To appear in Cognition

The Representation of Numbers

Jiajie Zhang Donald A. Norman Department of Psychology Apple Computer, Inc. The Ohio State University

ABSTRACT 1 for several examples of numeration This article explores the representational systems). This representational effect, that structures of numeration systems and the different representations of a common cognitive factors of the representational ef- abstract structure can cause dramatically fect in numerical tasks, focusing on external different cognitive behaviors, has had representations and their interactions with profound significance in the develop- internal representations. Numeration sys- ment of arithmetic and algebra in par- tems are analyzed at four levels: dimension- ticular and mathematics in general. The ally, dimensional representations, bases, and Arabic numeration system, remarkable symbol representations. The representa- as is its simplicity, has been regarded as tional properties at these levels affect the one of the greatest inventions of the processes of numerical tasks in different human mind. ways and are responsible for different as- The representational effect of pects of the representational effect. This hi- numeration systems is a cognitive phe- erarchical structure is also a cognitive tax- nomenon. However, early studies of onomy that can classify nearly all numera- numeration systems focused on their tion systems that have been invented across historical, cultural, mathematical, and the world. Multiplication is selected as an philosophical aspects (e.g., Brooks, 1876; example to demonstrate that complex nu- Cajori, 1928; Dantzig, 1939; Flegg, 1983; merical tasks require the interwoven Ifrah, 1987; Menninger, 1969). Recently, processing of information distributed across the representational efficiencies of dif- internal and external representations. ferent numeration systems have been Finally, a model of distributed numerical compared and analyzed in terms of their cognition is proposed and an answer to the cognitive properties (e.g., Nickerson, question of why Arabic numerals are so 1988; Norman, 1993), and the nature of special is provided. number representations has also been examined from a semiotic perspective We all know that Arabic1 numerals are (Becker & Varelas, 1993). In this paper, more efficient than Roman and many we attempt a systematic analysis of the other types of numerals for calculation representational structures of numera- (e.g., 73 ´ 27 is easier than LXXIII ´ tion systems and the cognitive factors XXVII), even though they all represent responsible for the representational ef- the same entitiesÑnumbers (see Figure fect in numerical tasks. Unlike the representational effect of numeration systems, there have been 1 Although the Arabic numerals were originally invented in India and are called Hindu-Arabic a large number of psychological studies numerals by some historians, we adopt the of numerical cognition over the last two conventional name for simplicity. decades (for a recent collection of re-

1 Zhang & Norman Number Representations views, see Dehaene, 1993). However, THE REPRESENTATIONAL most of these studies have entirely fo- STRUCTURES OF NUMERATION cused on internal representations: how SYSTEMS people perform numerical tasks in their heads, how numbers and arithmetic In this part, we first analyze the dimen- facts are represented in memory, and sionality of numeration systems. Next, what mental processes and procedures we analyze the representational proper- are involved in the comprehension, cal- ties of numeration systems at four dif- culation, and production of numbers. In ferent levels in terms of a hierarchical this present study, we focus on external structure. Then, based on the hierarchi- representations and their interactions cal structure, we propose . a cognitive with internal representations: how taxonomy of numeration systems. numbers are represented in the external environment, what structures in exter- The Dimensionality of Numeration nal representations are perceived and Systems processed, and how internal and exter- nal representations are integrated in 1 D Systems numerical tasks. Although our focus is One of the simplest ways to rep- on external representations, we are not resent numbers is to use stones: one denying the important roles of internal stone for one, two stones for two, and so representations. What we attempt to on. This Stone-Counting system only show is that external representations has a single dimension: the quantity of have much more important roles than stones. The Body-Counting system previously acknowledged. used by Torres Islanders is another one We start with a representational dimensional system, in which the single analysis of the hierarchical structures of dimension is represented by the posi- numeration systems. From this analysis, tions of different body parts (fingers, we develop a cognitive taxonomy of wrist, elbow, shoulder, toes, ankles, numeration systems. This taxonomy knees, and hips). The first numeration can not only classify most numeration systems invented in nearly all nations of systems but also can serve as a theoreti- antiquity were one dimensional systems cal framework for systematic studies of represented by simple physical objects, the representations of numbers and the such as stones, pebbles, sticks, tallies, processes in numerical tasks. In the sec- etc. One dimensional systems are de- ond part, we analyze how the dimen- noted as 1 D in this article. sional representations of numeration Many 1 D systems are very effi- systems are distributed across internal cient for small numbers. Moreover, they and external representations. In the make a number of numerical compari- third part, we use multiplication as an son tasks simpler than with other sys- example to examine the interwoven tems (e.g., Arabic), because the size of processing of internal and external in- the representation of the number is pro- formation in complex numerical tasks. portional to the numerical value. This In the last part, we outline a model of means that the system has analogical distributed numerical cognition and properties that make comparisons sim- provide an answer to the question of ple. In addition, the operations of addi- why Arabic numerals are so special. tion and subtraction are easy, requiring

2 Zhang & Norman Number Representations no knowledge of arithmetic properties or removal of notational marks (see of tables, but rather, simply the addition Norman, 1993, chapter 3).

Arabic Egyptian Babylonian Greek Roman Chinese Aztec Cretan Mayan 1 | 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 X 20 ÇÇ k XX ¥ 30 ÇÇÇ l XXX ¥ 40 ÇÇÇÇ m XXXX ¥¥ 50 ÇÇÇÇÇ n L ¥¥ 60 ÇÇÇÇÇ x LX ¥¥¥ Ç 70 ÇÇÇÇÇ o LXX ¥¥¥ ÇÇ 80 ÇÇÇÇÇ p LXXX ¥¥¥¥ ÇÇÇ 90 ÇÇÇÇÇ LXXXX ¥¥¥¥ ÇÇÇÇ 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.

