these parts into ten other equal parts, and so on, are called decimal fractions; because they decrease regularly by tens, or in a ten-fold ratio. (Art. 10. Obs. 2.)

177. Each order of integers or whole numbers, it has been shown, increases in value from units towards the left in a ten-fold ratio; (Art.. 9;) and, conversely, each order must decrease from left to right in the same ratio, till we come to units place again.

178. By extending this scale of notation below units towards the right hand, it is manifest that the first place on the right of units will be ten times less in value than units place; that the second will be ten times less than the first; the third, ten times less than the second, &c.

Thus we have a series of places or orders below units, which decrease in a ten-fold ratio, and exactly correspond in value with tenths, hundredths, thousandths, &c.

179. Hence, to express Decimal Fractions, or fractions whose denominator is 10, 100, 1000, &c.

Write simply the numerator with a point (.) before it, to distinguish the fractional parts from whole numbers. For example, may be written thus .1; 2 thus .2;

thus .3; &c. may be written thus .01, putting the 1 in hundredths place; thus .05; &c. That is, tenths are written in the first place on the right of units; hundredths in the second place; thousandths in the third place, &c.

180. The denominator of a decimal fraction is always I with as many ciphers annexed to it as there are figures in the numerator, and need not be expressed.

OBS. The point placed before decimals, is often called the Separatrix.

QUEST. 177. In 'what manner do whole numbers increase and decrease? 178. By extending this scale below units, what would be the value of the first place on the right of units? The second? The third? With what do these orders correspond? 179. How are decimal fractions expressed? 180. What is the denominator of a decimal fraction? Obs. What is the point placed before decimals called?

181. The names of the different orders of decimals or places below units, may be easily learned from the following

182. It will be seen from this Table that the value of each figure in decimals, as well as in whole numbers, depends upon the place it occupies, reckoning from units. Thus, if a figure stands in the first place on the right of units, it expresses tenths; if in the second, hundredths, &c.; each successive place or order towards the right, decreasing in value in a tenfold ratio. Hence,

183. Each removal of a decimal figure one place from units towards the right, diminishes its value ten times.

Prefixing a cipher, therefore, to a decimal diminishes its value ten times; for it removes the decimal one place farther from units' place. Thus .4=; but .04=ō• and .0041000, &c.; for the denominator to a decimal fraction is 1 with as many ciphers annexed to it as there are figures in the numerator. (Art. 180.)

Annexing ciphers to decimals does not alter their value; for each significant figure continues to occupy the same place from units as before. Thus, .5; so .50= , or, by dividing the numerator and denominator by 10; (Art. 116;) and .500 500 1000, or io, &c.

50

QUEST. 181. Repeat the Decimal Table, beginning units, tenths, &c. 182. Upon what does the value of a decimal depend? 183. What is the effect of removing a decimal one place towards the right? What then is the effect of prefixing ciphers to decimals? What, of annexing them?

OBS. 1. It should be remembered that the units' place is always the right hand place of a whole number. The effect of annexing and prefixing ciphers to decimals, it will be perceived, is the reverse of annexing and prefixing them to whole numbers. (Art. 58.)

2. A whole number and a decimal expressed together, are called a mixed number. (Art. 108.)

184. To read decimal fractions.

Beginning at the left hand, read the figures as if they were whole numbers, and to the last one add the name of its order.

Thus,

OBS. In reading decimals as well as whole numbers, the units" place should always be made the starting point. It is advisable for young pupils to apply to every figure the name of its order, or the place which it occupies, before attempting to read them. Beginning at the units' place, he should proceed toward the right, thus-units, tenths, hundredths, thousandths, &c., pointing to each figure as he pronounces the name of its order. In this way he will very soon be able to read decimals with as much ease as he can whole numbers.

Read the following numbers :

QUEST. Obs. Which is the units' place? What is a whole number and a decimal written together, called? 184. How are decimals read? Obs. In reading decimals, what should be made the starting point?

Note. Sometimes we pronounce the word decimal when we come to the separatrix, and then read the figures as if they were whole numbers; or, simply repeat them one after another. Thus, 125.427 is read, one hundred twenty-five, decimal four hundred twenty-seven; or, one hundred twenty-five, decimal four, two, seven.

Write the fractional parts of the following numbers in decimals:

13. Write 49 hundredths; 3 tenths; 445 ten thousandths.

14. Write 36 thousandths; 25 hundred thousandths; 1 millionth.

15. Write 7 hundredths; 3 thousandths; 95 ten thousandths; 63 millionths; 26 ten millionths.

185. Decimals are Added, Subtracted, Multiplied, and Divided in the same manner as whole numbers.

OBS. The only thing with which the learner is likely to find any difficulty, is pointing off the answer. To this part of the operation he should give particular attention..

ADDITION OF DECIMAL FRACTIONS.

186. Ex. 1. What is the sum of 2.5; 24.457; 123.4 and 2.369?

Operation.

2.5

24.457

123.4

Write the units under units, the tenths under tenths, hundredths under hundredths, &c.; then, beginning at the right hand or lowest. order, proceed thus: 9 (thousandths) and 7 (thousandths) are 16 (thousandths.) Write the 6 under the column added, and carry the 152.726 1 to the next column as in addition of whole numbers. 1 to carry to 6 (hundredths) makes 7 (hun

2.369

QUEST.-Note. What other method of reading decimals is mentioned?

dredths) and 5 are 12 (hundredths.) Set the 2 under the column, and carry the 1 as before. ] to carry to 3 (tenths) makes 4, and 4 are 8 (tenths) and 4 are 12 (tenths) and 5 are 17 (tenths) or 1 and 7 tenths. Set the 7 under the column, and carry the 1 to the next column. Finally, place the decimal point in the amount, directly under those in the numbers added.

187. Hence, we deduce the following general

RULE FOR ADDITION OF DECIMALS.

Write the numbers so that the same orders may stand under each other, placing tenths under tenths, hundredths under hundredths, &c. Brgin at the right hand or lowest order, and proceed in all respects as in adding whole numbers. (Art. 29.)

From the right hand of the amount, point off as many figures for decimals as are equal to the greatest number of decimal places in either of the given numbers.

PROOF. Addition of Decimals is proved in the same manner as Simple Addition. (Art. 28.)

Note.--The decimal point in the answer will always fall directly under the decimal points in the given numbers.

5. What is the sum of 2.5; 33.65 and 45.121?

6. What is the sum of 65.7; 43.09; 1.026 and 2.1765?

QUEST.-187. How are decimals added? How point off the answer? How is addition of decimals proved?