From which we see, that in each case the denominator gives denomination or name to the fraction; that is, determines whether the parts are tenths, hundredths, thousandths, &c.

Q. When the unit 1 is divided into 10 equal parts, what is each part called? What is each part called when it is divided in 100 equal parts? When into 1000? Into 10,000, &c.? How are decimal fractions formed? What gives denomination to the fraction?

§ 120. The denominators of decimal fractions are seldom set down. The fractions are usually expressed by means of a point, or comma, placed at the left of the

numerator.

Thus, four tenths,

forty-five hundredths,
125 thousandths,

1047 ten thousandths,

,4

,45

,125

,1047

The denominator of every decimal fraction, however, is always understood. It is a unit 1 with as many ciphers annexed as there are places of figures in the numerator.

The place next to the decimal point is called tenths; the next place to the right, the place of hundredths; the next, the place of thousandths; and so on for places further to the right, according to the following Table.

DECIMAL NUMERATION TABLE.

Decimal fractions are numerated from the left hand to the right, beginning with the tenths, hundredths, &c., as in the table.

Q. Are the denominators of decimal fractions generally set down? How are the fractions expressed? Is the denominator understood? What is it? What is the place next the decimal point called? The next? The third, &c.? Which way are decimals numerated?

§ 121. Let us now write and numerate the following decimals.

Four tenths,

Four hundredths,

Four thousandths,

Four ten thousandths,

Four hundred thousandths,

Four millionths,

Four ten millionths,

,4.

,0 4.
,0 0 4.
,0 0 0 4.

,0 0 0 0 4.

,0 0 0 0 0 4.

,0 0 0 0 0 0 4.

Here we see, that the same figure expresses different values, according to the place which it occupies.

Therefore, the value of the places diminish in a tenfold proportion from the left hand to the right.

Hence, ten of the parts in any one of the places, are ' equal to one of the parts in the place next to the left; that is, ten thousandths make one hundreth, ten hundreths make one tenth, and ten tenths a unit 1.

This law of increase from the right hand towards the left, is the same as in whole numbers. Therefore, whole numbers and decimal fractions may be united by placing the decimal point between them.

A number composed partly of a whole number and partly of a decimal, is called a mixed number.

Q. Does the value of a figure depend upon the place which it occupies? How does the value change from the left towards the right? What do ten parts of any one place make? How do they increase from the right towards the left? How may whole numbers be joined with decimals? What is a number called when composed partly of whole numbers and partly of decimals?

Write the following numbers in figures, and numerate

them.

1. Forty one, and three tenths.

2. Sixteen, and three millionths. 3. Five, and nine hundredths.

4. Sixty-five, and fifteen thousandths.

5. Eighty, and three millionths.

6. Two, and three thousand millionths.

41,3.

16,000003.

5,09.

65,015.

80,000003.

2,000000003.

7. Four hundred and ninety-two thousandths.

0,492.

8. Three thousand, and twenty-one ten thousandths. 9. Forty seven, and twenty-one ten thousandths. 10. Fifteen hundred and three millionths.

11. Thirty-nine, and six hundred and forty thousandths. 12. Three thousand, eight hundred and forty millionths. 13. Six hundred and fifty thousandths.

14. Fifty thousand, and four hundredths.

15. Six hundred, and eighteen ten thousandths. 16. Three millionths.

17. Thirty nine hundred thousandths.

§ 122. The denominations of Federal Money will correspond to the decimal division, if we regard 1 dollar as the unit. For, the dimes are tenths of the dollar, the cents are hundredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar.

EXAMPLES.

1. Express $16, 3 dimes 8 cents and 9 mills decimally. Ans. $16,389.

2. Express $95, 8 dimes 9 cents 5 mills decimally.

Ans. $95,895.

3. Express $107, 9 dimes 6 cents 8 mills decimally.

Ans. $107,968

4. Express $47 and 25 cents decimally. Ans. $47,25. 5. Express $39,39 cents and 7 mills decimally. Ans. $39,397. 6. Express $12 and 3 mills decimally. Ans. $12,003. 7. Express $147 and 4 cents decimally. Ans. $147,04. 8. Express $148, 4 mills decimally. Ans. $148,004. 9. Express four dollars, six mills decimally. Ans. $4,006. Q. If the denominations of Federal Money be expressed decimally, what is the unit? What part of a dollar is 1 dime? What part of a dime is a cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill?

§ 123. A cipher is annexed to a number, when it is placed on the right of it. If ciphers be annexed to the numerator of a decimal fraction, the same number of ciphers must also be annexed to the denominator ;. for there must be as many ciphers in the denominator as there are places of figures in the numerator (see § 120.) The numerator and denominator will therefore be multiplied by the same number, and consequently the value of the fraction will not be changed (see § 84.) Hence,

Annexing ciphers to a decimal fraction does not alter its value.

We may take as an example,3. If now we annex a cipher to the numerator, we must, at the same time, annex one to the denominator, which gives

,30

=

100

30 by annexing one cipher, ,300 = 300 by annexing two ciphers, ,3000=10000 3000 all of which are equal to =,3. Also, ,5,50=500=,500-5000. Also, ,8,80,800,8000=,80000.

Q. When is a cipher annexed to a number? Does the annexing of ciphers to a decimal alter its value? Why not? What does threetenths become by annexing a cipher? What by annexing two ciphers? Three ciphers? What does,8 become by annexing a cipher? By annexing two ciphers? By annexing three ciphers?

§ 124. Prefixing a cipher is placing it on the left of a number. If ciphers be prefixed to the numerator of a decimal fraction, that is, placed at the left hand of the

significant figures, the same number of ciphers must be annexed to the denominator. Now, the numerator will remain unchanged while the denominator will be increased ten times for every cipher which is annexed, and the value of the fraction will be decreased in the same proportion (see 82). Hence,

Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher prefixed.

Take the fraction,2= as an example.
,02-02 by prefixing one cipher :

,002-002 by prefixing two ciphers:
,00020002 by prefixing three ciphers:

in which the fraction is diminished ten times for every cipher prefixed.

Also, ,03 becomes ,003 by prefixing one cipher; and ,0003 by prefixing two.

Q. When is a cipher prefixed to a number? When prefixed to a decimal, does it increase the numerator? Does it increase the denominator? What effect then has it on the value of the fraction? What does,5 become by prefixing a cipher? By prefixing two ciphers? By prefixing three? What does ,07 become by prefixing a cipher? By prefixing two? By prefixing three? By prefixing four ?

ADDITION OF DECIMAL FRACTIONS.

§ 125. 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 then made in the same manner as that of whole numbers.

Add 37,04, 704,3 and ,0376 together. In this example, we place the tenths under tenths, the hundredths under hundredths, and this brings the decimal points and the like parts of the unit directly under each other. We then

add as in whole numbers.

OPERATION.

37,04

704,3

,0376

741,3776