3 Zhang & Norman Number Representations

1´1 D Systems The two dimensions of some 1´1 Although 1 D systems are effi- D systems are externally separable, cient for small numbers, they do not whereas those of some others are only work well for large numbers. For large internally separable. In Figure 2, the numbers, there must be more than one base and power dimensions of all but dimension, for example, one base dimen- the Greek system are externally separa- sion and one power dimension. The ble. For example, we can separate the power dimension decomposes a number shape and position of each digit in an into hierarchical groups on a base. The Arabic numeral via perceptual inspec- base can be raised to various powers on tion on the physical properties of the the power dimension. We denote two written symbols. In this case, shape and position are two separate physical di- dimensional systems as 1´ 1 D mensions. However, the two dimen- (base´power). A number in a 1´1 D sys- sions of the Greek system (see Figure 2) tem is represented as a polynomial: are not externally separable. Its base Saixi. Figure 2 shows the representa- and power dimensions are represented tional structures of six 1´1 D systems. by a single physical dimensionÑshape The two dimensions of 1´1D sys- (a one-to-two mapping): they can only tems can be represented by different be separated as two dimensions in the physical properties, which are usually mind by retrieving relevant information shape, quantity, and position. For ex- from memory. For example, the values ample, the Arabic system is a two di- on the base and power dimensions of t mensional system (Figure 2) with a base (300) are 3 and 102, which can not be dimension represented by the shapes of separated via perceptual inspection on the ten digits (0, 1, 2, ..., 9) and a power the physical property of the symbol "t." dimension represented by positions of The separation of the single physical the digits with a base ten. Thus, the dimension (shape) into a base and a middle 4 in 447 has a value 4 on the base power dimension in the mind is re- dimension and a position 1 (counting quired by the Greek multiplication algo- from the rightmost digit, starting from rithm, which needs to process the values zero) on the power dimension. The ac- on the base and power dimensions sepa- tual value it represents is forty (the rately (see Flegg, 1983; Heath, 1921). product of its values on the base and power dimensions, i.e., 4´101). As an- (1´1)´1 D Systems other example, the base dimension of Some numeration systems have the Egyptian system (see Figure 2) is three dimensions: one main power di- represented by quantities (the quantities mension, one sub-base dimension, and one sub-power dimension. The sub-base and of |'s, Ç's, 's, etc.), and the power di- mension is represented by shapes sub-power dimensions together form 0 1 2 the main base dimension. We denote this (|=10 , Ç=10 , =10 , etc.). Thus, type of three dimensional systems as ÇÇÇÇ||||||| also represents four (1´1)´1 D [(sub-base´sub-power)´main- hundred and forty-seven. The details of power]. A numeral in a (1´1)´1D sys- the representations of the base and yj)xi in its ab- power dimensions in several other 1´1D tem is expressed as SS(bij systems are shown in Figure 2. stract form. The representational struc-

4 Zhang & Norman Number Representations

Systems Example (447) Base Base Dimension Power Dimension Abstract S aixi xai xi Arabic 447 10 ai = shape xi = position

4´102+4´101+7´100 0, 1, 2, ..., 9 ... 102 101 100 Egyptian ÇÇÇÇ||||||| 10 ai = quantity xi = shape 4´102+4´101+7´100 The numbers of |'s, | Ç ...

Ç's, 's, etc. 100 101 102 ... Cretan 10 ai = quantity xi = shape 4´102+4´101+7´100 The numbers of 's, ...

's, 's, etc. 100 101 102 ... Greek umz 10 ai = shape xi = shape 4´102+4´101+7´100 a b g ... q i k l ...

1 2 3 ... 9 1´101 2´101 3´101...9´101 r s t ...

1´102 2´102 3´102...9´102 Aztec 20 ai = quantity xi = shape 1´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) Main Sub- Sub-base Sub-power Main Power Dimension Base base Dimension Dimension Abstract SS(bijyj)xi xy bij yj xi Babylonian 60 10 bij = quantity yj = shape xi = position

(0´101+7´100)601 The numbers = 100, = 101 ... 602 601 600 +(2´101+7´100)600 of Ôs and 's Mayan ¥ ¥¥ ¥¥ 20 5 bij = quantity yj = shape xi = position

(0´51+1´50)202 The numbers = 50, = 51 ... 202 201 200 +(0´51+2´50)201 of 's and +(1´51+2´50)200 's. Roman CCCCXXXXVII 10 5 bij = quantity yj = shape xi = shape

(0´51+4´50)102 The numbers I =100´50 V = 100´51 I = 50´100 V = 50´101

+(0´51+4´50)101 of I's, V's, X's, X = 101´50 L = 101´51X = 50´101 V = 51´101 +(1´51+2´50)100 L's, etc......

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

5 Zhang & Norman Number Representations

tures of three (1´1)´1 D systems are Hierarchical Structure and Cognitive shown in Figure 3. To illustrate this Taxonomy system, consider a Babylonian numeral (see Figure 3 for details): Based on the analysis of the dimension- ality of numeration systems, we can analyze number representations at four = (0´101+7´100)´601+(2´101+7´100)´600 levels: dimensionality, dimensional = 7´601+27´600 = 420 + 27 = 447 representation, bases, and symbol representation. The main power dimension of the Each level has an abstract structure Babylonian system is represented by po- that can be implemented in different sitions with a base 60. In this example, ways. The different representations at (the right component) is on position each level are isomorphic to each other 0 (600), and (the left component) is on in the sense that they all have the same position 1 (601). The value of (the left abstract structure at that particular level component) on the main base dimension (Figure 4). is 7 (= 0´101+7´100). The actual value it At the level of dimensionality, dif- represents is the product of its values on ferent numeration systems can have dif- the main base and main power dimen- ferent dimensionalities: 1D, 1´1D, 1 0 1 sions: (0´10 +7´10 )´60 = 420. The (1´1)´1D, and others. However, they main base dimension is composed of a are all isomorphic to each other at this sub-base dimension represented by level in the sense that they all represent quantity (the quantities of Ôs and 's) the same entitiesÑnumbers. This level and a sub-power dimension represented mainly affects the efficiency of informa- by shape ( = 100 and = 101) with a tion encoding. 1 D systems are linear, base 10. For example, can be decom- while 1´1 D and (1´1)´1 D systems are posed as 2´101+7´100. polynomial. Polynomial systems en- Similar to 1´1 D systems, the code information more efficiently than three dimensions in a (1´1)´1 D system can be externally separable or only in- linear systems: the number of symbols ternally separable. For example, the needed to encode a number in a poly- sub-base, sub-power, and the main nomial system is proportional to the power dimensions in the Babylonian logarithm of the number of symbols and the Mayan systems are externally needed to encode the same number in a separable with each other. In the linear system. level of dimensional represen- Roman system, although the sub-base At the tations dimension is externally separable from , isomorphic numeration systems the sub-power and the main power di- have the same dimensionality but dif- mensions, the sub-power and the main ferent dimensional representations. The power dimensions, which are repre- physical properties used to represent the sented by a single physical dimension dimensions of numeration systems are (shape), are only internally separable. usually quantity (Q), position (P), and For example, L (50) has a value 51 on the shape (S). For example, the base and sub-power dimension and a value 101 power dimensions of 1´1D systems can on the main power dimension, which be represented by shape and position can only be separated in the mind. (S´P, Arabic system), shape and shape

