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Counting, Computing, and the Representation of Numbers Raymond S. Nickerson Human Factors: The Journal of the Human Factors and Ergonomics Society 1988 30: 181 DOI: 10.1177/001872088803000206 The online version of this article can be found at: http://hfs.sagepub.com/content/30/2/181

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HUMAN

FACTORS,

1988,30(2),181-199

Counting, Computing, and the Representation of Numbers RAYMOND S. NICKERSON,I BBN Laboratories Incorporated, Cambridge, Massachusetts

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

INTRODUCTION Counting and computing are such common and fundamental activities in modern society that it is difficult to imagine what life would be like without them. It may be that the ability to count and to reckon are as old as humanity itself; however, there can be no doubt that the range of useful and interesting things that can be done with numbers has increased greatly, albeit gradually, over the history of humankind. Most of what we know about the development of number concepts and number manipulation skills comes from written records I Requests for reprints should be sent to Raymond S. Nickerson, BBN Laboratories. Inc., 10 Moulton St., Cambridge. MA02238.

and so goes back only as far as do those records. which is roughly to about 4000 to 5000 B.C. There is some evidence that the use of tokens to represent number concepts predated the use of written symbols-the oldest known remnants of which are attributed to the Sumerians of Mesopotamia-by perhaps as much as 5000 years (Schmandt-Besserat, 1978). The roots of counting and computing undoubtedly stretch farther back into antiquity than this. however, and are probably forever hidden in the mists of prehistory. Such evidence as we do have regarding the origin of number concepts suggests that for a very long time numbers were thought of as properties of the things with which they were associated in counting (Menninger, 1969). Concepts such as "3 sheep" and "3 goats," for

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example, appear to have predated the more abstract concept "3." Given an object-specific conceptualization of numerosity, one could understand that 4 sheep are more than 3 sheep and that 4 goats are more than 3 goats while being unable to make a numerical comparison between sheep and goats. Although Babylonian and Egyptian texts dating from the second millennium B.C. show that some aspects of arithmetic were well developed in ancient Babylon and Egypt (solutions of quadratic and cubic equations, tables of squares, cubes, and reciprocals), what existed at that time was not a mathematical theory of numbers but a collection of solutions for practical problems and rules for calculation (Aleksandrov, 1963). The idea of numbers as entities with interesting properties in their own right independently of the practical problems to which they could be applied was one that developed gradually over many centuries. The idea got a very significant push forward by the fertile minds of such Greeks as Archimedes, Euclid, and Pythagoras. One interesting aspect of the gradual increase in numerical sophistication relates to the evolution of the symbols and symbol systems by which n,umber concepts have been represented visually. The burden of this paper is that the relative ease with which one can count or compute depends to no small degree on the characteristics of the system that is used to represent number concepts. The so-called Arabic system, which is now used almost universally, is so familiar to us that we take it for granted. It is easy to see this way of representing numbers as the way to do it and to fail to recognize it as the convention it is. Why has this representational scheme proved to be so useful, and at what cost to the user has this usefulness been obtained? These are the questions that motivate this paper. The discussion begins with a fanciful but plausible account of how a represen-

FACTORS

tational scheme with many of the properties of the Arabic system might have evolved. How the Arabic notation instantiates the principles that are seen in this conjectural scheme is then considered. Some ancient systems are compared. Several properties desirable in any number system are identified, and how the Arabic system compares with its predecessors wi th respect to these properties is noted. These comparisons reveal that the Arabic system is more conducive to both counting and computing and is far more versatile than its predecessors. The power of the system stems primarily from the fact that it provides an efficient way to represent quantities of all magnitudes and simplifies the performance of mathematical manipulations. In several ways the Arabic system is more abstract than its predecessors, however, and the degree of correspondence between the symbols used to represent quantities and the quantities they represent is less direct. The abstractness of the system obscures some of the underlying principles on which it is based, such as the principle of one-for-many substitution and that of a base or radix of arbitrary size. In short, the system is far more powerful than any of its predecessors but may be somewhat more difficult to learn. A FANCIFUL ACCOUNT OF THE ORIGIN OF PLACE NOTATION Imagine that you lived in the days before numbers or the operation of counting had been invented. If the most demanding numerical problem with which you ever had to deal was to tell if all four of your children were home at night, you probably would not need to know how to count. Most people are able to apprehend directly a small number of items-perhaps as many as six-without counting. In McCulloch's (1961) terms: "The numbers from 1 through 6 are perceptibles; others, only countables" (p. 7). Psychologists

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refer to the direct apprehension of quantities as subitizing and distinguish it from counting or estimating. Suppose, however, that you were a prosperous shepherd with a sizable flock and you wanted to know whether you were losing or acquiring sheep from day to day. Large changes in the size of your flock would probably be detected without any counting operation, but you want to know when the size of your flock has been increased or diminished by even a single sheep. One thing you might do is get some stones and, as your sheep file into the sheepfold, put a stone in a special pile each time a sheep goes by. If you did this, you would be performing a one-for-one mapping operation, which is the essence of a tally system. You would be mapping the stones onto the sheep in such a way that the quantity of stones could be used to represent the quantity of sheep. The next time your sheep were put into the fold, you could take a stone away from your pile as each sheep went by and if, after the sheep were all in, you had stones left over, you would know that you had lost some sheep. If you ran out of stones before the sheep were all in. you would know that you had acquired some sheep in addition to those you had before. This scheme would not tell you how many sheep you had (presumably you could not count the stones if you could not count the sheep), but it would suffice to let you know whether you had lost or acquired sheep between successive tallies. One of the problems with this scheme is that if you had many sheep, you would need a pile of stones sufficiently large that it would be inconvenient to carry around. You might solve that problem by getting a fairly small pile of stones to serve as a standard pile, and use this pile to guide the construction of several other piles. You would want some way to distinguish the several piles you would make, and you might do that on the basis of, say,

color. To be specific, you might decide to use white stones, black stones, and red stones and to make your standard pile of speckled stones. You might then tally your sheep in the following way. Each time a sheep entered the fold, you would add a stone to the white pile. However, you would not let the pile of white stones get indefinitely large-in fact, you would never let it contain more stones than your standard pile. As soon as your white pile contained as many stones as the standard pile (which you would determine by using the one-for-one mapping procedure just mentioned), you would take them all away and add a stone to the pile of black stones. That is, you would let one black stone represent a whole pile of white stones. Then you would start again rebuilding the white pile and continue as before until it again contained as many stones as the standard pile, at which time you would again take it away and represent it with an additional stone in the black pile. Similarly, when the black pile eventually contained as many stones as the standard pile, you would take it away and represent it by an addition of a stone to the red pile. You would simply reverse this operation when "tallying down" instead of "tallying up." If you were really clever, you might not bother with that standard pile of speckled stones but instead use for a standard something that just happened to be readily available all the time, such as your fingers. If you did that, it would turn out that one red stone would represent ten black ones, each of which would, in turn, represent ten white ones. Further, if you were unable to find a sufficient quantity of colored stones, you might use an alternative scheme. You might, for example, attach some significance to the spatial arrangement of the piles. That is, you might decide to arrange the piles in a row and let one stone in any pile represent a "full set" of the stones in the pile immediately to

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its right or left. Although this account of how a number system might have evolved is a fanciful one, a method close to the conjectured stone tallying has been observed in use in relatively modern times (Conant, 1956, p. 436). It is a very short step conceptually from piles of stones to tally marks in the sand or on a piece of paper. It is a much larger step from a set of tally marks to a symbol that represents a quantity. However, the practicality of substituting a single symbol for a set of tally marks is clear; a symbol such as "8," for example, is far easier to write than is "11111111." If you invented a unique symbol to represent each of the possible quantities of stones (including the case of no stones) up to that of your standard and decided to use the position of that symbol to indicate the pile to which it refers, you would have what we refer to today as a place-notation system for representing numbers. This intellectual odyssey from the most primitive concept of quantity to a number system that uses place notation was not made, of course, by an individual. Indeed, the scheme for representing quantities that we use today-which we take very much for granted, and are likely to find singularly unimpressive-was many millennia in the making. Each of the principles on which it is based-one-for-one mapping, the use of a standard or base quantity, one-for-many substitution, the use of symbols to represent quantities, the notion of a symbol for representing an empty set, the use of position to carry information-constituted a major intellectual achievement. The history of the evolution of number systems is not best represented as a linear sequence of discoveries or innovations; different representational schemes have existed in different parts of the world at the same time, each reflecting some subset-but not necessarily the same subset -of the principles on which our current sys-

FACTORS

tem is based. But given that one system is today the lingua franca for number representation throughout the world, all the ancient systems may be thought of as way stations on some path-though not always the same path-to a common destination. I do not wish to suggest that the appearance of our current system marked the end of the evolution of number representation schemes, but it is of more than passing interest that this system has gained such wide acceptance and has not changed appreciably in a rather long time. NUMBERS AS ABBREVIATED POLYNOMIALS Although the system that we use for representing numbers is usually referred to as the Arabic system, its place of origin, though not known for certain, is believed to be India. It was introduced to Europe by the Arabs during the tenth century A.D. and for this reason became known to Europeans as the Arabic, or sometimes Hindu-Arabic, system. The scheme that this system uses for representing numbers is analogous in many ways to the rock-pile system that was presented earlier. In this scheme the value of a number is determined by four things: (1) The symbols (digits) that constitute the number. (2) The order in which the digits are arranged. (3) The base (or radix) of the system being used. (4) The position of the "point." (When the point is omitted, it is assumed to follow the rightmost digit.) The symbols correspond to the number of stones in the different piles; the positions of the digits serve to label the piles, as it were; and the base corresponds to the number of stones in the standard pile. The "point" has no analogue in the stone system. Its use makes this system more general and permits us to represent fractional as well as integer quantities. We should also note that the stone system described earlier is not really a

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number system; even given the use of a onefor-many substitution principle and of position as an information carrier, it remains only a tally system and does not by itself permit one to say how many of anything one has. Normally the third factor in the foregoing list is not an issue, inasmuch as we assume that the number is a decimal number, which is to say that it is written to the base ten. When there is the possibility of confusion, the base may be identified explicitly with a subscript. Thus although 743.1510 and 743.158 have the same digits in the same order and the point in the same position, the values of these numbers are different inasmuch as one is wri tten to the base ten and the other to the base eight. It will be helpful to abandon our rock-pile analogy at this point in favor of a general representation of what an Arabic number is and, in particular, for a representation that will accommodate numbers with fractional parts. Consider the string of digits 743.1510, What one means to signify with such a string is a quantity equal to 7 hundreds plus 4 tens plus 3 ones plus 1 one-tenth plus 5 one-hundredths; that is, (7 x 100) + (4 x 10) + (3 x 1) + (1 x .1) + (5 x .01), or equivalently, (7

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

Nr = anrn + an_Irn-1 + ... + alrl + aorO + a_lr-I + a_2r-2 + ... +a_mr-m. In other words, in the Arabic system a number is a kind of shorthand representation of a polynomial in r, where r is the base of the system. Specifically, from left to right, the digits are the coefficients of terms involving successively decreasing powers of the base. Representing a number as a coefficient of a polynomial in r is an enormous advance over a simple one-for-one tally system, no matter what r is. With r equal to 10, as it is in our familiar decimal system, it takes 7 digits to represent the quantity 1 million. If r equaled 20, it would take 5; if it equaled 2, it would take 20. With a one-for-one tally system, however, it would take 1 million. EGYPTIAN, GREEK, AND ROMAN NUMBER SYSTEMS The elegance and convenience of the system we use to represent numbers today is perhaps best appreciated when we compare this system with others that were used before it was invented. The ancient Egyptians, Greeks, and Romans all used systems that were similar to ours in some respects but differed significantly from it in others. Here is how the quantity that we represent as 2 4 3 3 2 would have been represented in each of those systems:

X

Similarly, 743.158 represents the quantity (7

Egyptian:

Hlm~~~IlM"

Greek: MMXXXXHHHM~II Roman: @@Q)Q)ill(DCCCXXXII

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

If we denote number by

the successive

digits

in our

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

The Egyptian, Greek, and old Roman systems were very similar in several respects. They all used a one-for-many substitution principle in much the same way: one symbol of a given type was used to represent the same quantity as was represented by several symbols of another type. The number of symbols in a lower register represented by a sym-

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186-April

HUMAN

1988

bol in the next higher register was lain aU cases. (The term "register" as used in this paper corresponds roughly to place: the third digit to the left of the decimal point in an Arabic number will be said to be in the third register; in an Egyptian, Greek, or Roman number, the set of symbols representing hundreds will be said to be in the third register.) Thus in the Egyptian system one" was equivalent to ten I 's, one ~ was equivalent to ten (\ 's, and so on. Table 1 shows the substitutionary equivalences for these three systems for groupings through 10,000. It is of some interest to note that aU three of these systems used a single vertical line to represent the quantity one, and that the first few numbers in each case were represented by a tally of such lines. In this regard the Egyptian, Greek, and Roman systems are prototypical of many if not most of the early number systems of the world. The ubiquity of the single-line representation of the quantity

TABLE 1 Symbols Used in the Egyptian, Greek, and (Old) Roman Number Systems Egyptian 1'5 10'5 100'5 1,000'5 10,000'5

I

Greek

Roman

I

I

6.

X

")

H

1

X

C CD

(

M

@)

n

Note: Some sources show the Egyptian symbols for 100's, 1000's end 1O,OOO'sas mirror images of those shown here. The Greeks, like the Hebrews, also used the letters of the alphabet to represent numbers. matching the letters to numbers in sequential fashion: A for 1, B for 2, r for 3. and so on. The system that evolved over time still made use of the letters but in a different way. When the one·formany substitution principle was adopted, lellers were used to represent the sets of different sizes and the letter chosen in each case was the first letter of the name of the associated number. Thus the letter for 10 was a. for a.EKA (from which the English decimal, decile. decathlon): for 100. H for HEKATON (from which hectometer. hectogram, hectare); for 1000. X for XIAIOI (from which kilometer. kilogram, kilowatt); and for 10,000, M for MYPIOI (from which myriad). The Egyptians wrote right to left. The Greeks. depending on the era, wrote right to left, left to right, or in boustrophedon style (left to right and right to left on alternate lines). For convenience, symbols are always ordered in this paper from left to right.

FACTORS

one (which in some cases is written vertically and in some, horizontally) and the prevalence of the use of a set of such symbols to represent the first few numbers are usually assumed to be consequences of the widespread practice of finger counting, which presumably predated other ways of representing quantities, perhaps by many millennia. Our use of the term "digits" to denote numerals as well as fingers and toes is suggestive of this close relationship between finger counting and the ways in which we represent (small) quanti ties symbolically. Independently of its resemblance to an extended finger, the use of a line to represent the quantity one, or of a few lines to represent other small quantities, has the distinct advantage of facilitating the process of writing numbers down, especially in media such as clay or stone. Remnants of the common ancient practice of representing the first few numbers as tallies of "ones" are seen in several systems, including Egyptian, Greek, Roman, Chinese (both ancient and modern scientific), Babylonian, Indian Kharosti, Indian Brahmi, and Mayan. Even the first few numerals of the Arabic system, it is sometimes assumed, are derivatives of this notation. Something close to 2 and 3 are what might be produced if one wrote = and == hurriedly without lifting the writing instrument from the writing surface between strokes. Some systems (e.g., Indian Brahmi, and some Chinese) do in fact represent the first three numbers as _, =, and The Egyptians actually had two number systems, the Hieroglyphic (described earlier) and the Hieratic (Aleksandrov, 1963), The symbols for two, three, and four in the Hiera-

=.

tic notation ( 'i' "'i' and ""1) also resemble what might be produced if one made two, three, or four tallies without removing one's pen from the paper. Another common feature of the Egyptian, Greek, and Roman systems is the fact that

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the tally principle was used with every symbol class. That is, for any given symbol, the number of times the quantity represented by that symbol was to be counted in the number was indicated by the number of times the symbol that represented that quantity appeared in the number's representation. Thus the appearance of .,':)~ in an Egyptian number indicated that the number contained three hundreds. Although the numbers were generally written with all the symbols of a certain type grouped spatially and with the groups arranged from higher order to lower order, the reading of a number as a quantity did not depend on this arrangement. That is, with an exception to be noted later, the positioning of the symbols with respect to one another carried no information. Thus although in Egyptian the number 2 4 1 3 would normally have been wri tten as

it would still be unambiguous, though somewhat more difficult to read, if written as

The Greek and Roman systems had a feature that the Egyptian system lacked: the use of special symbols to represent half-ten or five groupings, an innovation that decreased the number of symbols required to represent any number. In the Roman system the symbols for five groupings ( V for 5, L for 50, D for 500) did not relate to the other symbols in any consistent way, though in the Greek scheme they did. In the latter case groupings of five were represented by combining the lower-order symbol with a standard symbol that would be interpreted as "five of." Thus the symbol for 50 was a combination of r (five) and [:,.(ten) yielding 1"', which could be read as five tens. Similarly, the symbol for 500 was a combination of rand H yielding /" , which could be read as five hundreds.

Thus in the Greek system five-grouping representations could be generated by rule, whereas in the Roman one they had to be learned by rote. At some time the Roman system introduced another principle, which was to represent diminution of the value of a higher-order symbol by preceding it with a symbol of immediately lower order. Thus 9 became 10 diminished by one, or IX; 90 became 100 diminished by 10, or XC; the diminution principle was used also with five-grouping symbols, so that 4 was represented as ,40 was represented as XL, and so on. This innovation accomplished an economizing of the number of symbols that had to be written. It also made posi tion functional, as a carrier of information that is required to decode the symbol unambiguously: for example, IX and XI represented quite different quantities. THE BABYLONIAN SYSTEM The Babylonian cuneiform number system differed from the Egyptian, Greek, and Roman systems in several important ways. All numbers were represented by combinations of only two symbols: T and ( , both of which were easy to inscribe in soft clay with a wedge-shaped stylus. Unlike the Egyptian, Greek, and Roman systems, the Babylonian one was a true place-notation system: the value represented by a symbol depended on the location of that symbol in the collection of symbols representing the number. The system is usually referred to as a sexigesimal system because a T and a < in one register were equivalent in value to 60 T and 60 ( , respectively, in the next lower register. A 1-for-60 substitution was not used, however. Instead, a two-stage substitution principle was used: 1 (. was substituted for 10 T in the same register, and 1 T was substituted for 6 < in the next-lower register. Thus the Babylonian equivalents of the Arabic numbers 3, 24, and 175 would be as follows:

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HUMAN

188-April1988

3

HI

24

«THT

175

pose. Exactly when the 0 was introduced is not known, but the first-known Indian manuscript in which 0 appears dates from the latter part ofthe ninth century.

n«

nil

nI

IIm{J'RRHW

Any "carrying" operations can then be performed by the appropriate one-for-many substitutions in the sum. In this case, one I would be substituted for ten 9 and the sum rewritten as m:>"nll

l'?~nnn

lIn?RAR1N l~nnn Subtraction is equally easy. What we refer to, somewhat enigmatically, as "borrowing" is

accomplished by making appropriate manyfor-one substitutions in the minuend to accommodate those instances in which the number of symbols of a given type in the subtrahend is greater than the number of symbols of the same type in the minuend. After the necessary substitutions have been accomplished, the subtraction can be done by taking the difference independently for each symbol type. Not only are addition and subtraction straightforward in the Egyptian system (and relatively so in the Greek and Roman as well), but there is nothing mysterious about the operations of "carrying" and "borrowing," as there sometimes seems to be when students learn how to do arithmetic with the Arabic system. Many of the errors that children make when subtracting one multidigit Arabic number from another can be emulated by "buggy" algorithms that have been designed to apply operations in systematic ways that do not conform to the rules of subtraction (Brown and Burton, 1978; Brown and VanLehn, 1982). An example of a bug that accounts for some errors children make is to subtract always the smaller digit from the larger independently of whether the smaller digit is in the minuend or in the subtrahend. Another is to increment, rather than decrement, a digit when borrowing. An aspect of this research that is particularly interesting in the present context is the fact that extensive analyses of the systematic subtraction errors children make reveal only a subset of the bugs that it would be logically possible to invent. This finding raises the question of the extent to which the types of bugs that are invented depend on the properties of the representational system used. One plausible conjecture is that the more abstract the system-the longer and more circuitous the path from the characteristics of the symbols to the properties of the quantities they represent-the greater the room for inven-

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tion of bugs. This is because the more arbitrary the symbols, the easier it will be to treat them as symbols per se than as surrogates for quantities, and the less obvious will be the inappropriateness of specific buggy opera tions. The concepts of carrying and borrowingwhich really are misleading-are not needed and do not arise with some of the ancient systems, such as the Egyptian, Greek, and Roman. It is apparent from the symbology that the operations required are a one-formany substitution in one case and a manyfor-one substitution in the other. The terms "carrying" and "borrowing" were presumably introduced into the vocabulary of arithmetic to facilitate a child's learning how to add and subtract multidigit Arabic numbers. One can argue, however, that the terms themselves, while possibly facilitating the rote" learning of a procedure that works, may help to obscure the principles from which the operations derive. The term "carrying" provides no hint of the fact that what is involved is the substitution of one unit of one type for several units of another, and the term "borrowing" is worse than uninformative because it suggests a transaction that is incomplete until whatever is borrowed is paid back. Multiplication and division are not nearly as easy to accomplish with the Egyptian, Greek, or Roman system as with the Arabic. Conceptually, multiplication is not difficult in these systems. To multiply It?"nll by l??nnn, for example, one need only write down U9?51nl/ l~nnn times and then make the necessary one-for-many symbol substitutions. Mechanically, however, the process is prohibitively tedious even on paper, let alone on any less convenient medium. When multiplication and division were done, they presumably were accomplished by means of successive additions and subtractions. There were some shortcuts, however; the Egyptians, for example, some-

