The Representation of Numbers - Stanford University · 2009-09-18 · 1·1 D Systems Although 1 D systems are effi-cient for small numbers, they do not work well for large numbers

To appear in Cognition

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The Representation of Numbers

Jiajie ZhangDepartment of PsychologyThe Ohio State University

Donald A. NormanApple Computer, Inc.

ABSTRACTThis article explores the representationalstructures of numeration systems and thecognitive factors of the representational ef-fect in numerical tasks, focusing on externalrepresentations and their interactions withinternal representations. Numeration sys-tems are analyzed at four levels: dimension-ally, dimensional representations, bases, andsymbol representations. The representa-tional properties at these levels affect theprocesses of numerical tasks in differentways and are responsible for different as-pects of the representational effect. This hi-erarchical structure is also a cognitive tax-onomy that can classify nearly all numera-tion systems that have been invented acrossthe world. Multiplication is selected as anexample to demonstrate that complex nu-merical tasks require the interwovenprocessing of information distributed acrossinternal and external representations.Finally, a model of distributed numericalcognition is proposed and an answer to thequestion of why Arabic numerals are sospecial is provided.

We all know that Arabic1 numerals aremore efficient than Roman and manyother types of numerals for calculation(e.g., 73 ´ 27 is easier than LXXIII ´XXVII), even though they all representthe same entitiesÑnumbers (see Figure

1 Although the Arabic numerals were originallyinvented in India and are called Hindu-Arabicnumerals by some historians, we adopt theconventional name for simplicity.

1 for several examples of numerationsystems). This representational effect, thatdifferent representations of a commonabstract structure can cause dramaticallydifferent cognitive behaviors, has hadprofound significance in the develop-ment of arithmetic and algebra in par-ticular and mathematics in general. TheArabic numeration system, remarkableas is its simplicity, has been regarded asone of the greatest inventions of thehuman mind.

The representational effect ofnumeration systems is a cognitive phe-nomenon. However, early studies ofnumeration systems focused on theirhistorical, cultural, mathematical, andphilosophical aspects (e.g., Brooks, 1876;Cajori, 1928; Dantzig, 1939; Flegg, 1983;Ifrah, 1987; Menninger, 1969). Recently,the representational efficiencies of dif-ferent numeration systems have beencompared and analyzed in terms of theircognitive properties (e.g., Nickerson,1988; Norman, 1993), and the nature ofnumber representations has also beenexamined from a semiotic perspective(Becker & Varelas, 1993). In this paper,we attempt a systematic analysis of therepresentational structures of numera-tion systems and the cognitive factorsresponsible for the representational ef-fect in numerical tasks.

Unlike the representational effectof numeration systems, there have beena large number of psychological studiesof numerical cognition over the last twodecades (for a recent collection of re-

Zhang & Norman Number Representations

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views, see Dehaene, 1993). However,most of these studies have entirely fo-cused on internal representations: howpeople perform numerical tasks in theirheads, how numbers and arithmeticfacts are represented in memory, andwhat mental processes and proceduresare involved in the comprehension, cal-culation, and production of numbers. Inthis present study, we focus on externalrepresentations and their interactionswith internal representations: hownumbers are represented in the externalenvironment, what structures in exter-nal representations are perceived andprocessed, and how internal and exter-nal representations are integrated innumerical tasks. Although our focus ison external representations, we are notdenying the important roles of internalrepresentations. What we attempt toshow is that external representationshave much more important roles thanpreviously acknowledged.

We start with a representationalanalysis of the hierarchical structures ofnumeration systems. From this analysis,we develop a cognitive taxonomy ofnumeration systems. This taxonomycan not only classify most numerationsystems but also can serve as a theoreti-cal framework for systematic studies ofthe representations of numbers and theprocesses in numerical tasks. In the sec-ond part, we analyze how the dimen-sional representations of numerationsystems are distributed across internaland external representations. In thethird part, we use multiplication as anexample to examine the interwovenprocessing of internal and external in-formation in complex numerical tasks.In the last part, we outline a model ofdistributed numerical cognition andprovide an answer to the question ofwhy Arabic numerals are so special.

THE REPRESENTATIONALSTRUCTURES OF NUMERATION

SYSTEMS

In this part, we first analyze the dimen-sionality of numeration systems. Next,we analyze the representational proper-ties of numeration systems at four dif-ferent levels in terms of a hierarchicalstructure. Then, based on the hierarchi-cal structure, we propose . a cognitivetaxonomy of numeration systems.

The Dimensionality of NumerationSystems

1 D SystemsOne of the simplest ways to rep-

resent numbers is to use stones: onestone for one, two stones for two, and soon. This Stone-Counting system onlyhas a single dimension: the quantity ofstones. The Body-Counting systemused by Torres Islanders is another onedimensional system, in which the singledimension is represented by the posi-tions of different body parts (fingers,wrist, elbow, shoulder, toes, ankles,knees, and hips). The first numerationsystems invented in nearly all nations ofantiquity were one dimensional systemsrepresented by simple physical objects,such as stones, pebbles, sticks, tallies,etc. One dimensional systems are de-noted as 1 D in this article.

Many 1 D systems are very effi-cient for small numbers. Moreover, theymake a number of numerical compari-son tasks simpler than with other sys-tems (e.g., Arabic), because the size ofthe representation of the number is pro-portional to the numerical value. Thismeans that the system has analogicalproperties that make comparisons sim-ple. In addition, the operations of addi-tion and subtraction are easy, requiring


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no knowledge of arithmetic propertiesof tables, but rather, simply the addition

or removal of notational marks (seeNorman, 1993, chapter 3).

Arabic Egyptian Babylonian Greek Roman Chinese Aztec Cretan Mayan1 | a I ¥

2 || b II ¥¥

3 ||| g III ¥¥¥

4 |||| d IIII ¥¥¥¥

5 ||||| e V

6 |||||| VI ¥¥¥¥¥¥¥¥

7 ||||||| z VII ¥¥

8 |||||||| h VIII ¥¥¥

9 ||||||||| q VIIII ¥¥¥¥

10 Ç i X20 ÇÇ k XX ¥

30 ÇÇÇ l XXX ¥

40 ÇÇÇÇ m XXXX ¥¥

50 ÇÇÇÇÇ n L ¥¥

60 ÇÇÇÇÇ

Ç

x LX ¥¥¥

70 ÇÇÇÇÇ

ÇÇ

o LXX ¥¥¥

80 ÇÇÇÇÇ

ÇÇÇ

p LXXX ¥¥¥¥

90 ÇÇÇÇÇ

ÇÇÇÇ

LXXXX ¥¥¥¥

100 r C

200 s CC

Figure 1. Examples of numeration systems. In the Roman system, the subtraction forms for four (IV, XL,etc.) and nine (IX, XC, etc.) were later inventions. We only consider the original additive forms , that is,IIII, VIIII, XXXX, LXXXX, etc.


