 9].In Mesoamerica there emerged a concept of zero, at first as a placeholder(an ordinal zero), before entering into arithmetic (a cardinal zero) [2, 10]. Aswe shall discuss below, it is questioned whether the concept of zero was anothersuch inheritance from the Olmecs to the Maya [11]. It is possible that theconcept of zero has been discovered only twice: once in the Old World, whereit seems to have first appeared as a placeholder in Sumerian Mesopotamia fourto five thousand years ago, and once in the New. What seems certain is thatthe New World discovered zero on its own and that it was the Maya who fullydeveloped the idea into a cardinal zero, used for calculations.A significant characteristic of the Maya calendar is the concurrent use ofthree separate number systems: the Haab, the Tzolk’in and the Long Count;the former two formed the Calendar Round, in which all dates are repeatedevery 52 years [12, 6]. This combination of calendars is similar to our use today,3ithout a second thought, of a year-month-day calendrical system together withan incommensurate week system, where repetition comes after 28 years . TheMaya civil calendar, the Haab, represents an annual solar cycle of 365 days,composed of 18 months (winals) of 20 days (kins) each, plus — as in manycalendar systems — five extra epagomenal days at the end of the year, whichwere called unlucky days or days without name (wayeb, for the Maya) [13]. Onthe other hand, the divine calendar, the Tzolk’in, used to determine the timeof religious and ceremonial events and for divination, has a cycle of 260 days,composed of 20 weeks of 13 days each [14], possibly owing to the 260-day spanof time between zenithal sun positions at the latitude of 15 o N in Mesoamerica[15].Much debate has focused on who developed these calendrical systems. TheIsthmus of Tehuantepec has long been seen as an important area of elaborationand differentiation of the first calendar in Mesoamerica, although opinions differas to which side of the isthmus can claim precedence. Some scholars look tothe Olmec society on the north side of the isthmus on the coast of the Gulf ofMexico, in the modern Mexican states of Veracruz and Tabasco. Others look tothe south, to the Pacific coast, present-day Chiapas (Mexico) and Guatemala.And others indicate west, to modern-day Oaxaca [16]. Grove [17, 18] points outthat the numeral glyph found in the Olmec culture, with a 260-day count in itscalendrical inscription, may be the oldest. Similarly, Edmonson [19] proposesas the oldest calendrical record, one corresponding to the Olmec culture in theyear 679 BCE. Diehl [20] indicates that in the decadence of Olmec culture, theepi-Olmec period, in Chiapa de Corzo in Chiapas and Tres Zapotes in Veracruzstelae were erected with the earliest known inscriptions of the Long Count.Blume [2] notes that the earliest Mesoamerican Long Count is inscribed onStela 2 at Chiapa de Corzo with a date of 7.16.3.2.13, corresponding to 36BCE. In terms of the development of mathematical ideas, we may affirm thatthe epi-Olmec and the proto-Maya came together something over 2 000 yearsago in this fertile crescent and the Long Count, and zero, were the eventualresults.For everyday activities, the Maya used a pure vigesimal, base-20, numeralsystem (although there are no extant Maya documents showing this, and weknow it only from what bishop Diego de Landa told of cacao bean countingin sixteenth century Yucat´an) [22]. In their Long Count calendar (Figure 1),however, they would use a slightly modified version of this. The first and second Disregarding the complications introduced in the Gregorian Calendar where centuries arenot leap years unless divisible by 400. The date corresponding to the beginning of the Long Count, 0.0.0.0.0, is Monday, 11thAugust, 3114 BCE, according to the most accepted Goodman–Mart´ınez–Thompson correla-tion with