6 Zhang & Norman Number Representations

(S´S, Chinese system), quantity and systems are 1´1D systems, and the base shape (Q´S, Egyptian system), etc. This and power dimensions of both systems level is crucial for the representational are represented by quantity and shape. effect of numeration systems. The sec- However, the base of the Egyptian sys- ond part of this paper is devoted for the tem is ten while that of the Aztec system analysis of the representational proper- is twenty (see Figure 2). This level is ties at this level. important for tasks involving addition At the level of bases, isomorphic and manipulation tables: the larger a numeration systems have the same di- base is, the larger the addition and mul- mensionality, same dimensional repre- tiplication tables are and the harder they sentations, but different bases. For ex- can be memorized and retrieved. ample, both the Egyptian and the Aztec

ABSTRACT NUMBERS

Dimensionality 1 D 1´1 D (1´1)´1D

Dimensional S´P [S´S] S´S Q´S (Q´S)´P (Q´S)´S Representation P Q

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

Symbol Representation Body StoneArabic GreekChinese Egyptian Cretan Aztec Mayan Babylonian Roman

Figure 4. The hierarchical structure of number representations. This is a also a cognitive taxonomy of numeration systems. At the level of dimensionality, different systems have different dimensionalities. At the level of dimensional representations, the dimensions of different systems are represented by different physical 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, different systems may have different bases. At the level of symbol representations, different systems use different symbols.

7 Zhang & Norman Number Representations

At the level of symbol representa- consider a few more systems (see Ifrah, tions, isomorphic numeration systems 1987, for detailed descriptions of these have the same abstract structures at the systems). At the level of symbol repre- previous three levels. However, differ- sentations, the Hebrew alphabetic sys- ent symbols are used. For example, tem is in the same group as the Greek both the Egyptian and the Cretan sys- system, and the Greek acrophonic, tems are 1´1D systems, the two dimen- Dalmatian, and Etruscan systems are in sions of both systems are represented by the same group as the Roman system. quantity and shape, and both systems At the level of bases, the Chinese scien- have the base ten. However, in the tific system is in the same group as the Egyptian system, the symbols for 100, Mayan system. In addition to numeration sys- 101, and 102 are |, Ç, and , whereas in the Cretan system, the corresponding tems of written numerals, this taxonomy can also classify numeration systems of symbols are , , and . This level object numerals. The following are a mainly affects the reading and writing few examples (see Ifrah, 1987), in which of individual symbols. P = Position, Q = Quantity, and S = The hierarchical structure of Shape. The Peruvian knotted string sys- number representations in Figure 4 is in tem is a P´ Q (base 10) system; the fact a cognitive taxonomy of numeration Chimpu (knotted strings used by the systems. For example, the Egyptian and Indians of Peru and Bolivia) is a Q´Q Cretan systems are in the same group at (base 10) system; the knotted string sys- the level of symbol representations; the tem used by the German millers is a S´S Mayan and Babylonian systems are in (base 10) system; the Roman counting the same group at the level of bases; the board, the Chinese abacus, and the Arabic, Greek, Chinese, Egyptian, Japanese Soroban are (Q´P)´P (main Cretan, and Aztec systems are in the base 10 and sub-base 5) systems, and the same group at the level of dimensional Russian abacus is a Q´P (base 10) sys- representations; and all the systems in tem. Figure 4 are in the same group at the The hierarchical structures of the level of dimensionality. Under this tax- numeration systems discussed above are onomy, the lower the level at which two based on the representations of small systems are in the same group, the more numbers (usually less than 1000). For similar they are. For example, the large numbers, the representational Egyptian and the Cretan systems are structures of some systems may have more similar to each other than the different forms. The change is mainly Arabic and the Babylonian systems, be- due to the representations of power di- cause the former two are in the same mensions. The representational struc- group at the level of symbol representa- tures of numeration systems with power tions whereas the latter two at the level dimensions represented by positions of dimensionality. usually do not change, because positions Although this cognitive taxon- can be extended indefinitely. However, omy was derived from the eleven sys- for those systems with power dimen- tems in Figures 2 and 3, it can classify sions represented by shapes, their repre- nearly all numeration systems that have sentational structures sometimes have to been invented across the world. Let us be changed to represent large numbers,

8 Zhang & Norman Number Representations

because practically, shapes can not be the values of individual symbols (e.g., extended indefinitely. For example, the the value of the arbitrary symbol "7" is Roman system is a (Q´S)´S system for seven), the addition and multiplication numbers up to one thousand. For num- tables, arithmetic procedures, etc., bers above one thousand, one version of which have to be retrieved from mem- the Roman system becomes a ory; and the external representations are ((Q´S)´S)´S system. A new power di- the shapes and positions of the symbols, mension represented by shape is added: the spatial relations of partial products, a horizontal bar ¾ is used to represent etc., which can be perceptually in- 103, and an open frame for 105. For spected from the environment. example, CCXXXVIIIrepresents 238, 000, Zhang & Norman (1994a; Zhang, and DCCCXXVII represents 82, 700, 000. 1992, 1995) developed a theory of dis- tributed representations to account for THE DISTRIBUTED the behavior in distributed cognitive REPRESENTATION OF NUMBERS tasksÑtasks that involve both internal and external representations. In this The Theory of Distributed view, the representation of a distributed Representations cognitive task is neither solely internal nor solely external, but distributed as a In complex numerical tasks, as well as in system of distributed representations many other cognitive tasks, people need with internal and external representa- to process the information perceived tions as two indispensable parts. Thus, from external representations and the external representations are intrinsic information retrieved from internal rep- components and essential ingredients of resentations in an interwoven, integra- distributed cognitive tasks. They need tive, and dynamic manner. External not be re-represented as internal repre- representations are the representations sentations in order to be involved in dis- in the environment, as physical symbols tributed cognitive tasks: they can di- or objects (e.g., written symbols, beads rectly activate perceptual processes and of abacuses, etc.) and external rules, directly provide perceptual information constraints, or relations embedded in that, in conjunction with internal repre- physical configurations (e.g., spatial re- sentations, determine people's behavior. lations of written digits, visual and spa- External representations have rich struc- tial layouts of diagrams, physical con- tures. Without a means of accommodat- straints in abacuses, etc.). The informa- ing external representations in its own tion in external representations can be right, we sometimes have to postulate picked up by perceptual processes. In complex internal representations to ac- contrast, internal representations are the count for the structure of behavior, representations in the mind, as proposi- much of which, however, is merely a re- tions, productions, schemas, mental im- flection of the structure of external rep- ages, neural networks, or other forms. resentations. The information in internal representa- In next section we apply the prin- tions has to be retrieved from memory ciples of distributed representations to by cognitive processes. For example, in analyze the representation of informa- the task of 735 ´ 278 with paper and tion in the basic structure of numeration pencil, the internal representations are systemsÑdimensions. In the later part