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1´1 D SystemsAlthough 1 D systems are effi-

cient for small numbers, they do notwork well for large numbers. For largenumbers, there must be more than onedimension, for example, one base dimen-sion and one power dimension. Thepower dimension decomposes a numberinto hierarchical groups on a base. Thebase can be raised to various powers onthe power dimension. We denote twodimensional systems as 1´ 1 D(base´power). A number in a 1´1 D sys-tem is represented as a polynomial:Saixi. Figure 2 shows the representa-tional structures of six 1´1 D systems.

The two dimensions of 1´1D sys-tems can be represented by differentphysical properties, which are usuallyshape, quantity, and position. For ex-ample, the Arabic system is a two di-mensional system (Figure 2) with a basedimension represented by the shapes ofthe ten digits (0, 1, 2, ..., 9) and a powerdimension represented by positions ofthe digits with a base ten. Thus, themiddle 4 in 447 has a value 4 on the basedimension and a position 1 (countingfrom the rightmost digit, starting fromzero) on the power dimension. The ac-tual value it represents is forty (theproduct of its values on the base andpower dimensions, i.e., 4´101). As an-other example, the base dimension ofthe Egyptian system (see Figure 2) isrepresented by quantities (the quantitiesof |'s, Ç 's, 's, etc.), and the power di-mension is represented by shapes(|=100 , Ç=101 , =102 , etc.). Thus,

ÇÇÇÇ||||||| also represents fourhundred and forty-seven. The details ofthe representations of the base andpower dimensions in several other 1´1Dsystems are shown in Figure 2.

The two dimensions of some 1´1D systems are externally separable,whereas those of some others are onlyinternally separable. In Figure 2, thebase and power dimensions of all butthe Greek system are externally separa-ble. For example, we can separate theshape and position of each digit in anArabic numeral via perceptual inspec-tion on the physical properties of thewritten symbols. In this case, shape andposition are two separate physical di-mensions. However, the two dimen-sions of the Greek system (see Figure 2)are not externally separable. Its baseand power dimensions are representedby a single physical dimensionÑshape(a one-to-two mapping): they can onlybe separated as two dimensions in themind by retrieving relevant informationfrom memory. For example, the valueson the base and power dimensions of t(300) are 3 and 102, which can not beseparated via perceptual inspection onthe physical property of the symbol "t."The separation of the single physicaldimension (shape) into a base and apower dimension in the mind is re-quired by the Greek multiplication algo-rithm, which needs to process the valueson the base and power dimensions sepa-rately (see Flegg, 1983; Heath, 1921).

(1´1)´1 D SystemsSome numeration systems have

three dimensions: one main power di-mension, one sub-base dimension, and onesub-power dimension. The sub-base andsub-power dimensions together formthe main base dimension. We denote thistype of three dimensional systems as(1´1)´1 D [(sub-base´sub-power)´main-power]. A numeral in a (1´1)´1D sys-tem is expressed as SS(bijyj)xi in its ab-stract form. The representational struc-


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Systems Example (447) Base Base Dimension Power DimensionAbstract S aixi x ai xi

Arabic 447 10 ai = shape xi = position4´102+4´101+7´100 0, 1, 2, ..., 9 ... 102 101 100

Egyptian ÇÇÇÇ||||||| 10 ai = quantity xi = shape4´102+4´101+7´100 The numbers of |'s,

Ç's, 's, etc. | Ç ... 100 101 102 ...

Cretan 10 ai = quantity xi = shape4´102+4´101+7´100 The numbers of 's,

's, 's, etc. ...100 101 102 ...

Greek umz 10 ai = shape xi = shape4´102+4´101+7´100 a b g ... q

1 2 3 ... 9 i k l ... 1´101 2´101 3´101...9´101

r s t ... 1´102 2´102 3´102...9´102

Aztec 20 ai = quantity xi = shape1´202+2´201+7´200 The numbers of 's,

's, 's, etc. ...

200 201 202 ...Chinese 10 ai = shape xi = shape

4´102+4´101+7´100 ... 1 2 3 ... 9

...101 102 103 ...

Figure 2. The representational structures of 1´1 D numeration systems.

Systems Example (447) MainBase

Sub-base

Sub-baseDimension

Sub-powerDimension

Main Power Dimension

Abstract SS(bijyj)xi x y bij yj xi

Babylonian 60 10 bij = quantity yj = shape xi = position(0´101+7´100)601

+(2´101+7´100)600The numbersof Ôs and

's

= 100, = 101 ... 602 601 600

Mayan ¥¥¥ ¥¥ 20 5 bij = quantity yj = shape xi = position

(0´51+1´50)202

+(0´51+2´50)201

+(1´51+2´50)200

The numbersof 's and

's.

= 50, = 51 ... 202 201 200

Roman CCCCXXXXVII 10 5 bij = quantity yj = shape xi = shape(0´51+4´50)102

+(0´51+4´50)101

+(1´51+2´50)100

The numbersof I's, V's, X's,L's, etc.

I =100´50 V = 100´51

X = 101´50 L = 101´51

...

I = 50´100 V = 50´101

X = 50´101 V = 51´101

...

Figure 3. The representational structures of (1´1)´1 D numeration systems. In the Mayan syste. m, thesecond power of the main power dimension is not represented as 202, but as 20´18. This might be due tosome astronomical reasons, because 20´18 = 360, which is close to the number of days in a year.


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tures of three (1´1)´1 D systems areshown in Figure 3. To illustrate thissystem, consider a Babylonian numeral(see Figure 3 for details):

= (0´101+7´100)´601+(2´101+7´100)´600

= 7´601+27´600 = 420 + 27 = 447

The main power dimension of theBabylonian system is represented by po-sitions with a base 60. In this example,

(the right component) is on position0 (600), and (the left component) is onposition 1 (601). The value of (the leftcomponent) on the main base dimensionis 7 (= 0´101+7´100). The actual value itrepresents is the product of its values onthe main base and main power dimen-sions: (0´101+7´100)´601 = 420. Themain base dimension is composed of asub-base dimension represented byquantity (the quantities of Ôs and 's)and a sub-power dimension representedby shape ( = 100 and = 101) with abase 10. For example, can be decom-posed as 2´101+7´100.

Similar to 1´1 D systems, thethree dimensions in a (1´1)´1 D systemcan be externally separable or only in-ternally separable. For example, thesub-base, sub-power, and the mainpower dimensions in the Babylonianand the Mayan systems are externallyseparable with each other. In theRoman system, although the sub-basedimension is externally separable fromthe sub-power and the main power di-mensions, the sub-power and the mainpower dimensions, which are repre-sented by a single physical dimension(shape), are only internally separable.For example, L (50) has a value 51 on thesub-power dimension and a value 101

on the main power dimension, whichcan only be separated in the mind.