our Gregorian calendar. It is supposed that this initial value was decided a posterioriin a similar fashion to how the current calendar era was proposed by Dionysius Exiguus inthe sixth century and widely implemented by Charlemagne in the 9th century. Compare theearliest known numerical calendar, the Seleucid Era, which did begin with year 1 in 312/11BCE [21]. “Que su cuenta es de V en V, hasta XX, y de XX en XX hasta C, y de C en C hasta 400,y de CCCC en CCCC hasta VIII mil. Y desta cuenta se serv´ıan mucho para la contrataci´ondel cacao. Tienen otras cuentas muy largas, y que las protienden in infinitum , cont´andolas and 20 as usual, but the third was 20 ×
18. This ispresumably because 20 ×
18 = 360 represents much more accurately than 20 =400 the number of days in a year. All subsequent place values were multipliedby 20. Thus we have 1 kin (= 1 day), 1 winal = 20 kins, 1 tun = 18 winals, 1katun = 20 tuns and 1 baktun = 20 katuns. Accordingly, a number would beexpressed in this system as N = d k (18 × k − ) + . . . + d (18 × ) + d (18 ×
20) + d
20 + d . This is an example of a non-power positional number representation system.Since the digits go up beyond 9, to avoid using extra non-decimal digit symbolsto write Long Count numbers the convention is to use the following notationwith intercalated dots between digits written in decimal to avoid any confusionbetween numbers: N = d k . · · · .d .d . In a regular base-20 system, when a place value is completely filled, wesimply write 0 and carry a 1 to the next power. For instance, we can fill upthe units place with numbers 1 up to 19, but on reaching 20 we have to writeit as 1 . + 0 × , and similarly for higher powers. In the calendarcount, however, the third place being 18 ×
20 creates some difficulties. If thesecond place (20 ) is filled up, we would have 20 sets of 20 which cannot becarried over to the next power: only 18 ×
20 can, leaving 2 ×
20 back. Thesame happens if we have 19 ×
20 in the 20’s place: it can be carried over leaving1 ×
20 behind. So for example, 3 . . . . . .
11 = 8 . .
11. Thus, if thedigits can go up to 19 in the second place, this non-power positional numberrepresentation would not be unique starting from Maya Long Count numbersfrom 18 . . . .
19 = 1 . . , corresponding to the base-10 numbers 360to 399. In other words, the number system is partially nonunique, with thenonuniqueness affecting about 10% of numbers.So did the Maya ensure uniqueness in the Long Count by having the seconddigit go only to 17? That might make sense considering that this numberrepresentation system was used as a calendar (recall 18 20-day ‘months’ plusfive extra days made up the Maya year). In that context, one might naturallycarry directly into the larger units. Indeed, Freitas and Shell-Gellasch [23] didnot find any examples: “no Maya Long Count numbers with an 18 or 19 in thesecond place appear on known monuments or documents” they wrote. But thisis not so. Closs [24] notes an example in the Dresden codex of 390 expressedin the form 19.10. We show this instance in Figure 2a. Note that Closs wasexpecting the number to appear as 1.1.10, and was surprised to find it written VIII mil XX vezes que son C y LX mil, y tornando ´a XX duplican estas ciento y LX mil,y despu´es yrlo ass´ı XX duplicando hasta que hazen un incontable n´umero: cuentan en elsuelo ´o cosa llana” [22]. Landa’s account, written circa 1566, in which he does not mentionzero, demonstrates that Hindu-Arabic numerals were still being used little in Europe. Landa,who ordered the burning of almost all Maya texts, probably did not appreciate that he wasdestroying one zero in America just as another was struggling to emerge from the hegemonyof Roman numerals in Europe. (Ironically, Landa’s original manuscript is also lost, so wecannot be sure that the version we have with Roman numerals is how he wrote it.)
