Popular Science Monthly/Volume 75/November 1909/The Decimal System of Numbers
|THE DECIMAL SYSTEM OF NUMBERS|
By Dr. L. C. KARPINSKI
UNIVERSITY OF MICHIGAN
IS there a limitation placed upon our thought by the language which we use? Do the Germans take to philosophy more easily than other people because of some peculiarly philosophical bias of their language? These are speculative questions which can never be satisfactorily answered. It may, however, safely be asserted that the literature of a language is immediately dependent upon the written alphabet. It is impossible to conceive of a novel having been written in Babylonian cuneiform characters or in Egyptian hieroglyphics. Romance was the same, in its larger outlines, then as now, but writing was too serious a matter to be undertaken for such fleeting fancies. With a difficult alphabet and lack of facilities for writing, general culture was impossible. The Chinese, in modern times, furnish a striking illustration of the deadening effect of a difficult alphabet.
As literature and general culture are related to the alphabet and written language, so scientific advancement is related to the number system in use and to the system of writing numbers. A slight study of the Roman numerals gives the clue to the reason why the advancement along scientific lines lagged so far behind the general advancement achieved by the Roman peoples. The Greeks had a peculiar genius for arithmetical research,-but with them long division was a difficult operation, on account of the symbols. Only an Archimedes could overcome the clumsiness of an unscientific method, and even he could solve but comparatively simple problems.
In order to comprehend the essence of our own number system, it is necessary to distinguish between a number system and a place system. A ten system involves having symbols for 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 groups of objects, respectively, and beyond that separate symbols for the successive powers of 10—100, 1,000, 10,000, 100,000.... A five system would involve separate symbols for 1, 2, 3 and 4 groups of objects and further symbols for 5 and for the successive powers of 5—25, 125, 625, 3,125, 15,625.... A logically complete 5 system has not been developed among any people of the earth. In fact no other complete system, than a decimal system has ever been developed. Among the Mayas of Central America a 20 system was partially developed. Among the Babylonians there was in use a sixty system interwoven with a decimal system.
A decimal place system involves symbols for 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. The ideas of 10, 100, 1,000 and successive powers of ten are involved, but the symbols are given by combination of the symbols for 1 to 9, with the symbol for zero. As our development will show the symbol for nothing was the great stumbling block in the development of a scientific method of writing the numerals. A place system to the base five would require only the addition of a symbol for zero to the symbols for 1, 2, 3 and 4. Leibnitz occupied himself with the binary system, as this required only two characters, one for unity and one for non-entity. To illustrate a binary place system the numbers from 1 to 16 are written, using only 1 and 0.
Three written as 11, means one, two and one unit. Nine written as 1001 represents one cube of two, no squares of two, no first powers of
two, and one unit. The construction of the arithmetic universe out of the single unit afforded Leibnitz some philosophical satisfaction in connection with his system of monads. All the operations of ordinary arithmetic are possible in this system. We catch a glimpse of our slight comprehension of the infinite totality of numbers in noting that any number that can be expressed with our ordinary ten digits can also be expressed with these two digits, and that even though we used a thousand digits we could add no new numbers. Doubtless it would afford Leibnitz some gratification to know that his binary system is used in modern mathematical analysis in certain delicate proofs. The study of these number systems is not wholly foreign to the history of the decimal system, as traces of the binary and quinary systems appear among primitive peoples.
Among the South Australian tribes the binary system of numeration is almost universal. This is undoubtedly due to the fact that the hands and feet and eyes and ears occur in groups of two in each normal individual. These tribes are not advanced enough to have a system of symbols; such a development would imply a degree of intelligence which would proceed to a higher and more convenient number base. The system is seen in their words; three is given as two and one, four as two and two, five as two and two and one, and six as two, two, two. This system is found also among South American tribes. The quinary system is the most frequent of all the systems occurring in the numerals of American languages, although the twenty system is common along the Pacific. A study of the words of various American Indian tribes reveals traces of a five system in the formation of the words for six, seven and eight which are given as five and one, five and two, and five and three. The higher numbers, however, are formed on the decimal scale. The word for twenty signifies two tens and the higher tens are similarly constructed. Among some of the African tribes a partial five system is in use. Other tribes of northern Africa have borrowed the decimal notation from their civilized neighbors.
