numerals and numeral systems

▪ mathematics

Introduction

      a collection of symbols used to represent small numbers, together with a system of rules for representing larger numbers.

      Just as the first attempts at writing came long after the development of speech, so the first efforts at the graphical representation of numbers came long after people had learned how to count. Probably the earliest way of keeping record of a count was by some tally system involving physical objects such as pebbles or sticks. Judging by the habits of indigenous peoples today as well as by the oldest remaining traces of written or sculptured records, the earliest numerals were simple notches in a stick, scratches on a stone, marks on a piece of pottery, and the like. Having no fixed units of measure, no coins, no commerce beyond the rudest barter, no system of taxation, and no needs beyond those to sustain life, people had no necessity for written numerals until the beginning of what are called historical times. Vocal sounds were probably used to designate the number of objects in a small group long before there were separate symbols for the small numbers, and it seems likely that the sounds differed according to the kind of object being counted. The abstract notion of two, signified orally by a sound independent of any particular objects, probably appeared very late.

Number bases (base)

 When it became necessary to count frequently to numbers larger than 10 or so, the numeration had to be systematized and simplified; this was commonly done through use of a group unit or base, just as might be done today counting 43 eggs as three dozen and seven. In fact, the earliest numerals of which there is a definite record were simple straight marks for the small numbers with some special form for 10. These symbols appeared in Egypt (Egypt, ancient) as early as 3400 BC and in Mesopotamia (Mesopotamia, history of) as early as 3000 BC, long preceding the first known inscriptions containing numerals in China (c. 1600 BC), Crete (c. 1200 BC), and India (c. 300 BC). Some ancient symbols for 1 and 10 are given in the figure—>.

      The special position occupied by 10 stems from the number of human fingers, of course, and it is still evident in modern usage not only in the logical structure of the decimal number system but in the English names for the numbers. Thus, eleven comes from Old English endleofan, literally meaning “[ten and] one left [over],” and twelve from twelf, meaning “two left”; the endings -teen and -ty both refer to ten, and hundred comes originally from a pre-Greek term meaning “ten times [ten].”

      It should not be inferred, however, that 10 is either the only possible base or the only one actually used. The pair system, in which the counting goes “one, two, two and one, two twos, two and two and one,” and so on, is found among the ethnologically oldest tribes of Australia (Australian Aborigine), in many Papuan languages of the Torres Strait and the adjacent coast of New Guinea, among some African Pygmies (Pygmy), and in various South American (South America) tribes. The indigenous peoples of Tierra del Fuego and the South American continent use number systems with bases three and four. The quinary scale, or number system with base five, is very old, but in pure form it seems to be used at present only by speakers of Saraveca, a South American Arawakan language; elsewhere it is combined with the decimal or the vigesimal system, where the base is 20. Similarly, the pure base six scale seems to occur only sparsely in northwest Africa and is otherwise combined with the duodecimal, or base 12, system.

      In the course of history, the decimal system finally overshadowed all others. Nevertheless, there are still many vestiges of other systems, chiefly in commercial and domestic units, where change always meets the resistance of tradition. Thus, 12 occurs as the number of inches in a foot, months in a year, ounces in a pound ( troy weight or apothecaries' weight), and twice 12 hours in a day, and both the dozen and the gross measure by twelves. In English the base 20 occurs chiefly in the score (“Four score and seven years ago…”); in French it survives in the word quatre-vingts (“four twenties”), for 80; other traces are found in ancient Celtic, Gaelic, Danish, and Welsh. The base 60 still occurs in measurement of time and angles.

Numeral systems

      It appears that the primitive numerals were |, ||, |||, and so on, as found in Egypt and the Grecian lands (ancient Greek civilization), or −, =, ≡, and so on, as found in early records in East Asia, each going as far as the simple needs of people required. As life became more complicated, the need for group numbers became apparent, and it was only a small step from the simple system with names only for one and ten to the further naming of other special numbers. Sometimes this happened in a very unsystematic fashion; for example, the Yukaghirs (Yukaghir) of Siberia counted, “one, two, three, three and one, five, two threes, two threes and one, two fours, ten with one missing, ten.” Usually, however, a more regular system resulted, and most of these systems can be classified, at least roughly, according to the logical principles underlying them.

