Decimal 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. ”

“ Our neighbours in France, who are particularly fond of framing theories and experimenting on them for the edification of other nations, availed themselves of the general upturning of affairs in 1789, to introduce a universal decimal system, to be applied to everything whatever that could be counted, weighed, or measured. ”

“ It is incontrovertible that in dealing with the division of material substances, the binary scale comes into mental and practical play as easy and convenient. Not all the specious pleas of those who advocate the so-called reform can turn us from it. Its advantages over the decimal system in all the operations of every day life are apparent. ”

“ 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. ”

“ 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. ”

“ In this system, then, the only digit employed is 1. The 0 plays the same important part in it as in the decimal system. It multiplies the figure that immediately precedes it by the value of the radix. The symbol 40, in our denary scale, represents ten times four; in the octenary it would denote eight times four, and in the quinary five times four. These two symbols, 1 and 0, then, are the only ones that enter into calculations. It is evident that thought in arithmetical work is almost superseded, and that all numerical operations are reduced to the manual labor of writing. ”

“ Its place in the organization of society as well as in the regulation of other aspects of human life has been taken by a numerical system that still dominates our entire culture—the decimal system. Even prior to the age of Columbus, the decimal system made its appearance in certain more civilized parts of the Western world where the duodecimal system was never known. ”

“ The Dewey decimal system is undoubtedly the most complete and highly perfected in theory of any, and, as the name indicates, is a combination of figures and periods giving a symbol of almost unlimited expansion. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Now it seems to me that the two facts—first, that in early days men diverged from the decimal division into modes of division which furnished convenient aliquot parts, and, second, that where, as in America, the decimal system has been adopted for coinage, they have in the focus of business fallen into the use of aliquot parts in spite of the tacit governmental dictation—not only prove the need for this mode of division, but imply that, if the metric system were universally established, it would be everywhere traversed by other systems. ”

“ Ten is, theoretically, ill suited for the radix of a system of notation, because it permits of only one bisection. The half of it is five, an odd number. It also is incapable of any other division. On account of these defects the system is ill adapted to the operations of the shop and market. Although our calculations are universally made in the decimal system, none of our tables of weights and measures are decimal in any one of their subdivisions. In all departments of trade the current prices have been derived from a process of successive halvings. ”

“ 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 greatest objections to the change, apart from the difficulty of getting people to make it, which is doubtless much exaggerated, are that a decimal system does not admit to a sufficient extent of binary subdivision, and that it does not admit of ternary subdivision at all. The third part, for instance, of a pound, of a florin, of a cent, being 333^, 33J, 3J mils respectively, could not be exactly paid in decimal currency, while there is no difficulty in paying the third part of a pound, or of a shilling by our present coinage. ”

“ The additional mental labor required to compute in this scale is not very great, while the manual labor is somewhat less than in using the decimal system. The scale has always been a favorite one with those who object to the decimal notation. ”

“ A decimal coinage has for the last sixty years been proposed for England, and it is almost certain that if any one scheme could be pointed out as much preferable to any other it would be accepted. As it is, there are two or three proposals, each commanding some support, while many advocates of the decimal system prefer to wait till an international agreement for its adoption can be obtained. ”

“ Lowe fancies that when he pays one and sixpence, he pays in integers, and so he does, if his integer be a penny or a sixpence. Let him bring his mind to contemplate a mil as the integer, the lowest integer, and the seven cents five mils which he would pay under the new system would be payment in integers also. But, as it happens with some others, he looks up the present system, with Cocker, [299] and Walkingame, [300] and always looks down the proposed system. The word decimal is obstinately associated with fractions, for which there is no need. ”

“ 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. ”

“ 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. ”

“ What gives the metric system pre-eminence is the simplicity and uniformity of its decimal scale, forming part and parcel as it does of the decimal system of notation, and lending itself to a decimal coinage as in France, Germany, Italy, and Spain. The metre is organically related to all measures of length, surface, capacity, solidity, and weight. A cubic centimetre of water, taken as it melts in a vacuum, at 4° C., the temperature of maximum density, is the gram from which other weights are derived ”

“ It was more than a chance that produced the decimal system, for the universe is pentalogic, as all of the fundamental series discovered in nature are pentalogic by reason of the five concomitant properties. The origin of the decimal system was the recognition by primitive man of the reciprocal pentalogic systems involved in the two hands of the human body, and the pentalogic properties are always in pairs. ”

“ In Europe, before the introduction of the algorithm or full Indo-Arabic system with the zero, we find a transition system in which calculations were made on the decimal system with an abacus, but instead of unit counters there were placed in the columns ciphers, with values from one to nine, and of forms that are at bottom the Indian forms and agree most nearly with the numerals used by the Arabs of Africa and Spain. ”

“ The essentially vigesimal character of this system changes in the formation of some of the higher numerals, and a suspicion of the decimal enters. One hundred is boinic, 5 men; but 200, instead of being simply lahuh-inic, 10 men, is tsa-bo-inic, 2 × 100, or more strictly, 2 times 5 men. Similarly, 300 is 3 × 100, 400 is 4 × 100, etc. The word for 1000 is simple instead of compound, and the thousands appear to be formed wholly on the decimal base. A comparison of this scale with that of the Nahuatl shows how much inferior it is to the latter, both in simplicity and consistency. ”

“ But without opposing the metric system as threatening to stand in the way of a more perfect system, he opposes it as intrinsically undesirable, saying:—"I think that all that can be done is to make our coinage and measures as little decimal as possible, and our computation as decimal as may be." ”

“ 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. ”

“ These examples show that Napier was in possession of all the conventions and attributes that enable the decimal point to complete so symmetrically our system of notation, viz. (1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, &c., were required; (2) he used ciphers after the decimal point and preceding the first significant figure; and (3) he had no objection to a decimal standing by itself without any integer. Napier thus had complete command over decimal fractions and the use of the decimal point. ”

“ 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 ”

“ Decimal fractions were used by the early peoples of central Asia and were transmitted by them to the Babylonians. Their system was based, apparently, upon a sexagesimal scale. Simon Stevin (1548-1620) , a Belgian, made great improvements in [Pg 104] decimals. He adopted the plan of William Buckley, of England, and other mathematicians, and made the base 100,000, instead of 60.
John Napier (1550-1617) , a Scottish nobleman, invented logarithms. The story of this great mathematician's work is one of the most interesting in the history of science. ”

“ 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. ”