DECIMALS.

128. The numbers 1000, 100, 10, 1, fo, To, rooo, etc., increase from right to left, and decrease from left to right, by

the same uniform scale of ten. 100 is 10 times 10; 10 is ten Also, Too is one tenth of

Thus, 1000 is ten times 100; times 1; 1 is ten times, etc. ; To is one tenth of 1;

one tenth of 1; 1 is one tenth of 10, etc. decimal figure is always

the next place to the left.

is

The value of any

of the value of the same.figure in

1. If a unit or 1 is divided into ten equal parts, what is one part called? 2 parts? 5 parts?

2. If 1 tenth of 1 is divided into ten equal parts, what is one part called? 3 parts? 7 parts? 25 parts?

3. If 1 hundredth of 1 is divided into ten equal parts, what is one part called? 9 parts? 75 parts?

4. What is of of 1?ofofof 1? 5. What is the fractional unit of 3 tenths? dredths? Of 25 thousandths?

fo of Too?

Of 5 hun

NOTATION AND NUMERATION.

129. A decimal fraction or decimal is one or more of the decimal divisions of a unit.*

Thus, 16, 10, 180, 1850, etc., are decimal fractions.

The denominators of decimal fractions increase and decrease by the uniform scale of 10. Hence, the fractional units are always tenths, hundredths, thousandths, etc.

130. The decimal sign (.), or decimal point, is used to distinguish a decimal from an integer, and is always placed before the numerator of the decimal, when expressed alone.

The number of figures after the decimal sign indicates the denominator, and determines the value of the decimal expression.

131. Decimals may be written in two ways, viz.: either as common fractions, the denominator being expressed, or in decimal notation, the denominator being omitted. Thus,

1. The denominator of a decimal, when expressed, is always 1 with 28 many ciphers annexed as there are figures in the given decimal.

Thus, .4:

10, .09 180, .007 = 1000. Hence,

2. The numerator, when expressed alone must have as many deci. mal places as there are ciphers in the denominator.

Thus, fo

16,

.8, 106

1885 .012, Toooo

.0014.

*The terms "fraction” and “decimal" will hereafter be used to distinguish the common from the decimal form of expression. Thus, or, and .75 are two forms of expression for the same thing.

132. Express in the form of a decimal, and read,

7.

3. 10.

4. 10. 8. 1830.

Express in the form of a fraction, and read,

11.

12.

206

16.

1000

145 100000

21.

133. A mixed decimal consists of an integer and a decimal written together.

In reading a mixed number, the word "and" is always read in place of the point. Thus, 7.12 is read, 7 and 12 hundredths; 26.134 is read, 26 and 134 thousandths.

134. The relation of integers and decimals to each other is clearly shown in the following

NAMES.

TABLE.

ORDERs. ཀྱྀབྱཾ ཙ་ཏུ བྷུཊྛཊྛཾ བྷྲ

UNITS. 555 555 55 5.55 555 555

The number is read, 555 million 555 thousand 555, and 55 million 555 thousand 555 hundred-millionths.

135. As in integers, so in decimals, make the order of units the starting-point of notation and numeration, extending the scale to the left of units' place in writing integers, and to the right of 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 at the left is hundreds, and the second order to the right is hundredths, and so on. The orders on the left and the decimals on the right, equally distant from units' place, correspond in name, and are reciprocals of each other. Thus, the 2d order on the left is 1 ten, and the second order on the right is 1 tenth, and, or .1, is the reciprocal of 10; the 3d order on the left is 1 hundred, and the 3d order on the right is 1 hundredth, and ʊ, or .01, is the reciprocal of 100, etc.

The value of any decimal order is always ten times the value of the next lower order, or 1 tenth the value of the next higher order.

136. In writing decimals, as in writing integers, vacant orders must be filled with ciphers.

Since decimals are governed by the same laws of notation as integers, it follows that,

137. Each removal of a decimal figure one place to the right, causes it to represent units of one tenth the value it did before, and hence divides the number by 10. (17, III.)

Thus, .03 is the value of .3; .003 is the value of .03; or, .3÷10 = .03; .03÷10 = .003.

138. Each removal of a decimal figure one place to the left, causes it to represent units of ten times the value it did before, and hence multiplies the number by 10. (17, II.) Thus, .05 is 10 times .005; .5 is 10 times .05; or, .005 × 10 .05 × 10 = .5; .5 x 10 = 5.

=

.05;

139. Ciphers annexed or rejected at the right of any decimal do not change its value, since they do not change the place of the significant figures of the decimal.

Thus, .6 is the same as .60, or .600, or .6000.

WRITTEN

140. Express decimally :

EXERCISES.

1. Seventy-five thousandths. 138 = .075.

2. Fifteen hundredths.

3. Seven thousandths.

6. 22 ten-thousandths.

7. 245 ten-thousandths.

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

18. Four hundred thirty-seven thousand five hundred 49 millionths.

19. Three million forty thousand 12 ten-millionths.

20. Six hundred and 24 hundred-millionths.

21. Four hundred ninety-five million seven hundred five thousand and 43075 ten-millionths.

22. Four million seven hundred thirty-five thousand and 903624 hundred-millionths.

23. Three million forty thousand ten ten-millionths.

24. Twenty-four hundred-millionths.

25. Eight thousand six hundred forty-five hundred-thousandths.