Numeral system

“ Almost all civilized languages, however, except the Teutonic, and practically all uncivilized languages, begin their direct numeral combinations as soon as they have passed their number base, whatever that may be. To give an illustration, selected quite at random from among the barbarous tribes of Africa, the Ki-Swahili numeral scale runs as follows:53
The words for 11, 12, and 13, are seen at a glance to signify ten-and-one, ten-and-two, ten-and-three, and the count proceeds, as might be inferred, in a similar manner as far as the number system extends. ”

“ Wherever the numeral 20 is expressed by the term man, it may be expected that 40 will be 2 men, 60, 3 men, etc. This form of numeration is usually, though not always, carried as far as the system extends ”

“ It is never the principal base in any extended system.
To the statement just made respecting the origin of vigesimal counting, exception may, of course, be taken. In the case of numeral scales like the Welsh, the Nahuatl, and many others where the exact meanings of the numerals cannot be ascertained, no proof exists that the ancestors of these peoples ever used either finger or toe counting; and the sweeping statement that any vigesimal scale is the outgrowth of the use of these natural counters is not susceptible of proof. ”

“ In the selection of a base,—of a number from which he makes a fresh start, and to which he refers the next steps in his count,—the savage simply follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter of the greatest interest to find that other numbers have, in exceptional cases, been used for this purpose. Two centuries ago the distinguished philosopher and mathematician, Leibnitz, proposed a binary system of numeration. ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ But now the important fact, at present ignored, and to which I draw your attention, is that it is perfectly possible to have all the facilities which a method of notation like that of decimals gives, along with all the facilities which duodecimal division gives. It needs only to introduce two additional digits for 10 and 11 to unite the advantages of both systems. The methods of calculation which now go along with the decimal system of numeration would be equally available were 12 made the basic number instead of 10. ”

“ Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. ”

“ In savage languages it is sometimes possible to detect these meanings, and thus to obtain possession of the clue that leads to the development, in the barbarian's rude mind, of a count scale—a number system. But in languages like those of modern Europe, the pedigree claimed by numerals is so long that, in the successive changes through which they have passed, all trace of their origin seems to have been lost. ”

“ A most inviting field for research is, however, furnished by the primitive methods of counting and of giving visible expression to the idea of number. Our starting-point must, of course, be the sign language, which always precedes intelligible speech; and which is so convenient and so expressive a method of communication that the human family, even in its most highly developed branches, never wholly lays it aside. It may, indeed, be stated as a universal law, that some practical method of numeration has, in the childhood of every nation or tribe, preceded the formation of numeral words. ”

“ From this last arrangement seems to have come the fundamental law of the decimal notation in which its superior utility consists, and upon which quite recently has been based the metric system of weights and measures. By placing any of the digits in the place of the zero to make the numbers between ten and twenty, we have the law established. The science of arithmetic, like all other sciences, was very limited and imperfect at the beginning, and the successive steps by which it has reached its present extension and perfection have been taken at long intervals, and among different nations. ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”

“ If the strong man who had six fingers on each hand, and six toes on each foot, had invented a system of numeration, it would have gone in twelves, nearly like the duodecimals which our carpenters use; unless, indeed, he had been stupid after the manner of very strong men, and not gone beyond sixes. We see how the Romans, though they inherited from their Eastern ancestors a numeration by tens up to decem, and then beginning again undecim, &c., yet when they began to write a notation could get no farther than five—I., II., III., IV., V. ”

“ The systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present. This latter presentation may, in the absence of any accepted term, be called number-ideation; this word covering not only the perception or recognition of particular numbers, but also the formation of a number-concept.
13. Notation of Numbers.—The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system. ”

“ It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, the number carried being always the radix of the system. Thus, in the quinary system we carry fives instead of tens. I now give an example of multiplication and division: [Pg 170]
Another way of turning a number from one scale into another is as follows: Multiply the first digit by the old radix in the new scale, and add the next digit ”

“ A fresh mental effort is required of the ignorant child of nature; and the result is “all the fingers and one more,” “both hands and one more,” “one on another count,” or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. ”

“ Whether or not the principal number base of any tribe is to be 20 seems to depend entirely upon a single consideration; are the fingers alone used as an aid to counting, or are both fingers and toes used? If only the fingers are employed, the resulting scale must become decimal if sufficiently extended. If use is made of the toes in addition to the fingers, the outcome must inevitably be a vigesimal system. ”

