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1988 30: 181Human Factors: The Journal of the Human Factors and Ergonomics SocietyRaymond S. Nickerson

Counting, Computing, and the Representation of Numbers  

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HUMAN FACTORS, 1988,30(2),181-199

Counting, Computing, and the Representationof Numbers

RAYMOND S. NICKERSON,I BBN Laboratories Incorporated, Cambridge, Massachusetts

How easy it is to manipulate numbers depends in part on how they are represented visually.In this paper several ancient systems for representing numbers are compared with the Arabicsystem, which is used throughout the world today. It is suggested that the Arabic system is asuperior vehicle for computing, largely because of the compactness and extensibility of itsnotation, and that these features have been bought at the cost of greater abstractness.Numbers in the Arabic system bear a less obvious relationship to the quantities they repre-sent than do numbers in many earlier systems. Moreover, the elementary arithmetic opera-tions of addition and subtraction are also more abstract; in some of the earlier systems theaddition of two numbers is similar in an obvious way to the addition of two sets of objects,and the correspondence between subtraction with numbers and the subtraction of one set ofobjects from another is also relatively direct. The greater abstractness of the Arabic systemmay make it somewhat more difficult to learn and may obscure the basis for such elemen-tary arithmetic operations as carrying and borrowing. The power of the system lies in thefact that once it has been learned, it is the most efficient of any system yet developed forrepresenting and manipulating quantities of all magnitudes.

INTRODUCTION

Counting and computing are such commonand fundamental activities in modern societythat it is difficult to imagine what life wouldbe like without them. It may be that the abil-ity to count and to reckon are as old as hu-manity itself; however, there can be no doubtthat the range of useful and interesting thingsthat can be done with numbers has increasedgreatly, albeit gradually, over the history ofhumankind.

Most of what we know about the develop-ment of number concepts and number ma-nipulation skills comes from written records

I Requests for reprints should be sent to Raymond S.Nickerson, BBN Laboratories. Inc., 10 Moulton St., Cam-bridge. MA02238.

and so goes back only as far as do those rec-ords. which is roughly to about 4000 to 5000B.C. There is some evidence that the use oftokens to represent number concepts pre-dated the use of written symbols-the oldestknown remnants of which are attributed tothe Sumerians of Mesopotamia-by perhapsas much as 5000 years (Schmandt-Besserat,1978). The roots of counting and computingundoubtedly stretch farther back into antiq-uity than this. however, and are probably for-ever hidden in the mists of prehistory.

Such evidence as we do have regarding theorigin of number concepts suggests that for avery long time numbers were thought of asproperties of the things with which they wereassociated in counting (Menninger, 1969).Concepts such as "3 sheep" and "3 goats," for

© 1988,The Human Factors Society, Inc. All rights reserved.

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example, appear to have predated the moreabstract concept "3." Given an object-spe-cific conceptualization of numerosity, onecould understand that 4 sheep are more than3 sheep and that 4 goats are more than 3goats while being unable to make a numeri-cal comparison between sheep and goats.

Although Babylonian and Egyptian textsdating from the second millennium B.C. showthat some aspects of arithmetic were well de-veloped in ancient Babylon and Egypt (solu-tions of quadratic and cubic equations, tablesof squares, cubes, and reciprocals), what ex-isted at that time was not a mathematicaltheory of numbers but a collection of solu-tions for practical problems and rules for cal-culation (Aleksandrov, 1963). The idea ofnumbers as entities with interesting proper-ties in their own right independently of thepractical problems to which they could beapplied was one that developed graduallyover many centuries. The idea got a very sig-nificant push forward by the fertile minds ofsuch Greeks as Archimedes, Euclid, andPythagoras.

One interesting aspect of the gradual in-crease in numerical sophistication relates tothe evolution of the symbols and symbol sys-tems by which n,umber concepts have beenrepresented visually. The burden of thispaper is that the relative ease with which onecan count or compute depends to no smalldegree on the characteristics of the systemthat is used to represent number concepts.

The so-called Arabic system, which is nowused almost universally, is so familiar to usthat we take it for granted. It is easy to seethis way of representing numbers as the wayto do it and to fail to recognize it as the con-vention it is. Why has this representationalscheme proved to be so useful, and at whatcost to the user has this usefulness been ob-tained? These are the questions that motivatethis paper. The discussion begins with a fan-ciful but plausible account of how a represen-

HUMAN FACTORS

tational scheme with many of the propertiesof the Arabic system might have evolved.How the Arabic notation instantiates theprinciples that are seen in this conjecturalscheme is then considered. Some ancient sys-tems are compared. Several properties desir-able in any number system are identified,and how the Arabic system compares with itspredecessors wi th respect to these propertiesis noted.

These comparisons reveal that the Arabicsystem is more conducive to both countingand computing and is far more versatile thanits predecessors. The power of the systemstems primarily from the fact that it providesan efficient way to represent quantities of allmagnitudes and simplifies the performanceof mathematical manipulations. In severalways the Arabic system is more abstract thanits predecessors, however, and the degree ofcorrespondence between the symbols used torepresent quantities and the quantities theyrepresent is less direct. The abstractness ofthe system obscures some of the underlyingprinciples on which it is based, such as theprinciple of one-for-many substitution andthat of a base or radix of arbitrary size. Inshort, the system is far more powerful thanany of its predecessors but may be somewhatmore difficult to learn.

A FANCIFUL ACCOUNT OF THE ORIGINOF PLACE NOTATION

Imagine that you lived in the days beforenumbers or the operation of counting hadbeen invented. If the most demanding nu-merical problem with which you ever had todeal was to tell if all four of your childrenwere home at night, you probably would notneed to know how to count. Most people areable to apprehend directly a small number ofitems-perhaps as many as six-withoutcounting. In McCulloch's (1961) terms: "Thenumbers from 1 through 6 are perceptibles;others, only countables" (p. 7). Psychologists

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NUMBER REPRESENTATION

refer to the direct apprehension of quantitiesas subitizing and distinguish it from count-ing or estimating.

Suppose, however, that you were a pros-perous shepherd with a sizable flock and youwanted to know whether you were losing oracquiring sheep from day to day. Largechanges in the size of your flock would proba-bly be detected without any counting opera-tion, but you want to know when the size ofyour flock has been increased or diminishedby even a single sheep.

One thing you might do is get some stonesand, as your sheep file into the sheepfold, puta stone in a special pile each time a sheepgoes by. If you did this, you would be per-forming a one-for-one mapping operation,which is the essence of a tally system. Youwould be mapping the stones onto the sheepin such a way that the quantity of stonescould be used to represent the quantity ofsheep. The next time your sheep were putinto the fold, you could take a stone awayfrom your pile as each sheep went by and if,after the sheep were all in, you had stones leftover, you would know that you had lost somesheep. If you ran out of stones before thesheep were all in. you would know that youhad acquired some sheep in addition to thoseyou had before. This scheme would not tellyou how many sheep you had (presumablyyou could not count the stones if you couldnot count the sheep), but it would suffice tolet you know whether you had lost or ac-quired sheep between successive tallies.

