All Fractions are either Vulgar or Decimal.

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In Vulgar Fractions the Denominator may Number whatsoever, and is always fet under the Numerator with a Line between; fo two Thirds is thus expreffed ; three Fourths, thus; and eleven Fifteenths, thus I.

But in Decimal Fractions the Numerator only is expreffed, or wrote down; the Denominator being understood by Places, and is always 10, 100, 1000, &c. being an Unit with as many Cyphers annexed to it, as there are Figures in the Numerator. So the Denominator of .5 is 10; the Denominator of .47 is 100; and of .358 is 1000, &c.

A Decimal Fraction is known from a whole Number by a fmall crooked Dash before it, called a parting Line: Sometimes a Point or Dot is ufed inftead of it. Thus (58 or .397 are Decimals.

If a Number confifts of a whole Number and a Decimal, it is called a mixt Number. If a Decimal ends at a certain Number of Places, it is faid to be Finite: But if it runs on without terminating, it is faid to be Infinite. When one of the Figures in the Decimal is repeated, as .666, &c. it is called a single circulating or recurring Decimal. When two or more are repeated, as .602.602, &c. it is called a compound circulating or recurring Decimal. In Numbers where the fame Figure continually circulates, make a Dafh across the first, thus .866; but where every two or three Figures repeat, make a Dafh acrofs the first and last, thus.Bof; the reft may at prefent be omitted.

Motation of Decimals.

AS in Common Arithmetic whole Numbers increase towards the Left Hand in a tenfold Proportion; fo on the contrary, in this Kind of Arithmetic, Decimals decrease towards the Right Hand in the fame Proportion, as appears in the following Table.

From this Table it is evident, that the first Figure in the Decimals is fo many tenth Parts of the Unit or whole Number, as the Figure itself denotes: The Second is fo many hundredth Parts: The third so many thousandth Parts. Thus .2 is, or two tenth Parts; 23 is 23, or twenty-three hundredth Parts; .234 is 234, or two hundred and thirty-four thoufandth Parts.

This Table is the very Foundation of Decimals, and on that Account, ought to be attentively confidered.

Cyphers annexed or added on the Right Hand of any Decimal Fraction neither increase nor diminish its Value. Thus is equivalent to, or to .25 hundredth Parts.

But Cyphers, if placed before the Decimal, decrease its Value in a tenfold Proportion: As .3 tenths having a Cypher placed before it becomes; if two Cyphers are prefixed it is 100%; that is, only three thousandth Parts and fo of any others.

16

Hence it follows, that when you are to write down a Decimal Fraction, whofe Denominator has more Cyphers in it than there are Figures in the Numerator, they must be fupplied by prefixing fo many Cyphers before the Figures of the Numerator. As fuppofe was to be written down without its Denominator; in this Cafe, because there are three Cyphers in the Denominator, and but two Figures in the Numerator, we must therefore prefix a Cypher before the 16, and fet it down thus, .016.

A mixt Number is compofed of a whole Number and a Fraction, and is thus written 5.7, viz. five and seven Tenths; and 25.47, which is twenty-five, and forty-feven hundredth Parts of another.

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The more Places a Decimal (which does not terminate) confifts of, the nearer it expreffes the Truth; but in Practice we feldom ufe more than three Places; and in high and accurate Calculations not above 4, or 5, or 6 at the most and when the Decimal confifts of feveral Nines, we reject them, and make the next Figure on the Left Hand one more: Thus, for 5.199 we write 5.2; and for 9.99 wę write 10.

Addition of Decimals.

IN Addition of Decimals, carefully place the Numbers under each other according to their respective Places; that is, Units (in whole Numbers) under Units; and Tenths (in Decimals) under Tenths, &c. Then add as if they were all whole Numbers, cutting off as many Figures from the Sum towards the Right Hand for Decimal Parts as there are Decimals found in any of the Numbers to be added.

The chief Care to be taken here, is to keep the Dots, or feparating Points, exactly under one another; and to cut off in the whole Sum as many Figures towards the Right Hand, as there are Decimals in the greater Number. Thus, in the Second Example, because there are 4 Decimals in the third Line, four Figures are feparated or cut off towards the Right Hand in the total Sum.

Subtraction of Decimals.

IN this Rule, we muft alfo carefully fet the Units under Units, and the Tenths under Tenths, &c. and fubtract as in whole Numbers, always remembering to cut off as many Figures in the Remainder as there are Decimal Places in either of the other Numbers;

Note. If the Number of Places in the. Decimals be more in that which is to be fubtracted, than in that which, we fubtract from; we muft fuppofe Cyphers to make up the Number of void Places, as in the laft Example above, where three Cyphers are fuppofed to be added, and the Subtraction made accordingly.