5) b)Figure 2: Instances from the Dresden codex of Maya numbers written with an18 or a 19 in the second place. The dot and bar notation of the Maya seen herecomposes the digits 1–19 using zero to four dots to represent ones and zero tothree bars to represent fives, so that 1 is one dot, and 19 is three bars belowfour dots. Generally numbers were written vertically with the most significantfigure at the top. (a) 390 written as 19.10 rather than as 1.1.0 on page 72; (b)10.11.3.19.14 or 10.11.3.18.14, i.e., 1 520 654 or 1 520 674 in decimal, on page 70.Iit is unclear whether is there a dot missing in the second place digit owing towear of the codex, but the unequal dot spacing — compare with 19 as writtenin (a) — makes it plausible that there were originally four dots, with this digitthus reading 19 rather than 18. 6igure 3: Today one occasionally finds an example of Maya-style notational non-uniqueness in our representation of time. Information on restaurant openinghours at Tokyo Haneda airport; notice the opening hours and time for LastOrders of the second entry.in this other manner. Cauty and Hoppan [25] found this same example andnoted a further example in the Dresden codex, which we show in Figure 2b,where instead of 10.11.4.1.14 there is written 10.11.3.19.14. Again, they seethese instances as being irregular variants where the scribe has omitted to carryinto the larger units.We may note that many old units of money and measure functioned in thissame way. E.g., in the old British monetary system (imported from Charle-magne’s continental Frankish empire) of pounds, shillings and pence, 12 pen-nies made a shilling and there were 20 shillings in a pound: N = d (20 ×
12) + d
12 + d , where the digits d , d go only to 11, 19; that is, to b i +1 /b i − < amonth), months ( < a year) and years, and the same with hours, minutes ( < anhour) and seconds ( < a minute). We would not generally think of giving a dateas 13/13/2018, rather than 13/1/2019, nor a time as 12:65 rather than 13:05;but see Figure 3. One instance where we are perfectly comfortable with suchnonuniqueness today is in currency, where coins and notes in denominationsoften based on 1, 2, 5 permit us to pay a given sum of cash in multiple ways.It is notable that we are worse off today with our calendar with a jumble ofmonths with different lengths than the ancient Maya with their 20-day months,as we have to remember that “30 days hath September...” etc, in order toperform calendar calculations with our very irregular length months inherited7rom the Romans. Moreover, although we are happy to consider the first minuteof the hour, minute zero, and the last minute, 59, only in the 24-hour clock dowe condescend to have an hour zero, and we refuse to consider a day zero ora month zero, just as there is no year zero in today’s Gregorian calendar. Itshould also be noted that the Tzolk’in — perhaps the earliest Mesoamericancalendar [2] — has days 1 to 13, without a zero in the same way as the daysof our months lack a zero day. On the other hand, the days of the solar Haabcalendar use the same digit notation of dots and bars as we see in Figure 2 usedfor coefficients of Long Count quantities, so it is natural to ask how the Mayagot to be more logical than us, to arrive at a day zero. It is not widely appreciated that a positional number representation system doesnot need a zero. Instead of digits in the range from 0 to b −
1, we can simplyshift them by one to the range from 1 to b [27] (see the Appendix for a discussionof digit shifting). We have done away with zero — whose introduction is oftenheld to have been essential for the development of positional number systems— yet we can still represent all numbers uniquely. For instance, if we do thiswith base 10, we simply need a new digit symbol for ten; let us borrow fromthe Romans and use X. Then the digits are from 1 to X, and most numbersremain written as in normal base 10. Only those containing 0 are altered: 10becomes X, twenty, 1X, one hundred, 9X, and so on (Figure 4a). It is still thecase that 1 + 1 = 2, but now 9 + 1 = X [28]. This zero-less number systemhas sometimes been called bijective numeration [29]; it is bijective or one to onebecause in this system there is no possibility to have leading zeros in front ofa number. It keeps on being rediscovered [30, 31, 32] We see, then, that theintroduction of zero, although viewed historically as linked to the developmentof our Hindu-Arabic decimal positional number system, was not necessary fora positional number system.A key point in exploring the Maya number systems is this: given a posi-tional number system that uses an unfamiliar set of symbols, how may we knowwhether the symbols include a zero, or not? How do we know whether the Mayain their Long Count were writing their calendar using days from 0 to 19, or from1 to 20? Clearly this question is pertinent given that today our calendar doesnot include a day zero. That is to say, did the Maya really begin their monthswith day zero and end them with day nineteen, or did they begin with one, likeus, and end with twenty? If the meaning of the digit symbols is completely un-known a priori then only way to answer this question is to look at arithmeticaloperations with these symbols [30] . In the four surviving Maya codices, writtenon astronomical and calendrical themes between the classical Maya period and Note that one may combine bijective numeration with non-power representation systemsin the same way as with a fixed base system (Figure 4c). It reminds us of the passage in
Alice’s Adventures in Wonderland “I’ll try if I know allthe things I used to know. Let me see: four times five is twelve, and four times six is thirteen,and four times seven is — oh dear! I shall never get to twenty at that rate!”, which can be ) Bijective base 10 Decimal1 12 2... ...9 9 X X X X X XX . . . . . .
19 17 .
19 17 .
19 35917 .
20 1 . . .
20 or 18 . . . . . . . . . . . . . . . . . . .
20 1 . . .
20 or 19 . . .
20 1 . . . . . . . . . . . . . .
10 or 19 .
10 1 . .
10 or 19 .
10 1 . .
10 or 19 .
10 390... ... ... ...1 . .
20 or 19 .
20 1 . . .