One of the problems with this scheme isthat if you had many sheep, you would need apile of stones sufficiently large that it wouldbe inconvenient to carry around. You mightsolve that problem by getting a fairly smallpile of stones to serve as a standard pile, anduse this pile to guide the construction of sev-eral other piles. You would want some way todistinguish the several piles you would make,and you might do that on the basis of, say,

April 1988-183

color. To be specific, you might decide to usewhite stones, black stones, and red stonesand to make your standard pile of speckledstones. You might then tally your sheep inthe following way. Each time a sheep enteredthe fold, you would add a stone to the whitepile. However, you would not let the pile ofwhite stones get indefinitely large-in fact,you would never let it contain more stonesthan your standard pile. As soon as yourwhite pile contained as many stones as thestandard pile (which you would determine byusing the one-for-one mapping procedurejust mentioned), you would take them allaway and add a stone to the pile of blackstones. That is, you would let one black stonerepresent a whole pile of white stones. Thenyou would start again rebuilding the whitepile and continue as before until it again con-tained as many stones as the standard pile, atwhich time you would again take it away andrepresent it with an additional stone in theblack pile. Similarly, when the black pileeventually contained as many stones as thestandard pile, you would take it away andrepresent it by an addition of a stone to thered pile. You would simply reverse this oper-ation when "tallying down" instead of "tally-ing up."

If you were really clever, you might notbother with that standard pile of speckledstones but instead use for a standard some-thing that just happened to be readily avail-able all the time, such as your fingers. If youdid that, it would turn out that one red stonewould represent ten black ones, each ofwhich would, in turn, represent ten whiteones. Further, if you were unable to find asufficient quantity of colored stones, youmight use an alternative scheme. You might,for example, attach some significance to thespatial arrangement of the piles. That is, youmight decide to arrange the piles in a rowand let one stone in any pile represent a "fullset" of the stones in the pile immediately to

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its right or left. Although this account of howa number system might have evolved is afanciful one, a method close to the conjec-tured stone tallying has been observed in usein relatively modern times (Conant, 1956, p.436).

It is a very short step conceptually frompiles of stones to tally marks in the sand or ona piece of paper. It is a much larger step froma set of tally marks to a symbol that repre-sents a quantity. However, the practicality ofsubstituting a single symbol for a set of tallymarks is clear; a symbol such as "8," for ex-ample, is far easier to write than is"11111111." If you invented a unique symbolto represent each of the possible quantities ofstones (including the case of no stones) up tothat of your standard and decided to use theposition of that symbol to indicate the pile towhich it refers, you would have what we referto today as a place-notation system for repre-senting numbers.

This intellectual odyssey from the mostprimitive concept of quantity to a numbersystem that uses place notation was notmade, of course, by an individual. Indeed, thescheme for representing quantities that weuse today-which we take very much forgranted, and are likely to find singularly un-impressive-was many millennia in themaking. Each of the principles on which it isbased-one-for-one mapping, the use of astandard or base quantity, one-for-many sub-stitution, the use of symbols to representquantities, the notion of a symbol for repre-senting an empty set, the use of position tocarry information-constituted a major in-tellectual achievement. The history of theevolution of number systems is not best rep-resented as a linear sequence of discoveriesor innovations; different representationalschemes have existed in different parts of theworld at the same time, each reflecting somesubset-but not necessarily the same subset-of the principles on which our current sys-

HUMAN FACTORS

tem is based. But given that one system istoday the lingua franca for number represen-tation throughout the world, all the ancientsystems may be thought of as way stations onsome path-though not always the samepath-to a common destination. I do notwish to suggest that the appearance of ourcurrent system marked the end of the evolu-tion of number representation schemes, butit is of more than passing interest that thissystem has gained such wide acceptance andhas not changed appreciably in a rather longtime.

NUMBERS ASABBREVIATED POLYNOMIALS

Although the system that we use for repre-senting numbers is usually referred to as theArabic system, its place of origin, though notknown for certain, is believed to be India. Itwas introduced to Europe by the Arabs dur-ing the tenth century A.D. and for this reasonbecame known to Europeans as the Arabic, orsometimes Hindu-Arabic, system.

The scheme that this system uses for repre-senting numbers is analogous in many waysto the rock-pile system that was presentedearlier. In this scheme the value of a numberis determined by four things:

(1) The symbols (digits) that constitute thenumber.

(2)The order in which the digits are arranged.(3)The base (or radix) of the system being used.(4) The position of the "point." (When the point

is omitted, it is assumed to follow the right-most digit.)

The symbols correspond to the number ofstones in the different piles; the positions ofthe digits serve to label the piles, as it were;and the base corresponds to the number ofstones in the standard pile. The "point" hasno analogue in the stone system. Its usemakes this system more general and permitsus to represent fractional as well as integerquantities. We should also note that the stonesystem described earlier is not really a

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NUMBER REPRESENTATION

number system; even given the use of a one-for-many substitution principle and of posi-tion as an information carrier, it remainsonly a tally system and does not by itself per-mit one to say how many of anything one has.

Normally the third factor in the foregoinglist is not an issue, inasmuch as we assumethat the number is a decimal number, whichis to say that it is written to the base ten.When there is the possibility of confusion, thebase may be identified explicitly with a sub-script. Thus although 743.1510 and 743.158

have the same digits in the same order andthe point in the same position, the values ofthese numbers are different inasmuch as oneis wri tten to the base ten and the other to thebase eight.

It will be helpful to abandon our rock-pileanalogy at this point in favor of a general rep-resentation of what an Arabic number is and,in particular, for a representation that willaccommodate numbers with fractional parts.Consider the string of digits 743.1510, Whatone means to signify with such a string is aquantity equal to 7 hundreds plus 4 tens plus3 ones plus 1 one-tenth plus 5 one-hun-dredths; that is,

(7 x 100) + (4 x 10) + (3 x 1)+ (1 x .1) + (5 x .01),

or equivalently,

(7 X 102) + (4 x 101) + (3 x 10°)+ (1 x 10-1) + (5 x 10-2).

Similarly, 743.158 represents the quantity

(7 X 82) + (4 x 81) + (3 x 8°)+ (1 x 8-1) + (5 x 8-2).

If we denote the successive digits in ournumber by

and the radix of the system by r, we may gen-eralize this relationship in the following way:

April 1988-185

Nr = anrn + an_Irn-1 + ... + alrl + aorO+ a_lr-I + a_2r-2 + ... +a_mr-m.

In other words, in the Arabic system anumber is a kind of shorthand representationof a polynomial in r, where r is the base of thesystem. Specifically, from left to right, thedigits are the coefficients of terms involvingsuccessively decreasing powers of the base.

Representing a number as a coefficient of apolynomial in r is an enormous advance overa simple one-for-one tally system, no matterwhat r is. With r equal to 10, as it is in ourfamiliar decimal system, it takes 7 digits torepresent the quantity 1 million. If r equaled20, it would take 5; if it equaled 2, it wouldtake 20. With a one-for-one tally system,however, it would take 1 million.