Hierarchical Structure and CognitiveTaxonomy

Based on the analysis of the dimension-ality of numeration systems, we cananalyze number representations at fourlevels: dimensionality, dimensionalrepresentation, bases, and symbolrepresentation.

Each level has an abstract structurethat can be implemented in differentways. The different representations ateach level are isomorphic to each otherin the sense that they all have the sameabstract structure at that particular level(Figure 4).

At the level of dimensionality, dif-ferent numeration systems can have dif-ferent dimensionalities: 1D, 1´1D,(1´1)´1D, and others. However, theyare all isomorphic to each other at thislevel in the sense that they all representthe same entitiesÑnumbers. This levelmainly affects the efficiency of informa-tion encoding. 1 D systems are linear,while 1´1 D and (1´1)´1 D systems arepolynomial. Polynomial systems en-code information more efficiently thanlinear systems: the number of symbolsneeded to encode a number in a poly-nomial system is proportional to thelogarithm of the number of symbolsneeded to encode the same number in alinear system.

At the level of dimensional represen-tations, isomorphic numeration systemshave the same dimensionality but dif-ferent dimensional representations. Thephysical properties used to represent thedimensions of numeration systems areusually quantity (Q), position (P), andshape (S). For example, the base andpower dimensions of 1´1D systems canbe represented by shape and position(S´P, Arabic system), shape and shape


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(S´S, Chinese system), quantity andshape (Q´S, Egyptian system), etc. Thislevel is crucial for the representationaleffect of numeration systems. The sec-ond part of this paper is devoted for theanalysis of the representational proper-ties at this level.

At the level of bases, isomorphicnumeration systems have the same di-mensionality, same dimensional repre-sentations, but different bases. For ex-ample, both the Egyptian and the Aztec

systems are 1´1D systems, and the baseand power dimensions of both systemsare represented by quantity and shape.However, the base of the Egyptian sys-tem is ten while that of the Aztec systemis twenty (see Figure 2). This level isimportant for tasks involving additionand manipulation tables: the larger abase is, the larger the addition and mul-tiplication tables are and the harder theycan be memorized and retrieved.

S´P

Base 10

Arabic ChineseBody Stone Greek Egyptian Cretan Aztec Mayan Babylonian Roman

Base 10 Base 10 Base 10 Base 20 Base 20 Base 60 Base 10

[S´S] S´S Q´S

ABSTRACT NUMBERS

1 D

Q

1´1 D

P

(1´1)´1D

(Q´S)´P (Q´S)´S

Dimensionality

Dimensional Representation

Base

Symbol Representation

Figure 4. The hierarchical structure of number representations. This is a also a cognitive taxonomy ofnumeration systems. At the level of dimensionality, different systems have different dimensionalities. Atthe level of dimensional representations, the dimensions of different systems are represented by differentphysical properties. P = Position, Q = Quantity, S = Shape. The two dimensions of the Greek system([S´S]) are represented in the mind and only separable in the mind. At the level of bases, differentsystems may have different bases. At the level of symbol representations, different systems use differentsymbols.


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At the level of symbol representa-tions, isomorphic numeration systemshave the same abstract structures at theprevious three levels. However, differ-ent symbols are used. For example,both the Egyptian and the Cretan sys-tems are 1´1D systems, the two dimen-sions of both systems are represented byquantity and shape, and both systemshave the base ten. However, in theEgyptian system, the symbols for 100,101, and 102 are |, Ç, and , whereas inthe Cretan system, the correspondingsymbols are , , and . This levelmainly affects the reading and writingof individual symbols.

The hierarchical structure ofnumber representations in Figure 4 is infact a cognitive taxonomy of numerationsystems. For example, the Egyptian andCretan systems are in the same group atthe level of symbol representations; theMayan and Babylonian systems are inthe same group at the level of bases; theArabic, Greek, Chinese, Egyptian,Cretan, and Aztec systems are in thesame group at the level of dimensionalrepresentations; and all the systems inFigure 4 are in the same group at thelevel of dimensionality. Under this tax-onomy, the lower the level at which twosystems are in the same group, the moresimilar they are. For example, theEgyptian and the Cretan systems aremore similar to each other than theArabic and the Babylonian systems, be-cause the former two are in the samegroup at the level of symbol representa-tions whereas the latter two at the levelof dimensionality.

Although this cognitive taxon-omy was derived from the eleven sys-tems in Figures 2 and 3, it can classifynearly all numeration systems that havebeen invented across the world. Let us

consider a few more systems (see Ifrah,1987, for detailed descriptions of thesesystems). At the level of symbol repre-sentations, the Hebrew alphabetic sys-tem is in the same group as the Greeksystem, and the Greek acrophonic,Dalmatian, and Etruscan systems are inthe same group as the Roman system.At the level of bases, the Chinese scien-tific system is in the same group as theMayan system.

In addition to numeration sys-tems of written numerals, this taxonomycan also classify numeration systems ofobject numerals. The following are afew examples (see Ifrah, 1987), in whichP = Position, Q = Quantity, and S =Shape. The Peruvian knotted string sys-tem is a P´ Q (base 10) system; theChimpu (knotted strings used by theIndians of Peru and Bolivia) is a Q´Q(base 10) system; the knotted string sys-tem used by the German millers is a S´S(base 10) system; the Roman countingboard, the Chinese abacus, and theJapanese Soroban are (Q´P)´P (mainbase 10 and sub-base 5) systems, and theRussian abacus is a Q´P (base 10) sys-tem.

The hierarchical structures of thenumeration systems discussed above arebased on the representations of smallnumbers (usually less than 1000). Forlarge numbers, the representationalstructures of some systems may havedifferent forms. The change is mainlydue to the representations of power di-mensions. The representational struc-tures of numeration systems with powerdimensions represented by positionsusually do not change, because positionscan be extended indefinitely. However,for those systems with power dimen-sions represented by shapes, their repre-sentational structures sometimes have tobe changed to represent large numbers,


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because practically, shapes can not beextended indefinitely. For example, theRoman system is a (Q´S)´S system fornumbers up to one thousand. For num-bers above one thousand, one version ofthe Roman system becomes a((Q´S)´S)´S system. A new power di-mension represented by shape is added:a horizontal bar ¾ is used to represent103, and an open frame for 105. Forexample, CCXXXVIIIrepresents 238, 000,and DCCCXXVII represents 82, 700, 000.