9 Zhang & Norman Number Representations

for multiplication, we will further ana- ordinal scales: a movie ranked Ò1Ó is lyze how the information needed to better than a movie ranked Ò2Ó carry out multiplication is distributed (magnitude) and the quality of a movie across internal and external representa- ranked Ò5Ó is different from that of a tions. movie ranked Ò7Ó (category). However, the rankings themselves tell us nothing The Distributed Representation of about the differences and ratios between Dimensions the rankings. Interval scales have three formal properties: category, magnitude, Dimensions are the basic structures of and equal interval. Time is an example numeration systems. Thus, it is impor- of interval scales: 02:00 is different from tant to understand how dimensions are 22:00 (category), 14:00 is later than 09:00 represented in numeration systems. (magnitude), and the difference between 15:00 and 14:00 is the same as between Psychological Scales 09:00 and 08:00 (equal interval). Every dimension is on a certain However, time does not have an abso- type of scale, which is the abstract mea- lute zero (in a realistic sense). Thus, we surement property of the dimension. cannot say that 10:00 is twice as late as Stevens (1946) identified four major 05:00. Ratio scales have all of the four types of psychological scales: ratio, in- formal properties: category, magnitude, terval, ordinal, and nominal. Each type equal interval, and absolute zero. has one or more of the following formal Length is an example of ratio scales: 1 properties: category, magnitude, equal inch is different from 3 inches interval, and absolute zero. Category (category), 10 inches are longer than 5 refers to the property that the instances inches (magnitude), the difference be- on a scale can be distinguished from tween 10 and 11 inches is the same as each another. Magnitude refers to the the difference between 100 and 101 property that one instance on a scale can inches (equal interval), and 0 inch means be judged greater than, less than, or the nonexistence of length (absolute equal to another instance on the same zero). For length, we can say that 10 scale. Equal interval refers to the prop- inches are twice as long as 5 inches. erty that the magnitude of an instance represented by a unit on the scale is the Distributed Representation of Scale same regardless of where on the scale Information the unit falls. An absolute zero is a value The base and power dimensions which indicates that nothing at all of the of all numeration systems are abstract property being represented exists. dimensions with ratio scales. In differ- Nominal scales only have one ent numeration systems, these abstract formal property: category. Names of ratio dimensions are implemented by people are an example of nominal different physical dimensions with their scales: they only discriminate different own scale types, which are usually entities but have no information about quantity (ratio), shape (nominal), and magnitudes, intervals, and ratios. position (ratio). Figure 5 shows the rep- Ordinal scales have two formal proper- resentation of base dimension in the ties: category and magnitude. The rank- Arabic and Egyptian systems. In Figure ing of movie quality is an example of 5, the abstract information space for

10 Zhang & Norman Number Representations base dimension contains the four formal representation with category informa- properties of ratio scales, since base di- tion represented externally by shape mension is on a ratio scale. In the and other three types of information Arabic system, the base dimension is represented internally in memory. In represented by shapeÑan external the Egyptian system, the base dimension physical dimension with a nominal is represented by quantityÑan external scale. Because a nominal dimension physical dimension with a ratio scale. only has category information, shape Thus, in this case, all four types of in- can only represent the category infor- formation of the ratio base dimension mation of the ratio base dimension in are represented externally (Figure 5B). external representations: the other three Similar to quantity, position is also a ra- types of information have to be repre- tio dimension. Therefore, if the base or sented in internal representations, that power dimension of a numeration sys- is, memorized in the mind (Figure 5A). tem is represented by position, its scale Thus, the representation of base dimen- information is all represented externally. sion in the Arabic system is a distributed

432 |||| ||| ||

External Internal External Internal Representation Representation Representation Representation

category category magnitude magnitude equal interval equal interval absolute zero absolute zero

category category magnitude magnitude equal interval equal interval absolute zero absolute zero

Abstract Abstract Information Space Information Space (A) Arabic (B) Egyptian

Figure 5. The distributed representation of dimensions. The scale information of a dimension is in the abstract 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 base dimension is represented externally by shape (a nominal dimension) but the other three types of information are represented internally in memory. (B) The representation of base dimension in the Egyptian system. All four types of information of ratio base dimension are represented externally by quantity (also a ratio dimension).

11 Zhang & Norman Number Representations

The information in external rep- Magnitude Comparison resentations can be picked up by percep- Let us use magnitude comparison tual processes, whereas the information as an example to illustrate how the in internal representations has to be re- above analysis of dimensional represen- trieved from memory. For example, in tations can be applied to simple numeri- Figure 5A for the Arabic system, the cal tasks. Comparing the relative mag- three symbols "4", "3", and "2" can be nitudes of numbers only requires the perceptually identified as three different magnitude information of the numbers. entities (category information). For a 1 D system, if the dimension is However, the information about represented by an ordinal or a higher whether "4" is smaller or larger than "3" dimension, the magnitude information (magnitude), whether the difference be- needed for comparison is external. For tween "4" and "3" is the same as or dif- example, the quantity dimension in the ferent from that between "3" and "2" Stone-Counting system is an external (equal interval), and whether "4" is twice representation of magnitude informa- as large as "2" (absolute zero) has to be tion. For a 1´1 D system, if both the retrieved from memory. In contrast, in base and power dimensions are on ordi- Figure 5B for the Egyptian system, all nal or higher scales (e.g., Russian aba- four types of information can be picked cus), the magnitude information needed up by perceptual processes: ||||, |||, and || for comparison is external, and if both are different entities (category), |||| is dimensions are on nominal scales (e.g., Greek system), the information is inter- larger than ||| (magnitude), the difference nal. Otherwise, it depends on the repre- between |||| and ||| is the same as the dif- sentation of each dimension. For ex- ference between ||| and || (equal interval), ample, for the Arabic system (a and |||| is twice as large as || (absolute Position´Shape system, ratio power di- zero). mension and nominal base dimension), The scale information of a dimen- the relative magnitude information of sion, however, is only relational infor- two numbers is external if the two num- mation: it does not specify the absolute bers have different highest power values values on the dimension. Thus, the rep- (e.g., 75 vs. 436) and internal if they have resentation of the absolute values on a the same highest power values (e.g., 75 dimension needs to be considered sepa- vs. 43). rately. The absolute values on base and power dimensions (base values and The Interaction between Dimensions power values) are represented internally by shapes but externally by positions Because most numeration sys- and quantities. For example, even if "4", tems have more than one dimension, it "3", and "2" can be perceptually identi- is also important to understand how the fied as three different entities, the abso- dimensions interact with each other. lute values these symbols represent The dimensions of a multi-dimensional have to be retrieved from memory. In stimulus can be either separable or inte- contrast, the absolute values represented gral. Separable dimensions are those by "||||", "|||", and "||" are external and can whose component dimensions can be be picked up by perceptual processes. directly and automatically separated and perceived. Integral dimensions can