Simple grouping systems

      In its pure form a simple grouping system is an assignment of special names to the small numbers, the base b, and its powers b², b³, and so on, up to a power b^k large enough to represent all numbers actually required in use. The intermediate numbers are then formed by addition, each symbol being repeated the required number of times, just as 23 is written XXIII in Roman numerals (Roman numeral).

 The earliest example of this kind of system is the scheme encountered in hieroglyphs (hieroglyph), which the Egyptians used for writing on stone. (Two later Egyptian systems, the hieratic and demotic, which were used for writing on clay or papyrus, will be considered below; they are not simple grouping systems.) The number 258,458 written in hieroglyphics appears in the figure—>. Numbers of this size actually occur in extant records concerning royal estates and may have been commonplace in the logistics and engineering of the great pyramids.

Cuneiform numerals

 Around Babylon (Babylonia), clay was abundant, and the people impressed their symbols in damp clay tablets before drying them in the sun or in a kiln, thus forming documents that were practically as permanent as stone. Because the pressure of the stylus gave a wedge-shaped symbol, the inscriptions are known as cuneiform, from the Latin cuneus (“wedge”) and forma (“shape”). The symbols could be made either with the pointed or the circular end (hence curvilinear writing) of the stylus, and for numbers up to 60 these symbols were used in the same way as the hieroglyphs, except that a subtractive symbol was also used. The figure—> shows the number 258,458 in cuneiform.

      The cuneiform and the curvilinear numerals occur together in some documents from about 3000 BC. There seem to have been some conventions regarding their use: cuneiform was always used for the number of the year or the age of an animal, while wages already paid were written in curvilinear and wages due in cuneiform. For numbers larger than 60, the Babylonians used a mixed system, described below.

Greek numerals

      The Greeks (ancient Greek civilization) had two important systems of numerals, besides the primitive plan of repeating single strokes, as in ||| ||| for six, and one of these was again a simple grouping system. Their predecessors in culture—the Babylonians, Egyptians, and Phoenicians—had generally repeated the units up to 9, with a special symbol for 10, and so on. The early Greeks also repeated the units to 9 and probably had various symbols for 10. In Crete, where the early civilization was so much influenced by those of Phoenicia and Egypt, the symbol for 10 was −, a circle was used for 100, and a rhombus for 1,000. Cyprus also used the horizontal bar for 10, but the precise forms are of less importance than the fact that the grouping by tens, with special symbols for certain powers of 10, was characteristic of the early number systems of the Middle East.

      The Greeks, who entered the field much later and were influenced in their alphabet by the Phoenicians, based their first elaborate system chiefly on the initial letters of the numeral names. This was a natural thing for all early civilizations, since the custom of writing out the names for large numbers was at first quite general, and the use of an initial by way of abbreviation of a word is universal. The Greek system of abbreviations, known today as Attic numerals, appears in the records of the 5th century BC but was probably used much earlier.

Roman numerals (Roman numeral)

      The direct influence of Rome (ancient Rome) for such a long period, the superiority of its numeral system over any other simple one that had been known in Europe before about the 10th century, and the compelling force of tradition explain the strong position that the system maintained for nearly 2,000 years in commerce, in scientific and theological literature, and in belles lettres. It had the great advantage that, for the mass of users, memorizing the values of only four letters was necessary—V, X, L, and C. Moreover, it was easier to see three in III than in 3 and to see nine in VIIII than in 9, and it was correspondingly easier to add numbers—the most basic arithmetic operation.

      As in all such matters, the origin of these numerals is obscure, although the changes in their forms since the 3rd century BC are well known. The theory of German historian Theodor Mommsen (Mommsen, Theodor) (1850) has had wide acceptance. He argued that the use of V for five is due to the fact that it is a kind of hieroglyph representing the open hand with its five fingers. Two of these gave the X for 10. Three of the other symbols, he asserted, were modifications of Greek letters not needed in the Etruscan and early Latin alphabet. These were Χ (chi) for 50, which later became the L; θ (theta) for 100, which later changed to C under the influence of the word centum (“hundred”); and Φ (phi) for 1,000, which finally took the forms I and M, the last being chosen because of the word mille (“thousand”).