Without a single exception the ancient civilized peoples of all the world—Egyptians, Babylonians, Hebrews, Chinese, Greeks, Romans, Hindus—all used the decimal systems. Such striking uniformity among all the races of the earth requires a natural origin for the decimal number base. As Herodotus first suggested, man counts by tens because he has ten fingers. While there may be logical grounds for the advocates of a duo-decimal system, the ten system is too deep-rooted to be dislodged. Were we to acquire numbers as adults with mature minds, a duo-decimal system might be possible, but with children the acquisition of a twelve system may be said to be almost a psychological impossibility.
Among the Babylonians existed a sixty system mixed with a decimal system. Separate symbols and words are found for and and also for and and .
The ingenious hypothesis is advanced by M. Aures that the Babylonians having originally a decimal system, gradually changed from that system of numeration to the duo-decimal and then to the sexagesimal in order to make the number system accord with their systems of measurements. This is the reciprocal movement to that which is taking place with us to-day and that which was effected for France by the French Revolution, the change from duo-decimal and what not else systems of measurements to a decimal system in conformity with our number system. The hypothesis of Aures is justified by the existence of the special symbols and names for 10, 100 and 1,000, and many other curious mixtures of decimal, duo-decimal and sexagesimal systems in the Babylonian measures. There is some comfort to be found in the reflection that ours is not the first civilization to struggle with diverse systems of notation and measurement.
The most striking fact of Babylonian mathematics is that they were in possession of a sixty place system. The famous tablets of Senkereh, discovered by the English geologist, W. K. Loftus, give tables of square and cubic numbers in cuneiform characters. In these tables the numbers proceed regularly up to 82, which is given as 1.4, 92 is given as 1.21, as 1.36, as 6.40—naturally all in cuneiform characters. The only possible interpretation of this is that the 1 in the left hand place stands for 60. The table of cubic numbers bears out this interpretation as is given as 7.30, meaning or which makes the total of 27,000. Up to date no documents have been found which show the presence of the zero in this system. Even though a zero, and with it thus a full place system, had existed the unwieldiness of the large base would have operated against a universal adoption of the system; a number system must be adapted to child mind.
Our division of the day into 24 hours is probably a heritage from the Babylonians; the division of the hour and minute into sixty parts is certainly a survival from this hoary system. So also the division of the arc of the circle into 360° and the further subdivisions have come to us from this extinct civilization. Greek astronomers and through them all European astronomers borrowed much from the same source, and for over fifteen hundred years of the Christian era sexagesimal fractions were used in all arithmetical computation. The first tables of trigonometric functions were on the basis of a radius of 600,000, later 6,000,000, finally to be discarded by Regiomontanus in 1470 for the base 105, later for 1015, and then by the great Vieta, in 1579, for the base one with decimal values.
It is entirely within the bounds of possibility that the first development of the Hindu, commonly called Arabic, place system was due to some oriental scholar who was familiar with the writings of these ancient Babylonians. Abundant testimony exists tending to prove the communication between Europe and the east. Having special symbols, such as existed in India for 1, 2, 3, 4, 5, 6, 7, 8 and 9 as early as the second century, acquaintance with this advancement of the Babylonians may have suggested the step to a decimal place system and the innovation of a zero. The existence of a Babylonian zero symbol would strengthen this hypothesis; even a blank space may have been the first symbol.
The Egyptians were in possession of a complete decimal system, with separate symbols for 1, 10, 100, 1,000, 10,000 and higher powers of 10. The famous Papyrus Rhind of the British Museum gives us a practically complete Egyptian arithmetic. The striking peculiarity of their arithmetic consisted in the work in fractions which was confined almost entirely to unit fractions. The Ahmes Papyrus of date about 1700 B.C. gives a table for a conversion of fractions from 2 to 2⁄99 into unit fractions. The tremendous inertia of even the clumsiest system once established is seen in the fact that Greek manuscripts of date 700 a.d., at least 2,200 years later, contain this same bungling system of fractions. Aside from this malign influence European arithmetic was not affected by the Egyptian.
Among the ancient Semitic peoples we find separate symbols for 1, 10, 20 and 100. Noteworthy is the use of twenty in forming the higher powers of ten; sixty is written as three twenties. The use of twenty as a unit of higher order goes back to primitive counting on fingers and toes, which operation still exists among Pacific coast tribes of Indians, Mexicans and Esquimaux. Persistence of the unit twenty is seen in our word for score; more markedly in the French quatre-vingt for 80.0.
Some time before the Christian era, the Phœnicians changed to an alphabet system of numbers. The first nine letters of their alphabet were given the number values 1 to 9; to the second nine attach the values 10—90; and similarly with the hundreds. From the Phœnicians this method was taken by the Hebrews and the Greeks. In any numerical work the order hundreds, tens, units is strictly observed. Nevertheless, as to each word there was a definite number value the Hebrews indulged in secret writing by giving one name with the hint to the wise to substitute some other well-known name with the same number value. This near-punning occurs in the Book of the Revelations, "the number of the Beast is 666" referring to the Roman Emperor whose name written in Hebrew letters had the numerical value, 666.
A different type is presented by the Attic system of numbers in use among the ancient Greeks, in which the symbols are the first letters of the corresponding Greek words.
Combinations r A, r H, r x, r M were used for 50, 500, 5,000 and 50,000. The advantage in numerical computation of this system over the alphabet system is great as the connection between 50, 500 and 5,000 is brought out by "the symbols. Deceived by the apparent simplicity of the alphabet system, the Greeks abandoned the Attic in favor of the alphabet form.
are apparently simple, but they fail to show any trace of the underlying decimal system.
These second forms are organically connected, whereas the first forms exhibit no connection.
During the first thousand years of the Christian era, the alphabet system held full sway; then for a period of nearly five hundred years the Roman and the Greek systems vied with each other for popular favor among European arithmeticians.
The origin of the Roman numerals is lost in obscurity. Undoubtedly the symbols are from Etruscan sources changed gradually into the similar Roman letters. It is to be noted that such changes in the forms of letters were most easily effected by copyists previous to the invention of printing. Just as the Babylonians operated with the common denominator sixty, so the Romans confined themselves to the denominator 12 (and powers of 12). The twelfth represented at first a definite concrete unit of weight or length, the uncius, which later acquired a numerical sense.
List of Roman 12ths
The connection between the unciæ and our inches and ounces is evident. The Roman numerals like their prototypes in the Attic system of Greece and the more ancient Semitic systems, left no traces upon our current arithmetic. However, the Roman system of calculating upon a reckoning table was one of the vital factors in the development of the decimal place system. This system was not peculiarly Roman, as ancient Greek reckoning tables are found in several continental museums. The Chinese suan-pan, in popular use in Chinese laundries, is familiar to most readers. A similar instrument is found in Russian elementary schools.
A series of parallel grooved spaces and a goodly number of pebbles constitute the simplest form of one of these primitive calculating machines. Any right-hand column is chosen as the units column and the successive columns to the left are designated by the symbols for the successive powers of ten. Ten pebbles in any one column are replaced by one pebble in the next column to the left. Addition and subtraction are simple operations and even multiplication with small integers is not a difficult operation. Division was an accomplishment which only masters achieved; the complicated rules given by some medieval writers on the subject lead one to suspect that the writers were concealing ignorance in obscurity. On the Roman abacus the extreme right hand column represented twelfths (unciæ) and three smaller columns denoted 24ths, 48ths and 36ths respectively. On the Greek abacus also the right-hand columns were of mixed systems which serve to make the calculating more difficult.
A late development of the same nature was the reckoning on lines which continued into the sixteenth century. The essentials are similar. A glance at the accompanying diagram explains the connection between this system and the decimal place system. The upper part represents the number 4,063, the lower part the number 3,251. It seems such a slight step, after acquiring special symbols for the groups of one
to nine to construct a symbol to indicate a blank space, but that step took centuries to achieve.
Of all ancient peoples the Hindus occupied themselves most deeply with numbers. To some of their scholars came the conception of a connection between the infinite of the universe and the infinite of numbers. This longing for the infinite found expression in the construction of ever increasing numbers. Buddha calculates the number of grains of sand in a mile and shows how to compute the number in a sphere whose radius is the distance to one of the fixed stars. Not content with this, the Buddha goes on to show how even greater numbers may be expressed, arriving at the equivalent in modern exponential notation of . The numerals of the ancient Tamils, who, like the mountaineers of Appalachian America, conserve the traditions of a more remote civilization, show us that the Hindu peoples originally had special symbols not only for the first nine units, but also for the nine tens, the nine hundreds and even the nine thousands. The formation of the sequences of large numbers revealed the futility of having separate signs for the mixed tens and hundreds, with the consequent result that they dropped the separate symbols for 20 to 90, 200 to 900, and used the pure decimal units in connection with the symbols for one to nine. A similar development took place in quite early times—pre-Christian—among the Chinese but their clumsy notation obscured the realization of the possibility of a simpler place system.
