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therefore, is that which has for its Denominator an Unit, with a Cypher, or Cyphers, annexed to it, as 10, 100, 1000, &c. But the ufual Way of noting down Decimal Fractions, and which makes them more ready for Ufe, is to omit the Denominator, and prefix a Point or Comma before the Numerator to the left Hand, to diftinguish them from whole Numbers.

As, in whole Numbers, Cyphers to the right Hand increase the Value or Power of the Figures in a ten-fold Proportion; fo, in Decimal Fractions, Cyphers to the left Hand decreafe their Value or Power in a ten-fold Proportion. Thus .5 ftands for, or five tenth Parts; .05 for, or five hundredth Parts; .005 for, or five thoufandth Parts. And, as Cyphers on the left Hand of whole Numbers neither add or diminish, in the fame Manner, Cyphers on the right Hand of Decimals are of no Signification; for .8 and .80 and .8co are all the fame Fraction, fince they are, 80 and 800 And it has been already fhewn, at Art. 36, that, when there are Cyphers both in the Numerator and Denominator, the fame Number in both may be cut off or omitted: By which may be feen, that the difficult Work of reducing Vulgar Fractions to the fame Denominators, in order to their being added or fubtracted, is performed in the Decimals without any Trouble.

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Τότο

The Order of Places in Decimals proceeds from the left Hand to the right: The first next the Dot is called the Place of tenth Parts, the fecond the Place of hundredth Parts, the third the Place of thoufandth Parts, &c. because whatever Figure ftands in thofe Places ftands refpectively for fo many tenth, hundredth, thoufandth, &c. Parts of an Unit: As in this, .5473; here the 5, in the firft Place, is, the 4, in the fecond Place, is the 7, in the third Place, is TO; and the 3, in the laft Place, being the fourth, is TO; and may be feparately expreffed, .5, .04, .007, .co03. The Denominator of a Decimal Fraction is always known by the Number of Places in the Decimal, the Denominator confifting of as many Cyphers as there are decimal Places, with an Unit prefixed; always obferving, that Cyphers in the Decimal are reckoned as Places as well as Figures: Thus .05207 is a Decimal of five Places, whofe Denominator is rooooo.

For the better understanding the Notation of Decimals, the following Scheme of Numbers, increafing in a ten-fold Pro

portion

portion from Unity to the left Hand, and decreasing in the fame Proportion from Unity to the right Hand, will be of Service,

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792453

or

In this Scheme, or Table, you may obferve, that the Places of whole Numbers are feparated from the Places of decimal Parts by a Dot. The Number on the left Hand of the Dot is 4735286, which expreffes four Million, feven Hundred and thrity-five Thoufand, two Hundred and eighty-fix Units or Integers. The Number on the right Hand of the Dot expreffes feven Hundred and ninety-two Thousand, four Hundred and fifty-three Parts of an Unit, the whole being divided into one Million equal Parts; which is to be wrote thus, 7000000, according to the Method of Vulgar Fractions; thus, 792453, according to the Decimal Notation. So that, if ĺ were to read this Decimal Fraction, .7, I fhould call it feven Tenths of an Unit, the Unit being divided into 10 Parts. If I were to read this Decimal Fraction, .79, I fhould call it feven Tenths and nine Hundredth Parts of an Unit, or Seventynine Hundredth Parts of an Unit, the Unit being now divided into 100 equal Parts. And .792 is feven Tenths, nine Hundredths, and two Thousandth Parts, or feven Hundred and ninety two Thousandth Parts of an Unit, the Unit being now divided into rooo Parts. And .7924 is feven Tenths, nine Handredths, tuo Thoufandths, and four Ten-thoufandth Parts

of

of an Unit, or seven Thoufand, nine Hundred and twenty-four Ten-thousandth Parts, the Unit being now divided into 10000 Parts; and the fame of any other Decimal Fraction,

Addition of Decimal Fractions.

Article 40. To add Decimal Fractions together.

Rule. Place them one under the other, as in Addition of whole Numbers, obferving to place Tenths under Tenths, Hundredths under Hundredths, Thoufandths under Thoufandths, and fo on; then add them, as in whole Numbers, fetting down the odd ones, and carrying the Tens to the next Place. If the laft Place, or Place of Tenths, next the Dots, amounts to any Number of Tens, they must be carried to the Place of Units, on the left Hand of the Dot.

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Exam. 1. Having placed the feveral Fractions one under another, according to their feveral Denominations, and added them, as in whole Numbers, the Sum is .719, or feven Tenths, one Hundredth, and nine Thousandth Parts of an Integer or Unit, or Seven Hundred and nineteen Thousandth Parts of an Integer.

The other three Examples are added in the fame Manner,

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Exam. 5. There being no Fraction in the middle Row of the fame Denomination with the 3 in the upper and the 4 in

the

the lower Rows, I add the 4 and the 3 together, and then proceed as before; only, in adding the Place of Tenths, the Sum is 17, therefore I fet down the 7 in the Place of Tenths, with the Dot before it; and the 1, which is to be carried, I place to the left of the Dot: So is the Sum compofed of one Unit, and the Fraction .7587.

The fixth and feventh Examples are added in the fame Manner.

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Exam. 8. The Fractions are added as before; but, when you come to the Place of Tenths, the Sum being 16, the 6 must be placed down, and a Dot before it; then carry the 1 to the Integers, and add them, placing them to the left of the Dot, as is done in the Example. In the ninth Example, there is 2 to be carried to the Integers. In the fame Manner may any other Addition in the fame Kind be done, only taking Care that the Figures, whether Integers or Fractions, be placed under the Figures of a like Denomination; for, if I am to add 47.52 to .42, the Abfurdity would be evident, were they to be placed and added in this Manner, viz.

47.52 .42

89.52

But, by taking Care to place the Figures of the fame Denomination one under another, the Work will eafily be observed to be true, thus,

47.52
.42

Subtra&tion

Subtraction of Decimal Fractions.

Article 41. To fubtract Decimal Fractions.

Rule. Place the Numbers one under another, according to their Denomination, as in Addition; then fubtract them, as in Subtraction of whole Numbers.

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This being done, in all Refpects, as in Subtraction of whole Numbers, there can be no Neceffity of multiplying Examples: But it may be proper to obferve, that, when the Number to be fubtracted has more decimal Places than the Number from which it is to be fubtracted, you must take fuch Figures from 10, for which I must be carried to the next Place, as will appear by the following Examples.

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Exam. 5. There being nothing to fubtract the 2 from, I take it from 10, and the Remainder is 8; then I carry the I that I borrowed to the 4, which makes 5, and, there being nothing over the 4, I take 5 from 10, and the Remainder is 5; then I carry the I to the 9, which makes 10; now 10 from 8 I cannot, but 10 from 18, as in whole Numbers, and there remains 8; and the I must be carried to the 6, as in Subtraction of whole Numbers. performed in the fame Manner. as in whole Numbers, that is, add the Remainder to the Sum fubtracted, and it makes the fame as the Number it was fubtracted from,

I

The other Examples are And the Proof is the fame

Multiplication

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