4160. Which way are decimals numerated? Does the value of a figure depend upon the place which it occupies?

161. How does the value change from the left towards the right? What do ten parts of any one place make? How may whole numbers be joined with decimals? What is a number called when composed partly of whole numbers and partly of decimals?

§ 162. When will the denominations of Federal Money correspond to decimal fractions?

163. If ciphers are annexed to a decimal, is the value changed? Why not?

§ 164. If ciphers be prefixed, is the value changed? How much for each cipher? Why?

ADDITION OF DECIMAL FRACTIONS.

§ 165. It must be recollected that only like parts of unity can be added together, and therefore in setting down the numbers for addition the figures occupying places of the same value must be placed directly under each other.

The addition of decimal fractions is made in the same manner as that of whole numbers.

Hence we have the following

RULE.

I. Set down the numbers to be added so that tenths shall fall under tenths, hundredths under hundredths, &c. This will bring all the decimal points directly under each other.

II. Then add as in simple numbers and point off in the sum, from the right hand, so many places for decimals as are equal to the greatest number of places in any of the given numbers.

Ex. 1. Add 37,04, 704,3 and ,0376 together.

In this example the ober of decimal in the sum is 4.

37,04

704,3

,0376

Sum 741,3776

2. Add 4,035, 763,196, 445,3741 and 91,3754 together.

Ans. 1303, 9805.

3. Add 365,103113, ,76012, 1,34976, ,3549 and 61,11 together.

Ans. 428,677893.

4. Required the sum of twenty-nine and 3 tenths, four hundred and sixty-five, and two hundred and twenty one thousandths.

Ans. 494, 521.

5. Required the sum of two hundred dollars one dime three cents and nine mills, four hundred and forty dollars nine mills, and one dollar one dime and one mill.

Ans. $641,249.

or 641 dolls. 2 dimes 4 cents 9 mills. 6. What is the sum of one tenth, one hundredth, and one thousandth.

Ans.,111.

7. What is the sum of 4 and 6 ten thousandths. Ans. 4,0006.

SUBTRACTION OF DECIMAL FRACTIONS.

§ 166. Subtraction of Decimal Fractions teaches how to find the difference between two decimal numbers.

RULE.

Set down the less number under the greater, so that figures occupying places of the same value shall fall directly under each other: then subtract as in simple numbers and point off the decimal places as in dition.

Ez. 1. From 3,275 take,0979.

In this example a cipher is annexed to the minuend to make the number of decimal places equal to the number in the subtrahend. This does not alter the value of the minuend, § 163.

2. From 3295 take,0879.

3,2750

,0879 3,1871

Ans. 3294,9121.

3. From 291,10001 take 41,375.

Ans. 249,72501.

4. From 10,000001 take,111111.

Ans. 9,888890.

5. From three hundred and ninety-six, take 8 ten thousandths.

6. From 1 take one thousandth.

Ans. 295,9992.

Ans. ,999.

MULTIPLICATION OF DECIMAL FRACTIONS.

Ex. 1. Let it be required to multiply,37 by,8. The two decimals may be expressed and : then,,37,8=7ׂ&=3=,296 answer. 2. Multiply,3 by,02.

,3×,02===,006 answer.

§ 167. It is seen from these examples that the product of the denominators will contain as many ciphers as there are places of decimals in both the numerators. The product, therefore, of the numerators should contain so many places of decimals as there are in both the numbers, and when there are not so many the deficiency must be supplied by prefixing ciphers; for otherwise, we could not in the product reject the denominator and express the nu merator decimally. Therefore to multiply one deci fraction by another we have the following

RULE.

Multiply as in simple numbers, and point off in the product, from the right hand, as many figures for decimals as are equal to the number of decimal places in the multiplicand and multiplier; and if there be not so many in the product, supply the deficiency by prefixing ciphers.

Ex. 1. Multiply 3,049 by ,012.

4. Multiply one and one millionth by one thousandth.

Ans. ,001000001. 5. Multiply one hundred and forty seven millionths, by one millionth.

Ans. ,000000000147. 6. Multiply three thousand, and twenty seven hundredths by 31.

Ans. 93008,37.

NOTE. 168. When a decimal fraction is to be multiplied by 10, 100, 1000, &c. the multiplication may be made by removing the decimal point as many places to the right hand as there are ciphers in the multiplier, and if there be not so many figures on the right of the decimal point, supply the deficiency by annexing ciphers.

DIVISION OF DECIMAL FRACTIONS.

$169. Division of Decimal Fractions is similar to that of simple numbers.

We have seen § 47, that when there is no remainder, the divisor multiplied by the quotient will produce the dividend. But when decimal fractions are multiplied together there will be as many decimal places in the product as there are in the multiplier and multiplicand § 167.

Therefore the dividend must contain as many decimal places as the divisor and quotient together. The number of decimal places in the quotient must therefore be equal to the difference between the number of places in the dividend and the number of places in the divisor. Hence, for division of deciinal fractions we have the following

RULE.

Divide as in simple numbers, and point off in the quotient, from the right hand, so many places for decimals as the decimal places in the dividend exceed those in the divisor; and if there are not so many, supply the deficiency by prefixing ciphers.

Ex. Divide 1,38483 by 60,21.

There are five decimal places in the dividend, and two in the divisor: there

60,21)1,38483(23

1 2042

18063

18063

Ans. ,023.

fixede 23, and the decimal point placed before it.

Ans. 1,11.