      The oldest noteworthy inscription containing numerals representing very large numbers is on the Columna Rostrata, a monument erected in the Roman Forum to commemorate a victory in 260 BC over Carthage during the First Punic War. In this column a symbol for 100,000, which was an early form of (((I))), was repeated 23 times, making 2,300,000. This illustrates not only the early Roman use of repeated symbols but also a custom that extended to modern times—that of using (I) for 1,000, ((I)) for 10,000, and (((I))) for 100,000, and ((((I)))) for 1,000,000. The symbol (I) for 1,000 frequently appears in various other forms, including the cursive ∞. All these symbols persisted until long after printing became common. In the Middle Ages a bar (known as the vinculum or titulus) was placed over a number to multiply it by 1,000, but this use is not found in the Roman inscriptions. When the bar appeared in early manuscripts, it was merely for the purpose of distinguishing numerals from words. Also used in the Middle Ages were such forms as |X| or |X| for 1,000,000 and |M| for 100,000,000.

      Of the later use of the numerals, a few of the special types are as follows:

● c∙lxiiij∙ccc∙l∙i for 164,351, Adelard Of Bath (c. 1120)

● II.DCCC.XIIII for 2,814, Jordanus Nemorarius (c. 1125)

● MCLVI for 1,656, in San Marco, Venice

● cI.I.Ic for 1,599, Leiden edition of the work of Martianus Capella (Capella, Martianus Minneus Felix) (1599)

● IIIIxx et huit for 88, a Paris treaty of 1388

● four Cli.M for 451,000, Humphrey Baker's The Well Spryng of Sciences Whiche Teacheth the Perfecte Woorke and Practise of Arithmeticke (1568)

● vj.C for 600 and CCC.M for 300,000, Robert Recorde (Recorde, Robert) (c. 1542)

      Item (1) represents the use of the vinculum; (2) represents the place value as it occasionally appears in Roman numerals (D represents 500); (3) illustrates the not infrequent use of , like D, originally half of (I), the symbol for 1,000; (4) illustrates the persistence of the old Roman form for 1,000 and 500 and the subtractive principle so rarely used by the Romans for a number like 99; (5) shows the use of quatre-vingts for 80, commonly found in French manuscripts until the 17th century and occasionally later, the numbers often being written like iiij^xx, vij^xx, and so on; and (6) represents the coefficient method, “four C” meaning 400, a method often leading to forms like ijM or IIM for 2,000, as shown in (7).

      The subtractive principle is seen in Hebrew number names, as well as in the occasional use of IV for 4 and IX for 9 in Roman inscriptions. The Romans also used unus de viginti (“one from twenty”) for 19 and duo de viginti (“two from twenty”) for 18, occasionally writing these numbers as XIX (or IXX) and IIXX, respectively. On the whole, however, the subtractive principle was little used in the numerals of the Classical period.

Multiplicative grouping systems

 In multiplicative systems, special names are given not only to 1, b, b², and so on but also to the numbers 2, 3, …, b − 1; the symbols of this second set are then used in place of repetitions of the first set. Thus, if 1, 2, 3, …, 9 are designated in the usual way but 10, 100, and 1,000 are replaced by X, C, and M, respectively, then in a multiplicative grouping system one should write 7,392 as 7M3C9X2. The principal example of this kind of notation is the Chinese (China) numeral system, three variants of which are shown in the figure—>. The modern national and mercantile systems are positional systems, as described below, and use a circle for zero.

Ciphered numeral systems

      In ciphered systems, names are given not only to 1 and the powers of the base b but also to the multiples of these powers. Thus, starting from the artificial example given above for a multiplicative grouping system, one can obtain a ciphered system if unrelated names are given to the numbers 1, 2, …, 9; X, 2X, …, 9X; C, 2C, …, 9C; M, 2M, …, 9M. This requires memorizing many different symbols, but it results in a very compact notation.

 The first ciphered system seems to have been the Egyptian hieratic (literally “priestly”) numerals, so called because the priests were presumably the ones who had the time and learning required to develop this shorthand outgrowth of the earlier hieroglyphic numerals. An Egyptian arithmetical work on papyrus, employing hieratic numerals, was found in Egypt about 1855; known after the name of its purchaser as the Rhind papyrus, it provides the chief source of information about this numeral system (see figure—>). There was a still later Egyptian system, the demotic, which was also a ciphered system.

 As early as the 3rd century BC, a second system of numerals, paralleling the Attic numerals, came into use in Greece that was better adapted to the theory of numbers, though it was more difficult for the trading classes to comprehend. These Ionic, or alphabetical, numerals, were simply a cipher system in which nine Greek letters were assigned to the numbers 1–9, nine more to the numbers 10, …, 90, and nine more to 100, …, 900. Thousands were often indicated by placing a bar at the left of the corresponding numeral. (See figure—>.)

      Such numeral forms were not particularly difficult for computing purposes once the operator was able automatically to recall the meaning of each. Only the capital letters were used in this ancient numeral system, the lowercase letters being a relatively modern invention.

 Other ciphered numeral systems include Coptic, Hindu Brahmin, Hebrew, Syrian, and early Arabic. The last three, like the Ionic, are alphabetic ciphered numeral systems. The Hebrew system is shown in the figure—>.

Positional numeral systems

      The decimal number system is an example of a positional system, in which, after the base b has been adopted, the digits 1, 2, …, b − 1 are given special names, and all larger numbers are written as sequences of these digits. It is the only one of the systems that can be used for describing large numbers, since each of the other kinds gives special names to various numbers larger than b, and an infinite number of names would be required for all the numbers. The success of the positional system depends on the fact that, for an arbitrary base b, every number N can be written in a unique fashion in the form

N = a_nbⁿ + a_{n − 1}b^{n − 1} + ⋯ + a₁b + a₀

where a_n, a_{n − 1}, …, a₀ are digits; i.e., numbers from the group 0, 1, …, b − 1. Then N to the base b can be represented by the sequence of symbols a_na_{n − 1}…a₁a₀. It was this principle which was used in the multiplicative grouping systems, and the relation between the two kinds of systems is immediately seen from the earlier noted equivalence between 7,392 and 7M3C9X2; the positional system derives from the multiplicative simply by omitting the names of the powers b, b², and so on and by depending on the position of the digits to supply this information. It is then necessary, however, to use some symbol for zero to indicate any missing powers of the base; otherwise 792 could mean, for example, either 7M9X2 (i.e., 7,092) or 7C9X2 (792).

 The Babylonians (Babylonia) developed (c. 3000–2000 BC) a positional system with base 60—a sexagesimal system. With such a large base, it would have been awkward to have unrelated names for the digits 0, 1, …, 59, so a simple grouping system to base 10 was used for these numbers, as shown in the figure—>.

      In addition to being somewhat cumbersome because of the large base chosen, the Babylonian system suffered until very late from the lack of a zero symbol; the resulting ambiguities may well have bothered the Babylonians as much as later translators.

 In the course of early Spanish expeditions into Yucatan, it was discovered that the Maya, at an early but still undated time, had a well-developed positional system, complete with zero. It seems to have been used primarily for the calendar rather than for commercial or other computation; this is reflected in the fact that, although the base is 20, the third digit from the end signifies multiples not of 20² but of 18 × 20, thus giving their year a simple number of days. The digits 0, 1, …, 19 are, as in the Babylonian, formed by a simple grouping system, in this case to base 5 (see figure—>); the groups were written vertically.

      Neither the Mayan nor the Babylonian system was ideally suited to arithmetical computations, because the digits—the numbers less than 20 or 60—were not represented by single symbols. The complete development of this idea must be attributed to the Hindus, who also were the first to use zero in the modern way. As was mentioned earlier, some symbol is required in positional number systems to mark the place of a power of the base not actually occurring. This was indicated by the Hindus by a dot or small circle, which was given the name sunya, the Sanskrit word for “vacant.” This was translated into the Arabic ṣifr about AD 800 with the meaning kept intact, and the latter was transliterated into Latin about 1200, the sound being retained but the meaning ignored. Subsequent changes have led to the modern cipher and zero.

      A symbol for zero appeared in the Babylonian system about the 3rd century BC. However, it was not used consistently and apparently served to hold only interior places, never final places, so that it was impossible to distinguish between 77 and 7,700, except by the context.

Development of modern numerals and numeral systems

The Hindu-Arabic system

 Several different claims, each having a certain amount of justification, have been made with respect to the origin of modern Western numerals, commonly spoken of as Arabic but preferably as Hindu-Arabic. These include the assertion that the origin is to be found among the Arabs, Persians, Egyptians, and Hindus. It is not improbable that the intercourse among traders served to carry such symbols from country to country, so that modern Western numerals may be a conglomeration from different sources. However, as far as is known, the country that first used the largest number of these numeral forms is India. The 1, 4, and 6 are found in the Ashoka inscriptions (rock edicts) (3rd century BC); the 2, 4, 6, 7, and 9 appear in the Nana Ghat inscriptions about a century later; and the 2, 3, 4, 5, 6, 7, and 9 in the Nasik caves of the 1st or 2nd century AD—all in forms that have considerable resemblance to today's, 2 and 3 being well-recognized cursive derivations from the ancient = and ≡. None of these early Indian inscriptions gives evidence of place value or of a zero that would make modern place value possible. Hindu literature gives evidence that the zero may have been known earlier, but there is no inscription with such a symbol before the 9th century.

      The first definite external reference to the Hindu numerals is a note by Severus Sebokht, a bishop who lived in Mesopotamia about 650. Since he speaks of “nine signs,” the zero seems to have been unknown to him. By the close of the 8th century, however, some astronomical tables of India are said to have been translated into Arabic at Baghdad, and in any case the numeral became known to Arabian scholars about this time. About 825 the mathematician al-Khwārizmī (Khwārizmī, al-) wrote a small book on the subject, and this was translated into Latin by Adelard Of Bath (c. 1120) under the title of Liber Algorismi de numero Indorum. The earliest European manuscript known to contain Hindu numerals was written in Spain in 976.

      The advantages enjoyed by the perfected positional system are so numerous and so manifest that the Hindu-Arabic numerals and the base 10 have been adopted almost everywhere. These might be said to be the nearest approach to a universal human language yet devised; they are found in Chinese, Japanese, and Russian scientific journals and in every Western language.

The binary system (binary number system)

       Decimal numerals represented by digitsThere is one island, however, in which the familiar decimal system is no longer supreme: the electronic computer. Here the binary positional system has been found to have great advantages over the decimal. In the binary system, in which the base is 2, there are just two digits, 0 and 1; the number two must be represented here as 10, since it plays the same role as does ten in the decimal system. The first few binary numbers are displayed in the table (Decimal numerals represented by digits).

      A binary number is generally much longer than its corresponding decimal number; for example, 256,058 has the binary representation 111 11010 00001 11010. The reason for the greater length of the binary number is that a binary digit distinguishes between only two possibilities, 0 or 1, whereas a decimal digit distinguishes among 10 possibilities; in other words, a binary digit carries less information than a decimal digit. Because of this, its name has been shortened to bit; a bit of information is thus transmitted whenever one of two alternatives is realized in the machine. It is of course much easier to construct a machine to distinguish between two possibilities than among 10, and this is another advantage for the base 2; but a more important point is that bits serve simultaneously to carry numerical information and the logic of the problem. That is, the dichotomies of yes and no, and of true and false, are preserved in the machine in the same way as 1 and 0, so in the end everything reduces to a sequence of those two characters.

David Eugene Smith William Judson LeVeque

Additional Reading

An accessible history of ancient and non-Western number systems is David Eugene Smith, History of Mathematics, 2 vol. (1923–25, reissued 1968). Graham Flegg (ed.), Numbers Through the Ages (1989), explores the history of counting systems, including basic techniques of calculation. Karl Menninger, Number Words and Number Symbols: A Cultural History of Numbers (1969, reissued 1992; trans. from German 2nd ed., 1957–58), is considered a classic on the origins of numbers. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (1998, reissued 2000; originally published in French, 1981), is encyclopaedic in its scope.

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