THE DISTRIBUTEDREPRESENTATION OF NUMBERS

The Theory of DistributedRepresentations

In complex numerical tasks, as well as inmany other cognitive tasks, people needto process the information perceivedfrom external representations and theinformation retrieved from internal rep-resentations in an interwoven, integra-tive, and dynamic manner. Externalrepresentations are the representationsin the environment, as physical symbolsor objects (e.g., written symbols, beadsof abacuses, etc.) and external rules,constraints, or relations embedded inphysical configurations (e.g., spatial re-lations of written digits, visual and spa-tial layouts of diagrams, physical con-straints in abacuses, etc.). The informa-tion in external representations can bepicked up by perceptual processes. Incontrast, internal representations are therepresentations in the mind, as proposi-tions, productions, schemas, mental im-ages, neural networks, or other forms.The information in internal representa-tions has to be retrieved from memoryby cognitive processes. For example, inthe task of 735 ´ 278 with paper andpencil, the internal representations are

the values of individual symbols (e.g.,the value of the arbitrary symbol "7" isseven), the addition and multiplicationtables, arithmetic procedures, etc.,which have to be retrieved from mem-ory; and the external representations arethe shapes and positions of the symbols,the spatial relations of partial products,etc., which can be perceptually in-spected from the environment.

Zhang & Norman (1994a; Zhang,1992, 1995) developed a theory of dis-tributed representations to account forthe behavior in distributed cognitivetasksÑtasks that involve both internaland external representations. In thisview, the representation of a distributedcognitive task is neither solely internalnor solely external, but distributed as asystem of distributed representationswith internal and external representa-tions as two indispensable parts. Thus,external representations are intrinsiccomponents and essential ingredients ofdistributed cognitive tasks. They neednot be re-represented as internal repre-sentations in order to be involved in dis-tributed cognitive tasks: they can di-rectly activate perceptual processes anddirectly provide perceptual informationthat, in conjunction with internal repre-sentations, determine people's behavior.External representations have rich struc-tures. Without a means of accommodat-ing external representations in its ownright, we sometimes have to postulatecomplex internal representations to ac-count for the structure of behavior,much of which, however, is merely a re-flection of the structure of external rep-resentations.

In next section we apply the prin-ciples of distributed representations toanalyze the representation of informa-tion in the basic structure of numerationsystemsÑdimensions. In the later part


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for multiplication, we will further ana-lyze how the information needed tocarry out multiplication is distributedacross internal and external representa-tions.

The Distributed Representation ofDimensions

Dimensions are the basic structures ofnumeration systems. Thus, it is impor-tant to understand how dimensions arerepresented in numeration systems.

Psychological ScalesEvery dimension is on a certain

type of scale, which is the abstract mea-surement property of the dimension.Stevens (1946) identified four majortypes of psychological scales: ratio, in-terval, ordinal, and nominal. Each typehas one or more of the following formalproperties: category, magnitude, equalinterval, and absolute zero. Categoryrefers to the property that the instanceson a scale can be distinguished fromeach another. Magnitude refers to theproperty that one instance on a scale canbe judged greater than, less than, orequal to another instance on the samescale. Equal interval refers to the prop-erty that the magnitude of an instancerepresented by a unit on the scale is thesame regardless of where on the scalethe unit falls. An absolute zero is a valuewhich indicates that nothing at all of theproperty being represented exists.

Nominal scales only have oneformal property: category. Names ofpeople are an example of nominalscales: they only discriminate differententities but have no information aboutmagnitudes, intervals, and ratios.Ordinal scales have two formal proper-ties: category and magnitude. The rank-ing of movie quality is an example of

ordinal scales: a movie ranked Ò1Ó isbetter than a movie ranked Ò2Ó(magnitude) and the quality of a movieranked Ò5Ó is different from that of amovie ranked Ò7Ó (category). However,the rankings themselves tell us nothingabout the differences and ratios betweenthe rankings. Interval scales have threeformal properties: category, magnitude,and equal interval. Time is an exampleof interval scales: 02:00 is different from22:00 (category), 14:00 is later than 09:00(magnitude), and the difference between15:00 and 14:00 is the same as between09:00 and 08:00 (equal interval).However, time does not have an abso-lute zero (in a realistic sense). Thus, wecannot say that 10:00 is twice as late as05:00. Ratio scales have all of the fourformal properties: category, magnitude,equal interval, and absolute zero.Length is an example of ratio scales: 1inch is different from 3 inches(category), 10 inches are longer than 5inches (magnitude), the difference be-tween 10 and 11 inches is the same asthe difference between 100 and 101inches (equal interval), and 0 inch meansthe nonexistence of length (absolutezero). For length, we can say that 10inches are twice as long as 5 inches.

Distributed Representation of ScaleInformation

The base and power dimensionsof all numeration systems are abstractdimensions with ratio scales. In differ-ent numeration systems, these abstractratio dimensions are implemented bydifferent physical dimensions with theirown scale types, which are usuallyquantity (ratio), shape (nominal), andposition (ratio). Figure 5 shows the rep-resentation of base dimension in theArabic and Egyptian systems. In Figure5, the abstract information space for


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base dimension contains the four formalproperties of ratio scales, since base di-mension is on a ratio scale. In theArabic system, the base dimension isrepresented by shapeÑan externalphysical dimension with a nominalscale. Because a nominal dimensiononly has category information, shapecan only represent the category infor-mation of the ratio base dimension inexternal representations: the other threetypes of information have to be repre-sented in internal representations, thatis, memorized in the mind (Figure 5A).Thus, the representation of base dimen-sion in the Arabic system is a distributed

representation with category informa-tion represented externally by shapeand other three types of informationrepresented internally in memory. Inthe Egyptian system, the base dimensionis represented by quantityÑan externalphysical dimension with a ratio scale.Thus, in this case, all four types of in-formation of the ratio base dimensionare represented externally (Figure 5B).Similar to quantity, position is also a ra-tio dimension. Therefore, if the base orpower dimension of a numeration sys-tem is represented by position, its scaleinformation is all represented externally.

ExternalRepresentation

InternalRepresentation

AbstractInformation Space

(A) Arabic

4 3 2

categorymagnitudeequal intervalabsolute zero




ExternalRepresentation

InternalRepresentation

(B) Egyptian

AbstractInformation Space

|||| ||| ||

Figure 5. The distributed representation of dimensions. The scale information of a dimension is in theabstract information space, which is distributed across an internal and an external representation. (A)The representation of base dimension in the Arabic system. The category information of ratio basedimension is represented externally by shape (a nominal dimension) but the other three types ofinformation are represented internally in memory. (B) The representation of base dimension in theEgyptian system. All four types of information of ratio base dimension are represented externally byquantity (also a ratio dimension).


