| II. | ·01003 is | 1003 | or | 1 | + | 3 |
| 100000 | 100 | 100000 | ||||
| ·1003 is | 1003 | or | 1 | + | 3 | |
| 10000 | 10 | 10000 | ||||
| 10·03 is | 1003 | or | 10 | + | 3 | |
| 100 | 100 | |||||
| 100·3 is | 1003 | or | 100 | + | 3 | |
| 10 | 10 |
| III. | ·1238 | = | 1 | + | 2 | + | 8 | + | 3 |
| 10 | 100 | 1000 | 10000 | ||||||
| = | ·1 | + | ·02 | + | ·008 | + | ·0003 | ||
| = | ·1 | + | ·0283 | = | ·12 | + | ·0083 | ||
| = | ·128 | + | ·0003 | = | ·108 | + | ·0203 | ||
| = | ·1003 | + | ·028 | = | ·1203 | + | ·008 |
| IV. | In 1234·56789 inches the |  | 1 is | 1000 | inches |
| 2 is | 200 | ||||
| 3 is | 30 | ||||
| 4 is | 4 | ||||
| 5 is | ⁵/₁₀ | of an inch | |||
| 6 is | ⁶/₁₀₀ | ||||
| 7 is | ⁷/₁₀₀₀ | ||||
| 8 is | ⁸/₁₀₀₀₀ | ||||
| 9 is | ⁹/₁₀₀₀₀₀ |
136. The ciphers on the right hand of the decimal point serve the same purpose as the ciphers in (10). They are not counted as any thing themselves, but serve to shew the place in which the accompanying numbers stand. They might be dispensed with by writing the numbers in ruled columns, as in the first section. They are distinguished from the numbers which accompany them by calling the latter significant figures. Thus, ·0003747 is a decimal of seven places with four significant figures, ·346 is a decimal of three places with three significant figures, &c.
137. The value of a decimal is not altered by putting any number of ciphers on its right. Take, for example, ·3 and ·300. The first (135) is ³/₁₀, and the second ³⁰⁰/₁₀₀₀, which is made from the first by multiplying both its numerator and denominator by 100, and (108) is the same quantity.
138. To reduce two decimals to a common denominator, put as many ciphers on the right of that which has the smaller number of places as will make the number of places in both fractions the same. Take, for example, ·54 and 4·3297. The first is ⁵⁴/₁₀₀, and the second ⁴³²⁹⁷/₁₀₀₀₀. Multiply the numerator and denominator of the first by 100 (108), which reduces it to ⁵⁴⁰⁰/₁₀₀₀₀, which has the same denominator as ⁴³²⁹⁷/₁₀₀₀₀. But ⁵⁴⁰⁰/₁₀₀₀₀ is ·5400 (135). In whole numbers, the decimal point should be placed at the end: thus, 129 should be written 129·. It is, however, usual to omit the point; but you must recollect that 129 and 129·000 are of the same value, since the first is 129 and the second ¹²⁹⁰⁰⁰/₁₀₀₀.
139. The rules which were given in the last chapter for addition, subtraction, multiplication, and division, apply to all fractions, and therefore to decimal fractions among the rest. But the way of writing decimal fractions, which is explained in this chapter, makes the application of these rules more simple. We proceed to the different cases.
Suppose it required to add 42·634, 45·2806, 2·001, and 54. By (112) these must be reduced to a common denominator, which is done (138) by writing them as follows: 42·6340, 45·2806, 2·0010, and 54·0000. These are decimal fractions, whose numerators are 426340, 452806, 20010, and 540000, and whose common denominator is 10000. By (112) their sum is
| 426340 + 452806 + 20010 + 540000 | , which is | 1439156 |
| 10000 | 10000 |
or 143·9156. The simplest way of doing this is as follows: write the decimals down under one another, so that the decimal points may fall under one another, thus: