Human Factors: The
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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
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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
© 1988, The Human Factors Society, Inc. All rights reserved.
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182-Apri11988
HUMAN
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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April 1988-183
NUMBER REPRESENTATION
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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HUMAN
184-Apri11988
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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April 1988-185
NUMBER REPRESENTATION
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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April 1988-187
NUMBER REPRESENTATION
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«<<<nTH
The equivalent of Arabic 95,657 would be
represented as;
It takes three registers to represent this
number. The Arabic equivalent of this
number is derived from the contents of the
individual registers as follows:
<TTT\«TTI.J'TTT
<m <TT '-mT
(a) 26 x 602
(b) 34 x 601
FACTORS
=
93,600
2,040
(c) 17 x 60° =
-----11
=
95,657
This use of location to convey information
that was essential to the interpretation of a
symbol required the development of a way to
represent the absence of a symbol. This could
be done by simply leaving a space, but such
would be risky given the natural variability
in the spacing of symbols; there would also
be the problem of distinguishing between a
space within a number and a space between
numbers. (Imagine the problems that would
arise in our use of the Arabic system if instead of zeros we used spaces.) The Babylonians addressed this problem in much the
same way as it is addressed in Arabic notation, namely by developing a special symbol
that could serve as a place or "vacancy"
marker. A way to represent the null set becomes important when the same symbols can
appear in different registers. Although the
Babylonians made use of a rudimentary vacancy marker in their late cuneiform writing,
it was the Hindus who first used 0 in a systematic way. Apparently the idea of a place
holder for a missing digit was known in India
as early as the sixth century, but for two or
three centuries a dot was used for this pur-
THE MAYANSYSTEM
An especially interesting way of representing numbers was used by the Maya, who occupied much of what is now Central America
and maintained a remarkable culture that
lasted from about the fifth to the twelfth century A.D. (Lambert, Ownbey-McLaughlin,
and McLaughlin, 1980; Thompson, 1954).
The Mayan system was similar to the Babylonian cuneiform system in several respects
and differed in important ways from the systems of Egypt, Greece, and Rome. Like the
Babylonian system, it represented all
numbers with only two symbols and was a
true place-notation scheme. Unlike all the
other systems we have considered, it used a
base not of 10 or 60 but of 20. (The third register in the Mayan system sometimes was interpreted as having a base of 18 instead of 20.
This peculiarity was introduced for calendric
purposes. A unit in this register, instead of
representing 202, represented 18 x 20, or
360, the number of days in a Mayan year. In
what follows this aspect of the system is ignored.)
The two symbols used by the Maya were a
dot to represent the quantity 1and a horizontal line or bar to represent 5. As in the case of
the Babylonian system, the Mayan used a
two-stage substitution principle: 5 dots were
replaced by a bar in the same register, and 4
bars in a register were replaced by one dot in
the next higher one. By convention, the Maya
wrote numbers vertically, so the equivalent
of Arabic 1,549 would appear as follows:
_
=a
3 X 202 = 1,200
17 x 20· = 340
9 X 20° = __ 9
1,549
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April 1988-189
NUMBER REPRESENTATION
Why did the Maya choose 5 and 20 to play
the special roles they did? Was it because we
have five fingers on each hand and a total of
twenty digits? Aleksandrov (1963) points out
that among certain peoples the names for five
and twenty are respectively
"hand" and
"wholeman," which presumably translates
loosely as a man with aU his fingers and toes.
Why did the Maya select a dot and a bar as
the symbols to represent one and five? Was it
because they are so convenient to write? Did
the inspiration
come from the shapes of
beans and bean pods that might have been
used as tokens in counting and computing?
The answers to these questions
are not
known.
Both the Mayan and Babylonian systems
are in some ways more similar to our own
than are any of the others considered. They
have several advantages over the Egyptian,
Greek, and Roman systems, not the least of
which is the ease with which they permit
computation to be done. (More on this point
later.) In their total reliance on only two
symbols, they are parsimonious in the ex-
It is highly doubtful that any of the number
systems that have been considered in this
paper were designed in the usual sense of
that word; rather, each evolved over a considerable period of time. What has guided
their evolution? Clearly the characteristics of
these systems are not completely arbitrary,
and the similarities among systems are too
great to be coincidental. The use of the principle of one-for-many substitution, for example, seems to be common to an number systems.
Is the nearly universal acceptance of the
Arabic system a consequence of its superiority over other systems that have been used in
various parts of the world in the past? And if
so, what is it about this system that gives it
this edge?
One way to approach some of these questions is to imagine being given the task of designing a system for representing numbers,
or the slightly less ambitious task of specifying the characteristics or properties that the
desired system should have. This would be a
formidable
undertaking,
and one could
treme. Moreover, the choice of symbols-
hardly hope to approach it free of the biases
especially that of the Maya-is
ingenious;
they are readily distinguishable from one another, certainly easy to learn, and easy to
produce, even in stone.
that familiarity
with an existing system
would assure. Nevertheless,
comparisons
among systems such as those that have been
briefly considered here suggest some ideas as
to what certain of those characteristics might
be: in particular, ease of interpretation, ease
of writing, ease of learning, extensibility,
compactness of notation, and ease of computation.
DESIGN CRITERIA FOR A
NUMBER SYSTEM
Even a cursory inspection of ways in which
number concepts have been represented is
sufficient to convince one that some of these
representations
would be easier to use, at
least for certain purposes, than others. Comparison of one number system with another
raises some intriguing questions and provides the basis for some conjectures not only
about number systems but about representational schemes in general. Both their similarities and differences ten us some things of interest.
Ease
of Interpretation
One wants a number to be easy to read.
The quantity it represents should be apparent at a glance and errors should not be
overly easy to make. One of the coding conventions of the Egyptian, Greek, and Roman
systems seems consistent with this objective
-namely,
the fact that each to-grouping is
represented
by a different symbol. This
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190-April1988
HUMAN
means that one can tell the 10-group to
which any symbol belongs without paying
attention to its location. In the Babylonian,
Mayan, and Arabic systems the same-shaped
symbol can occur in different registers; to tell
the register of a particular symbol, one must
note its position. The price the Egyptian,
Greek, and Roman systems pay for having
the to-group coded by symbol shape is the
need to have the same symbol repeated numerous times, which leads to a relatively
large number of symbols per number represented. Inasmuch as a particular symbol can
occur in a register as many as nine times in
those systems, one would sometimes have to
resort to counting to determine how many
occurrences of a given symbol there were.
The Babylonian and Mayan systems also
have the feature that the same symbol can
occur several times within the same register.
In contrast to all of these systems, the Arabic system gains economy of expression by
virtue of the fact that it never has more than
a single symbol in a given register. From this
it follows that on the average, any given
number is represented by fewer symbols in
the Arabic system than in the others we have
considered. Another advantage that follows
from the fact that an Arabic number always
has a single symbol in every register is that
the order of magnitude of the number is apparent from the number of digits that constitute it. None of the other systems we have
considered has this property. With the Arabic
system, counting digits can be a useful way
to produce order-of-magnitude estimates of
the results of multiplication
and division.
The number of digits in the product of two
numbers will be close to the sum of the digits
in the multiplier and the multiplicand; similarly, the number of digits in a quotient will
be close to the difference
between the
number in the dividend and in the divisor.
The fact that the Arabic system uses the
same symbols in different registers leads to
FACTORS
difficulties only with numbers that have sufficiently many digits that one cannot readily
tell, without counting, how many there are.
To facilitate determination of the number of
digits in a number, the convention typically
used is that of grouping the digits in threes
and separating the groupings with commas.
The value of this convention is readily apparent when one compares the ease of reading
236,172,345,924
with that of reading
16279543617.
For very large numbers even the Arabic
system becomes cumbersome. One innovation addressed to this problem is the use of
exponential notation, in which very large (or
very small) numbers are expressed as powers
of 10. With this notation 10 trillion is written
as 1013 instead of as 10,000,000,000,000. The
exponential notation sacrifices accuracy, but
ofen this is of little consequence because very
large numbers typically are used in contexts
for which order of magnitude accuracy is all
that one can hope to obtain.
Ease
of Writing
One would like a number system not to
unreasonably burden an individual writing
numbers down. What constitutes an unreasonable burden may differ depending on
whether one is chiseling numbers in stone or
marking them with a piece of graphite on
paper, but even for a given method of writing
one can distinguish among systems on the
basis of how much effort they require on the
part of the scribe.
Both the Babylonian and Mayan systems
were based on symbols that are easy to write
on practically any medium, but the writer
must use many of those symbols in order to
represent a number of even modest size. The
Egyptian and Greek symbols are poorly
suited for inscription in stone or other hard
media. Whether because of the number of
symbols required to represent a number or
the relative complexity of the individual
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April 1988-191
NUMBER REPRESENTATION
symbols, the time required to write numbers
in any of these systems would be quite long
relative to the time required to write the
same numbers in Arabic notation.
Ease
of Learning
One would prefer that one's number system be easy to learn. What determines ease of
learning is not entirely clear; one would
guess, however, that-other
things being
equal-the
more direct the correspondence
between any representational
scheme and
whatever is being represented, the easier it
should be to learn that scheme. Pictorial or
iconographic representations
are undoubtedly easier to learn to recognize than are
more abstract representations
of the same
things. If the Piagetians are right, the ability
to handle abstract concepts and operations
requires a higher level of cognitive development in general than does the ability to handle concrete concepts and operations.
The learning required by various systems
differs considerably. To master the Egyptian,
Greek, and Roman systems one must learn
the meanings of many symbols. Learning the
Babylonian and Mayan systems requires that
one learn only two symbols and a set of rules
for combining these symbols. Arabic requires
that one learn to symbols and the rules for
their combination.
Where the Arabic system differs from most
if not all earlier systems is in its greater level
of abstractness. The correspondence between
the representation of a number and the quantity represented by that number is less direct
in the Arabic system than in its predecessors.
The other systems that have been discussed
in this paper preserve more of the aspects of a
tally than does the Arabic system; in the latter any remnants of a tally are obscure indeed. Because of its greater abstractness, one
might guess that this system poses a somewhat greater learning challenge to the child
than do the others. This is a difficult hypoth-
esis to test, given the ubiquity of the Arabic
system today and the importance that is attached to learning it early. A fair test would
require an effort to teach some children alternative systems with the same amount of
emphasis that is normally devoted to the
teaching of the Arabic system while precluding exposure of these children to the .Arabic
system itself. The possibility of conducting
such an experiment is unlikely, and the advisability of doing so even if it were possible
is highly doubtful.
Extensibility
Closely related to the issue of ease of learning is that of extensibility. One wants to be
able to write any number on the basis of
knowledge of how numbers in general are
written. That is, one does not want to have to
learn something new whenever one has occasion to recognize or to write a number for the
first time.
A system that used a unique symbol for
every number concept would obviously fail
on this criterion because one would have to
memorize as many symbols as there were
numbers that one wanted to be able to use.
The Egyptian, Greek, and Roman systems
also fail to a degree in this regard because
whenever one has occasion to use a number
that is an order of magnitude larger than the
largest number with which one is familiar,
one must learn the symbol that is used to represent a to-grouping at the next-higher level.
Thus knowledge of how to represent quantities from 1 through 999 in Egyptian, for example, does not suffice if one wants to represent the quantity 1,000 because quantities
less than 1,000 can be represented by combinations of the three symbols I , (\. and ~
whereas those between 1,000 and 9,999 cannot. They require that one learn a new symbol: 1 . More generally, no finite amount of
knowledge about these number systems will
suffice to guarantee the ability to represent
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192-April1988
HUMAN
all numbers. No matter how many symbols
one knows how to write, it is easy to specify a
number that cannot be written with only
those symbols.