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The information in external rep-resentations can be picked up by percep-tual processes, whereas the informationin internal representations has to be re-trieved from memory. For example, inFigure 5A for the Arabic system, thethree symbols "4", "3", and "2" can beperceptually identified as three differententities (category information).However, the information aboutwhether "4" is smaller or larger than "3"(magnitude), whether the difference be-tween "4" and "3" is the same as or dif-ferent from that between "3" and "2"(equal interval), and whether "4" is twiceas large as "2" (absolute zero) has to beretrieved from memory. In contrast, inFigure 5B for the Egyptian system, allfour types of information can be pickedup by perceptual processes: ||||, |||, and ||are different entities (category), |||| islarger than ||| (magnitude), the differencebetween |||| and ||| is the same as the dif-ference between ||| and || (equal interval),and |||| is twice as large as | | (absolutezero).

The scale information of a dimen-sion, however, is only relational infor-mation: it does not specify the absolutevalues on the dimension. Thus, the rep-resentation of the absolute values on adimension needs to be considered sepa-rately. The absolute values on base andpower dimensions (base values andpower values) are represented internallyby shapes but externally by positionsand quantities. For example, even if "4","3", and "2" can be perceptually identi-fied as three different entities, the abso-lute values these symbols representhave to be retrieved from memory. Incontrast, the absolute values representedby "||||", "|||", and "||" are external and canbe picked up by perceptual processes.

Magnitude ComparisonLet us use magnitude comparison

as an example to illustrate how theabove analysis of dimensional represen-tations can be applied to simple numeri-cal tasks. Comparing the relative mag-nitudes of numbers only requires themagnitude information of the numbers.For a 1 D system, if the dimension isrepresented by an ordinal or a higherdimension, the magnitude informationneeded for comparison is external. Forexample, the quantity dimension in theStone-Counting system is an externalrepresentation of magnitude informa-tion. For a 1´1 D system, if both thebase and power dimensions are on ordi-nal or higher scales (e.g., Russian aba-cus), the magnitude information neededfor comparison is external, and if bothdimensions are on nominal scales (e.g.,Greek system), the information is inter-nal. Otherwise, it depends on the repre-sentation of each dimension. For ex-ample, for the Arabic system (aPosition´Shape system, ratio power di-mension and nominal base dimension),the relative magnitude information oftwo numbers is external if the two num-bers have different highest power values(e.g., 75 vs. 436) and internal if they havethe same highest power values (e.g., 75vs. 43).

The Interaction between Dimensions

Because most numeration sys-tems have more than one dimension, itis also important to understand how thedimensions interact with each other.The dimensions of a multi-dimensionalstimulus can be either separable or inte-gral. Separable dimensions are thosewhose component dimensions can bedirectly and automatically separatedand perceived. Integral dimensions can


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only be perceived in a holistic fashion:they can not be separated without a sec-ondary process that is not automaticallyexecuted (Garner, 1974).

Although the concept of separa-bility usually refers to external physicaldimensions, we use it to refer to internalcognitive dimensions as well. If the in-formation needed to separate two di-mensions can be picked up by percep-tual processes from external representa-tions, we say that these two dimensionsare externally separable. For example,the shape (base dimension) and position(power dimension) of the Arabic systemare two physical dimensions that are ex-ternally separable via perceptual pro-cesses. If the information needed toseparate two dimensions is not availablein external representations and can onlybe retrieved from internal representa-tions, we say that these two dimensionsare externally inseparable and only in-ternally separable. In the Greek system,for example, the base and power di-mensions are represented by a singlephysical dimension (shape). They canonly be separated internally.

Since many numerical tasks (e.g.,multiplication and addition under poly-nomial representations) need to processthe base and power dimensions sepa-rately, whether the dimensions of a nu-meration system are externally separa-ble or not can dramatically affect taskdifficulties. We will see this effect innext section.

A NUMERICAL TASK:MULTIPLICATION

In this section, we apply the principlesof distributed representations to analyzehow representational formats affectcomplex numerical tasks. To make ourdemonstration simple and clear, we

choose multiplication on the task sideand 1´1 D numeration systems on therepresentation side. Since numerationsystems are represented at four levels,the representational properties at differ-ent levels can affect numerical tasks indifferent ways. Here we only focus onthe representational properties at thelevel of dimensional representations,since whether the dimensions of numer-ation systems are represented externallyand whether they are externally separa-ble can dramatically affect the difficultyof a numerical task.

The Hierarchical Structure ofMultiplication

Multiplication can be analyzed at threelevels (Figure 6): algebraic structures,algorithms, and number representa-tions. At the level of algebraic struc-tures, different multiplication methodshave different algebraic structures. Thealgebraic structures of three typicalmethods, simple addition, binary, andpolynomial methods, are as follows:

Simple Addition:N1Ń2 = N1+N1+N1+...+N1

(add N2 times)

Binary:N1Ń2 = N1´åai2i = åN1ai2i

Polynomial:N1Ń2 = åaixi´åajxj = ååaixiajxj = ååaiajxi+j

The simple addition method issimply adding the multiplicand thenumber of times indicated by the multi-plier. For the doubling method, multi-plication is performed by (a) decompos-ing the multiplier into its binary repre-sentation, (b) multiplying the multipli-


14

cand with each term of the binary repre-sentation of the multiplier, and (c)adding the partial products. Becausethe terms of the binary representationare either in the form of 1´2n or 0´2n,the multiplication of the multiplicandwith each term of the binary representa-tion of the multiplier can be performedby successively doubling the multipli-cand. For the polynomial method, mul-tiplication is performed by multiplyingevery term of the multiplicand with ev-ery term of the multiplier and thenadding the partial products together

At the level of algorithms, differ-ent algorithms can be applied to thesame algebraic structure. For example,the binary method can be realized by theEgyptian algorithm and the Russian al-gorithm (for details of these two algo-rithms, see Zhang, 1992), and the poly-nomial method can be realized by theStandard algorithm and the Greek algo-rithm, which are described in a later sec-tion. At the level of number representa-tions, multiplication can be performedunder different numeration systems.

Here we only analyze the poly-

nomial method, the one that includesthe Standard algorithm we are using to-day. The simple addition and the binarymethods, which can be reduced to addi-tion, were analyzed in Zhang (1992).

The Polynomial Method ofMultiplication

A numeral in a 1´1 D system isrepresented as a polynomial: åaixi.Multiplication by the polynomialmethod is performed by multiplying ev-ery term of the multiplicand with everyterm of the multiplier and then addingthe partial products together, regardlessof which particular algorithm (e.g., thestandard or the Greek algorithm, seenext section) is used. Thus, the two ba-sic components of polynomial multipli-cation are the multiplication of individ-ual terms and the addition of partialproducts. Here we only show how rep-resentational formats affect the pro-cesses of term multiplication. (A de-tailed analysis involving both termmultiplication and partial product addi-tion can be found in Zhang, 1992.)