12 Zhang & Norman Number Representations only be perceived in a holistic fashion: choose multiplication on the task side they can not be separated without a sec- and 1´1 D numeration systems on the ondary process that is not automatically representation side. Since numeration executed (Garner, 1974). systems are represented at four levels, Although the concept of separa- the representational properties at differ- bility usually refers to external physical ent levels can affect numerical tasks in dimensions, we use it to refer to internal different ways. Here we only focus on cognitive dimensions as well. If the in- the representational properties at the formation needed to separate two di- level of dimensional representations, mensions can be picked up by percep- since whether the dimensions of numer- tual processes from external representa- ation systems are represented externally tions, we say that these two dimensions and whether they are externally separa- are externally separable. For example, ble can dramatically affect the difficulty the shape (base dimension) and position of a numerical task. (power dimension) of the Arabic system are two physical dimensions that are ex- The Hierarchical Structure of ternally separable via perceptual pro- Multiplication cesses. If the information needed to separate two dimensions is not available Multiplication can be analyzed at three in external representations and can only levels (Figure 6): algebraic structures, be retrieved from internal representa- algorithms, and number representa- tions, we say that these two dimensions tions. At the level of algebraic struc- are externally inseparable and only in- tures, different multiplication methods ternally separable. In the Greek system, have different algebraic structures. The for example, the base and power di- algebraic structures of three typical mensions are represented by a single methods, simple addition, binary, and physical dimension (shape). They can polynomial methods, are as follows: only be separated internally. Since many numerical tasks (e.g., Simple Addition: multiplication and addition under poly- nomial representations) need to process N1´N2 = N1+N1+N1+...+N1 the base and power dimensions sepa- (add N2 times) rately, whether the dimensions of a nu- Binary: meration system are externally separa- N ´N = N ´åa 2i = åN a 2i ble or not can dramatically affect task 1 2 1 i 1 i difficulties. We will see this effect in Polynomial: i j i j next section. N1´N2 = åaix ´åajx = ååaix ajx i+j = ååaiajx A NUMERICAL TASK: MULTIPLICATION The simple addition method is simply adding the multiplicand the In this section, we apply the principles number of times indicated by the multi- of distributed representations to analyze plier. For the doubling method, multi- how representational formats affect plication is performed by (a) decompos- complex numerical tasks. To make our ing the multiplier into its binary repre- demonstration simple and clear, we sentation, (b) multiplying the multipli-

13 Zhang & Norman Number Representations cand with each term of the binary repre- nomial method, the one that includes sentation of the multiplier, and (c) the Standard algorithm we are using to- adding the partial products. Because day. The simple addition and the binary the terms of the binary representation methods, which can be reduced to addi- are either in the form of 1´2n or 0´2n, tion, were analyzed in Zhang (1992). the multiplication of the multiplicand with each term of the binary representa- The Polynomial Method of tion of the multiplier can be performed Multiplication by successively doubling the multipli- cand. For the polynomial method, mul- A numeral in a 1´1 D system is i tiplication is performed by multiplying represented as a polynomial: åaix . every term of the multiplicand with ev- Multiplication by the polynomial ery term of the multiplier and then method is performed by multiplying ev- adding the partial products together ery term of the multiplicand with every At the level of algorithms, differ- term of the multiplier and then adding ent algorithms can be applied to the the partial products together, regardless same algebraic structure. For example, of which particular algorithm (e.g., the the binary method can be realized by the standard or the Greek algorithm, see Egyptian algorithm and the Russian al- next section) is used. Thus, the two ba- gorithm (for details of these two algo- sic components of polynomial multipli- rithms, see Zhang, 1992), and the poly- cation are the multiplication of individ- nomial method can be realized by the ual terms and the addition of partial Standard algorithm and the Greek algo- products. Here we only show how rep- rithm, which are described in a later sec- resentational formats affect the pro- tion. At the level of number representa- cesses of term multiplication. (A de- tions, multiplication can be performed tailed analysis involving both term under different numeration systems. multiplication and partial product addi- Here we only analyze the poly- tion can be found in Zhang, 1992.)

A´B = C

Algebraic Binary Polynomial Structures Simple Addition

Algorithms Egyptian Russian Standard Greek

Number Representations Number Represenations

Figure 6. The hierarchical structure of multiplication.

14 Zhang & Norman Number Representations

For all 1´1 D systems, term mul- are certain constraints: Step 1 must be i j the first step and Step 4 must be the last tiplication (aix ´bjy ) has the same set of six basic steps (see Figure 7): one step step, but Steps 2a and 2b can either pro- (Step 1) to separate the power and base ceed or follow Steps 3a and 3b. The in- dimensions, two steps (Steps 2a and 2b) formation needed for each step can be to multiply the base values, two steps either perceived from external represen- (Steps 3a and 3b) to add the power val- tations or retrieved from internal repre- ues, and one step (Step 4) to combine the sentations. Figure 7 and the following power and base values of the final descriptions show the analyses of the six product. Although the six steps need steps for the Arabic, Greek, and not be followed in the exact order, there Egyptian systems.

