| What is | 12362 - 274·22107 + ·5; | |
| 9976·2073942 - ·00143976728; | ||
| and | 1·2 + ·03 + ·004 - ·0005? | |
| Answer, | 12088·27893, 9976·20595443272; and 1·2335. | |
141. The multiplication of a decimal by 10, 100, 1000, &c., is performed by merely moving the decimal point to the right. Suppose, for example, 13·2079 is to be multiplied by 100. The decimal is ¹³²⁰⁷⁹/₁₀₀₀₀, which multiplied by 100 is (117) ¹³²⁰⁷⁹/₁₀₀, or 1320·79. Again, 1·309 × 100000 is ¹³⁰⁹/₁₀₀₀ × 100000, or (116) ¹³⁰⁹⁰⁰⁰⁰⁰/₁₀₀₀ or 130900. From these and other instances we get the following rule: To multiply a decimal fraction by a decimal number (126), move the decimal point as many places to the right as there are ciphers in the decimal number. When this cannot be done, annex ciphers to the right of the decimal (137) until it can.
142. Suppose it required to multiply 17·036 by 4·27. The first of these decimals is ¹⁷⁰³⁶/₁₀₀₀, and the second ⁴²⁷/₁₀₀. By (118) the product of these fractions has for its numerator the product of 17036 and 427, and for its denominator the product of 1000 and 100; therefore this product is ⁷²⁷⁴³⁷²/₁₀₀₀₀₀, or 72·74372. This may be done more shortly by multiplying the two numbers 17036 and 427, and cutting off by the decimal point as many places as there are decimal places both in 17·036 and 4·27, because the product of two decimal numbers will contain as many ciphers as there are ciphers in both.
143. This question now arises: What if there should not be as many figures in the product as there are decimal places in the multiplier and multiplicand together? To see what must be done in this case, multiply ·172 by ·101, or ¹⁷²/₁₀₀₀ by ¹⁰¹/₁₀₀₀. The product of these two is ¹⁷³⁷²/₁₀₀₀₀₀₀, or ·017372 (135). Therefore, when the number of places in the product is not sufficient to allow the rule of the last article to be followed, as many ciphers must be placed at the beginning as will make up the deficiency.
ADDITIONAL EXAMPLES.
| ·001 × ·01 is | ·00001 |
| 56 × ·0001 is | ·0056. |
EXERCISES.
Shew that
| 3·002 × 3·002 | = | 3 × 3 + 2 × 3 × ·002 + ·002 × ·002 |
| 11·5609 × 5·3191 | = | 8·44 × 8·44 - 3·1209 × 3·1209 |
| 8·217 × 10·001 | = | 8 × 10 + 8 × ·001 + 10 × ·217 + ·001 × ·217. |
| Fraction. | Square. | Cube. |
|---|---|---|
| 82·92 | 6875·7264 | 570135·233088 |
| ·0173 | ·00029929 | ·000005177717 |
| 1·43 | 2·0449 | 2·924207 |
| ·009 | ·000081 | ·000000729 |