SECTION IX.

DECIMAL FRACTIONS.

308. Fractions which decrease in a tenfold ratio, or which express simply tenths, hundredths, thousandths, &c., are called DECIMAL FRACTIONS.

They arise from dividing a unit into ten equal parts, then dividing each of these parts into ten other equal parts, and so on. Thus, if a unit is divided into 10 equal parts, 1 of those parts is called a tenth. (Art. 178.) If a tenth is divided into 10 equal parts, 1 of those parts will be a hundredth; for,÷10=1. If a hundredth is divided into 10 equal parts, 1 of the parts will be a thousandth; for, 110=T, &c. (Art. 227.)

OBS. Fractions of this class are called decimals, because they regularly decrease in a tenfold ratio. (Art. 37. Obs. 2.)

Decimal fractions are said to have been invented by Lord Napier, in 1602.

309. Each order of whole numbers, we have seen, increases in value from units towards the left in a tenfold ratio; and, conversely, each order must decrease from left to right in the same ratio, till we come to units' place again. (Art. 37.)

310. 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 orders below units, which decrease in a tenfold ratio, and exactly correspond in value with tenths, hundredths, thousandths, &c. (Art. 308.)

QUEST.-308. What are Decimal Fractions? From what do they arise? Obs. Why called decimals? 309. In what manner do whole numbers increase and decrease? 310. 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 in value?

311. Decimal Fractions are commonly expressed by writing the numerator with a point ( . ) before it.

.

The point placed before decimals is called the Decimal Point, or Separatrix. Its object is to distinguish the fractional parts from whole numbers.

If the numerator does not contain so many figures as there are ciphers in the denominator, the deficiency must be supplied by prefixing ciphers to it. For example, is written thus .1; thus .2; thus .3; &c. T is written thus .01, putting the 1 in hundredths' place; 15 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.

312. The denominator of a decimal fraction is always 1 with as many ciphers annexed to it, as there are figures in the given numerator. (Art. 308.)

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

314. 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

QUEST.-311. How are decimal fractions expressed? What is the point placed before decimals called? 312. What is the denominator of a decimal fraction? 313. Repeat the Decimal Table, beginning units, tenths, &c. 314. Upon what does the value of a decimal depend?

the second, hundredths, &c.; each successive place or order towards the right, decreasing in value in a tenfold ratio. Hence,

315. 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; and .004 TT, &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. 312.)

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. 191,) and .500, or, &c.

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. 98.)

2. A whole number and a decimal, written together, is called a mixed numLer. (Art. 183.)

316. 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 the learner to apply to

QUEST.-315. 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. Which is the units' place? What is a whole number and a decimal written to gether, called? 316. How are decimals read? Obs. In reading decimals, what should be made the starting point?

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 towards 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 be able to read decimals with as much ease as he can whole numbers.

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 part of the following numbers in decimals:

13. Write 9 tenths; 25 hundredths; 45 thousandths.

14. Write 6 hundredths; 7 thousandths; 132 ten thousandths.

15. Write 462 thousandths; 2891 ten thousandths.

16. Write 25 hundred thousandths; 25 millionths.

17. Write 1637246 ten millionths; 65 hundred millionths.

18. Write 71 thousandths; 7 millionths.

19. Write 23 hundredths; 19 ten thousandths.

20. Write 261 hundred thousandths; 65 hundredths; 121 millionths; 751 trillionths.

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

317. Decimal Fractions, it will be perceived, differ from Common Fractions both in their origin and in the manner of expressing them.

Common Fractions arise from dividing a unit into any nuniber of equal parts; consequently, the denominator may be any number whatever. (Art. 182.) Decimals arise from dividing a unit into ten equal parts, then subdividing each of those parts into ten other equal parts, and so on; consequently, the denominator is always 10, 100, 1000, &c. (Arts. 308, 312.)

Again, Common Fractions are expressed by writing the numerator over the denominator; Decimals are expressed by writing the numerator only, with a point before it, while the denominator is understood. (Arts. 182, 311.)

318. 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.

319. Ex. 1. What is the sum 28.35; 345.329; 568.5; and 6.485?

Operation. 28.35 345.329

568.5

Write the units under units, tenths under tenths, hundredths under hundredths, &c.; then, beginning at the right hand or lowest order, proceed thus: 5 thousandths and 9 6.485 thousandths are 14 thousandths. Write the 948.664 Ans. 4 under the column added, and carrying the 1 to the next column, proceed through all the orders in the same manner as in simple addition. (Art. 54.) Finally, place the decimal point in the amount directly under that in the numbers added.

QUEST.-317. How do decimals differ from common fractions? From what do common fractions arise? From what do decimals arise? How are common fractions expressed? Bow are decimals?.