i j Step Abstract: aix ´bjx Greek: l´d Egyptian: ÇÇÇ´|||| Arabic: 30´4 1 separate power & internal: external: external: base dimensions internally separable externally separable externally separable 2a get base values of internal: external: internal: i j aix & bjx ai, bj = shapes ai, bj = quantities ai, bj = shapes i B(aix ) = ai B(l) = g B(ÇÇÇ) = ||| B(30) = 3 j B(bjx ) = bj B(d) = d B(||||) = |||| B(4) = 4 2b multiply base internal: internal: internal: values multiplication table multiplication table multiplication table

ai´bj = cij g ´ d = ib |||´|||| = Ç|| 3´4 = 12 3a get power values internal: internal: external: i j of aix & bjx i, j = shapes i, j = shapes i, j = positions i P(aix ) = i P(l) = 1 P(ÇÇÇ) = 1 P(30) = 1 j P(bjx ) = j P(d) = 0 P(||||) = 0 P(4) = 0 3b add power values internal: internal: external: addition table addition table positions i j P(aix bjx ) P(l´d) P(ÇÇÇ´||||) P(30´4) i j = P(aix )+P(bjx ) = P(l)+P(d) = P(ÇÇÇ)+P(||||) = P(30)+P(4) = i+j = pij = 1+0 = 1 = 1+0 = 1 = 1+0 = 1 4 attach power internal: internal: external: values shapes shapes positions i j aix ´bjx l´d ÇÇÇ´|||| 30´4 pij 1 1 1 = cij´x = (ib)´10 = (Ç||)´10 = 12´10 = rk = ÇÇ = 120

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

15 Zhang & Norman Number Representations

Step 1: Separate power and base step is external. In the Greek and dimensions. The power and base di- Egyptian systems, since the sums of mensions are externally separable for power values have to be retrieved inter- the Arabic system (position and shape) nally from the mental addition table, the and the Egyptian system (shape and information needed for this step is in- quantity) but only internally separable ternal. for the Greek system (a single shape di- Step 4: Attach power values to i j mension for both power and base di- the product of base values: aix ´bjy = mensions). Thus, the information pij cij ´ x . The sum of power values of the needed for this step is external for the two terms need to be attached to the Arabic and Egyptian systems but inter- product of the base values of the two nal for the Greek system. terms to get the final result. The infor- Step 2a: Get base values of a xi i mation needed for this step is external and b yj: a , b . The base values are rep- j i j for the Arabic system because the sum resented internally in the Greek system of power values can be attached to the (shape) and the Arabic system (shape) product of base values by shifting posi- but externally in the Egyptian system tions. For the Greek and Egyptian sys- (quantity). Thus the information needed tems, the information needed to this for this step is internal for the Greek and step is internal because power values in Arabic systems but external for the both systems are represented internally Egyptian system. by shapes. Step 2b: Multiply base values: a i From the above analysis, we see ´ bj = cij. The information needed for that the information needed for the six this step is internal for all three systems, basic steps of term multiplication is dis- which need to be retrieved from an in- tributed across internal and external ternal multiplication table2. representations. Figure 8 shows the dis- i Step 3a: Get power values of aix tributed representations of term multi- j and bjy : i, j. The power values are rep- plication for the Greek, Egyptian, and resented externally in the Arabic system Arabic systems. If we assume that with (position) but internally in the Greek all other conditions kept the same, the system (shape) and the Egyptian system more information needs to be retrieved (shape). Thus, the information needed from internal representations, the harder for this step is external for the Arabic the task (e.g., due to working memory system but internal for the Greek and load), then for term multiplication, the Egyptian systems. Greek system (six internal steps) is Step 3b: Add power values: i + j harder than the Egyptian system (four = pij. In the Arabic system, since adding internal steps), which in turn is harder the power values of two terms can be than the Arabic system (two internal performed perceptually by shifting posi- steps)3. We need to note that difficulty tions, the information needed for this

3 A complete multiplication task involves not 2 The multiplication table can be externalized. just the multiplication of individual terms but For example, Napier's Bone is an external also the addition of partial products. When the representation of the multiplication table (see addition of partial products is considered, the Zhang, 1992). same difficulty order remains: Greek > Egyptian > Arabic (see Zhang, 1992).

16 Zhang & Norman Number Representations can be measured along many dimen- the Greek algorithm starts from the sions, some of which may have trade-off highest term of the multiplicand. The between each other. The amount of in- Standard algorithm groups the partial ternal and external information is only products that have common powers by one of the difficulty measures. their spatial relations (positions), which makes the addition of partial products Multiplication Algorithms for the easier. The Greek algorithm, however, Polynomial Method does not group partial products that have common powers and does not The polynomial method of multiplica- make use of spatial relations for the ad- tion can be realized by different algo- dition of partial products. rithms. Although different algorithms Under the same algorithm, the have the same set of six basic steps of Arabic system is more efficient than the term multiplication, they usually have Greek system because the former is different orders for the multiplication of more efficient than the latter for term individual terms and different group- multiplication. Under the same nu- ings for the partial products. Figure 9 meration system, the Standard algo- compares two algorithms: the Greek al- rithm is more efficient than the Greek gorithm used in ancient times by Greeks algorithm because the spatial grouping and the Standard algorithm used today of partial products in the Standard algo- by people in nearly all cultures. rithm makes the addition of partial The Standard algorithm starts the products easier. multiplication of individual terms from the lowest term of the multiplier, while

External Internal External Internal External Internal

1 1 1 2a 2a 2a 2b 2b 2b 3a 3a 3a 3b 3b 3b 4 4 4

1 1 1 2a 2a 2a 2b 2b 2b 3a 3a 3a 3b 3b 3b 4 4 4

Abstract Abstract Abstract

(A) Greek System(B) Egyptian System (C) Arabic System

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

17 Zhang & Norman Number Representations

Procedures Abstract Form Greek system Arabic system 1 0 Greek a1x a0x i z 10 7 1 0 Algorithm b1x b0x i g 10 3 ÑÑÑÑÑÑÑÑÑÑÑÑÑÑ ÑÑÑÑÑÑ ÑÑÑÑÑÑ 2 1 a1b1x a1b0x r l 100 30 1 0 a0b1x a0b0x o ka 70 21 ÑÑÑÑÑÑÑÑÑÑÑÑÑÑ ÑÑÑÑÑÑ ÑÑÑÑÑÑ 2 1 1 0 (a1b1x +a0b1x ) + (a1b0x +a0b0x ) ro na ska 170 51 221 1 0 Standard a1x a0x i z 1 7 1 0 Algorithm b1x b0x i g 1 3 ÑÑÑÑÑÑÑÑÑÑÑÑÑ ÑÑÑÑÑÑ ÑÑÑÑÑÑ 0 a0b0x k a 2 1 1 a1b0x l 3 1 a0b1x o 7 2 a1b1x r 1 ÑÑÑÑÑÑÑÑÑÑÑÑÑ ÑÑÑÑÑÑ ÑÑÑÑÑÑ 2 1 1 0 a1b1x + (a1b0x +a0b1x ) + a0b0x s k a 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 different aspects of the representational effect in numerical tasks. These levels In this article, we have explored the provide a cognitive taxonomy that can cognitive aspects of numeration sys- classify most numeration systems that tems, focusing on external number rep- have been invented across the world. resentations and their interaction with We suggest that the hierarchical struc- internal representations. We show that ture can be considered as a theoretical the information that needs to be pro- framework for systematic studies of cessed in complex numerical tasks is number representations. Among the distributed across internal and external four levels, the level of dimensional rep- representations. resentations is most important: whether We analyzed numeration systems the dimensions and their absolute val- at four levels: dimensionality, dimen- ues are represented externally or inter- sional representations, bases, and sym- nally and whether the dimensions are bol representations. The representa- externally or internally separable can tional properties at these levels can af- dramatically affect the difficulty of a fect the processes of numerical tasks in numerical task. Thus, when we analyze different ways and are responsible for the polynomial multiplication method

