6. What part of 1 tenth is 1 hundredth? How many hundredths in 1 tenth?

7. If 1 hundredth of a unit be divided into 10 equal parts, what is each part called? What are 3 parts? What are 8 parts? 9 parts? 15 parts?

8. What is of of ?

of too? to of bo? 9. If a unit be divided into 1000 equal parts, or each hundredth into 10 equal parts, what are the parts called? What are 12 parts? 26 parts? 42 parts?

10. What part of 1 hundredth is 1 thousandth? How many 1 thousandths is 1 hundredth ?

NOTATION AND NUMERATION. 248. A Decimal Fraction is one or more of the decimal divisions of a unit. Thus, fo, fo, 150, 1850, etc., are decimal fractions.

Decimal Fractions are commonly called Decimals.* (16.)

249. Decimals are like other Fractions, except that their denominators increase and decrease by the uniform scale of 10. The fractional units are, therefore, always tenths, hundredths, thousands, etc.

250. The Decimal Sign (.), called the decimal point, is used to distinguish a decimal from an integer, and must always be placed before the numerator of the decimal.

* The terms fraction and decimal will hereafter be used to distinguish the common from the decimal form of expression. Thus, 75 , and .75, are two forms of expressing the same thing. For convenience we shall call the first form a fraction, and the other a decimal.

251. The position of the decimal sign indicates the denominator, and determines the value of the decimal expression.

Thus,
is expressed .7.

12 is expressed .126. 188%

252. The Denominator of a decimal fraction is always 10, 100, 1000, etc., or 1 with as many ciphers annexed as there are figures in the given decimal. Thus, .4; .09180; .007 = 10%, etc.

=

253. The Numerator of a decimal fraction when expressed alone, must have as many decimal places as there are ciphers in the denominator. Thus, = .8; 10% = .12; 12.125, etc.

If the numerator does not contain as many figures as there are ciphers in the denominator, prefix ciphers until the number of places is equal to the number of ciphers in the denominator, and prefix the decimal point. Thus, 1.07; Too = .009, etc.

254. Decimal fractions may be written in two ways; either as other fractions, the denominator being expressed, or, in decimal notation, the denominator being omitted. Thus,

255. The value of any decimal figure is always of the value of the same figure in the next place to the left.

256. When an integer and decimal are written together, the expression is a Mixed Number (195). Thus, 7.12 and 26.134 are mixed numbers.

257. The relation of decimals and integers to each other is clearly shown by the following

INTEGERS.

3 2 1

DECIMALS. The number is read 987 million 654 thousand 321, and 23 million 456 thousand 789 hundred-millionths.

A decimal takes the name of its right-hand order.

258. In decimals, as in integers, make the order of units the starting-point of notation and of numeration, extending the scale to the left of the units' place in writing integers, and to the right of the units' place in writing decimals.

The first order to the left of units is tens, and the first order to the right of units is tenths; the second order to the left of units is hundreds, and the second order to the right is hundredths; the third order to the left is thousands, and the third order to the right is thousandths, and so on, the integers on the left, and the decimals on the right, equally distant from the units' place, corresponding in name.

259. Hence, both in integers and in decimals, the value of any figure is determined by the position of that figure, and is always ten times the value of the same figure in the next lower order, or 1 tenth the value of the same figure in the next higher order. Hence,

260. In writing decimals, vacant orders must be filled with ciphers. (36, 2.)

Dictation exercises, both oral and written, should be given, until the pupil can write and read decimals with rapidity and correctness. Orai, thus, Ques. The denominator of a fraction is 100, the numerator 7; what will express the decimal?" The prompt response should be, "Point, naught, seven, read, seven-hundredths" (.07). Ques. "The denominator is 1000, the numerator 35." Ans. "Point, naught, three, five, read thirty-five thousandths" (.035), etc.

Also the converse; thus, Ques. "Point, naught, eight; what will express the fraction?" Ans. "The numerator is eight, the denominator one hundred, and the fraction is eight-hundredths” (180). Ques. "Point, naught, one, five?" Ans. "The numerator is fifteen, the denominator is one thousand, and the fraction fifteen-thou sandths" (115), etc.

262. Prefixing a cipher to a decimal multiplies the denominator by 10, and hence divides the decimal by 10 (200, 2). Thus, .5; .05 = 10; .005 = 1000; or, .5 ÷ 10 = .05; .05 ÷ 10 = .005, etc.

263. Rejecting a cipher from the left of a decimal divides the denominator by 10, and hence multiplies the decimal by 10 (200, 1). Thus, .0071000; .07 = Too; .7 = ; or, .007 × 10 = .07; .07 × 10 = .7.

264. Annexing a cipher to a decimal multiplies both numerator and denominator by 10, and hence reduces the fraction to higher terms (200, 3). Thus, .3 = † ; .30% 300 = 30%.

1000

265. Rejecting a cipher from the right of a decimal divides both numerator and denominator by 10, and hence reduces to lower terms (200, 3). Thus, = .600; 18% = .60; = .6.

600

1000

From the foregoing explanations are deduced the following

266. PRINCIPLES.-1. Decimals are governed by the same laws of notation as integers. Hence,

2. The value of any decimal figure depends upon the place it occupies at the right of the decimal sign. (258.) 3. Every removal of a decimal figure one place to the right diminishes its value tenfold. (262.)

4. Every removal of a decimal figure one place to the left increases its value tenfold. (263.)

5. Ciphers may be annexed or rejected at the right of any decimal, without changing its value. (264, 265.)

WRITTEN EXERCISES.

267. Express in figures and decimally;

1. Seventy-five thousandths. 1780.075.

2. Fifteen hundredths.

3. Seven thousandths.

6. 22 ten-thousandths.

7. 245 ten-thousandths.

4. Fifty-three thousandths. 8. 1042 hundred-thousandths.

RULE.-I. Write the numerator of the decimal as if an integer, writing ciphers in the place of vacant orders to give each significant figure its proper value, and place the decimal point before tenths.