The reading of a large number in Hindu style reveals how close their nomenclature brought them to the place system.
|In modern notation, 8,443,682,155, 8 billion, 443 million, 682 thousand, 155.|
|In Hindu, 8 padmas, 4 vyarbondas, 4 kotis, 3 prayoutas, 6 lakchas, ayoutas, 2 sahasra, 1 gata 5 dagan 5.|
|In Arabic and in later German, eight thousand thousand thousand and four hundred thousand thousand and forty-three thousand thousand and six hundred thousand and eighty-two thousand and one hundred fifty-five.|
|In Greek, eighty-four myriads of myriads and four thousand three hundred sixty-eight myriads and two thousand one hundred fifty-five.|
It is well established that in different parts of India the names for some of the higher powers took different forms, even the order was interchanged. However, as the significance of the name was further given by the order in reading, the variations did not lead to error. Indeed, the variation itself may have necessitated the introduction of a word to signify a vacant place or a lacking unit, with the ultimate introduction of a zero symbol for the word. The use of a special word to indicate absence of a unit is not hypothesis, but is found in verses in the ancient Indian book on astronomy, the Sourya-Siddhanta, and in numerous other ancient Hindu writings.
Brockhaus has well said that if there was any invention for which the Hindus by all their philosophy and religion were well fitted it was the invention of a symbol for zero. This making of nothingness the crux of a tremendous achievement was a step in complete harmony with the genius of the Hindu. The exact date of the birth of the zero symbol is not known, doubtless never will be known. The burden of proof points to a use of this symbol towards the beginning of the fifth century of our era. Wide-spread use in India did not occur until towards the ninth century. With nations as with individuals, the complete significance of great idea is not achieved in a moment; even as this idea itself in its unfolding required the labor of master minds of many centuries, so the appreciation and application of this advance required centuries for its completion.
The intellectual awakening of the Arabs beginning about the middle of the eighth century, manifested itself in the appearance of numerous translations of Greek, Syrian and Hindu works. Barbarians as they undoubtedly were at the period of their first conquests, the Arabs distinguished themselves by their desire for the further conquests of the science and literature of the subjugated peoples. The Persian invasion brought them close to the civilization of the Hindus and here the scholars went further than the flag. Hindu astronomy and astrology accompanied by the Hindu arithmetic were given to the scientific public in translations made at the command of one of the first great Mahometan patrons of learning, the Caliph Almansur, who reigned during the second half of the eighth century.
For a period of five hundred years the intellectual activity of the Mediterranean countries was well nigh confined to the Arabs. With what extraordinary diligence the pursuit of foreign learning was made by these erstwhile wanderers is evidenced by the thousands upon thousands of Arabic manuscripts. The library of Hakam at Cordova in Spain contained 400,000 manuscripts; the catalogue alone is in 44 volumes. Original work in science and mathematics did not come from Arabic hands, but the debt of civilization is none the less great as they were long the conservors of the learning of the Greeks and Hindus. The revival of Euclid was brought about by translations made from the Arabic; indeed many important Greek works in all the sciences have come to us only from Arabic sources.
The points of contact of Europeans and the Ottomans were numerous. From Asia Minor at the east to Greece was a well-traveled route; Sicily, Sardinia and Africa were in constant communication with Italy. Moorish Spain was for centuries a meeting place of English, French, Polish and German scholars.
The church played an important role in the spread of the Hindu numerals over Europe, and at the beginning of the thirteenth century in England, France, Germany, Italy and Poland, the arithmetic of the far east was explained by churchmen who had learned of Moorish teachers. However, it remained for a commercial traveler (line he handled is not known) to write the epoch-making work explaining the new doctrine. Leonard of Pisa traveled for business purposes in Africa, Syria, Egypt, Greece and Sicily and incidentally he acquired enough mathematics to make him the greatest mathematician since Archimedes. His Liber Abaci, or book of the abacus, first edition written in 1202, gave the first masterful exposition of the better way to reckon. It was for four centuries the great work of reference in this field.
With the knowledge of the Hindu method spread over all Europe at the beginning of the thirteenth century the acceptance of the improvement might be presupposed, but as late as 1520 arithmetics were published entirely in Roman numerals. The logically self-evident step to the right, to decimal fractions, required further centuries for its completion. The step up, to exponents to base ten, was made rather quickly, but has not yet taken its proper place in commercial work.
It is not too much to say that the present development of modern science would be impossible without our number system, yet how slow the world was to accept the reform. Is not the same story being repeated in the United States and England with the decimal system of weights and measures? But the optimistic soul regards chiefly the final acceptance with the comfortable assurance that the forward movement is as sure as it is slow.