BinarySimple Addition Polynomial

Standard

Number Represenations

A´B = C

GreekEgyptian Russian

AlgebraicStructures

Algorithms

NumberRepresentations

Figure 6. The hierarchical structure of multiplication.


15

For all 1´1 D systems, term mul-tiplication (aixi´bjyj) has the same set ofsix basic steps (see Figure 7): one step(Step 1) to separate the power and basedimensions, two steps (Steps 2a and 2b)to multiply the base values, two steps(Steps 3a and 3b) to add the power val-ues, and one step (Step 4) to combine thepower and base values of the finalproduct. Although the six steps neednot be followed in the exact order, there

are certain constraints: Step 1 must bethe first step and Step 4 must be the laststep, but Steps 2a and 2b can either pro-ceed or follow Steps 3a and 3b. The in-formation needed for each step can beeither perceived from external represen-tations or retrieved from internal repre-sentations. Figure 7 and the followingdescriptions show the analyses of the sixsteps for the Arabic, Greek, andEgyptian systems.

Step Abstract: aixi´bjxj Greek: l´d Egyptian: ÇÇÇ´|||| Arabic: 30´4

1 separate power &base dimensions

internal:internally separable

external:externally separable

external:externally separable

2a get base values ofaixi & bjxj

internal:ai, bj = shapes

external:ai, bj = quantities

internal:ai, bj = shapes

B(aixi) = ai

B(bjxj) = bj

B(l) = gB(d) = d

B(ÇÇÇ) = |||B(||||) = ||||

B(30) = 3B(4) = 4

2b multiply basevalues

internal:multiplication table



ai´bj = cij g ´ d = ib |||´|||| = Ç|| 3´4 = 12

3a get power valuesof aixi & bjxj

internal:i, j = shapes

internal:i, j = shapes

external:i, j = positions

P(aixi) = iP(bjxj) = j

P(l) = 1P(d) = 0

P(ÇÇÇ) = 1P(||||) = 0

P(30) = 1P(4) = 0

3b add power values internal:addition table

internal:addition table

external:positions

P(aixibjxj)= P(aixi)+P(bjxj)= i+j = pij

P(l´d)= P(l)+P(d)= 1+0 = 1

P(ÇÇÇ´||||)= P(ÇÇÇ)+P(||||)= 1+0 = 1

P(30´4)= P(30)+P(4)= 1+0 = 1

4 attach powervalues

internal:shapes

internal:shapes

external:positions

aixi´bjxj

= cij´xpij

l´d

= (ib)´101

= rk

ÇÇÇ´||||

= (Ç||)́ 101

= ÇÇ

30´4= 12´101

= 120

Figure 7. The six basic steps of term multiplication for the polynomial method. See text for details.


16

Step 1: Separate power and basedimensions. The power and base di-mensions are externally separable forthe Arabic system (position and shape)and the Egyptian system (shape andquantity) but only internally separablefor the Greek system (a single shape di-mension for both power and base di-mensions). Thus, the informationneeded for this step is external for theArabic and Egyptian systems but inter-nal for the Greek system.

Step 2a: Get base values of aixi

and bjyj: ai , bj. The base values are rep-resented internally in the Greek system(shape) and the Arabic system (shape)but externally in the Egyptian system(quantity). Thus the information neededfor this step is internal for the Greek andArabic systems but external for theEgyptian system.

Step 2b: Multiply base values: ai

´ bj = cij. The information needed forthis step is internal for all three systems,which need to be retrieved from an in-ternal multiplication table2.

Step 3a: Get power values of aixi

and bjyj: i, j. The power values are rep-resented externally in the Arabic system(position) but internally in the Greeksystem (shape) and the Egyptian system(shape). Thus, the information neededfor this step is external for the Arabicsystem but internal for the Greek andEgyptian systems.

Step 3b: Add power values: i + j= pij. In the Arabic system, since addingthe power values of two terms can beperformed perceptually by shifting posi-tions, the information needed for this

2 The multiplication table can be externalized.For example, Napier's Bone is an externalrepresentation of the multiplication table (seeZhang, 1992).

step is external. In the Greek andEgyptian systems, since the sums ofpower values have to be retrieved inter-nally from the mental addition table, theinformation needed for this step is in-ternal.

Step 4: Attach power values tothe product of base values: aixi´bjyj =cij ´ xpij. The sum of power values of thetwo terms need to be attached to theproduct of the base values of the twoterms to get the final result. The infor-mation needed for this step is externalfor the Arabic system because the sumof power values can be attached to theproduct of base values by shifting posi-tions. For the Greek and Egyptian sys-tems, the information needed to thisstep is internal because power values inboth systems are represented internallyby shapes.

From the above analysis, we seethat the information needed for the sixbasic steps of term multiplication is dis-tributed across internal and externalrepresentations. Figure 8 shows the dis-tributed representations of term multi-plication for the Greek, Egyptian, andArabic systems. If we assume that withall other conditions kept the same, themore information needs to be retrievedfrom internal representations, the harderthe task (e.g., due to working memoryload), then for term multiplication, theGreek system (six internal steps) isharder than the Egyptian system (fourinternal steps), which in turn is harderthan the Arabic system (two internalsteps)3. We need to note that difficulty

3 A complete multiplication task involves notjust the multiplication of individual terms butalso the addition of partial products. When theaddition of partial products is considered, thesame difficulty order remains: Greek > Egyptian> Arabic (see Zhang, 1992).


17

can be measured along many dimen-sions, some of which may have trade-offbetween each other. The amount of in-ternal and external information is onlyone of the difficulty measures.

Multiplication Algorithms for thePolynomial Method

The polynomial method of multiplica-tion can be realized by different algo-rithms. Although different algorithmshave the same set of six basic steps ofterm multiplication, they usually havedifferent orders for the multiplication ofindividual terms and different group-ings for the partial products. Figure 9compares two algorithms: the Greek al-gorithm used in ancient times by Greeksand the Standard algorithm used todayby people in nearly all cultures.

The Standard algorithm starts themultiplication of individual terms fromthe lowest term of the multiplier, while

the Greek algorithm starts from thehighest term of the multiplicand. TheStandard algorithm groups the partialproducts that have common powers bytheir spatial relations (positions), whichmakes the addition of partial productseasier. The Greek algorithm, however,does not group partial products thathave common powers and does notmake use of spatial relations for the ad-dition of partial products.

Under the same algorithm, theArabic system is more efficient than theGreek system because the former ismore efficient than the latter for termmultiplication. Under the same nu-meration system, the Standard algo-rithm is more efficient than the Greekalgorithm because the spatial groupingof partial products in the Standard algo-rithm makes the addition of partialproducts easier.

