M \ I HIM I Pica ARITHMETIC. 



[HE. in i LS. 



8)19-000 

 2-375 



11)6 



It is pretty evident that whatever whole number be 

 prefixed to a decimal, the same may be profile- . 

 numerator of the fraction which replaces that decimal : 

 thus, Uking the previous values, we hare 



6-W; 13664->i8JJ;73641-*iV<AJf ;2-07-?8J- 

 for this u uiily reducing the foregoing tnixcd 

 to improper fractions. 



7*0 rWwe* a Proptr Fraction to a Decimal. 



RrtE. Annex a zero to the numerator, and then 

 actually divide by the denominator : if there be a re- 

 mainder, annex another zero, and continue tlio division, 

 still annexing a zero, either till the division terminates 

 without remainder, or till as many decimals as are 

 considered Decenary are obtained ; tho quotient, with 

 the decimal point before it, will be the value of the 

 fraction in ! . mials. 



For example : let it be required 

 to express } in decimals ; the opera- 

 n that in the margin. That f 

 -375 U easily proved ; for f J JJJ; 

 consequently, dividing numerator and denominator by 

 8, we have | f$fo "375, from the very nature of 

 decimals. If au t'nproper fraction 

 had been chosen, the operation would 

 clearly have been just the same, only 

 there would have been an integer 

 prefixed to the decimal : thus, the 

 operation for "^ would have been as here annexed, show- 

 ing that y 2375. We need not take the trouble of 

 actually annexing the zeros, as here : 

 it is enough that we proceed as if 

 they were inserted, as in the mar- 

 ginal example, for reducing -ff to a 

 decimal ; where it is plain, from the 

 remainders, that 54 would recur continually ; so that -,", 

 is equal to a recurring decimal ; the recurring period 

 being 54. As a final example, let 

 it be required to convert yfj into a 

 decimal When one is annexed to 

 the 8, the divisor 113 will go no 

 timn; then-fore, the first decimal 

 place U to be occupied with a 0. 

 Annexing now a second 0, the next 

 decimal figure is 7, and the work 

 i : i-i .1- in t'.e in u (in : tip. 

 noughit being suppressed, though 

 conceived to bo annexed to the 8, 

 and brought down one at a time, as 

 in ordinary division. The quotient shows 

 07079, Ac. : the decimals may be carried out to 'any 

 extent ; but if we stop the work here, the error cannot 

 be so great as O0001 ; that is, it is less than -nnforo : bl t 

 it is obvious that, by continuing the work, we can make 

 the error as small as we please. 



The following are a few examples for exercise : 

 (1.) V-1875. (2.) A-* 1 -:.--. 



(3-) 14= 4) i- -08135. 



::. A-c. (6.) ^--00488, <tc. 



Adititinn mill s H it>rnrtii,n i,f J>ecintaU.~Tho rules for 

 these fundamental operations are in 

 reality the same as those for integers. 

 We must here be careful not only to 

 place units under units, tens under 

 tens, and so on, but also place 

 under t: n/li.1, hiiiulr-'ilthn unde: 



. --. : that is, the decimal 

 point* must all range under one an- 

 other in the same vertical line. This 28-5015 



attended to, the operations are just 



the same as those with integers. (See 

 the operations in the marn 



HtUlipUcatum <>f J>,ci,n,il*. Multiplication re<i\iires 

 no special rule. The multiplier is to be placed under 

 the multiplicand, just as if both were integers, no regard 

 being paid to the decimal points. The only thing to be 

 attended to is the marking off the proper number of 

 decimal places in the product ; and that is a very easy 

 matter. We have teen that a number involving decimals 



6454, <fec. 



113)8 ( 07079, <tc. 

 791 



Addition. 



i -in.-..; 

 01:17 

 004i 

 75 



Subtraction. 



11 :i>.-, 

 8-MB] 



107709 



12743 



23-402 

 1,7-W 



23402 

 70880 



if,', 



40012722 



is, in fact, a fraction with that numl>ur. the d- 

 point being suppressed, for numerator, and 1, fu!i 

 by as many ciphers as there a- <r de- 



nominator. Two such fractions ni-.iltiplied together, 

 being the product of the numerators divided 

 by the product of the denominators, will 

 therefore be a fraction of which the- deno- 

 minator is 1, followed by as many ciphers as 

 there are in both factors. Consequently, in tlio 

 multiplication of decimals, as many d 

 places are to be marked off in the product as 

 there are decimal places in both factors. 



The example in the margin will suffice for 

 illustration. As there are three decim.-ils in 

 the multiplicand, and two in the multiplier, 

 Jice are marked otf in the product. 



liirismn (if Derimuls. This operation, like that of 

 multiplication, is the same for decimals as for integers ; 

 and the way to estimate the number of decimal places 

 in the quotient is suggested by the plan adopted in 

 multiplication. 



All the decimals employed in the dividend, including, 

 of course, whatever ciphers may have been added to it 

 to carry on the division, are to be count. 1. \Ve have 

 then only to provide so many in the quotient, that when 

 added to the number of them in the 

 divisor, we may have just as many 

 as in the dividend. 



If the quotient figures, though 

 all be considered as decimals, be 

 too few in number to make up, 

 with those in the divisor, the num- 

 ber in the dividend, then ciphers 

 sufficient for this purpose are to be 

 prefixed to the quotient figures,. and 

 the decimal point to be placed before 

 them. (See the second example in 

 the margin.) In the first of these 

 examples, six have been used in the 

 dividend, and as there are two in 

 the divisor, there must be four in 

 the quotient, which is therefore 

 10 0315. The last decimal, 5, is a 

 little too great; but it is easy to see 

 that if we had made it 4, the error 

 in defect would have exceeded the 

 present error in excess; and in 

 limiting the number of decimals, 

 we always make the lad figure as 

 near the truth as possible. In the second example, ./ire 

 decimals have been used in the dividend ; and as there 

 is but one in the divisor, four are required in the quo- 

 tient ; and to make up this niimlier, a cipher is prefixed. 

 The quotient is, therefore, -ui.'ii.">. as far as the Jen 

 have been carried : that is, to four places. 



The following examples will serve for exercise in these 

 two rules : 



1.) -321000 x -2405 =-0791501 64. 

 2.) 404 3 X -00521 = 2 419003. 

 LOeSS-t- 17 371-3-23. 

 4.) 2-419003-1-4643-00321. 



Extraction uf the Square lil. If a number be mul- 

 tiplied by itself, the product is called the ttcoinl / 

 or the square, of that number. If this also be multiplied 

 by the same number, the product is called tlir ' 

 power, or the cube of that numlwr : and so on for tho 

 fourth power, fifth power, 4c. This raising of powers, 

 which is called involuting, is therefore nothing more 

 than the multiplication together of equal factors, and is 

 easy enough. But the reverse operation that is, to find 

 the factor which, involved in this manner, shall produce 

 a given number is a problem not so readily disposed of. 

 The factor referred to is called the root of the power ; so 

 that the reverse problem spoken of is tho problem of the 

 extraction of roots. To extract the square root of a given 

 number, is to find a number which, when squared, or 

 raised to the second power, or, which is the same thing, 

 when multiplied by itself, shall reproduce tho gi\en 

 number. The rule for this operation is as follows : 



2-3T>)23-621(10-0315 

 296 



121 



1175 



35 

 235 



115 

 1170 



324)86 (-0265 

 648 



212 

 1944 



176 



1020 

 140 