FACTORS

times multiplied by successive doublings and then by summing the appropriate subset of those doublings (Newman, 1956). To multiply 456 by 26 using this method, one would proceed as follows. First, calculate a succession of doublings until the sum of the doubling indices is at least as large as the multiplier, thus:

Doublings

Doubling Indices

456

1

912 1824 3648 7296

2 4 8 16

Cumulative Sum of Doubling Indices 1

3 7 15 31 (>26)

Then add only those terms associated with the subset of doubling indices that sums exactly to the multiplier-that is, 2 + 8 + 16 = 26. 456 912

--t

t8Z4

-4 8

3648

2

7296

16

11856

26

The easy way to find the subset of indices that sums to the multiplier is to begin with the largest index and add successively smaller indices, eliminating any that would cause the sum to exceed the multiplier. Although fractions were known to the ancients, they presented special problems, and the symbol systems did not facilitate their use. Various methods were evolved for dealing with them. With the exception of 213, for which they had a special symbol. the Egyptians expressed all fractions as a series of fractions having 1 as a numerator. A sign indicating addition was not represented explicitly; however, the sum of the fractions in the series represented the fraction of interest. Thus, for example, % would be written '12, '14

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NUMBER REPRESENTATION

(except, of course, with Egyptian symbols rather than Arabic). The first part of the Rhind Papyrus (Newman, 1956) gives a table showing how to represent 2 divided by odd numbers from 3 to 101. The Greeks also expressed all fractions as sums of those with 1 as the numerator, and the Romans expressed all fractions as twelfths (Jourdain, 1956). None of these methods comes close to providing the versatility of the notation we use today. The thought of trying to do algebra or higher mathematics with systems such as the Egyptian, Greek, or Roman is somewhat daunting. Indeed, it seems safe to assume that mathematics, and those sciences heavily dependent on mathematics, could not have progressed if systems with greater versatility and power had not evolved. Progress required systems that not only provided more convenient and compact ways of representing the number concepts that the ancients understood but that would also facilitate the invention of new concepts-such as those of negative, imaginary, and complex numbers -that have proved so important in both pure and applied mathematics. In the Babylonian and Mayan systems, addition and subtraction also were straightforward. As in the Egyptian, Greek, and Roman systems, the sum of two numbers could be formed by first taking the union of the symbols representing the numbers and then performing whatever one-for-many substitutions were required to get the sum in standard form. Figure I, for example, shows how the two decimal numbers 2773 and 2256 could be added with the Mayan notation. In Step 1 the union of the addends is taken register by register: thus ~ + =i= in the l's registers of the addends becomes § in the 1's register of the sum. After each of the unions has been taken, the one-for-many substitutions are made so there are no more than three bars or four dots in any register. Four of the

April 1988-195

bars in the 1's register, in our example, would be replaced by a single dot in the 20's register. In making the one-for-many substitutions, it is convenient to begin by substituting a bar for any set of five dots within registers, and then-working from the lower- to the higher-order register-substituting for any set of four bars within a register a dot in the next-higher register. Figure 1 shows the one-for-many substitution step broken down into a sequence of substeps. Sometimes it may be necessary to iterate one or more of these substeps, as when, for example, the substitution of a dot in the nth register for four bars in the (n - 1)th register brings the number of dots in the nth register to five, which would require another within-register substitution of a bar for five dots. Alternatively, one can perform the operation on a register-by-register basis. In this case one makes the necessary one-for-many substitutions immediately after (or in the process of) adding the contents of two registers. Thus the sum of == + would be written as ~ and a . would be "added" to the next higher register. The fact that the latter step, which is equivalent to the carry operation with which all users of Arabic notation are familiar, is a one-for-many substitution is apparent. Subtraction with the Mayan notation is, of course, the inverse of addition and is nearly as straightforward. The "borrowing" operation, which must be performed whenever the quantity represented by a register in the subtrahend is greater than that represented by the corresponding register in the minuend, is simply a many-for-one substitution whereby either a bar is replaced by five dots in the same register or a dot is replaced by four bars in the next lower-order register. Apparently there is no compelling evidence that the Maya knew how to multiply and divide; however, the representational system lends itself readily to these operations (Lam-

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=

HUMAN

196-April1988

Step 1 (unionof addends)

•••

a

20's

400's

• -

•••

• ===

I•••••

b

••

= +

l's=

Step 2 (one-for-many substitution)

-•

400's~

-

•••••

•••• -

-

••

-

• ••••

•

••

~---

•

---

20'5--

•••• l's

FACTORS

====

••••

••••

• •••

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

bert, Ownbey-McLaughlin, and McLaughlin, 1980). In this respect it is quite unlike the systems used by the Egyptians, Greeks, and Romans. Multiplication requires three simple product rules and rules for determining the register of a product from the registers of its factors. The product rules are (1) dot x dot = dot; (2) dot x bar = bar; and (3) bar x bar = bar in one register and dot in the nexthigher register. The rules for determining the register of a product from the registers of its factors can be combined with these three product rules to yield the following set, where the symbols in parentheses represent the registers of the factors and product:

+ n - 1) (m + n - 1) (m + n) + - (m

(1) • (m) x • (n) = • (m

(2) • (m) x - (n) (3) - (m) x - (n)

=-

=•

+n-

1)

Lambert, Ownbey-McLaughlin, and McLaughlin (1980), from whom this notation was adapted, describe the process this way: In calculating the product, each element of the multiplicand must be multiplied by each element of the multiplier, register by register. For each dot-times-dot operation, add together the number of the registers occupied by each of the dots, and place a dot in the product register that is one beneath this total. For each bar-times-dot operation, do the same thing, but place a bar in the product stack. For each bar-times-bar operation, add the number of the register of each of the bars, and place a dot in the register corresponding to this sum and a bar in the register just beneath it. When 5 dots accumulate in a given register, they are replaced by a bar in the same register; when 4 bars accumulate in a register, they are replaced by a dot in the next higher register. (p. 253)

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April 1988-197

NUMBER REPRESENTATION

The procedure is quite simple and a few practice multiplications are likely to be enough to make one comfortable with it. Division, which, of course, is the inverse of multiplication, is also relatively straightforward and easily learned. Most of what has been said here about the Mayan system could also be said (with appropriate substitutions) about the Babylonian one. It too makes addition and subtraction trivially easy and multiplication and division quite manageable. Moreover, the Babylonians are known to have developed considerable computational skills (Menninger, 1969). The greater suitability of the Babylonian and Mayan systems to computation beyond addition and subtraction results, at least in part, from the fact that they are place-notation systems. One of the consequences of this way of representing numbers is that the rules of multiplication, division, and other operations can be relatively succinct and general; that is, a few of them suffice. The possibility of simple rules follows from the fact that the same symbols are used in every register. To do multiplication, for example, one need only know the products of all possible combinations of the individual symbols and how the place(s) of a product of two of these symbols depends on the places of those symbols in the multiplier and the multiplicand. How does the Arabic system compare, with respect to computational convenience, with the others we have considered? Except perhaps for simple addition and subtraction by a beginner, it is clearly superior to the Egyptian, Greek, and Roman systems, for much the same reasons that the Babylonian and Mayan systems are. It too is a place-notation system and shares the advantages of other place-notation systems in this regard. A comparison of the Arabic system with the Babylonian and Mayan systems with respect to computational convenience, however, re-

veals some interesting trade-offs. It may be easier to learn the rules of multiplication and division in the Babylonian and Mayan systems than in the Arabic: there are fewer symbols to contend with, and the rationale for the rules determining the registers of products and quotients is straightforward. By contrast, the algorithms for multiplication and (especially) division that many of us learned in our early school years equipped us to find products and quotients but unhappily did not, in many cases, leave us with an understanding of the reasons for the steps involved. The reader who doubts this statement is encouraged to work a long division problem before an inquisitive observer who has been primed to demand a clear and complete explanation for every step in the process. When it comes to the doing of computational problems by someone who has mastered the system, there can be little doubt that the Arabic system has significant advantages over the others considered here, the more so the greater the complexity of the computational problem involved. What makes the Arabic system superior for computing purposes is probably a combination of factors. That it is a place-notation system is certainly one important factor, although, as we have noted, this alone does not distinguish it from some other systems. Another major advantage for computation is the relatively economical way in which the Arabic system represents numbers, as was noted in the preceding section. CONCLUSIONS Aristotle attached considerable significance to the fact that human beings can count: it is this ability, he claimed, that demonstrates our rationality. However that may be, it is difficult to imagine what life would be like if we had never learned to count or compute. It is perhaps unthinkable that