External Internal

(B) Egyptian System

12a2b3a3b4

Abstract

(A) Greek System (C) Arabic System

External Internal

Abstract

External Internal

Abstract

12a2b3a3b4

12a

2b3a3b4

12a2b3a3b4

12a2b3a3b4

1

3a3b4

2a2b

Figure 8. The distributed representation of the information needed for the six basic steps of termmultiplication.


18

Procedures Abstract Form Greek system Arabic system

GreekAlgorithm

a1x1 a0x0

b1x1 b0x0

ÑÑÑÑÑÑÑÑÑÑÑÑÑÑ a1b1x2 a1b0x1

a0b1x1 a0b0x0

ÑÑÑÑÑÑÑÑÑÑÑÑÑÑ(a1b1x2+a0b1x1) + (a1b0x1+a0b0x0)

i z i gÑÑÑÑÑÑ r l o kaÑÑÑÑÑÑ ro na ska

10 7 10 3ÑÑÑÑÑÑ 100 30 70 21ÑÑÑÑÑÑ 170 51 221

StandardAlgorithm

a1x1 a0x0

b1x1 b0x0

ÑÑÑÑÑÑÑÑÑÑÑÑÑ a0b0x0

a1b0x1

a0b1x1

a1b1x2

ÑÑÑÑÑÑÑÑÑÑÑÑÑ a1b1x2 + (a1b0x1+a0b1x1) + a0b0x0

i z i g ÑÑÑÑÑÑ k a l o r ÑÑÑÑÑÑ s k a

1 7 1 3 ÑÑÑÑÑÑ 2 1 3 7 1 ÑÑÑÑÑÑ 2 2 1

Figure 9. The Greek and the Standard algorithms of multiplication under the Greek and Arabic systems.The example is 17´13.

GENERAL DISCUSSION

In this article, we have explored thecognitive aspects of numeration sys-tems, focusing on external number rep-resentations and their interaction withinternal representations. We show thatthe information that needs to be pro-cessed in complex numerical tasks isdistributed across internal and externalrepresentations.

We analyzed numeration systemsat four levels: dimensionality, dimen-sional representations, bases, and sym-bol representations. The representa-tional properties at these levels can af-fect the processes of numerical tasks indifferent ways and are responsible for

different aspects of the representationaleffect in numerical tasks. These levelsprovide a cognitive taxonomy that canclassify most numeration systems thathave been invented across the world.We suggest that the hierarchical struc-ture can be considered as a theoreticalframework for systematic studies ofnumber representations. Among thefour levels, the level of dimensional rep-resentations is most important: whetherthe dimensions and their absolute val-ues are represented externally or inter-nally and whether the dimensions areexternally or internally separable candramatically affect the difficulty of anumerical task. Thus, when we analyzethe polynomial multiplication method


19

for three 1´ 1 D systems (Arabic,Egyptian, and Greek) using the amountof information that has to be retrievedfrom internal representations as a mea-sure of problem difficulty, the Greeksystem is found to be the hardest amongthe three systems and the Arabic systemthe easiest.

A Cognitive Model of DistributedNumerical Cognition

Any distributed cognitive taskcan be analyzed into three aspects:formal structures, representations, andprocesses. Formal structures specifythe information that has to be processedin a task; representations specify howthe information to be processed isrepresented across internal and external

representations; and processes specifythe actual mechanisms of informationprocessing. These three aspects areclosely interrelated: the same formalstructure can be implemented bydifferent representations, and differentrepresentations can activate differentprocesses. Any study of distributedcognitive tasks has to consider all ofthese three aspects. Our present studyof numeration systems and numericaltasks, however, only focused on theirformal structures and representations.In this section, we use the termmultiplication of Arabic numerals as aspecial case to outline a cognitive modelof distributed numerical cognition withseveral assumptions about the process-ing mechanisms.

External Representation

Internal Representation

Abstract Task Space

Perceptual Processes Memorial Processes

identifying positions

separating dimensions

shifting positionsshifting positions

retrieving multiplication tableretrieving base values

Central Control

coordinating perceptual and memorial processes, executing arithmetic procedures, allocating attentional resources, etc.

12a2b3a3b4

1

3a3b4

2a2b

Figure 10. The model of distributed numerical cognition. See text for details.


20

The model is shown in Figure 10.The abstract task space specifies theformal structure of the taskÑthe six ba-sic steps of term multiplication, and theinformation that has to be processedÑthe information needed for the six basicsteps of term multiplication. The infor-mation needed for the six basic steps isdistributed across internal and externalrepresentations. For Arabic numerals,the information for Steps 1, 3a, 3b, and 4is represented externally in the envi-ronment and the information for Steps2a and 2b is represented internally inmemory (see also Figures 7 and 8).External information is picked up fromexternal representations by perceptualprocesses. For Arabic numerals, theseperceptual processes include separatingthe base and power dimensions (Step 1),identifying the positions to get powervalues (Step 3a), shifting positions toadd power values (Step 3b), and shiftingpositions to combine the base andpower values of the final product (Step4). Internal information is retrievedfrom internal representations by memo-rial processes. For Arabic numerals,these memorial processes include re-trieving the base values from memory(Step 2a) and retrieving multiplicationfacts from the multiplication table inmemory (Step 2b). The central controlcoordinates the perceptual and memo-rial processes, executes arithmetic pro-cedures, allocates attentional resources,and performs other processes that arenecessary for the completion of the task.

The first assumption of ourmodel is for representations: the infor-mation needed for a numerical task isdistributed across internal and externalrepresentations. It has been shown(Zhang & Norman, 1994a; Zhang, 1995)that manipulating the information dis-

tributed across internal and externalrepresentations can dramatically affectthe difficulty of a task. The second as-sumption is for processes: external rep-resentations activate perceptual pro-cesses and internal representations acti-vate memorial processes. Since percep-tual processes can be direct, automatic,unconscious, and efficient, their in-volvement can reduce the difficulty of atask. The third assumption is for thecentral control, whose major function isthe coordination of perceptual andmemorial processes. Although therehave been several studies on the inter-play between perception and memoryin terms of attention switching (e.g.,Carlson, Wenger, & Sullivan, 1993;Dark, 1990; Weber, Byrd, & Noll, 1986),it is still unclear how perceptual andmemorial processes interact with eachother. For instance, the reason we didnot mention working memory in thecentral control is because it is not clearto us how the framework of workingmemory (Baddeley, 1986; Baddeley &Hitch, 1974) can fit here. Can percep-tion, especially the automatic and directkind, bypass working memory to di-rectly participate in complex distributedcognitive tasks? In addition to the in-ternal working memory, is there a sepa-rate external working memory? If yes,what are its functions and what is itsrelation to the internal working mem-ory? These questions are central to theunderstanding of the nature of dis-tributed cognitive tasks. They deservesystematic studies, both empirically andtheoretically.

Is the Arabic System Special?