EGYPTIAN, GREEK, AND ROMANNUMBER SYSTEMS

The elegance and convenience of the sys-tem we use to represent numbers today isperhaps best appreciated when we comparethis system with others that were used beforeit was invented. The ancient Egyptians,Greeks, and Romans all used systems thatwere similar to ours in some respects but dif-fered significantly from it in others. Here ishow the quantity that we represent as 2 4 3 32 would have been represented in each ofthose systems:

Egyptian: Hlm~~~IlM"

Greek: MMXXXXHHHM~II

Roman: @@Q)Q)ill(DCCCXXXII

The Egyptian, Greek, and old Roman sys-tems were very similar in several respects.They all used a one-for-many substitutionprinciple in much the same way: one symbolof a given type was used to represent thesame quantity as was represented by severalsymbols of another type. The number of sym-bols in a lower register represented by a sym-

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186-April 1988

bol in the next higher register was lain aUcases. (The term "register" as used in thispaper corresponds roughly to place: the thirddigit to the left of the decimal point in an Ar-abic number will be said to be in the thirdregister; in an Egyptian, Greek, or Romannumber, the set of symbols representinghundreds will be said to be in the third regis-ter.) Thus in the Egyptian system one" wasequivalent to ten I 's, one ~ was equivalentto ten (\ 's, and so on. Table 1 shows the sub-stitutionary equivalences for these three sys-tems for groupings through 10,000.

It is of some interest to note that aU threeof these systems used a single vertical line torepresent the quantity one, and that the firstfew numbers in each case were representedby a tally of such lines. In this regard theEgyptian, Greek, and Roman systems areprototypical of many if not most of the earlynumber systems of the world. The ubiquity ofthe single-line representation of the quantity

TABLE 1

Symbols Used in the Egyptian, Greek, and (Old)Roman Number Systems

Egyptian Greek Roman

1'5 I I I10'5 n 6. X100'5 ") H C1,000'5 1 X CD10,000'5 ( M @)

Note: Some sources show the Egyptian symbols for 100's, 1000'send 1O,OOO'sas mirror images of those shown here. The Greeks, likethe Hebrews, also used the letters of the alphabet to representnumbers. matching the letters to numbers in sequential fashion: Afor 1, B for 2, r for 3. and so on. The system that evolved over timestill made use of the letters but in a different way. When the one·for-many substitution principle was adopted, lellers were used to repre-sent the sets of different sizes and the letter chosen in each casewas the first letter of the name of the associated number. Thus theletter for 10 was a. for a.EKA (from which the English decimal, decile.decathlon): for 100. H for HEKATON (from which hectometer. hecto-gram, hectare); for 1000. X for XIAIOI (from which kilometer. kilo-gram, kilowatt); and for 10,000, M for MYPIOI (from which myriad).

The Egyptians wrote right to left. The Greeks. depending on theera, wrote right to left, left to right, or in boustrophedon style (left toright and right to left on alternate lines). For convenience, symbolsare always ordered in this paper from left to right.

HUMAN FACTORS

one (which in some cases is written verticallyand in some, horizontally) and the preva-lence of the use of a set of such symbols torepresent the first few numbers are usuallyassumed to be consequences of the wide-spread practice of finger counting, which pre-sumably predated other ways of representingquantities, perhaps by many millennia. Ouruse of the term "digits" to denote numeralsas well as fingers and toes is suggestive ofthis close relationship between finger count-ing and the ways in which we represent(small) quanti ties symbolically. Indepen-dently of its resemblance to an extendedfinger, the use of a line to represent the quan-tity one, or of a few lines to represent othersmall quantities, has the distinct advantageof facilitating the process of writing numbersdown, especially in media such as clay orstone.

Remnants of the common ancient practiceof representing the first few numbers as tal-lies of "ones" are seen in several systems, in-cluding Egyptian, Greek, Roman, Chinese(both ancient and modern scientific), Baby-lonian, Indian Kharosti, Indian Brahmi, andMayan. Even the first few numerals of the Ar-abic system, it is sometimes assumed, are de-rivatives of this notation. Something close to2 and 3 are what might be produced if onewrote = and == hurriedly without lifting thewriting instrument from the writing surfacebetween strokes. Some systems (e.g., IndianBrahmi, and some Chinese) do in fact repre-sent the first three numbers as _, =, and =.The Egyptians actually had two number sys-tems, the Hieroglyphic (described earlier)and the Hieratic (Aleksandrov, 1963), Thesymbols for two, three, and four in the Hiera-tic notation ( 'i' "'i' and ""1) also resemblewhat might be produced if one made two,three, or four tallies without removing one'spen from the paper.

Another common feature of the Egyptian,Greek, and Roman systems is the fact that

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NUMBER REPRESENTATION

the tally principle was used with every sym-bol class. That is, for any given symbol, thenumber of times the quantity represented bythat symbol was to be counted in the numberwas indicated by the number of times thesymbol that represented that quantity ap-peared in the number's representation. Thusthe appearance of .,':)~ in an Egyptiannumber indicated that the number containedthree hundreds.

Although the numbers were generally writ-ten with all the symbols of a certain typegrouped spatially and with the groups ar-ranged from higher order to lower order, thereading of a number as a quantity did not de-pend on this arrangement. That is, with anexception to be noted later, the positioning ofthe symbols with respect to one another car-ried no information. Thus although in Egyp-tian the number 2 4 1 3 would normally havebeen wri tten as

it would still be unambiguous, though some-what more difficult to read, if written as

The Greek and Roman systems had a fea-ture that the Egyptian system lacked: the useof special symbols to represent half-ten orfive groupings, an innovation that decreasedthe number of symbols required to representany number. In the Roman system the sym-bols for five groupings ( V for 5, L for 50, Dfor 500) did not relate to the other symbols inany consistent way, though in the Greekscheme they did. In the latter case groupingsof five were represented by combining thelower-order symbol with a standard symbolthat would be interpreted as "five of." Thusthe symbol for 50 was a combination of r(five) and [:,.(ten) yielding 1"', which could beread as five tens. Similarly, the symbol for500 was a combination of rand H yielding/" , which could be read as five hundreds.

April 1988-187

Thus in the Greek system five-grouping rep-resentations could be generated by rule,whereas in the Roman one they had to belearned by rote.

At some time the Roman system intro-duced another principle, which was to repre-sent diminution of the value of a higher-ordersymbol by preceding it with a symbol of im-mediately lower order. Thus 9 became 10 di-minished by one, or IX; 90 became 100 di-minished by 10, or XC; the diminutionprinciple was used also with five-groupingsymbols, so that 4 was represented as ,40was represented as XL, and so on. This inno-vation accomplished an economizing of thenumber of symbols that had to be written. Italso made posi tion functional, as a carrier ofinformation that is required to decode thesymbol unambiguously: for example, IX andXI represented quite different quantities.