20 or 20 . . .
20 1 . . . . . . . . . . . .
20 or 20 .
20 1 . . .
20 or 1 . .
20 1 . . . . . . . . Oliva [2] — represents zero. Thatit does represent zero is clear from, for example, multiplication tables in theDresden codex that would be incorrect if we tried to interpret the shell symbolas 20. (Although it should be noted that the arithmetic in the codices containserrors.) So, certainly by their post-classical period during which the codiceswere written, we do have a zero-based positional system, with digits 0–19.There is a variety of evidence pointing towards an earlier bijective system,with digits 1–20, both within the codices themselves and in the stone inscrip-tions on the earlier classical stelae. Within the codices there is evidence foran earlier, non-positional system with an explicit symbol for twenty. In theDresden Codex, distance numbers between 20 and 39 are frequently expressedby prefixing with the dot and bar notation a number between 0 and 19 to amoon glyph representing twenty, and similar “x+20” notation was used earlieron stelae, as has been discussed by Thompson [33]. Although an explicit zero isfirst used for a Long Count inscription of 8.16.0.0.0 (357 CE), at Uaxactun onStelae 18 and 19, a later 711 CE inscription at Stela 5 at Pixoy is not written as9.14.0.0.0, but as 9.13.20.0.0, as Closs has pointed out [34]. And at the Templeof the Cross (Palenque, Chiapas) there are the forms 20 Mol (in 13 Ik 20 Mol= 13 Ik 0 Ch’en) and 0 Zac (in 9 Ik 0 Zac) entered in two Haab dates writtenside by side [25].
The route to zero that Mesoamerica took must be teased out from the sparseevidence available. Mathematics can help with this task. Initially there was understood if Alice is counting in a varying base: she is expressing 4 n in base 3 n + 3, and shecannot get to 20, as many people have pointed out, since after 4 ×
12 = 19 , 4 ×
13 = 1 X .
10 non-positional number system with digits from 1–13, without a zero, in theTzolk’in calendar. Archeological evidence of this calendar has been found inOlmec cave paintings dated 800–500 BCE [17, 18].Then there is the Haab, again a non-positional number system. The Haabmay possibly have been set up around 500 BCE [35]; there is archeologicalevidence from 500–400 BCE from Monte Alb´an, Oaxaca [36]. As we have in-dicated, the days of the Haab generally run not from 1–20, but from an initialday, followed by the 1st, 2nd, etc, up to the 19th day. And as we have pointedout above, in the Haab on occasion a glyph for the end of a month, i.e., for day20, was used instead of that for the beginning of the month. Of course then it isnatural today to translate the glyph for the initial day, often referred to as chum ,by zero. However, it is not one of the same glyphs as the zero later found in theLong Count. Maya scholars have debated for many years about the meaning ofchum for the Maya [2]. Some have thought it to be the end of the precedingmonth, i.e., a species of 20, and others the beginning of the new month, i.e., atype of zero. Some translate it in a non-numerical way as the “seating of” themonth [37, 38, 33]. What we can put forward for our purposes about chum withrelative certainty, however, is twofold: (1) that it does not perform the samefunction as the zero in the Long Count, being an ordinal, not a cardinal zero;and, however, (2) that it may well have influenced the development of the zeroplaceholder to come in the Long Count.Now let us move to the Long Count, for which the earliest evidence is severalhundred years later than the two preceding calendars. It is tantalizing that innone of the eight known inscriptions that show the earliest development ofpositional notation in the Long Count can we find the full range of digits thatwere necessarily then in use , which would enable us to understand whetherthe calculations behind the Long Count were being performed with digits 1–20, 0–19, or 0–20. Neither 0 nor 20 appears in any of the earliest exampleswritten from 36 BCE to 162 CE. We know that there necessarily has to havebeen one of these three systems in use in order to satisfy the mathematics ofthe Long Count; in order to be able to write all numbers (Figure 4c). Withfewer digits with the same multipliers not all numbers can be represented andso some calculations simply could not be performed.Then there appear in the archeological record examples with an implicit zerodenoted by the lack of a digit, before finally we get examples with the explicitwritten cardinal zero, as well as the example with an explicit twenty on Stela5 at Pixoy. (Since the Tzolk’in and Haab continued to be used alongside theLong Count, we can use this calendrical redundancy for error checking, to makesure that we understand correctly whether a glyph is a zero or a twenty.) As inthe case of the solar calendar glyph chum , within