Very few things in the world are as uni-versal as the Arabic numeration system.


21

It has often been argued that the place-value notation (position) of the Arabicsystem is one of the most ingenious in-ventions of the human mind and it iswhat makes the Arabic system so spe-cial. Our analysis of the representa-tional properties of numeration systemsallows us to examine the question ofwhy the Arabic numeration system is sospecial.

Based on our analysis, the place-value notation is not sufficient to makethe Arabic system so special, since posi-tions were also used to represent thepower dimensions in many other sys-tems, including the Babylonian, Chinesescientific, and Mayan written numera-tion systems, and the abacus, countingboard, Peruvian knotted string, andother object numeration systems. Then,what properties of the Arabic systemmake it so special? Let us first considerthe properties of the Arabic system at itsfour levels of representations.

At the level of dimensionality, theArabic system is a 1´1 D system, whichis more efficient for information encod-ing than 1 D systems. Although (1´1)´1D systems (e.g., Babylonian, Mayan,Roman, etc.) are as efficient as 1´1 Dsystems for information encoding, theextra third dimension only introducesunnecessary complexity.

At the level of dimensional repre-sentations, the base and power dimen-sions of the Arabic system are externallyseparable, and the power dimension andits power values are represented exter-nally by positions. Positions can handlelarge numbers efficiently. However, thebase values and three of the four typesof scale information of the base dimen-sion are represented internally byshapes. If we are only concerned withwhether the base and power dimensionsand their values are represented inter-

nally or externally, the Russian abacus,whose base dimension (quantity) as wellas power dimension (position) are bothrepresented externally, are superior. Infact, abacuses are always superior towritten numerals for simple calculationsuch as addition and subtraction. But ifwe consider other factors such as ease ofreading and writing symbols, the Arabicsystem is still superior, because aftersufficient learning Arabic symbols canbe easily read and written.

At the level of bases, the base 10of the Arabic system is a manageablesize. There is a trade-off in base size:larger bases require more symbols to belearned but are more efficient in han-dling large numbers. The addition andmultiplication tables that need to bememorized for the Arabic system is rela-tively small in comparison with theBabylonian system (base 60) and theMayan system (base 20), which also usepositions to represent their main powerdimensions.

At the level of symbols, the tensymbols of the Arabic system are easy towrite, which makes calculation with pa-per and pencil efficient.

Combining the representationalproperties at these four levels, theArabic system is better than many othersystems in terms of representation.Though the Arabic system is also betterthan many other systems (e.g., Greek,Egyptian, etc.) in terms of calculation, itis not always the most efficient one. Asdiscussed above, the abacus, which wasstill widely used in Japan, China, andRussia before electronic calculators be-came popular, is more efficient than theArabic system for addition and subtrac-tion.

So what makes the Arabic systemso special? Numerals have two majorfunctions: representation and calcula-


22

tion. In many cultures, these two func-tions are achieved by two separate sys-tems. For example, in China, calculationwas carried out by abacuses and sticks,whereas representation was realized bywritten numerals. A more interestingcase is the Roman counting board, inwhich counters were used for calcula-tion and Roman numerals were used forrepresentation, both in the same physi-cal device. We propose that what makesthe Arabic system so special and widelyaccepted is that it integrates representa-tion and calculation into a single system,in addition to its other nice features ofefficient information encoding, com-pactness, extendibility, spatial represen-tation, small base, effectiveness of calcu-lation, and especially important, ease ofwriting. Thus, Arabic numerals are aspecial type of object-symbols (Hutchins& Norman, 1988; Norman, 1991)Ñsym-bols that serve the functions of both rep-resentation and manipulation. The inte-gration of representation and calculationinto a single system, however, would beuseless without appropriate media andtools such as paper and pencil: a niceexample of technological constraints.Interestingly, calculation and the Arabicnumerals were so integrated that theword algorithm is merely a corruption ofAl Kworesmi, the name of the Arabicmathematician of the ninth centurywhose book on Arabic numerals was thefirst one reaching Western Europe.

Algebra, in the broad sense inwhich the term is used today, deals withoperations upon symbolic forms, includ-ing numerals, unknowns, and arithmeticoperators. It has been argued (e.g.,Dantzig, 1939) that the invention ofArabic numerals was instrumental inthe emergence and development of al-gebra. Our analysis of Arabic numeralssupports this argument: Arabic numer-

als are not only an efficient representa-tion of numbers but also a symbolicform that can be easily operated upon.It is also interesting to note that Greeks,who were highly advanced in geometry,did not develop an algebra in the mod-ern sense. This is probably becauseGreek alphabetical numerals, thougheasy to manipulate, were neither effi-cient for representation nor for calcula-tion.

Implications for OtherRepresentational Systems

Although our present study has focusedon numeration systems, the methodol-ogy of representational analysis and theprinciples of distributed representationsillustrated here and elsewhere (Zhang &Norman, 1994a; Zhang, 1992, 1995) canbe applied to other domains as well. Forexample, relational information dis-plays, which include graphs, charts,plots, diagrams, tables, lists, and othertypes of visual displays that representrelational information, are distributedrepresentations; and the tasks for rela-tional information displays are dis-tributed cognitive tasks. Zhang &Norman (1994b) analyzed relation in-formation displays in terms of a hierar-chical structure similar to that of nu-meration systems and developed a cog-nitive taxonomy that can classify nearlyall types of relational information dis-plays and can serve as a theoreticalframework for systematic studies of thecognitive properties in relational infor-mation displays. As another example,several cockpit instrument displays,such as different types of altimeters andnavigation displays, were also studiedfrom the same perspective (Zhang,1992).

In addition to numeration sys-


23

tems, numerous representational sys-tems (usually called notational systems)have been invented over the last fewthousand years in the development ofmathematics (see the two volume bookby Cajori, 1928, on the history of math-ematical notations). Of these systems,only a fraction survived. Although po-litical, economic, social, and cultural fac-tors certainly played some roles in theevolution of representational systems,cognitive factors might have played themost important role since the activitiesof individuals in mathematics are es-sentially cognitive activities. How theforms of representations affect the cog-nitive activities of scientists and whatroles they played in the evolution ofmathematics in particular and science ingeneral are interesting issues worth ofthe attention of not just historians andphilosophers but also psychologists andcognitive scientists.

ACKNOWLEDGMENT

We are very grateful to two anonymousreviewers for valuable comments andconstructive criticisms. Request forreprints should be sent to Jiajie Zhang,Department of Psychology, The OhioState University, 1885 Neil Avenue,Columbus, OH, 43210, U.S.A. Email:[email protected],[email protected].

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The Representation of Numbers - Stanford University · 2009-09-18 · 1·1 D Systems Although 1 D systems are effi-cient for small numbers, they do not work well for large numbers