18 Zhang & Norman Number Representations for three 1´ 1 D systems (Arabic, representations; and processes specify Egyptian, and Greek) using the amount the actual mechanisms of information of information that has to be retrieved processing. These three aspects are from internal representations as a mea- closely interrelated: the same formal sure of problem difficulty, the Greek structure can be implemented by system is found to be the hardest among different representations, and different the three systems and the Arabic system representations can activate different the easiest. processes. Any study of distributed cognitive tasks has to consider all of A Cognitive Model of Distributed these three aspects. Our present study Numerical Cognition of numeration systems and numerical tasks, however, only focused on their Any distributed cognitive task formal structures and representations. can be analyzed into three aspects: In this section, we use the term formal structures, representations, and multiplication of Arabic numerals as a processes. Formal structures specify special case to outline a cognitive model the information that has to be processed of distributed numerical cognition with in a task; representations specify how several assumptions about the process- the information to be processed is ing mechanisms. represented across internal and external

Central Control

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

External Internal Representation Representation

separating dimensions 1 2a retrieving base values 2b retrieving multiplication table identifying positions 3a shifting positions 3b shifting positions 4 Perceptual Processes Memorial Processes

1 2a 2b 3a 3b 4

Abstract Task Space

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

19 Zhang & Norman Number Representations

The model is shown in Figure 10. tributed across internal and external The abstract task space specifies the representations can dramatically affect formal structure of the taskÑthe six ba- the difficulty of a task. The second as- sic steps of term multiplication, and the sumption is for processes: external rep- information that has to be processedÑ resentations activate perceptual pro- the information needed for the six basic cesses and internal representations acti- steps of term multiplication. The infor- vate memorial processes. Since percep- mation needed for the six basic steps is tual processes can be direct, automatic, distributed across internal and external unconscious, and efficient, their in- representations. For Arabic numerals, volvement can reduce the difficulty of a the information for Steps 1, 3a, 3b, and 4 task. The third assumption is for the is represented externally in the envi- central control, whose major function is ronment and the information for Steps the coordination of perceptual and 2a and 2b is represented internally in memorial processes. Although there memory (see also Figures 7 and 8). have been several studies on the inter- External information is picked up from play between perception and memory external representations by perceptual in terms of attention switching (e.g., processes. For Arabic numerals, these Carlson, Wenger, & Sullivan, 1993; perceptual processes include separating Dark, 1990; Weber, Byrd, & Noll, 1986), the base and power dimensions (Step 1), it is still unclear how perceptual and identifying the positions to get power memorial processes interact with each values (Step 3a), shifting positions to other. For instance, the reason we did add power values (Step 3b), and shifting not mention working memory in the positions to combine the base and central control is because it is not clear power values of the final product (Step to us how the framework of working 4). Internal information is retrieved memory (Baddeley, 1986; Baddeley & from internal representations by memo- Hitch, 1974) can fit here. Can percep- rial processes. For Arabic numerals, tion, especially the automatic and direct these memorial processes include re- kind, bypass working memory to di- trieving the base values from memory rectly participate in complex distributed (Step 2a) and retrieving multiplication cognitive tasks? In addition to the in- facts from the multiplication table in ternal working memory, is there a sepa- memory (Step 2b). The central control rate external working memory? If yes, coordinates the perceptual and memo- what are its functions and what is its rial processes, executes arithmetic pro- relation to the internal working mem- cedures, allocates attentional resources, ory? These questions are central to the and performs other processes that are understanding of the nature of dis- necessary for the completion of the task. tributed cognitive tasks. They deserve The first assumption of our systematic studies, both empirically and model is for representations: the infor- theoretically. mation needed for a numerical task is distributed across internal and external Is the Arabic System Special? representations. It has been shown (Zhang & Norman, 1994a; Zhang, 1995) Very few things in the world are as uni- that manipulating the information dis- versal as the Arabic numeration system.

20 Zhang & Norman Number Representations

It has often been argued that the place- nally or externally, the Russian abacus, value notation (position) of the Arabic whose base dimension (quantity) as well system is one of the most ingenious in- as power dimension (position) are both ventions of the human mind and it is represented externally, are superior. In what makes the Arabic system so spe- fact, abacuses are always superior to cial. Our analysis of the representa- written numerals for simple calculation tional properties of numeration systems such as addition and subtraction. But if allows us to examine the question of we consider other factors such as ease of why the Arabic numeration system is so reading and writing symbols, the Arabic special. system is still superior, because after Based on our analysis, the place- sufficient learning Arabic symbols can value notation is not sufficient to make be easily read and written. the Arabic system so special, since posi- At the level of bases, the base 10 tions were also used to represent the of the Arabic system is a manageable power dimensions in many other sys- size. There is a trade-off in base size: tems, including the Babylonian, Chinese larger bases require more symbols to be scientific, and Mayan written numera- learned but are more efficient in han- tion systems, and the abacus, counting dling large numbers. The addition and board, Peruvian knotted string, and multiplication tables that need to be other object numeration systems. Then, memorized for the Arabic system is rela- what properties of the Arabic system tively small in comparison with the make it so special? Let us first consider Babylonian system (base 60) and the the properties of the Arabic system at its Mayan system (base 20), which also use four levels of representations. positions to represent their main power At the level of dimensionality, the dimensions. Arabic system is a 1´1 D system, which At the level of symbols, the ten is more efficient for information encod- symbols of the Arabic system are easy to ing than 1 D systems. Although (1´1)´1 write, which makes calculation with pa- D systems (e.g., Babylonian, Mayan, per and pencil efficient. Roman, etc.) are as efficient as 1´1 D Combining the representational systems for information encoding, the properties at these four levels, the extra third dimension only introduces Arabic system is better than many other unnecessary complexity. systems in terms of representation. At the level of dimensional repre- Though the Arabic system is also better sentations, the base and power dimen- than many other systems (e.g., Greek, sions of the Arabic system are externally Egyptian, etc.) in terms of calculation, it separable, and the power dimension and is not always the most efficient one. As its power values are represented exter- discussed above, the abacus, which was nally by positions. Positions can handle still widely used in Japan, China, and large numbers efficiently. However, the Russia before electronic calculators be- base values and three of the four types came popular, is more efficient than the of scale information of the base dimen- Arabic system for addition and subtrac- sion are represented internally by tion. shapes. If we are only concerned with So what makes the Arabic system whether the base and power dimensions so special? Numerals have two major and their values are represented inter- functions: representation and calcula-

