Decimal

“ In the case of decimal fractions the psychological process is more complex still; for the pupil has given him one of the terms, the numerator, from which 20 he must mentally deduce the denominator from the position of the decimal point. This doubles the work of reading and recognizing the fractional number. But it makes addition and subtraction of fractions nearly as easy as that of simple numbers and assists also in multiplication of fractions. But division of decimals is a much more complex operation than that of common fractions. ”

“ The Tract on Decimal Arithmetic, which introduced the idea of decimal fractions and a notation for them, was published in 1585 in Dutch, under the title of ‘De Thiende,’ and in the same year in French, under the title of ‘La Disme.’
We find Briggs, in his ‘Remarks on the Appendix,’ while sometimes employing the point, also using the notation 25118865 for , distinguishing the fractional part by retaining the line separating the numerator and denominator, but omitting the latter. The form 2|5118865 has also been used. ”

“ To Simon Stevin of Bruges in Belgium (1548–1620) , a man who did a great deal of work in most diverse fields of science, we owe the first systematic treatment of decimal fractions. In his La Disme (1585) he describes in very express terms the advantages, not only of decimal fractions, but also of the decimal division in systems of weights and measures. Stevin applied the new fractions "to all the operations of ordinary arithmetic." [25] What he lacked was a suitable notation. In place of our decimal point, he used a cipher; to each place in the fraction was attached the corresponding index. ”

“ The first Appendix, on Computation, and the sixth, on Decimal Money, should be read and practised by every student with as much attention as any part of the work. The mastery of the rules for instantaneous conversion of the usual fractions of a pound sterling into decimal fractions, gives the possessor the greater part of the advantage which he would derive from the introduction of a decimal coinage. ”

“ 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. ”

“ 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. ”

“ Of all the simplifications of commercial arithmetic, none is comparable to that of expressing shillings, pence, and farthings as decimals of a pound. The rules are thereby put almost upon as good a footing as if the country possessed the advantage of a real decimal coinage. ”

“ 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. ”

“ As the 247th remainder cannot be so great as 247, it must be one of these which have preceded. From the step where the remainder becomes the same as a former remainder, it is evident that former figures of the quotient must be repeated in the same order.
131. You will here naturally ask, What is the use of decimal fractions, if the greater number of fractions cannot be reduced at all to decimals? The answer is this: The addition, subtraction, multiplication, and division of decimal fractions are much easier than [Pg 69] those of common fractions ”

“ Decillion, de-sil′yun, n. a million raised to the tenth power: in the French and American notation, a thousand raised to the eleventh power.
Decimal, des′i-mal, adj. numbered or proceeding by tens.—n. a fraction having ten or some power of ten for its denominator.—v.t. Dec′imalise, to reduce to the decimal system.—ns. Dec′imalism; Dec′imalist.—adv. Dec′imally.—Decimal notation, a system of writing numbers based on ten and powers of ten, like our ordinary system; Decimal system, a system whose units are tens and powers of tens, esp. in the French metric system of weights and measures. ”

“ Devices and illustrations of a material kind are necessary to build up in the pupil’s mind at the beginning a clear concept of a tenth, etc., etc., and then to show that one-tenth written as a decimal is only a shorthand way of writing 1/10 as a common fraction, and so on. He sees very soon that the decimal is only a shorthand common fraction, and this notion he must hold to. This is the vital point in decimals. The idea that they can be changed into common fractions and the reverse at will establishes the fact in the pupil’s mind that they are common fractions and not uncommon ones. ”

“ The result is 085·9 = 85·9, the position of the decimal point being determined by counting the figures before the decimal points in the original numbers.
112. Division.—In the same way, in performing approximate division, we can at a certain stage begin to abbreviate the divisor, taking off one figure (but with correction of the final figure of the partial product) at each stage. Thus, to divide 85·9 by 3·1415927 to two places of decimals, we in effect divide ·0859 by ·31415927 to four places of decimals. ”

“ 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. ”

“ But even there, where the decimal measures have been introduced, much time is lost in school simply because the teachers are not accustomed to the idea that every measure is only approximate, and that it is absurd to calculate with the exactitude of one gramme, or of one metre, when the measuring itself does not give the elements of such an exactitude. Whereas in France, where the decimal system of measures and money is a matter of daily life, even those workers who have received the plainest elementary education are quite familiar with decimals. ”

“ 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. ”

“ If ever the fraction of a farthing be wanted, remember that the coinage-result is larger than the decimal of a pound, when we use only three places. From 1000 times the decimal take 4 per cent, and we get the exact number of farthings, and we need only look at the decimal then left to set the preceding right. ”

“ A vulgar fraction is reduced to a decimal by dividing the numerator (increased sufficiently with ciphers) by the denominator. Thus, 1⁄2 = ·5, 1⁄8 = ·125, &c.
The value of a decimal, of any denomination, is found by multiplying it by the number of parts in the next less denomination, and cutting off as many places to the right hand as there are decimals, and so on until the terms are exhausted. Thus, ·634 oz. ”

“ 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. ”

“ 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. ”

“ Here four ciphers have been annexed to the numerator, and one has been taken from the denominator. Make five decimal places in the quotient, which then becomes 2·84192, and this is the quotient of 17·762 divided by 6·25.
150. The rule for division of one decimal by another is as follows: Equalise the number of decimal places in the dividend and divisor, by annexing ciphers to that which has fewest places. Then, further, annex as many ciphers to the dividend [18] as it is required to have decimal places, throw away the decimal point, and operate as in common division. ”

“ Decimal Fractions are so called because the fractions are always tenths. They differ from [90] Vulgar Fractions in this, that the denominator is not written, but a point before it is used instead.
A STRONG TITHE.
Decimals are best illustrated by tithes, which are general and universal tenths extracted in every part of "merry England." They are added, subtracted, multiplied, and divided like any other numbers, but to designate their value a point is prefixed.
In tithes, as in decimals, the denominator does not appear; that is to say, the incumbent rarely lives at his incumbency. ”

“ To have a really correct apprehension of the principle of decimal notation, for instance, to understand that it is purely arbitrary, and that we might in the same way take any other number than ten as the base of a numerical scale,—that we might increase for instance by fives, or eights, or nines, or twelves, just as well as by tens—all this requires considerable maturity of intellect, and some subtlety of reasoning. ”

“ 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. ”

“ Simple as decimal fractions appear to us, the invention of them is not the result of one mind or even of one age. They came into use by almost imperceptible degrees. The first mathematicians identified with their history did not perceive their true nature and importance, and failed to invent a suitable notation. The idea of decimal fractions makes its first appearance in methods for approximating to the square roots of numbers. ”

“ 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 decimal scale Is no less simple in its structure than the quinary, and the savage, as he extends the limits of his scale from 5 to 6, may call his new number 5-1, or, with equal probability, give it an entirely new name, independent in all respects of any that have preceded it. ”

“ In whatever light it is viewed, the decimal notation must be regarded as one of the most striking monuments of human ingenuity, and its beneficial influence on the progress of science and the arts, on commerce and civilization, must win for its unknown author the everlasting admiration and gratitude of mankind. ”

“ The notation of the decimal form is such that the content of DL 10 can be passed out onto a card in the usual way and if the card is then listed the digits of the numbers α,m will then appear on the listing paper in the usual way. Or in other words only the first 10 minor cycles of DL 10 are used, and a decimal digit is represented by the minor cycle in which a pulse occurs, and its significance by the position of it within the minor cycle. ”

“ 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. ”