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 371 A R I T H M E T I C R. units, tens under tens, Sc,.: but if either or both of the Contractions, and fwiple ways of working multiplicafactors have ciphers on the right hand, fet their firft tion of integers. lignificant figures under other. The 01 der prefcribed in this rule is not abfolutely ne- 1. To multiply any number by 10, by 100, by 1000, cefiary, but very convenient as will appear in the ex- he. to the given number annex one,. two, three ciphers, amples. Thus, 23X10=230; and 384X100=38400; II. Beginning at the right hand, multiply each figure he. and 745 X 1000=745000. of the multiplier into the whole multiplicand, carrying, 2. To multiply any number by 9, by 99, by 999, 'he. as in addition, and placing the right-hand figure of each the given number firft by 10, by 100, by 1000, particular produ& dire&ly under the multiplying figure. multiply he. that is, annex one, two, three, he., ciphers to it; III. Add the particular produdls, and their fum will from this fubtradt the given number, and the remainder be the total produdl. examples. Examp. I. Having placed the multiplier under the is the produft ; as in the following Ex. 2. Ex. 3. multiplicand, as directed in Rule I. proceed to the ope- Mult. Ex. 471. Mult. 627 Mult. 999 ration, and fay, 7 times 4 make 28; fet the 8 beby 99 62700 by 999 999000 Faiftofs 947 multiplicand, multipler. low in the place of units, bySub.9 470 Sub. 627 Sub. 47 999 and carry the 2 tens to 658 produfh the next place, as dirdted prod. 423 Prod. Prod. 998001 in Rule H. faying 7 times From Ex. 3. we may learn,62073 Aat to multi9 make 63, and 2 that I carried make 65; fet 5 below any number confiding entirelyin genera!, of 9’s by itfelf, is to in the place qf tens, and the 6, which belongs to the next ply x in the place of units, then as many ciphers, fave place, fet on its left hand, there being no further place let one, as there are p’s in the given number; then 8, and to which it can be carried; fo the produdt is 658. left hand of 8 as many 9’s as there are ciphers on II. Here firft multiply the right-hand figure 8 into onits the the whole multiplicand, as in 3.right. To multiply any number by 5; firft multiply it by former example; then pro742 multiplicand. the that is, annex a cipher to it, and then halve it: and 68 multiplier. ceed, and multiply likewife the to10,multiply any number by 15, ufe the fame method; and 6 tens into the whole multipliboth numbers together, as in the following examples.' cand, faying 6 times 2 make addMultiply 5936"? particular 7439 Multiply 9856 12; fet the 2 below under the 4452 5 products. the multiplying figure, viz. in by 1S- 985607 dd 50456 total produdt. the place of tens, and carry by 5 - 74390 492803 the x to the next place, as direfled in Rule II. The reafon why the 2 is fet under Produft 37195 Produdt 147840 the multiplying figure, or in the place of tens, is, becaufe the multiplying figure 6 is really 6 tens or 60, and 60 4. To multiply any number by 11, 12, 13, 14, 15, times 2 make 120; fo that by carrying the 1 to the next 16, he. multiply by the unit’s figure, and add the backof the multiplicand to the produdt; and to multiply place, and fetting down 20, the o would fall into the figure 21, 22, 23, 24, 25, 26, 27, he. add the double of place of units, and throw the 2 into the place of tens ; but by back-figure; and to multiply by 31, 32, 33, 34, he. as o can make no alteration in the addition of the par- the tial produ&s, the fetting of it down is fofely and juftly add the triple of it; and to multiply by 112, x 13, 114, he. add the two back-figures ; and to multiply by 101, omitted. III. When the multiplier has ciphers on the right hand, 102, I03,xi04, he. add the next back-figure fave one: as in the following examples. as it would be evidently loft labour to mul853 Ex. 2. by the ciphers, their only ufe being to Ex. 1. 72000 tiply throw figures on their left hand into higher 876 or multiply by 876 or thus, 876 694 11 xx thus 876 876 14 places, fet the firft fignificant figures of the 1706 fatfors tinder other; and, after the ope5771 9716 9636 9636 ration is finiftied, annex the ciphers of 9636 Ex. 4. multiplier to the right hand of the proJZx. 3. 614:6000 the duct 241 435 IV. When the multiplier has ciphers intermixed with 27 34 figniHcant figures, omit the ciphers, 8194 29601847 becaufe the multiplying by them would 1'745 Ex. 6. E». 7. 300905 only produce fo many lines of ciphers Ex. 5. — and lb be labour in vain ; wherefore 7234 or thus, .7234 ^63 745 119 148009235 multiply by,the fignificant figures only; 112 7234 10.3 266416623 but take care to place the right-hand 7234 31297 88805541’ figure of each particular product di- 81020S 7234. 76735 8907343771535, reftly under the multiplying figure. §10208