“ This is so only because we have become conditioned to the decimal system, which was even more hopeless to us as youngsters. We may have forgotten, with the contempt of familiarity, that our number system is built on the idea of powers. In grade school we learned that starting at the right we had units, tens, hundreds, thousands, and so on. In the decimal number 111, for example, we mean 1 times 102, plus 1 times 101, plus 1. We have handled so many numbers so many times we have usually forgotten just what we are doing, and how. ”

“ The use of figures and symbols offers an infinite number of possible combinations, but the straight decimal system is in no way practical for general use, chiefly because a voluminous key is necessary even to interpret the symbol. ”

“ The merit of our actual system of numeration consists in including the expression of all numbers in a single idea, making the value of each figure ten times that to the right, and filling all intervals with zeros. ”

“ In nearly all the examples given the decimal becomes the leading element in the formation of all units above 100, just as was the case in the Celtic scales already noticed.
Among the northern tribes of Siberia the numeral scales appear to be ruder and less simple than those just examined, and the counting to be more consistently vigesimal than in any scale we have thus far met with. The two following examples are exceedingly interesting, as being among the best illustrations of counting by twenties that are to be found anywhere in the Old World. ”

“ Here is a subtraction sum. We cannot take 9 units from 3 units, so we get change from the next row, that gives 13 units, from which we take 9, and have 4 left. We have nothing to take from our 8 remaining tens, so we write 8. We have no hundreds, so we cannot take away 9, but we change one of our thousands into 10 hundreds, and take away 9, leaving 1; lastly we take away 2 from our 4 thousands, and get 2—altogether 2184.
Decimal fractions.Now would come in naturally the extension of this system of notation to decimal fractions, marking the unit by a full stop. ”

“ It was this pentadactyl peculiarity of the frog, inherited by men and women through the reptiles and mammals, that gave rise to the decimal system of numbers and other unhandy facts in human life. The decimal system arose out of the practice of early men performing their calculations on their fingers. This method of calculating is still used by primitive peoples all over the world. The sum of the digits of the two hands came, in the course of arithmetical evolution, to be used as a unit, and from this simple beginning grew up the complicated system of tens found among civilised peoples. ”

“ The uniform employment of twelve as the divisor and multiplier in tables of weight, distance, time, and space, as well as in arithmetical notation, has all the conveniences of the decimal system of France, and some others besides due to the greater convenience of twelve as a base. ”

“ Examples of this poverty of number knowledge are found among the forest tribes of Brazil, the native races of Australia and elsewhere, and they are considered in some detail in the next chapter. At first thought it seems quite inconceivable that any human being should be destitute of the power of counting beyond 2. But such is the case; and in a few instances languages have been found to be absolutely destitute of pure numeral words. The Chiquitos of Bolivia had no real numerals whatever,1 but expressed their idea for “one” by the word etama, meaning alone. ”

“ By so doing they perfected the system of place value, since they could now show that even when there was no one of the nine numerals in a particular place, the value of the place remained, and the values of the adjoining places could be maintained. The invention of a symbol for nothing is the crowning, transcendent achievement in the perfection of the decimal system, and lies at the base of all subsequent arithmetical progress. It is the peculiar triumph of the Hindu mathematicians to have made this contribution to the science of number. ”

“ Still another instance of quaternary counting, this time carrying with it a suggestion of binary influence, is furnished by the Mocobi207 of the Parana region. Their scale is exceedingly rude, and they use the fingers and toes almost exclusively in counting; only using their spoken numerals when, for any reason, they wish to dispense with the aid of their hands and feet. Their first eight numerals are:
There is probably no recorded instance of a number system formed on 6, 7, 8, or 9 as a base. ”

“ Etymological research into the derivation of numeral words thus hardly goes with safety beyond showing in the languages of the lower culture frequent instances of digit-numerals, words taken from direct description of the gestures of counting on fingers and toes. Beyond this, another strong argument is available, which indeed covers almost the whole range of the problem. The numerical systems of the world, by the actual schemes of their arrangement, extend and confirm the opinion that counting on fingers and toes was man’s original method of reckoning, taken up and represented in language. ”

“ Hence, while one race may use a decimal, another a quinary-vigesimal, and another a sexagesimal scale, and while one system may actually be inherently superior to another, no user of one method of reckoning need ever think of any other method as possessing practical inconveniences, of which those employing it are ever conscious. ”

“ How they indicated any number greater than one is a point still requiring investigation. In all other known instances we find actual number systems, or what may for the sake of uniformity be dignified by that name. In many cases, however, the numerals existing are so few, and the ability to count is so limited, that the term number system is really an entire misnomer.
Among the rudest tribes, those whose mode of living approaches most nearly to utter savagery, we find a certain uniformity of method. The entire number system may consist of but two words, one and many ”