21 Zhang & Norman Number Representations tion. In many cultures, these two func- als are not only an efficient representa- tions are achieved by two separate sys- tion of numbers but also a symbolic tems. For example, in China, calculation form that can be easily operated upon. was carried out by abacuses and sticks, It is also interesting to note that Greeks, whereas representation was realized by who were highly advanced in geometry, written numerals. A more interesting did not develop an algebra in the mod- case is the Roman counting board, in ern sense. This is probably because which counters were used for calcula- Greek alphabetical numerals, though tion and Roman numerals were used for easy to manipulate, were neither effi- representation, both in the same physi- cient for representation nor for calcula- cal device. We propose that what makes tion. the Arabic system so special and widely accepted is that it integrates representa- Implications for Other tion and calculation into a single system, Representational Systems in addition to its other nice features of efficient information encoding, com- Although our present study has focused pactness, extendibility, spatial represen- on numeration systems, the methodol- tation, small base, effectiveness of calcu- ogy of representational analysis and the lation, and especially important, ease of principles of distributed representations writing. Thus, Arabic numerals are a illustrated here and elsewhere (Zhang & special type of object-symbols (Hutchins Norman, 1994a; Zhang, 1992, 1995) can & Norman, 1988; Norman, 1991)Ñsym- be applied to other domains as well. For bols that serve the functions of both rep- example, relational information dis- resentation and manipulation. The inte- plays, which include graphs, charts, gration of representation and calculation plots, diagrams, tables, lists, and other into a single system, however, would be types of visual displays that represent useless without appropriate media and relational information, are distributed tools such as paper and pencil: a nice representations; and the tasks for rela- example of technological constraints. tional information displays are dis- Interestingly, calculation and the Arabic tributed cognitive tasks. Zhang & numerals were so integrated that the Norman (1994b) analyzed relation in- word algorithm is merely a corruption of formation displays in terms of a hierar- Al Kworesmi, the name of the Arabic chical structure similar to that of nu- mathematician of the ninth century meration systems and developed a cog- whose book on Arabic numerals was the nitive taxonomy that can classify nearly first one reaching Western Europe. all types of relational information dis- Algebra, in the broad sense in plays and can serve as a theoretical which the term is used today, deals with framework for systematic studies of the operations upon symbolic forms, includ- cognitive properties in relational infor- ing numerals, unknowns, and arithmetic mation displays. As another example, operators. It has been argued (e.g., several cockpit instrument displays, Dantzig, 1939) that the invention of such as different types of altimeters and Arabic numerals was instrumental in navigation displays, were also studied the emergence and development of al- from the same perspective (Zhang, gebra. Our analysis of Arabic numerals 1992). supports this argument: Arabic numer- In addition to numeration sys-

22 Zhang & Norman Number Representations

tems, numerous representational sys- illustrations from early mathematical tems (usually called notational systems) cognition. Psychological Review, 100 have been invented over the last few (3), 420-431. thousand years in the development of Brooks, E. (1876). The philosophy of arith- mathematics (see the two volume book metic. Philadelphia, PA: Sower, by Cajori, 1928, on the history of math- Potts & Company. ematical notations). Of these systems, Cajori, F. (1928). A history of mathematical only a fraction survived. Although po- notations (Volumes 1 & 2). La Salle, litical, economic, social, and cultural fac- IL: The Open Court Publishing tors certainly played some roles in the Company. evolution of representational systems, Carlson, R. A., Wenger, J. L., & Sullivan, cognitive factors might have played the M. A. (1993). Coordinating informa- most important role since the activities tion from perception and working of individuals in mathematics are es- memory. Journal of Experimental sentially cognitive activities. How the Psychology: Human Perception and forms of representations affect the cog- Performance, 19 (3), 531-548. nitive activities of scientists and what Dantzig, T. (1939). Number: The language roles they played in the evolution of of science. New York: The mathematics in particular and science in MacMillan Company. general are interesting issues worth of Dark, V. J. (1990). Switching between the attention of not just historians and memory and perception: Moving at- philosophers but also psychologists and tention or memory retrieval? Memory cognitive scientists. & Cognition, 18 (2), 119-127. Dehaene, S. (1993). Numerical cognition. ACKNOWLEDGMENT Blackwell Publishers. Flegg, G. (1983). Numbers: their history We are very grateful to two anonymous and meaning. New York: Schocken reviewers for valuable comments and Books. constructive criticisms. Request for Garner, W. R. (1974). The processing of reprints should be sent to Jiajie Zhang, information and structure. Potomac, Department of Psychology, The Ohio Md.: Lawrence Erlbaum Associates. State University, 1885 Neil Avenue, Heath, T. (1921). A history of Greek math- Columbus, OH, 43210, U.S.A. Email: ematics (Vol. 1): From Thales to Euclid. [email protected], Oxford: The Clarendon Press. [email protected]. Hutchins, E., & Norman, D. A. (1988). Distributed cognition in aviation: a con- REFERENCES cept paper for NASA (Contract No. NCC 2-591). San Diego: University Baddeley, A. D. (1986). Working memory. of California, Department of Oxford: Oxford University Press. Cognitive Science. Baddeley, A. D., & Hitch, G. (1974). Ifrah, G. 1987. From one to zero: A uni- Working memory. In G. Bower (Ed.), versal history of numbers . New York: Recent advances in learning and moti- Penguin Books. vation. New York: Academic Press. Menninger, K. W. (1969). Number words Becker, J., & Varelas, M. (1993). Semiotic and number symbols: A cultural history aspects of cognitive development - of numbers. Cambridge, MA: The

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