Decimal fraction

“ In transforming a vulgar fraction, say 1/17, to a decimal [Pg 229] fraction, we proceed as below, adding as many noughts to the dividend as we like until there is no remainder, or until we get a recurring series of figures, or until we have carried it as far as we require, since every additional figure in a never-ending decimal carries us nearer and nearer to exactitude.
Now, since all powers of 10 can only contain factors of the powers of 2 and 5, it clearly follows that your decimal never will come to an end if any other factor than these occurs in the denominator of your vulgar fraction. ”

“ The true definition of a fraction may be thus summed up: It is the division of its numerator by its denominator; as seven-eighths are equivalent to the whole number 7 divided by 8—whence a fraction is obtained. Decimal fractions simplify calculations greatly, as they are constructed on the principle of having one common denominator—a multiple of 10; and thus fractions can be added, subtracted, and divided without repeating over the tedious process of bringing down to a common denominator. See Arithmetic; Decimal Fraction. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Recurring Decimals are a particular kind of series, which arise from the expression of a fraction as a decimal. If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur. ”

“ 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. ”

“ 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 ”

“ We will remark lastly that if the factors of the product are incommensurable, and if it is necessary to convert them into decimals before effecting the multiplication, we should not know how far we should carry the approximation of the factors before applying the above rule. It will therefore be necessary to teach the pupils the abridged methods by which we succeed, at the same time, in using fewer figures and in knowing the real approximation of the result at which we arrive.
Periodical decimal fractions are of no use. ”

“ The late Professor Newcomb made the matter plain even to the non-mathematical mind, when he said that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." ”

“ Take a mathematical line of one foot in length, divide it into ten parts, each of those parts into ten parts, and so on continually. If a point A be taken at hazard in the line, it does not appear self-evident that if the decimal division be continued ever so far, one of the points of division must at last fall exactly on A: neither would the same appear necessarily true if the division were made into sevenths, or elevenths, or in any other way. There may then possibly be a part of a foot which is no exact numerical fraction whatever of the foot ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. ”

“ First, he will have it that we design to bring the uneducated into contact with decimal fractions. If it be so, it will only be as M. Jourdain was brought into contact with prose. In fact, Quoi! quand je dis, Nicole, apportez-moi mes pantoufles, c'est de la prose? [305] may be rendered: "What! do you mean that ten to the florin is a cent a piece must be called decimal reckoning?" If we had to comfort a poor man, horror-struck by the threat of decimals, we should tell him what manner of fractions had been inflicted upon him hitherto ”

“ Fractions on the en body are quite indistinct in tabular work, for which fractions on the em body should be preferred. Piece-fractions on two bodies, each one half of the en body of the text type, are often required for vulgar fractions, but they are almost unreadable when cast for the smaller sizes. As these piece-fractions differ in size and cut from the solid fractions of the font, the two forms should not be used in the same table.
Decimal fractions are most intelligibly stated in figures, with the decimal point placed on the line of the figure, as it is in .638. ”

“ 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. ”

“ 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. ”

“ When the simplicity of decimal calculations is so evident, it is to be regretted that all our measures are not subdivided into decimal parts, as our currency is, and why our government should set so good an example in one particular, and neglect all he rest, it is not easy to determine. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 pound, which was not a coin, but a note promising to pay 20 shillings to the bearer, is also disused for the present, and the sovereign supplies its place; but the name pound is still given to 20 shillings.
[44] Farthings are never written but as parts of a penny. Thus, three farthings being 3⁄4 of a penny, is written 3⁄4, or 3⁄4. One halfpenny may be written either as 2/4 or 1⁄2; the latter is most common.
[45] When a decimal follows a whole number, the decimal is always of the same unit as the whole number. ”

“ 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. ”