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198-April1988

HUMAN

human beings could have been around as long as they have without having developed these abilities. It is not at all unthinkable, however, that a system for representing numbers might have evolved to be something quite different from the one we use. How such a system might have been better is a little hard for us to see; perhaps future generations will discover that. If one accepts the idea that the Arabic system is in general the best way of representing numbers

that has yet been developed,

one

need not believe that it is clearly superior with respect to all the design goals that one might establish for an ideal system. It may be, in fact, that simultaneous realization of all such goals is not possible. Perhaps tradeoffs are necessary. One trade-off that the Arabic system seems to represent is that between ease of learning by the neophyte and ease of use in computing by the expert. All of the earlier number systems considered in this paper were in certain ways more obviously analogous to the things they represent than is the Arabic system. There was, for example, a more direct correspondence between the number of symbols in the number and the number of objects in the set represented by the number. The addition of two numbers was more directly analogous to the addition of the sets represented by those numbers. The correspondence was especially direct when the numbers were represented by physical tokens such as pebbles or sticks, as may sometimes have been the case. A similar point may be made with respect to subtraction. Moreover, the analogues of carrying and borrowing, for all their mysteriousness to the modern-day neophyte mathematician, have straightforward analogues in these systems. In short, a price of the increased abo stractness of the Arabic system is an obscuring of some of the key principles on which numbers are based. To an ancient Egyptian,

FACTORS

the fact that 3 + 4 = 7 was apparent from the relationship between the Egyptian number representing 7 and those representing 3 and 4. There is no hint of such a relationship, however, when this equation is expressed in Arabic notation. The price has been worth the paying, however. The convenience of the Arabic system for computing has played an indispensable role in the development of higher mathematics and of science and technology, which are so heavily dependent on mathematics.

More-

over, it has made it possible for the average person to attain a far greater degree of mathematical competence than would have been feasible with the more ancient systems. As Brainerd (1973) has pointed out, in most Western nations today we expect students, by the time they reach their early teens, to be much more competent with numbers than was an educated adult Greek or Roman of 2,000 years ago. This fact arises in no small measure from the elegance and power of the scheme that we now use to represent numbers. This scheme was a long time evolving. Moreover, what we know of the evolution obscures the distinction between cause and effect. We may assume, however, that it was not the case that from the beginning people wanted to do higher mathematics and therefore sought a representational scheme to make that possible. Much of what we think of as higher mathematics could not have been conceived had there not already existed representational schemes that facilitated its conception. The development of representational schemes and of new mathematical concepts has been mutually reinforcing: an effective way of representing existing concepts has been instrumental in extending those concepts, and those extensions have led to the need for and development of new representational schemes.

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NUMBER

April 1988-199

REPRESENTATION

ACKNOWLEDGMENTS The writing of this paper was supported by the National Institute of Education under Contract No. 400-80-0031. I am grateful to John Swets and Wallace Feurzeig for helpful comments on an earlier draft.

REFERENCES Aleksandrov, A. D. (1963). A general view of mathematics. In A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev (Eds.), Mathematics: Its content, methods, and meaning (pp. 1-64). Cambridge, MA: MIT Press. Brainerd, C. J. (1973). The origins of number concepts. Scientific American, 228(3), 101-109. Brown, J. S., and Burton, R. R. (1978). Diagnostic model for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155-192. Brown, J. S., and VanLehn, K. (1982). Towards a generative theory of "bugs." In T. P. Carpenter and J. M. Moser (Eds.), Addition and subtraction: A cognitive perspective (pp. 117-135). Hillsdale, NJ: Erlbaum. Conant, L. L. (1956). Counting. In J. R. Newman (Ed.), The

world of mathematics (Vol. 1, pp. 432-441). New York: Simon and Schuster. Jourdain, P. E. B. (1956). The nature of mathematics. In J. R. Newman (Ed.), The world of mathematics (Vol. I, pp. 4-72). New York: Simon and Schuster. Lambert, J. B., Ownbey-McLaughlin, B., and McLaughlin, C. D. (1980). Maya arithmetic. American Scientist, 68(3),249-255. McCulloch, W. S. (1961). What is a number that a man may know it, and a man that he may know a number? The Ninth Alfred Korzybski Memorial Lecture, General Semantics Bulletin, Nos. 26 and 27. Lakeville, CT: Institute of General Semantics, 1961; reprinted in McCulloch, W. S. (1965). Embodiments of mind (pp. 1-17). Cambridge, MA: MIT Press. Menninger, K. (1969). Number words and number symbols: A cultural history of numbers. Cambridge, MA: MIT Press. Newman, J. R. (1956). The Rhind Papyrus. In J. R. Newman (Ed.), The world of mathematics (Vol. I, pp. 170-178). New York: Simon and Schuster. Schmandt-Besserat, D. (1978). The earliest precursors of writing. Scientific American, 238(6), 50-59. Thompson, J. E. S. (1954). The rise and fall of the Maya civilization. Norman: University of Oklahoma Press.

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