THE BABYLONIAN SYSTEM

The Babylonian cuneiform number systemdiffered from the Egyptian, Greek, andRoman systems in several important ways.All numbers were represented by combina-tions of only two symbols: T and ( , both ofwhich were easy to inscribe in soft clay witha wedge-shaped stylus. Unlike the Egyptian,Greek, and Roman systems, the Babylonianone was a true place-notation system: thevalue represented by a symbol depended onthe location of that symbol in the collectionof symbols representing the number. The sys-tem is usually referred to as a sexigesimalsystem because a T and a < in one registerwere equivalent in value to 60 T and 60 ( ,respectively, in the next lower register. A1-for-60 substitution was not used, however.Instead, a two-stage substitution principlewas used: 1 (. was substituted for 10 T in thesame register, and 1 T was substituted for 6< in the next-lower register. Thus the Baby-lonian equivalents of the Arabic numbers 3,24, and 175 would be as follows:

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188-April1988 HUMAN FACTORS

The equivalent of Arabic 95,657 would berepresented as;

It takes three registers to represent thisnumber. The Arabic equivalent of thisnumber is derived from the contents of theindividual registers as follows:

<TTT\«TTI.J'TTT<m <TT '-mT(a) 26 x 602 = 93,600(b) 34 x 601 = 2,040(c) 17 x 60° = -----11

95,657This use of location to convey information

that was essential to the interpretation of asymbol required the development of a way torepresent the absence of a symbol. This couldbe done by simply leaving a space, but suchwould be risky given the natural variabilityin the spacing of symbols; there would alsobe the problem of distinguishing between aspace within a number and a space betweennumbers. (Imagine the problems that wouldarise in our use of the Arabic system if in-stead of zeros we used spaces.) The Babylo-nians addressed this problem in much thesame way as it is addressed in Arabic nota-tion, namely by developing a special symbolthat could serve as a place or "vacancy"marker. A way to represent the null set be-comes important when the same symbols canappear in different registers. Although theBabylonians made use of a rudimentary va-cancy marker in their late cuneiform writing,it was the Hindus who first used 0 in a sys-tematic way. Apparently the idea of a placeholder for a missing digit was known in Indiaas early as the sixth century, but for two orthree centuries a dot was used for this pur-

3

24

175

HI

«THT

n«<<<nTH

pose. Exactly when the 0 was introduced isnot known, but the first-known Indian manu-script in which 0 appears dates from the lat-ter part ofthe ninth century.

THE MAYANSYSTEM

An especially interesting way of represent-ing numbers was used by the Maya, who oc-cupied much of what is now Central Americaand maintained a remarkable culture thatlasted from about the fifth to the twelfth cen-tury A.D. (Lambert, Ownbey-McLaughlin,and McLaughlin, 1980; Thompson, 1954).The Mayan system was similar to the Baby-lonian cuneiform system in several respectsand differed in important ways from the sys-tems of Egypt, Greece, and Rome. Like theBabylonian system, it represented allnumbers with only two symbols and was atrue place-notation scheme. Unlike all theother systems we have considered, it used abase not of 10or 60 but of 20. (The third reg-ister in the Mayan system sometimes was in-terpreted as having a base of 18instead of 20.This peculiarity was introduced for calendricpurposes. A unit in this register, instead ofrepresenting 202, represented 18 x 20, or360, the number of days in a Mayan year. Inwhat follows this aspect of the system is ig-nored.)

The two symbols used by the Maya were adot to represent the quantity 1and a horizon-tal line or bar to represent 5. Asin the case ofthe Babylonian system, the Mayan used atwo-stage substitution principle: 5 dots werereplaced by a bar in the same register, and 4bars in a register were replaced by one dot inthe next higher one. By convention, the Mayawrote numbers vertically, so the equivalentof Arabic 1,549would appear as follows:

3 X 202 = 1,200_ 17 x 20· = 340=a 9 X 20° = __ 9

1,549

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NUMBER REPRESENTATION

Why did the Maya choose 5 and 20 to playthe special roles they did? Was it because wehave five fingers on each hand and a total oftwenty digits? Aleksandrov (1963) points outthat among certain peoples the names for fiveand twenty are respectively "hand" and"wholeman," which presumably translatesloosely as a man with aU his fingers and toes.Why did the Maya select a dot and a bar asthe symbols to represent one and five? Was itbecause they are so convenient to write? Didthe inspiration come from the shapes ofbeans and bean pods that might have beenused as tokens in counting and computing?The answers to these questions are notknown.

Both the Mayan and Babylonian systemsare in some ways more similar to our ownthan are any of the others considered. Theyhave several advantages over the Egyptian,Greek, and Roman systems, not the least ofwhich is the ease with which they permitcomputation to be done. (More on this pointlater.) In their total reliance on only twosymbols, they are parsimonious in the ex-treme. Moreover, the choice of symbols-especially that of the Maya-is ingenious;they are readily distinguishable from one an-other, certainly easy to learn, and easy toproduce, even in stone.

DESIGN CRITERIA FOR ANUMBER SYSTEM

Even a cursory inspection of ways in whichnumber concepts have been represented issufficient to convince one that some of theserepresentations would be easier to use, atleast for certain purposes, than others. Com-parison of one number system with anotherraises some intriguing questions and pro-vides the basis for some conjectures not onlyabout number systems but about representa-tional schemes in general. Both their similar-ities and differences ten us some things of in-terest.

April 1988-189

It is highly doubtful that any of the numbersystems that have been considered in thispaper were designed in the usual sense ofthat word; rather, each evolved over a con-siderable period of time. What has guidedtheir evolution? Clearly the characteristics ofthese systems are not completely arbitrary,and the similarities among systems are toogreat to be coincidental. The use of the prin-ciple of one-for-many substitution, for exam-ple, seems to be common to an number sys-tems.

Is the nearly universal acceptance of theArabic system a consequence of its superior-ity over other systems that have been used invarious parts of the world in the past? And ifso, what is it about this system that gives itthis edge?

One way to approach some of these ques-tions is to imagine being given the task of de-signing a system for representing numbers,or the slightly less ambitious task of specify-ing the characteristics or properties that thedesired system should have. This would be aformidable undertaking, and one couldhardly hope to approach it free of the biasesthat familiarity with an existing systemwould assure. Nevertheless, comparisonsamong systems such as those that have beenbriefly considered here suggest some ideas asto what certain of those characteristics mightbe: in particular, ease of interpretation, easeof writing, ease of learning, extensibility,compactness of notation, and ease of compu-tation.