By contrast, the advantages of the Babylonian, Mayan, and Arabic systems in this regard are apparent. If one knows the basic
symbols and understands the system's onefor-many substitution rules, one can write
any number without learning anything new.
It might be easier for a child to learn to use
the Egyptian, Greek, or Roman system well
enough to serve the purposes of counting up
to numbers of modest size than to learn the
Babylonian, Mayan, or Arabic system to the
same degree of usefulness.
But having
learned the principles on which numbers in
any of the latter three systems are constructed, one has the necessary knowledge to
extend the number sequence indefinitely. In
general, any system that uses the position of
a symbol to indicate its register may be more
difficult to learn than one that uses perceptually different symbols in different registers,
but this disadvantage is likely to be more
than offset by the advantages a place-notation system provides to the user once the
principles
of its construction
have been
learned.
Compactness
of Notation
It should be possible to represent most of
the numbers that one is likely to have to use
-including
relatively large numbers-compactly. The superiority of the Arabic system
over its predecessors in this regard has already been noted. This feature is worth highlighting, however, because it is one key to the
uniqueness of the Arabic system and a basis
for some of its other desirable properties.
In general, early number systems were not
efficient for representing very large quantities. The equivalent of 99,999 in ancient
Egyptian, for example, would look like this:
FACTORS
(((((
nm ':?'''J,,) nnnnn 1111I
((((un '~99 mnn II/I
The largest number for which the early
Romans had a single symbol was 100,000,
which they represented as eeeIJJ.?; thus to
write the number 2,000,000 they would have
to write the symbol for 100,000 twenty times.
Eventually the symbol n was used to represent 100,000 and the convention was adopted
of representing multiples of 100,000 by combining this symbol with the symbol for the
multiple, so that, for example, 1,000,000 was
then represented by a combination of the
symbols for 100,000 and 10: thus IXI (Menninger, 1969).
The ancient number systems were generally not conducive to the representation
of
large quantities, and calculations involving
many even moderately large numbers must
have been tedious indeed. To express the
number 73,584 requires the writing of 27, 15,
17, 16, and 19 symbols in Egyptian, Greek,
Roman, Babylonian,
and Mayan, respectively. (These counts are based on representations of Greek and Roman systems in which
five-groupings-and
in the latter case, the
diminution principle-are
used.) The advantages that the Arabic system provides over its
predecessors
derive in no small measure
from the compactness of its notation.
This compactness derives in turn from the
greater degree of abstractness of the system
and in particular from its having shed the
last vestiges of a tally system. Whereas the
Egyptian, Greek, Roman, Babylonian, and
Mayan systems all used the principle of onefor-many substitution, the Arabic system applies it at a higher level of abstraction than
do any of the others: not only does a unit in a
register represent 10 units in the next lowerorder register, but the number of units in any
given register is itself represented by a single
symbol.
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April 1988-193
NUMBER REPRESENTATION
Ease of Computation
Counting predates computing both developmentally and historically. Children learn
to count before they learn to compute, and
the evidence is compelling that for human
beings as a species these abilities were acquired in that order. Empirical evidence
aside, it could hardly have been otherwise: it
is certainly possible to be able to count without being able to compute, whereas-though
in principle one might do certain types of
computing without being able to count-it
is
difficult to imagine that the ability to compute could have been developed very far in
the absence of the ability to count. We may
assume that number systems were invented
to accommodate
counting, and that only
after having been used for this purpose for
some time were they applied to computational tasks. This being so, it is not surprising
that the earliest systems were better adapted
to counting than to computing.
The difficulty of doing any very complex
mathematics with the Egyptian, Greek, or
Roman systems is apparent. What may be
less obvious is the fact that these systems are
well suited to the fundamental operations of
addition and subtraction. The sum of two
Egyptian numbers, for example, can be represented literally as the union of the sets of
symbols representing the addends. Thus
n~~'RRII
· 1 ~?,
1'1'1>
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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194-Apri11988
HUMAN
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.
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