Maya scholarship there hasbeen a great deal of debate about how to read the Long Count glyphs that are The eight earliest known Long Count inscriptions (which are generally designated epi-Olmec rather than Maya [20]) are 7.16.3.2.13 (Chiapa de Corzo Stela 2), 7.16.6.16.18 (TresZapotes Stela C), 7.18.9.7.12 or 7.19.15.7.12 (El Ba´ul Stela 1), 8.3.2.10.15 and 8.4.5.17.11(Takalik Abaj Stela 5), 8.5.3.3.5 and 8.5.16.9.7 (La Mojarra Stela 1) and 8.6.2.4.17 (the Tuxtlastatuette) [39]. . All these digits and multipliers could beused together because the Maya mathematicians were happy with the flexibilityof their number representations leading to non-uniqueness, to redundancy, sothat they could write both a zero and a twenty side by side in a number suchas 9.13.20.0.0 at Pixoy. That is to say, their positional number system brokeboth sufficient conditions for uniqueness listed in the Appendix: neither arethe positional weights powers of a base b , nor are the digits limited to a rangefrom 0 to b −
1, or to a shift of that range. The Long Count with zero andtwenty gradually gave way to a Long Count with zero. By the time the extantcodices were written, when Maya civilization was on the wane, the Long Countwas worked almost always with digits with the explicit zero, using the shellglyph which is characteristic of the codices, and the use of twenty as a digit hadbecome vestigial. It is notable that Justeson [11], who comes at this question from a completely differentapproach, arrives at a similar conclusion on this point. Ex Nihilo Nihil Fit?
In the Old World, zero first emerged from the development in Sumerian Mesopotamiaof a sexagesimal positional number system for accounting purposes. This wasto begin with an implicit zero, by which we mean that at first it had no sym-bol associated with it, but was simply a lack of a digit. This makes perfectsense within the scheme of tallying goods: a lack of something corresponds to amissing number in its corresponding column. But it became awkward that thelack of something could be misinterpreted when writing tallies in columns, in apositional notation, and so after some time the implicit zero was given its ownsymbol and became explicit as a a placeholder in the base-60 notation. So it wasnatural, from this Old World point of view of counting goods, that a positionalnumber system and a zero should go together. In the New World, however, theimpulse for the positional number system came not from counting goods, butfrom the calendar, from counting days. And in counting days, what would bethe lack of a day? It is much more natural to use the counting numbers, 1, 2,3. We see this today in our our Gregorian calendar with no day, month, or yearzero. So from the American point of view, it made sense that there could be apositional notation without a zero.Mesoamerica did not have to discover zero at the same time as inventing po-sitional notation; the two are independent concepts. With a bijective positionalnumber system one can represent all numbers, and one does not need a zero.One has to ask whether the Maya used it; that is, was the symbol they useda 0 or a 20? That the Maya discovered the concept of zero beginning from abijective non-power number system is a plausible hypothesis when one considerson one hand the available historical evidence, which has led Maya scholars todebate whether chum meant zero or twenty, and on the other hand the easewith which one can move from a bijective system to a system with a zero. Weinfer that initially they used 20 and only later 0, and they shifted via an inter-mediate Long Count with both a twenty and a zero that we see in the historicalrecord. To go between these different systems affects only the numbers whoserepresentation contains the digit (20), in which one replaces a (20) with a (0),at the same time adding 1 to the digit in the superior position . If one adds tothis mixture of mathematics and anthropology the point that the Maya, owingto the redundancies built in to their mixed-based system, were used to the ideaof the same number being represented by more than one different sequencesof symbols, one can understand that they could make the momentous concep-tual leap to using a cardinal zero owing to their familiarity with multiple andredundant number representation systems. Of course as the number gets larger, the probability of it containing a (20) tends to one.However, although on occasion the Maya did represent numbers with a large number of digits,most of the numbers written have five or fewer digits. PPENDIX
The history of mathematics has an intrinsically interdisciplinary character.In order to make matters clear for a diverse readership, and to provide a reason-ably self-contained argument, in this Appendix we spend a little time runningthrough the mathematics involved.