Ease of Interpretation

One wants a number to be easy to read.The quantity it represents should be appar-ent at a glance and errors should not beoverly easy to make. One of the coding con-ventions of the Egyptian, Greek, and Romansystems seems consistent with this objective-namely, the fact that each to-grouping isrepresented by a different symbol. This

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means that one can tell the 10-group towhich any symbol belongs without payingattention to its location. In the Babylonian,Mayan, and Arabic systems the same-shapedsymbol can occur in different registers; to tellthe register of a particular symbol, one mustnote its position. The price the Egyptian,Greek, and Roman systems pay for havingthe to-group coded by symbol shape is theneed to have the same symbol repeated nu-merous times, which leads to a relativelylarge number of symbols per number repre-sented. Inasmuch as a particular symbol canoccur in a register as many as nine times inthose systems, one would sometimes have toresort to counting to determine how manyoccurrences of a given symbol there were.The Babylonian and Mayan systems alsohave the feature that the same symbol canoccur several times within the same register.

In contrast to all of these systems, the Ara-bic system gains economy of expression byvirtue of the fact that it never has more thana single symbol in a given register. From thisit follows that on the average, any givennumber is represented by fewer symbols inthe Arabic system than in the others we haveconsidered. Another advantage that followsfrom the fact that an Arabic number alwayshas a single symbol in every register is thatthe order of magnitude of the number is ap-parent from the number of digits that consti-tute it. None of the other systems we haveconsidered has this property. With the Arabicsystem, counting digits can be a useful wayto produce order-of-magnitude estimates ofthe results of multiplication and division.The number of digits in the product of twonumbers will be close to the sum of the digitsin the multiplier and the multiplicand; simi-larly, the number of digits in a quotient willbe close to the difference between thenumber in the dividend and in the divisor.

The fact that the Arabic system uses thesame symbols in different registers leads to

HUMAN FACTORS

difficulties only with numbers that have suf-ficiently many digits that one cannot readilytell, without counting, how many there are.To facilitate determination of the number ofdigits in a number, the convention typicallyused is that of grouping the digits in threesand separating the groupings with commas.The value of this convention is readily appar-ent when one compares the ease of reading236,172,345,924 with that of reading16279543617.

For very large numbers even the Arabicsystem becomes cumbersome. One innova-tion addressed to this problem is the use ofexponential notation, in which very large (orvery small) numbers are expressed as powersof 10. With this notation 10 trillion is writtenas 1013 instead of as 10,000,000,000,000. Theexponential notation sacrifices accuracy, butofen this is of little consequence because verylarge numbers typically are used in contextsfor which order of magnitude accuracy is allthat one can hope to obtain.

Ease of Writing

One would like a number system not tounreasonably burden an individual writingnumbers down. What constitutes an unrea-sonable burden may differ depending onwhether one is chiseling numbers in stone ormarking them with a piece of graphite onpaper, but even for a given method of writingone can distinguish among systems on thebasis of how much effort they require on thepart of the scribe.

Both the Babylonian and Mayan systemswere based on symbols that are easy to writeon practically any medium, but the writermust use many of those symbols in order torepresent a number of even modest size. TheEgyptian and Greek symbols are poorlysuited for inscription in stone or other hardmedia. Whether because of the number ofsymbols required to represent a number orthe relative complexity of the individual

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NUMBER REPRESENTATION

symbols, the time required to write numbersin any of these systems would be quite longrelative to the time required to write thesame numbers in Arabic notation.

Ease of Learning

One would prefer that one's number sys-tem be easy to learn. What determines ease oflearning is not entirely clear; one wouldguess, however, that-other things beingequal-the more direct the correspondencebetween any representational scheme andwhatever is being represented, the easier itshould be to learn that scheme. Pictorial oriconographic representations are undoubt-edly easier to learn to recognize than aremore abstract representations of the samethings. If the Piagetians are right, the abilityto handle abstract concepts and operationsrequires a higher level of cognitive develop-ment in general than does the ability to han-dle concrete concepts and operations.

The learning required by various systemsdiffers considerably. To master the Egyptian,Greek, and Roman systems one must learnthe meanings of many symbols. Learning theBabylonian and Mayan systems requires thatone learn only two symbols and a set of rulesfor combining these symbols. Arabic requiresthat one learn to symbols and the rules fortheir combination.

Where the Arabic system differs from mostif not all earlier systems is in its greater levelof abstractness. The correspondence betweenthe representation of a number and the quan-tity represented by that number is less directin the Arabic system than in its predecessors.The other systems that have been discussedin this paper preserve more of the aspects of atally than does the Arabic system; in the lat-ter any remnants of a tally are obscure in-deed. Because of its greater abstractness, onemight guess that this system poses a some-what greater learning challenge to the childthan do the others. This is a difficult hypoth-

April 1988-191

esis to test, given the ubiquity of the Arabicsystem today and the importance that is at-tached to learning it early. A fair test wouldrequire an effort to teach some children al-ternative systems with the same amount ofemphasis that is normally devoted to theteaching of the Arabic system while preclud-ing exposure of these children to the .Arabicsystem itself. The possibility of conductingsuch an experiment is unlikely, and the ad-visability of doing so even if it were possibleis highly doubtful.

Extensibility

Closely related to the issue of ease of learn-ing is that of extensibility. One wants to beable to write any number on the basis ofknowledge of how numbers in general arewritten. That is, one does not want to have tolearn something new whenever one has occa-sion to recognize or to write a number for thefirst time.

A system that used a unique symbol forevery number concept would obviously failon this criterion because one would have tomemorize as many symbols as there werenumbers that one wanted to be able to use.The Egyptian, Greek, and Roman systemsalso fail to a degree in this regard becausewhenever one has occasion to use a numberthat is an order of magnitude larger than thelargest number with which one is familiar,one must learn the symbol that is used to rep-resent a to-grouping at the next-higher level.Thus knowledge of how to represent quanti-ties from 1 through 999 in Egyptian, for ex-ample, does not suffice if one wants to repre-sent the quantity 1,000 because quantitiesless than 1,000 can be represented by combi-nations of the three symbols I , (\. and ~whereas those between 1,000 and 9,999 can-not. They require that one learn a new sym-bol: 1 . More generally, no finite amount ofknowledge about these number systems willsuffice to guarantee the ability to represent

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all numbers. No matter how many symbolsone knows how to write, it is easy to specify anumber that cannot be written with onlythose symbols.

By contrast, the advantages of the Baby-lonian, Mayan, and Arabic systems in this re-gard are apparent. If one knows the basicsymbols and understands the system's one-for-many substitution rules, one can writeany number without learning anything new.It might be easier for a child to learn to usethe Egyptian, Greek, or Roman system wellenough to serve the purposes of counting upto numbers of modest size than to learn theBabylonian, Mayan, or Arabic system to thesame degree of usefulness. But havinglearned the principles on which numbers inany of the latter three systems are con-structed, one has the necessary knowledge toextend the number sequence indefinitely. Ingeneral, any system that uses the position ofa symbol to indicate its register may be moredifficult to learn than one that uses percep-tually different symbols in different registers,but this disadvantage is likely to be morethan offset by the advantages a place-nota-tion system provides to the user once theprinciples of its construction have beenlearned.

Compactness of Notation

It should be possible to represent most ofthe numbers that one is likely to have to use-including relatively large numbers-com-pactly. The superiority of the Arabic systemover its predecessors in this regard has al-ready been noted. This feature is worth high-lighting, however, because it is one key to theuniqueness of the Arabic system and a basisfor some of its other desirable properties.

In general, early number systems were notefficient for representing very large quanti-ties. The equivalent of 99,999 in ancientEgyptian, for example, would look like this:

HUMAN FACTORS

(((((nm ':?'''J,,) nnnnn 1111I

((((un '~99 mnn II/I

The largest number for which the earlyRomans had a single symbol was 100,000,which they represented as eeeIJJ.?; thus towrite the number 2,000,000 they would haveto write the symbol for 100,000 twenty times.Eventually the symbol n was used to repre-sent 100,000 and the convention was adoptedof representing multiples of 100,000 by com-bining this symbol with the symbol for themultiple, so that, for example, 1,000,000 wasthen represented by a combination of thesymbols for 100,000 and 10: thus IXI (Men-ninger, 1969).

The ancient number systems were gener-ally not conducive to the representation oflarge quantities, and calculations involvingmany even moderately large numbers musthave been tedious indeed. To express thenumber 73,584 requires the writing of 27, 15,17, 16, and 19 symbols in Egyptian, Greek,Roman, Babylonian, and Mayan, respec-tively. (These counts are based on representa-tions of Greek and Roman systems in whichfive-groupings-and in the latter case, thediminution principle-are used.) The advan-tages that the Arabic system provides over itspredecessors derive in no small measurefrom the compactness of its notation.

This compactness derives in turn from thegreater degree of abstractness of the systemand in particular from its having shed thelast vestiges of a tally system. Whereas theEgyptian, Greek, Roman, Babylonian, andMayan systems all used the principle of one-for-many substitution, the Arabic system ap-plies it at a higher level of abstraction thando any of the others: not only does a unit in aregister represent 10 units in the next lower-order register, but the number of units in anygiven register is itself represented by a singlesymbol.

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NUMBER REPRESENTATION

Ease of Computation

Counting predates computing both devel-opmentally and historically. Children learnto count before they learn to compute, andthe evidence is compelling that for humanbeings as a species these abilities were ac-quired in that order. Empirical evidenceaside, it could hardly have been otherwise: itis certainly possible to be able to count with-out being able to compute, whereas-thoughin principle one might do certain types ofcomputing without being able to count-it isdifficult to imagine that the ability to com-pute could have been developed very far inthe absence of the ability to count. We mayassume that number systems were inventedto accommodate counting, and that onlyafter having been used for this purpose forsome time were they applied to computa-tional tasks. This being so, it is not surprisingthat the earliest systems were better adaptedto counting than to computing.

The difficulty of doing any very complexmathematics with the Egyptian, Greek, orRoman systems is apparent. What may beless obvious is the fact that these systems arewell suited to the fundamental operations ofaddition and subtraction. The sum of twoEgyptian numbers, for example, can be rep-resented literally as the union of the sets ofsymbols representing the addends. Thus

n~~'RRII11'1'1> nil· ~?,n I

IIm{J'RRHWAny "carrying" operations can then be per-formed by the appropriate one-for-many sub-stitutions in the sum. In this case, one Iwould be substituted for ten 9 and the sumrewritten as

m:>"nll l'?~nnnlIn?RAR1N l~nnn

Subtraction is equally easy. What we refer to,somewhat enigmatically, as "borrowing" is

April 1988-193

accomplished by making appropriate many-for-one substitutions in the minuend to ac-commodate those instances in which thenumber of symbols of a given type in the sub-trahend is greater than the number of sym-bols of the same type in the minuend. Afterthe necessary substitutions have been accom-plished, the subtraction can be done by tak-ing the difference independently for eachsymbol type.

Not only are addition and subtractionstraightforward in the Egyptian system (andrelatively so in the Greek and Roman aswell), but there is nothing mysterious aboutthe operations of "carrying" and "borrow-ing," as there sometimes seems to be whenstudents learn how to do arithmetic with theArabic system. Many of the errors that chil-dren make when subtracting one multidigitArabic number from another can be emu-lated by "buggy" algorithms that have beendesigned to apply operations in systematicways that do not conform to the rules of sub-traction (Brown and Burton, 1978; Brownand VanLehn, 1982). An example of a bugthat accounts for some errors children makeis to subtract always the smaller digit fromthe larger independently of whether thesmaller digit is in the minuend or in the sub-trahend. Another is to increment, rather thandecrement, a digit when borrowing.

An aspect of this research that is particu-larly interesting in the present context is thefact that extensive analyses of the systematicsubtraction errors children make reveal onlya subset of the bugs that it would be logicallypossible to invent. This finding raises thequestion of the extent to which the types ofbugs that are invented depend on the proper-ties of the representational system used. Oneplausible conjecture is that the more abstractthe system-the longer and more circuitousthe path from the characteristics of the sym-bols to the properties of the quantities theyrepresent-the greater the room for inven-

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194-Apri11988 HUMAN FACTORS

Then add only those terms associated withthe subset of doubling indices that sums ex-actly to the multiplier-that is, 2 + 8 + 16= 26.

times multiplied by successive doublings andthen by summing the appropriate subset ofthose doublings (Newman, 1956). To multi-ply 456 by 26 using this method, one wouldproceed as follows. First, calculate a succes-sion of doublings until the sum of the dou-bling indices is at least as large as the multi-plier, thus:

The easy way to find the subset of indicesthat sums to the multiplier is to begin withthe largest index and add successivelysmaller indices, eliminating any that wouldcause the sum to exceed the multiplier.

Although fractions were known to the an-cients, they presented special problems, andthe symbol systems did not facilitate theiruse. Various methods were evolved for deal-ing with them. With the exception of 213, forwhich they had a special symbol. the Egyp-tians expressed all fractions as a series offractions having 1 as a numerator. A sign in-dicating addition was not represented explic-itly; however, the sum of the fractions in theseries represented the fraction of interest.Thus, for example, % would be written '12, '14

tion of bugs. This is because the more arbi-trary the symbols, the easier it will be totreat them as symbols per se than as surro-gates for quantities, and the less obvious willbe the inappropriateness of specific buggyopera tions.

The concepts of carrying and borrowing-which really are misleading-are not neededand do not arise with some of the ancientsystems, such as the Egyptian, Greek, andRoman. It is apparent from the symbologythat the operations required are a one-for-many substitution in one case and a many-for-one substitution in the other. The terms"carrying" and "borrowing" were presum-ably introduced into the vocabulary of arith-metic to facilitate a child's learning how toadd and subtract multidigit Arabic numbers.One can argue, however, that the termsthemselves, while possibly facilitating therote" learning of a procedure that works, mayhelp to obscure the principles from which theoperations derive. The term "carrying" pro-vides no hint of the fact that what is involvedis the substitution of one unit of one type forseveral units of another, and the term "bor-rowing" is worse than uninformative becauseit suggests a transaction that is incompleteuntil whatever is borrowed is paid back.

Multiplication and division are not nearlyas easy to accomplish with the Egyptian,Greek, or Roman system as with the Arabic.Conceptually, multiplication is not difficultin these systems. To multiply It?"nll byl??nnn, for example, one need only writedown U9?51nl/ l~nnn times and thenmake the necessary one-for-many symbolsubstitutions. Mechanically, however, theprocess is prohibitively tedious even onpaper, let alone on any less convenient me-dium. When multiplication and divisionwere done, they presumably were accom-plished by means of successive additions andsubtractions. There were some shortcuts,however; the Egyptians, for example, some-

Doublings456912

182436487296

DoublingIndices

1248

16

456912

t8Z436487296

11856

Cumulative Sum ofDoubling Indices

137

1531 (>26)

--t2

-48

1626

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NUMBER REPRESENTATION

(except, of course, with Egyptian symbolsrather than Arabic). The first part of theRhind Papyrus (Newman, 1956) gives a tableshowing how to represent 2 divided by oddnumbers from 3 to 101. The Greeks also ex-pressed all fractions as sums of those with 1as the numerator, and the Romans expressedall fractions as twelfths (Jourdain, 1956).None of these methods comes close to pro-viding the versatility of the notation we usetoday.

The thought of trying to do algebra orhigher mathematics with systems such as theEgyptian, Greek, or Roman is somewhatdaunting. Indeed, it seems safe to assumethat mathematics, and those sciences heavilydependent on mathematics, could not haveprogressed if systems with greater versatilityand power had not evolved. Progress re-quired systems that not only provided moreconvenient and compact ways of represent-ing the number concepts that the ancientsunderstood but that would also facilitate theinvention of new concepts-such as those ofnegative, imaginary, and complex numbers-that have proved so important in bothpure and applied mathematics.

In the Babylonian and Mayan systems, ad-dition and subtraction also were straightfor-ward. As in the Egyptian, Greek, and Romansystems, the sum of two numbers could beformed by first taking the union of the sym-bols representing the numbers and then per-forming whatever one-for-many substitu-tions were required to get the sum instandard form. Figure I, for example, showshow the two decimal numbers 2773 and 2256could be added with the Mayan notation. InStep 1 the union of the addends is taken reg-ister by register: thus ~ + =i= in the l's regis-ters of the addends becomes § in the 1's reg-ister of the sum. After each of the unions hasbeen taken, the one-for-many substitutionsare made so there are no more than threebars or four dots in any register. Four of the

April 1988-195

bars in the 1's register, in our example,would be replaced by a single dot in the 20'sregister. In making the one-for-many substi-tutions, it is convenient to begin by substi-tuting a bar for any set of five dots within reg-isters, and then-working from the lower- tothe higher-order register-substituting forany set of four bars within a register a dot inthe next-higher register. Figure 1 shows theone-for-many substitution step broken downinto a sequence of substeps. Sometimes itmay be necessary to iterate one or more ofthese substeps, as when, for example, thesubstitution of a dot in the nth register forfour bars in the (n - 1)thregister brings thenumber of dots in the nth register to five,which would require another within-registersubstitution of a bar for five dots.

Alternatively, one can perform the opera-tion on a register-by-register basis. In thiscase one makes the necessary one-for-manysubstitutions immediately after (or in theprocess of) adding the contents of two regis-ters. Thus the sum of == + = would be writ-ten as ~ and a . would be "added" to thenext higher register. The fact that the latterstep, which is equivalent to the carry opera-tion with which all users of Arabic notationare familiar, is a one-for-many substitution isapparent.

Subtraction with the Mayan notation is, ofcourse, the inverse of addition and is nearlyas straightforward. The "borrowing" opera-tion, which must be performed whenever thequantity represented by a register in the sub-trahend is greater than that represented bythe corresponding register in the minuend, issimply a many-for-one substitution wherebyeither a bar is replaced by five dots in thesame register or a dot is replaced by four barsin the next lower-order register.

Apparently there is no compelling evidencethat the Maya knew how to multiply and di-vide; however, the representational systemlends itself readily to these operations (Lam-

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196-April1988 HUMAN FACTORS

Step 1 Step2(unionof (one-for-manyaddends) substitution)

400's~ • ••-••• •••••

+ • • -a 20's= - •

••• • ••••l's= - --- ••••-

I••••• ~---

•400's===b 20'5-----

••••l's ====

•

••••••••

••

•

• •••Figure 1. Addition of Mayan numbers (the Arabic equivalent is 2773 + 2256 = 5029). (a) Shows the twosteps of combining addends and then making one-for-many substitutions. (b) Shows the one-for-many substi-tution step in detail.

bert, Ownbey-McLaughlin, and McLaughlin,1980). In this respect it is quite unlike thesystems used by the Egyptians, Greeks, andRomans. Multiplication requires three sim-ple product rules and rules for determiningthe register of a product from the registers ofits factors. The product rules are (1) dot xdot = dot; (2) dot x bar = bar; and (3)bar xbar = bar in one register and dot in the next-higher register. The rules for determining theregister of a product from the registers of itsfactors can be combined with these threeproduct rules to yield the following set,where the symbols in parentheses representthe registers of the factors and product:

(1) • (m) x • (n) = • (m + n - 1)(2) • (m) x - (n) = - (m + n - 1)(3) - (m) x - (n) = • (m + n) + - (m + n - 1)

Lambert, Ownbey-McLaughlin, andMcLaughlin (1980), from whom this notationwas adapted, describe the process this way:

In calculating the product, each elementof the multiplicand must be multiplied byeach element of the multiplier, register byregister. For each dot-times-dot operation,add together the number of the registers oc-cupied by each of the dots, and place a dot inthe product register that is one beneath thistotal. For each bar-times-dot operation, dothe same thing, but place a bar in the prod-uct stack. For each bar-times-bar operation,add the number of the register of each of thebars, and place a dot in the register corre-sponding to this sum and a bar in the regis-ter just beneath it. When 5 dots accumulatein a given register, they are replaced by abar in the same register; when 4 bars accu-mulate in a register, they are replaced by adot in the next higher register. (p. 253)

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NUMBER REPRESENTATION

The procedure is quite simple and a fewpractice multiplications are likely to beenough to make one comfortable with it. Di-vision, which, of course, is the inverse of mul-tiplication, is also relatively straightforwardand easily learned.

Most of what has been said here about theMayan system could also be said (with ap-propriate substitutions) about the Babylo-nian one. It too makes addition and subtrac-tion trivially easy and multiplication anddivision quite manageable. Moreover, theBabylonians are known to have developedconsiderable computational skills (Men-ninger, 1969).

The greater suitability of the Babylonianand Mayan systems to computation beyondaddition and subtraction results, at least inpart, from the fact that they are place-nota-tion systems. One of the consequences of thisway of representing numbers is that the rulesof multiplication, division, and other opera-tions can be relatively succinct and general;that is, a few of them suffice. The possibilityof simple rules follows from the fact that thesame symbols are used in every register. Todo multiplication, for example, one need onlyknow the products of all possible combina-tions of the individual symbols and how theplace(s) of a product of two of these symbolsdepends on the places of those symbols in themultiplier and the multiplicand.

