Number system

“ If we look to natural reasons for the development of number systems, we might decide that the binary, or two-valued system, did not attain much prominence in naïve civilizations because 98there are so few one-legged, two-toed animals! Only when man built himself a machine uniquely suited to two-valued mathematics did the binary system come into its own.
Numbers are merely conventions, rigorous conventions to be sure with no semantic vagueness. God did not ordain that we use the decimal system, as evidenced in the large number of other systems that work just fine. ”

“ 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. ”

“ 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. ”

“ 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.... ”

“ This is due to his habit of counting by pairs, which enables him to discover whether any number within reasonable limit is odd or even. Some of the negro tribes of Africa, and of the Indian tribes of America, have the same habit. Progression by pairs may seem to some tribes as natural as progression by single units. It certainly is not at all rare; and in Australia its influence on spoken number systems is most apparent.
Any number system which passes the limit 10 is reasonably sure to have either a quinary, a decimal, or a vigesimal structure. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ That the first numbers of a numeral scale are usually derived from other sources, we have some reason to believe; but that all above 2, 3, or at most 4, are almost universally of digital origin we must admit. Exception should properly be made of higher units, say 1000 or anything greater, which could not be expected to conform to any law of derivation governing the first few units of a system. ”

“ Occasionally some exception to this rule is found; or some variation, such as is presented by the forest tribes of Brazil, who, instead of counting on the fingers themselves, count on the joints of their fingers.5 As the entire number system of these tribes appears to be limited to three, this variation is no cause for surprise.
The variety in practical methods of numeration observed among savage races, and among civilized peoples as well, is so great that any detailed account of them would be almost impossible. In one region we find sticks or splints used ”

“ The foregoing examples will show with considerable fulness the wide dispersion of the quinary scale. Every part of the world contributes its share except Europe, where the only exceptions to the universal use of the decimal system are the half-dozen languages, which still linger on its confines, whose number base is the vigesimal. Not only is there no living European tongue possessing a quinary number system, but no trace of this method of counting is found in any of the numerals of the earlier forms of speech, which have now become obsolete. ”

“ 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. ”

“ The process just indicated may be gone through with at 5, or at 20, thus giving us a quinary or a vigesimal, or, more probably, a mixed system; and, in rare instances, some other number may serve as the point of departure from simple into compound numeral terms. But the general idea is always the same, and only the details of formation are found to differ.
Without the establishment of some base any system of numbers is impossible. The savage has no means of keeping track of his count unless he can at each step refer himself to some well-defined milestone in his course. ”

“ This limit is often extended to 20, and not infrequently to 100. Again, we find 1000 as the limit; or perhaps 10,000; and sometimes the savage carries his number system on into the hundreds of thousands or millions. Indeed, the high limit to which some savage races carry their numeration is far more worthy of remark than the entire absence of the number sense exhibited by others of apparently equal intelligence. If the life of any tribe is such as to induce trade and barter with their neighbours, a considerable quickness in reckoning will be developed among them. ”

“ 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. ”

“ It is not to be supposed from the observations that have been made on the formation of savage numeral scales that all, or even the majority of tribes, proceed in the awkward and faltering manner indicated by many of the examples quoted. Some of the North American Indian tribes have numeral scales which are, as far as they go, as regular and almost as simple as our own. But where digital numeration is extensively resorted to, the expressions for higher numbers are likely to become complex, and to act as a real bar to the extension of the system. ”

“ In the even tens it is quite regularly decimal, unless 80 is of the structure suggested above. On the other hand, the odd tens are formed in the ordinary vigesimal manner. The reason for this anomaly is not obvious. I know of no other number system that presents the same peculiarity, and cannot give any hypothesis which will satisfactorily account for its presence here. ”

“ 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. ”

“ Nor, on the other hand, is the development of a numeral system an infallible index of mental power, or of any real approach toward civilization. A continued use of the trading and bargaining faculties must and does result in a familiarity with numbers sufficient to enable savages to perform unexpected feats in reckoning. Among some of the West African tribes this has actually been found to be the case ”

“ In the system of counting every place, or column, counting from the right, has a value ten times greater than the one in the place or column nearest on the right. Thus in the number 36,542 the first figure on [71] the right represents “ones,” the next ten times as much or “tens,” the next ten times as much again or “hundreds,” and so on. We really read this number backward when we name it, for in handling it in any way we have to start with the last figure, representing the “ones.” The number really means two ones, four tens, five hundreds, six thousands and three ten thousands. ”

“ 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. ”

“ 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 ”

“ Every system of numeration is thus founded on the employment of units of different orders, each of which contains the preceding one a certain number of times, or, in other words, upon a geometrical progression, the ratio of which is called the base of the system, Aristotle observed that the number four might take the place of ten, and Weigel, in 1687, published a plan of a quaternary arithmetic. ”

“ Given that tally systems are contingent with regard to number, the logically true system of arithmetic is not only useful but essential. This might not be immediately apparent because it might seem that tally systems can be identified as systems of counting by the fact that the totals they produce stand in a one-to-one relation with objects in the real world. ”

“ A binary scale could, as it is developed in primitive languages, hardly extend to 20, or even to 10, without becoming exceedingly cumbersome. A binary scale inevitably suggests a wretchedly low degree of mental development, which stands in the way of the formation of any number scale worthy to be dignified by the name of system. Take, for example, one of the dialects found among the western tribes of the Torres Straits, where, in general, but two numerals are found to exist. ”

“ But despite its simplicity, the binary system is far from being inferior to the decimal system. You can prove this by doing some counting on your fingers.
Normally we count, or tally, by bending down a finger for each new unit we want to record. With both hands, then, we can add up only ten units, a quite limited range. We can add a bit of sophistication, and assign a different number to each finger; thus 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Now, believe it or not, we can tally up to 55 with our hands! As each unit is counted, we raise and lower the correct finger in turn. ”

“ 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. ”

“ But such men experience no difficulty whatever in performing their mental computations in the decimal system; and they acquire through constant practice such quickness and accuracy of calculation, that it is difficult to see how octonary reckoning would materially assist them. Altogether, the reasons that have in the past been adduced in favour of this form of arithmetic seem trivial. There is no record of any tribe that ever counted by eights, nor is there the slightest likelihood that such a system could ever meet with any general favour. ”