A Number Representation Systems “The mirror of civilization” is what Hogben termed mathematics in his
Math-ematics for the Million [40], “interlocking with man’s common culture, his in-ventions, his economic arrangements, his religious beliefs”. It may be that theinitial use of the symbolic management of numbers through visual signs corre-sponded to utilitarian needs, for example, for the exchange of goods; the firstform of commerce. However, numbers became part of the human endeavourfor knowledge very early. Perhaps astronomy — the counting of the elapsedtime between recurring events of day–night, winter–summer, relative positionsof planets and stars, eclipses, and so on — was the earliest ‘scientific’ applica-tion of number systems . Geodesy also has represented an important practicalaspect in early civilizations that led rapidly into geometric developments. Wefind the apex of number in the Pythagorean doctrine that the entire universeis governed by numbers; for the Pythagoreans, that meant integer and rationalnumbers.The felicitous choice of a numeration system is relevant for solving specificproblems and also for developing and improving mathematical models and al-gorithms [41, 26]. We can see from our position of hindsight that civilizationsthat used inconvenient number systems were held back in their developmentof mathematical knowledge. Today’s decimal number system is a positionalsystem, but there are many historical examples of non-positional numerationsystems [42]. A familiar example is that of Roman numerals. In the Romansystem the values of the symbols are in general independent of their position;numerals are written from left to right in descending order, writing the biggestnumeral possible at each stage. There is only a relative positional dependencethat determines whether a particular number should be added to or subtractedfrom its neighbour for obtaining the represented number. For example, I repre-sents 1 and V, 5, and there are two ways to write 4, IV and IIII (with the latterversion generally seen on clocks; the subtractive rules leading to forms like IVwere alternatives that only became usual in later Roman periods). Thus Romannumerals constitute primarily a non-place-value system, but because of the useof the subtractive principle — e.g., IV represents four while VI represents six—, the Roman system may be classified as a mixed system. Arithmetical opera-tions are very difficult to implement with non-positional systems. Cultures that Until the scientific revolution of the 15th–16th centuries, astronomers were also astrologersor priests, and astronomical data were used for astrological or religious rather than what wewould think of as scientific purposes. Nonetheless, as numbers were used to document, explainand predict natural phenomena, we may consider this a proto-scientific application.
Liber Abaci (the Bookof Calculations) [43], and gradually replaced the cumbersome Roman system.In Mesoamerica a positional number system was in use much earlier.Most of the numeration systems that we know and use are univocal, thatis, they are not redundant; univocity means that any symbol represents onlyone number and, conversely, any number is represented by only one symbol(except for the minor point of leading zeros that we shall discuss below). Thisis usually achieved using a power positional system, defined by a small set ofintegers given symbols, called digits, that, depending on their position alonga representation string, are multiplied by the powers of a given base or radix.However, there have existed in the past, and there continue to exist, non-powerpositional number representation systems in which the multipliers are not thepowers of a given base. Such mixed-radix systems were studied by Cantor[44]. One historical example in which the necessity to describe more adequatelyannual timescales led to a non-power number representation system is the MayaLong Count calendar, depicted in Fig. 1. Others are the old systems of money,weights and measures from around the world not based on multiples of ten.While these systems were often employed in such a way as to preserve univocity,non-power number representation systems, mainly employed in computing fortransmission and storage of data, are today used so as to be redundant. Thereason for this redundancy is to have error detection, and so the possibility forerror correction, built in to the system [45] . It is thus most interesting that inDNA, the biological molecule of information transmission and storage, we findin the genetic code the structure of a non-power number representation system[49, 50]. B Positional Number Systems
A counting system is said to be positional if each digit is weighted with a differentvalue according to its location in the string. The most common positionalnumeral systems are power representation systems where the positional weightsare powers of some number b , called the base or radix, and the digits are allowed The redundancy of overspecifying calendrical information performs precisely this func-tion of error detection in our present calendar, as for example when we write Saturday 17thNovember 2018. (Algorithms to determine the day of the week for any given date have beendevised by mathematicians from Gauss [46], through Lewis Carroll [47] to John Horton Con-way [48].) We can presume that Maya scribes understood and used this same redundancy forerror checking purposes when they wrote dates using the three Maya calendars. The samedate in the Long Count is 13.0.5.17.17, 3 kab’an, 10 Keh.
15o take any value from 0 to b −
1. The main advantage of such a system is thatany integer N has a unique representation of the form N = d k b k + d k − b k − + . . . + d b , ≤ d i ≤ b − ∀ i. The decimal system, base 10, is undoubtedly the most familiar and widespreadexample, but it is not the only one. The first place-value system, developed bythe Mesopotamians, was sexagesimal, base 60, which is why we still measureangles and time in units of 60. In more recent times, the binary, base 2, systemhas become a fundamental tool in informatics; base 16, hexadecimal, and base8, octal, are also important in computing.
B.1 Uniqueness of the representation
As previously stated, in a positional system with base b each number has aunique representation. This can be proved by contradiction, i.e., assuming thatan integer N can be written in two different ways: N = d b + d b + . . . + d k b k , ≤ d i ≤ b − N = a b + a b + . . . + a k b k , ≤ a i ≤ b − ∃ i such that d i (cid:54) = a i . In particular, assume a i > d i , and ∀ j > i, d i = a i . Subtracting (2) from (1),0 = ( d − a ) b + ( d − a ) b + . . . + ( d i − a i ) b i ⇒ ( a i − d i ) b i = ( d i − − a i − ) b i − + . . . + ( d − a ) b . By assumption, a i − d i > b i ≤ ( a i − d i ) b i ⇒ b i ≤ ( d i − − a i − ) b i − + . . . + ( d − a ) b . Since d j is a digit, d j ≤ b − d j − a j ≤ b − ∀ j . So one obtains b i ≤ ( d i − − a i − ) b i − + . . . + ( d − a ) b ≤ ( b − b i − + . . . + ( b − b ⇒ b i ≤ ( b − b i − + . . . + b ) . But ( b i − + . . . + b ) is a geometric series, and we know that i − (cid:88) j =0 b i = b i − b − , and so we obtain b i ≤ b i − , which is a contradiction; the proof is complete.16 Signed-Digit Representation
It is clear from the above proof that the uniqueness of the power representationis given by two key assumptions:1. the digits are limited to a range varying from 0 to b −
1, and2. the positional weights are the powers of b .We can develop non-standard positional numeral systems, where one of thesetwo conditions is not fulfilled.A first relaxation of the above conditions is where digits are allowed to gobeyond the prescribed range. A particular example is the signed-digit repre-sentation, where each digit is given a positive or negative sign, hence its name.Uniqueness cannot be guaranteed anymore; it is easy to see that this represen-tation is in fact redundant. Let us take for instance the binary signed-digit case:the positional weights are powers of 2, just like the usual binary, but the digitscan now take values − , − balanced form of the representation. Given a base b , the allowed digits are b − d i taken from the range − (cid:22) b (cid:23) ≤ d i ≤ ( b − − (cid:22) b (cid:23) where the floor function (cid:98) x (cid:99) maps x to the largest integer smaller or equal to it.For simplicity, we will only consider the case where b is an odd integer, whichimplies (cid:22) b (cid:23) = b − − b − ≤ d i ≤ b − . We can prove that the balanced form is unique in the following way. We haveshown above that every integer has a unique representation in base b of the form N = d n b n + d n − b n − + . . . + d b , ≤ d i ≤ b − ∀ i. (3)First consider the coefficients d k = b −
1. Noting that d k b k = ( b − b k = b k +1 − b k d k b k into (3) to get N = d n b n + d n − b n − + . . . + d k +1 b k +1 + b k +1 + ( − b k + . . . + d b = d n b n + d n − b n − + . . . + ( d k +1 + 1) b k +1 + ( − b k + . . . + d b . If d k +1 +1 = b −
1, we repeat the previous step until there are no more coefficientsequal to b −
1. Then, we seek to eliminate every coefficient of the form d t = b − b − b t = b t +1 − b t . Finally, we get to the digits of the form (cid:18) b − b − (cid:19) b s = b s +1 − (cid:18) b − (cid:19) b s and we plug this expression into N . Therefore, with this process we have founda unique representation for N with digits drawn from the range − b − ≤ d i ≤ b − = 3 + 3 + 2 × , where the subscript refers to the base. In order to reduce it to a balanced formrepresentation, we seek to have only − , , + 3 + 2 × = 3 + 3 + (3 − = 3 + 3 − + 3 = 11¯11 , as desired. C.1 An application of signed-digit representation
An interesting example of how the balanced form of signed-digit representationwas used in the past can be found in the work of Fibonacci. With his
LiberAbaci , Fibonacci spread Hindu-Arabic numbers in Europe together with prac-tical applications, generally of a commercial nature. Zero and the positionalnumber system developed in the Old World on the back of trade and bookkeep-ing, while in America the calendar was the driving force, and we argue that thisdifference was to prove crucial.The problem is presented as follows:18 certain man in his trade had four weights with which he couldweigh integral pounds from one up to 40; it is sought how manypounds was each weight.Fibonacci provides a solution, stating that the four weights are 1lb, 3lb, 9lb and27lb respectively. Clearly, each weight can be used on either side of the balance.We can then give a weight three possible values as a digit: -1 if it is used onthe pan with the unknown weight on, 1 if used on the other pan and 0 if theweight is not used at all. The most natural choice for a counting system is thenthe balanced ternary system In this way, the highest number we can expressis 40 (written as 1111 = 27 + 9 + 3 + 1) and the smallest is -40 (¯1¯1¯1¯1). Everynumber within the range -40 to 40 has a unique representation. Observe that,from a practical point of view, the weights cannot be negative, so the system isuseful only for representing one half of the integer set, that is, the positive onesfrom 0 to 40.
C.2 The shifting property of digits
There is another way to think about the foregoing scales problem: 4 weights with3 possible positions each give rise to 3 = 81 combinations. On the scale, thesecombinations would read as the 81 numbers from -40 to 40. If we were to usea standard ternary system with digits 0, 1, 2, we could still express exactly 81consecutive numbers; however these numbers would go from 0 to 2222 = 80 (recall the subscript refers to the base). This idea can easily be generalized:if all digits are shifted by the same quantity, the representation remains non-redundant but the interval of represented numbers is also shifted. Note that ourreasoning proves not only the uniqueness of the representation, but its converseas well; every number is guaranteed to be representable. When using strings ofa given length, if we shift all the digits, the interval of representable numbersmoves up or down accordingly, but leaves no gaps. Consider an ordinary base- b system and a string of digits of length k , d k − d k − . . . d = d k − b k − + d k − b k − . . . + d b , ≤ d i ≤ b − ∀ i. Clearly, the smallest representable number is 0 and thelargest is ( b − b k − + . . . + b ) = ( b − (cid:18) b k − b − (cid:19) = b k − , thus defining an interval of b k numbers. Again, keeping in mind that we have b possibilities for k positions, the number of combinations is indeed b k . Suppose It is interesting that this balanced form of representation minimizes the number of carriesin addition, at least when the base is an odd prime [51]. Note, however that, as in the case of the Fibonacci weights (below), the useful representednumbers are the positive set. In power systems, negative numbers are externally defined by aminus sign. We can interpret the external minus sign as the additional possibility of makingall signs of all digits negative for a given represented number. s ∈ Z , i.e., s ≤ d i ≤ b − s ∀ i. As a result, the lower bound of the interval of representable numbers is s ( b k − + . . . + b ) = s (cid:18) b k − b − (cid:19) and similarly the upper bound is s ( b k − + . . . + b ) = ( b − s − (cid:18) b k − b − (cid:19) . Thus, a shift by s in the digits range results in a shift by s ( b k − / ( b −
1) inthe range of represented numbers.
D Non-power Representation Systems
As we have said, there are two ways to obtain redundant representation systems.Having seen the signed-digit case, we now move to the other case, that of a non-power representation system. This means that instead of having a given base orradix, the positional weights are numbers of a sequence that grows more slowlythan the powers of some number. If this is the case, then every number can berepresented and generally has more than one expression within the system. We can pick any sequence of numbers that grows more slowly than do powersof two. A famous choice is to use the Fibonacci numbers [26, 52]. Fibonaccinumbers are the elements of a sequence where each number F n is the sum ofthe previous two; F n = F n − + F n − , with F = F = 1. The first terms ofthe sequence are then 1, 1, 2, 3, 5, 8, 13, 21, . . . . These grow more slowly thanpowers of two. If one uses these as positional weights, and 0, 1 as digits, it canbe seen that every number is representable. Moreover, this representation isunique provided that there are no two consecutive 1’s [53]. In fact, since everyterm is the sum of the previous two, it is easy to see that any string of the form . . . . . . can be replaced with . . . . . . .Another, more ancient example is a Maya calendrical counting system, theLong Count, which is the subject of this work. References [1] J. E. S. Thompson. Maya astronomy.
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