How does the Arabic system compare, withrespect to computational convenience, withthe others we have considered? Except per-haps for simple addition and subtraction by abeginner, it is clearly superior to the Egyp-tian, Greek, and Roman systems, for muchthe same reasons that the Babylonian andMayan systems are. It too is a place-notationsystem and shares the advantages of otherplace-notation systems in this regard.

Acomparison of the Arabic system with theBabylonian and Mayan systems with respectto computational convenience, however, re-

April 1988-197

veals some interesting trade-offs. It may beeasier to learn the rules of multiplication anddivision in the Babylonian and Mayan sys-tems than in the Arabic: there are fewer sym-bols to contend with, and the rationale forthe rules determining the registers of prod-ucts and quotients is straightforward. Bycontrast, the algorithms for multiplicationand (especially) division that many of uslearned in our early school years equipped usto find products and quotients but unhappilydid not, in many cases, leave us with an un-derstanding of the reasons for the steps in-volved. The reader who doubts this state-ment is encouraged to work a long divisionproblem before an inquisitive observer whohas been primed to demand a clear and com-plete explanation for every step in the pro-cess.

When it comes to the doing of computa-tional problems by someone who has mas-tered the system, there can be little doubtthat the Arabic system has significant advan-tages over the others considered here, themore so the greater the complexity of thecomputational problem involved. Whatmakes the Arabic system superior for com-puting purposes is probably a combination offactors. That it is a place-notation system iscertainly one important factor, although, aswe have noted, this alone does not distin-guish it from some other systems. Anothermajor advantage for computation is the rela-tively economical way in which the Arabicsystem represents numbers, as was noted inthe preceding section.

CONCLUSIONS

Aristotle attached considerable signifi-cance to the fact that human beings cancount: it is this ability, he claimed, that dem-onstrates our rationality. However that maybe, it is difficult to imagine what life wouldbe like if we had never learned to count orcompute. It is perhaps unthinkable that

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198-April1988

human beings could have been around aslong as they have without having developedthese abilities. It is not at all unthinkable,however, that a system for representingnumbers might have evolved to be somethingquite different from the one we use. Howsuch a system might have been better is a lit-tle hard for us to see; perhaps future genera-tions will discover that.

If one accepts the idea that the Arabic sys-tem is in general the best way of representingnumbers that has yet been developed, oneneed not believe that it is clearly superiorwith respect to all the design goals that onemight establish for an ideal system. It maybe, in fact, that simultaneous realization ofall such goals is not possible. Perhaps trade-offs are necessary. One trade-off that the Ara-bic system seems to represent is that betweenease of learning by the neophyte and ease ofuse in computing by the expert.

All of the earlier number systems consid-ered in this paper were in certain ways moreobviously analogous to the things they repre-sent than is the Arabic system. There was, forexample, a more direct correspondence be-tween the number of symbols in the numberand the number of objects in the set repre-sented by the number. The addition of twonumbers was more directly analogous to theaddition of the sets represented by thosenumbers. The correspondence was especiallydirect when the numbers were representedby physical tokens such as pebbles or sticks,as may sometimes have been the case. Asimi-lar point may be made with respect to sub-traction. Moreover, the analogues of carryingand borrowing, for all their mysteriousnessto the modern-day neophyte mathematician,have straightforward analogues in these sys-tems. In short, a price of the increased abostractness of the Arabic system is an obscur-ing of some of the key principles on whichnumbers are based. To an ancient Egyptian,

HUMAN FACTORS

the fact that 3 + 4 = 7 was apparent fromthe relationship between the Egyptiannumber representing 7 and those represent-ing 3 and 4. There is no hint of such a rela-tionship, however, when this equation is ex-pressed in Arabic notation.

The price has been worth the paying, how-ever. The convenience of the Arabic systemfor computing has played an indispensablerole in the development of higher mathemat-ics and of science and technology, which areso heavily dependent on mathematics. More-over, it has made it possible for the averageperson to attain a far greater degree of math-ematical competence than would have beenfeasible with the more ancient systems. AsBrainerd (1973) has pointed out, in mostWestern nations today we expect students, bythe time they reach their early teens, to bemuch more competent with numbers thanwas an educated adult Greek or Roman of2,000 years ago. This fact arises in no smallmeasure from the elegance and power of thescheme that we now use to representnumbers.

This scheme was a long time evolving.Moreover, what we know of the evolution ob-scures the distinction between cause and ef-fect. We may assume, however, that it wasnot the case that from the beginning peoplewanted to do higher mathematics and there-fore sought a representational scheme tomake that possible. Much of what we think ofas higher mathematics could not have beenconceived had there not already existed rep-resentational schemes that facilitated itsconception. The development of representa-tional schemes and of new mathematicalconcepts has been mutually reinforcing: aneffective way of representing existing con-cepts has been instrumental in extendingthose concepts, and those extensions have ledto the need for and development of new rep-resentational schemes.

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NUMBER REPRESENTATION

ACKNOWLEDGMENTS

The writing of this paper was supported by the NationalInstitute of Education under Contract No. 400-80-0031. Iam grateful to John Swets and Wallace Feurzeig for help-ful comments on an earlier draft.

REFERENCES

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Brainerd, C. J. (1973). The origins of number concepts.Scientific American, 228(3), 101-109.

Brown, J. S., and Burton, R. R. (1978). Diagnostic modelfor procedural bugs in basic mathematical skills. Cog-nitive Science, 2, 155-192.

Brown, J. S., and VanLehn, K. (1982). Towards a genera-tive theory of "bugs." In T. P. Carpenter and J. M.Moser (Eds.), Addition and subtraction: A cognitive per-spective (pp. 117-135). Hillsdale, NJ: Erlbaum.

Conant, L. L. (1956). Counting. In J. R. Newman (Ed.), The

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world of mathematics (Vol. 1, pp. 432-441). New York:Simon and Schuster.

Jourdain, P. E. B. (1956). The nature of mathematics. InJ. R. Newman (Ed.), The world of mathematics (Vol. I,pp. 4-72). New York: Simon and Schuster.

Lambert, J. B., Ownbey-McLaughlin, B., and McLaughlin,C. D. (1980). Maya arithmetic. American Scientist,68(3),249-255.

McCulloch, W. S. (1961). What is a number that a manmay know it, and a man that he may know a number?The Ninth Alfred Korzybski Memorial Lecture, Gen-eral Semantics Bulletin, Nos. 26 and 27. Lakeville, CT:Institute of General Semantics, 1961; reprinted inMcCulloch, W. S. (1965). Embodiments of mind (pp.1-17). Cambridge, MA: MIT Press.

Menninger, K. (1969). Number words and number symbols:A cultural history of numbers. Cambridge, MA: MITPress.

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