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 15·625 × 64 = 1000 1·5625 × ·64 = 1 ·015625 × ·0064 = ·0001 ·15625 × ·64 = ·1 1562·5 × ·064 = 100 15625000 × ·064 = 1000000 144. The division of a decimal by a decimal number, such as 10, 100, 1000, &c., is performed by moving the decimal point as many places to the left as there are ciphers in the decimal number. If there are not places enough in the dividend to allow of this, annex ciphers to the beginning of it until there are. For example, divide 1734·229 by 1000: the decimal fraction is ¹⁷³⁴²²⁹/₁₀₀₀, which divided by 1000 (123) is ¹⁷³⁴²²⁹/₁₀₀₀₀₀₀, or 1·734229. If, in the same way, 1·2106 be divided by 10000, the result is ·00012106. 145. Before proceeding to shorten the rule for the division of one decimal fraction by another, it will be necessary to resume what was said in (128) upon the reduction of any fraction to a decimal fraction. It was there shewn that ⁷/₁₆ is the same fraction as ⁴³⁷⁵/₁₀₀₀₀ or ·4375. As another example, convert ³/₁₂₈ into a decimal fraction. Follow the same process as in (128), thus: