DECIMAL FRACTIONS.

[NOTE. The denominations of Federal Money are purely decimal; dollars being units or whole numbers, dimes tenths of a dollar, cents hundredths of a dollar, and mills thousandths of a dollar; consequently, Federal Money and Decimal Fractions are subject to the same methods of operation.]

A DECIMAL FRACTION is a fraction whose denominator is a unit with so many ciphers annexed as the numerator has places of figures.

As the denominator of a decimal fraction is always 10, 100, 1000, &c. the denominators need not be expressed For the numerator only may be made to express the true value. For this purpose it is only required to write the numerator with a point before it at the left hand, to distinguish it from a whole number, when it consists of so many figures, as the denominator hath ciphers annexed to unity, or 1: So is written .5; 33, .33; 73,735, &c.

The point prefixed is called a Separatrix.

But if the numerator has not so many places as the denominator has ciphers, put so many ciphers before it, viz. at the left hand, as will make up the defect: So write thus, .05; and T thus, .006, &c. And thus do these fractions receive the form of whole numbers.

We may consider unity as a fixed point, from whence whole numbers proceed infinitely increasing toward the left hand, and decimals infinitely decreasing toward the right hand to 0, as in the following

TABLE.

168.

What are Decimal Fractions ?-169. Explain the principles on which deci mals are founded?— -170. What is said of unity ?-171. Explain the Table.

M

All the figures at the left band of the decimal point are whole numbers. The 5 in the 1st place at the right hand of the point, represents 5 tenths; 5 in the 2nd place, 5 hundredths; 5 in the 3rd place, 5 thousandths; 5 in the 4th place, 5 ten-thousandths; 5 in the 5th place, 5 hundred thousandths; 5 in the 6th place, 5 millionths; 5 in the 7th place, 5 ten-millionths; 5 in the 8th place, 5 hundredmillionths; and all of them taken together, are read thus :-Fifty-five million, five hundred fifty-five thousand, five hundred and fifty-five hundred millionths.*

Ciphers, placed at the right hand of a decimal fraction, do not alter its value, since every significant figure contínues to possess the same place: So .5, .50, and .500, are all of the same value, and each equal to .

But ciphers placed at the left hand of a decimal, do alter its value, every cipher depressing it to of the value it had before, by removing every significant figure one place further from the place of units. So .5, .05, .005, all express different decimals, viz. .5, ō ; .05, To; .005, TOO.

Decimal fractions of unequal denominators are reduced to one common denominator, when there are annexed to the right hand of those, which have fewer places, so many ciphers as make them equal in places with that which has the most. So these decimals, .5, 06, .455, may be reduced to the decimals .500, .060, .455, which have all 1000 for their denominator.

Of decimals, that is the greatest, whose highest figure is greatest, whether they consist of an equal or unequal number of places. Thus, .5 is greater than .459, for if it be reduced to the same denominator with .459, it will be .500.

*It is evident from the Table, that since the decimal parts decrease in a tenfold proportion from the place of units towards the right hand, they must increase in a tenfold proportion from the right hand towards the left, which proves that decimal fractions are subject to the same law of notation, and consequently of operation, as whole numbers.

To read decimal figures.—Begin at the left hand, and read them in the same manner as we read whole numbers, and add to the whole the name of the place of the right hand figure.

EXAMPLES.

.24 is read 24 hundredths;

.035 is read 35 thousandths;

.0050 is read 50 ten thousandths;

.000750 is read 750 millionths;

.00000099 is read 99 hundred millionths.

172. What are all figures at the left hand of the point, or separatrix ?- -173. What does the first figure at the right of the decimal point represent?—174. What the second, third, fourth, &c. ?- -175. What effect have ciphers placed on either hand of a decimal? -176. How are decimal fractions of unequal denominators reduced to a common denominator?

A mixed number, viz. a whole number with a decimal annexed, is equal to an improper fraction, whose numerator is all the figures of the mixed number, taken as one whole number, and the denominator that of the decimal part. So 45-309 is equal to 45309, as is evident from the method given to reduce a mixed number to an improper fraction: Thus, 45×1000+.309-45309 as above. 1000

10009

ADDITION OF DECIMALS.

RULE.

1. Place the numbers, whether mixed, or pure decimals, under each other, according to the value of their places.

2. Find their sum, as in whole numbers, and point off so many places for decimals, as are equal to the greatest number of decimal places in any of the given numbers.

EXAMPLES.

1. Find the sum of 19.073+2·3597+223+0197581+3478·1+ 12.358.

19.073
2.3597

223.

0197581

3478.1

12.358

3734-9104581 the sum.

2. Required the sum of 429+21·37+355·003+1·07+1·7 ?

Ans. 808.143

3. Required the sum of 973+19+1.75+93.7164+9501 ?

Ans. 1088-4165.

4. Required the sum of 5.3+11·973+49+9+1·7314+34·3 ?

177. What is the rule for addition of decimals?

Ans. 103-2044.

SUBTRACTION OF DECIMALS.

RULE.

Place the numbers according to their value: Then subtract as in whole numbers, and point off the decimals as in Addition.

EXAMPLES.

1. Find the difference of 1793-13 and 817-05693.

1. Whether they be mixed numbers, or pure decimals, place the factors and multiply them as in whole numbers.

2. Point off so many figures from the product as there are decimal places in both the factors; and if there be not so many places in the product, supply the defect by prefixing ciphers.

NOTE. The reason of the rule for pointing off the figures for decimals, is evident from the notation of decimals. Thus, 5X.5=25 ; for 5X15 or once-five tenths. But as the multiplier is less than unity, or tenths, multiplying is only taking tenths of tenths, and so many tenths of tenths, are evidently so many hundredths. So also, tenths of hundredths would be thousandths; hundredths of hundredths would be ten thousandths; and so on.

178. What is the rule for subtracting decimal fractions ?——179. What is the rule for multiplication of decimals ?

To multiply by 10, 100, 1000, &c. remove the separating point so many places to the right hand, as the multiplier has ciphers.

1. The places of decimal parts in the divisor and quotient, counted together, must always be equal to those in the dividend; therefore, divide as in whole numbers, and, from the right hand of the quotient, point off so many places for decimals, as the decimal places in the dividend exceed those in the divisor.

2. If the places of the quotient be not so many as the rule requires, supply the defect by prefixing ciphers to the left hand. 3. If at any time there be a remainder, or the decimal places in the divisor be more than those in the dividend, ciphers may be annexed to the dividend or to the remainder, and the quotient carried on to any degree of exactness.

180. What is the rule for division of decimals?