ARITHMETIC.

(SECTION 8.)

DECIMALS.

Remark.-A knowledge of decimals is of the utmost importance to all who are required to make calculations of any kind. The subject is easy to learn, and for this reason the student is somewhat inclined to study it too hastily, the result being that he afterwards has trouble that might have been entirely avoided had he given the text the proper attention in the beginning. Decimals are much easier to use than common fractions, which they replace; at the same time it is frequently more expedient to use common fractions in certain operations, and, hence, they cannot be wholly dispensed with. Particular attention should be paid to the rules for multiplication and division--especially to the locating of the decimal point-and. to the operations of changing a common fraction to a decimal and vice versa.

132. Decimals are tenth fractions; that is, the parts of a unit are expressed on the scale of ten, as tenths, hundredths, thousandths, etc.

133. The denominator, which is always ten or a multiple of ten, as 10, 100, 1,000, etc., is not expressed, as it would be in common fractions, by writing it under the For notice of copyright, see page immediately following the title page.

numerator with a line between them, as, 180, 1000, but is expressed by placing a period (.), which is called a decimal point, to the left of the figures of the numerator, so as to indicate that the number on the right is the numerator of a fraction whose denominator is 10, 100, 1,000, etc.

134. The reading of a decimal number depends upon the number of decimal places in it, or the number of figures to the right of the decimal point.

One decimal place expresses tenths.
Two decimal places express hundredths.
Three decimal places express thousandths.
Four decimal places express ten-thousandths.
Five decimal places express hundred-thousandths.
Six decimal places express millionths.

Thus:

.3

=

10

.03

=

8 100

.003

3

.0003 =

1000
3

10000

.00003 = 100000

3

= 3 tenths.

3 hundredths.

= 3 thousandths.

3 ten-thousandths.
= 3 hundred-thousandths.

.000003 = 1000000 = 3 millionths.

We see in the above that the number of decimal places in a decimal equals the number of ciphers to the right of the figure 1 in the denominator of its equivalent fraction. This fact kept in mind will be of much assistance in reading and writing decimals.

Whatever may be written to the left of a decimal point is a whole number. The decimal point merely separates the fraction on the right from the whole number on the left.

When a whole number and decimal are written together, the expression is a mixed number. Thus, 8.12 and 17.25 are mixed numbers.

The relation of decimals and whole numbers to each other is clearly shown by the following table:

The figures to the left of the decimal point represent whole numbers; those to the right are decimals.

In both the decimals and whole numbers, the units place is made the starting point of notation and numeration. Both whole numbers and decimals decrease on the scale of ten to the right, and both increase on the scale of ten to the left. The first figure to the left of units is tens, and the first figure to the right of units is tenths. The second figure to the left of units is hundreds, and the second figure to the right is hundredths. The third figure to the left is thousands, and the third to the right is thousandths, and so on; the whole numbers on the left and the decimals on the right. The figures equally distant from units place correspond in name, the decimals having the ending ths, to distinguish them from whole numbers. The following is the numeration. of the number in the above table: nine hundred eighty-seven million, six hundred fifty-four thousand, three hundred twenty-one and twenty-three million, four hundred fifty-six thousand, seven hundred eighty-nine hundred-millionths.

The decimals increase to the left, on the scale of ten, the same as whole numbers; for, if you begin at the 4 in thousandths place in the above table, the next figure to the left is hundredths, which is ten times as great, and the next tenths, or ten times the hundredths, and so on through both decimals and whole numbers.

135. Annexing, or taking away, a cipher at the right of a decimal, does not affect its value.

.5 is; .50 is 50%, but

50.

=; therefore, .5 .50.

136. Inserting a cipher between a decimal and the decimal point, divides the decimal by 10.

.5=;÷10.05.

137. Taking away a cipher from the left of a decimal, multiplies the decimal by 10.

138. In some cases it is convenient to express a mixed decimal fraction in the form of a common (improper) fraction. To do so it is only necessary to write the entire number, omitting the decimal point, as the numerator of the fraction, and the denominator of the decimal part as the denominator of the fraction. Thus, 127.483 = 127483; for, 127.483 = 1274836 127000+483 = 127483.

1000

1000

1000

ADDITION OF DECIMALS.

139. Addition of decimals is similar in all respects to addition of whole numbers-units are placed under units, tens under tens, etc.; this, of course, brings the decimal points in line, directly under one another. Hence, in placing the numbers to be added, it is only necessary to take care that the decimal points are in line. In adding whole numbers, the right-hand figures are always in line; but in adding decimals, the right-hand figures will not be in line unless each decimal contains the same number of figures.