In the following examples the first two lines are the multiplicand and multiplier; and the number of decimals to be retained will be seen from the results. ·4471618 33·166248 3·4641016 3·7719214 1·4142136 1732·508 ========= ========== ============ 37719214 033166248 346410160 8161744 63124141 8052371 -------- ---------- ------------ 15087686 3316625 346410160 1508768 1326650 242487112 264034 33166 10392305 3772 13266 692820 2263 663 173205 38 33 2771 30 10 ------------ --------- 2 6001·58373 1·6866591 -------- 46·90415 Exercises may be got from article (143). 155. With regard to division, take any two numbers, for example, 16·80437921 and 3·142, and divide the first by the second, as far as any required number of decimal places, for example, five. This gives the following: 3·142)16·80437921(5·34830 15·710 ------- 1·0943 | 9426 | -----| 15177| (A) 12568| ---- -----|- 2609 2609|9 2514 2513|6 ---- ----|-- 95 96|32 94 94|26 -- --|--- 1 2|061 Now cut off by a vertical line, as in (153), all the figures which come on the right of the first figure 2, in the last remainder 2061. As in multiplication, we may obtain all that is on the left of the vertical line by an abbreviated method, as represented at (A). After what has been said on multiplication, it is useless to go further into the detail; the following rule will be sufficient: To divide one decimal by another, retaining only _n_ places: Proceed one step in the ordinary division, and determine, by (150), in what place is the quotient so obtained; proceed in the ordinary way, until the number of figures remaining to be found in the quotient is less than the number of figures in the divisor: if this should be already the case, proceed no further in the ordinary way. Instead of annexing a figure or cipher to the remainder, cut off a figure from the divisor, and proceed one step with this curtailed divisor as usual, remembering, however, in multiplying this divisor, to carry the _nearest ten_, as in (154), from the figure which was struck off; repeat this, striking off another figure of the divisor, and so on, until no figures are left. Since we know from the beginning in what place the first figure of the quotient is, and also how many decimals are required, we can tell from the beginning how many figures there will be in the whole quotient. If the divisor contain more figures than the quotient, it will be unnecessary to use them: and they may be rejected, the rest being corrected as in