4 I 



MATHEMATICS. ARITHMETIC. 



[ADDITION. 



ingle unit, number* too great to be even conceived or 



i . ... 



Ik-fore concluding UMM remarks on numeration, it 

 ny be M well to show the beginner bow a number ex- 

 pnmd in word* may be translated inlo Jitjurtt. Thin U 



jtiite <> easy at to translate figures into words ; the 

 plan is as follows : 



down a row of nought*, or cipher*, and, as if 



these blanks were numbers, mark off the period*: thru, 



.ng at the first cipher on the left, put under 



each tin- ; 'i the nuinlxjr proposed, taking 



eare that it U- in it.- proper place : if any vacancict appear 



under the corr. [ihers, fill them up with noMgAfe 



Th UK, let it be required to put into figures the number 



fire hundred and nix million, thirty-four thousand, and 



We know that the place of millions has tix 



places to the right of it ; we there-fore put a nought for 



millions, and write six noughts after it, and, as we 

 see, from hundred* being the leading word in the written 

 expression, that the first period will be a complete period, 

 wo prtfix two noughts more. The requisite row of 



s iliviili-il us proposed, is as in 



the margin ; and under these we now 000 000 000 

 have to write, in tlicir proper places, 606 034 048 

 the figures 6, 6, 3, 4, 4, 8, and then to 

 fill up the gaps with noughts ; we thus find the number, 

 when written in figures, to be 506,034,048. The learner 

 will be able to do without such helps as these after a little 

 practice; he should accustom himself to express in words 

 the numbers he uses, when these are of moderate extent, 

 and not content himself with merely looking at them. 



We shall now proceed to the four fundamental opera- 

 tions of arithmetic : these arc atldition, subtraction, mul- 

 tiplication, an<l diritiun. There are no calculations, 

 however Ion.; an<l intricate, that are not composed of one 

 or more of tlir-e four. 



a. Addition teaches us how to add 

 numbers together, and so to find the mm of all. It is 

 called simple addition, when the numbers to be added 

 fit her have no reference to particular tilings or object*, or 

 when t lie things referred to are aU of the tame denomi- 

 nation: thus, if 24 pounds, 37 pounds, 82 pounds, <tc., 

 were all to be added together, the operation would be 

 that of simple addition ; but if 24 pounds 7 shillings, 

 37 pounds 2 shillings, 82 pounds 12 shillings, etc., were 

 to be added, then, as pounds and shillings are different 

 things, the operation would not be simple, but compound 

 addition ; one of the first set of things being called a 

 nmjilr quantity, and one of the other set a compound 

 quantity. The rule for performing simple addition is 

 M follows : 



Arrange the numbers to be added one under 

 another, so that the first column of figures on the right 

 may be unit*, the next column on its left tens, the next 

 hundred*, and so on. This is nothing more than pre- 

 serving each figure in its proper place. Add up the 

 uniU' column : if it amount to a sum expressed by only 

 one figure, put this figure down under the units' column. 

 But if it be a number of more than one figure, the last 

 figure only of that number the units' figure is to be 

 put down, and tho number expressed by what is left, 

 after ruhl.iiiK out the figure thus put down, is to be 

 carried to the next, or tens' column, and added in with 

 that column. 



sum of tho tens' column be a number of a single 

 figure, it in to be put down under that column ; but if 



a number of more than one figure, then, as before, 

 only the last, or units' figure,' of that number, is to be 

 wn, ami the nuinU-r which U expressed, after the 

 figure put dow M is rubbed out, is to be carried 

 to. and added in with, the figures in the next 246 

 column, and o on ; observing, that when the last 357 

 column is reached, the entire sum of that column 26 



U to be put down. Kup|>ose, for example, the 148 

 following numbers are to be added together 6 



nameh I4H, and 6; then, writing 



the numbers, one under another, as in the margin, 7KI 

 so that the first column on the right may be a 

 column of unit*, the next a column of ten*, and the next 



a column of hundred*, we proceed, under the dire 

 of the rule, as follows : 6 and 8 are 14, and 6 are 20, and 

 7 are 27, and 6 are 33 ; there are, therefore, in the first 

 column, 33 units ; that is to say, 3 ttns, and 3 unit* : the 

 3 units we put, of course, under the column of units, 

 but we carry the 3 tens to the next, or tens' column, and 

 say 3 and 4 are 7, and 2 are 9, and 5 are 14, and 4 are 

 hat is 18 ten*: the 8 we put down, but the number 

 left, after rubbing this out, namely 1, we carry to the 

 next column, as it is clear we ought to do ; for this 1 is 

 one more place to the left ; it stands for one hundred, 

 and therefore belongs to the hundreds' column : any figure 

 next, on the left, to a figure that stands for tens, must, 

 from the principles of numeration, stand for hundred*. 

 Carrying, therefore, the 1 to the hundreds' column, we 

 say, 1 and 1 are 2, and 3 are 5, and 2 are 7 ; that is, 7 

 hundreds : so that the sum of the proposed numbers is 

 7*3 ; that is, seven hundred and eighty- three. From 

 this operation you see that the figures of the sum are all 

 carefully put in their proper places, so that each has its 

 own local value; the numbers carried from one column 

 to the next, are so carried because they really // to 

 the place, one in advance, to tho left. Arranging these 

 numbers one under another, as before, taking care not 

 to disturb their local positions, we proceed 

 thus : 3 and 2 are 5, and 8 are 13, and 3 are 8462 

 16, and 2 are 18 ; 8 and carry 1 : 1 and 5 are 

 6, and 7 are 13, and 6 are 19; 9 and carry 1 : 

 1 and 7 are 8, and 7 are 15, and 8 are 23, and 4 47' '- 

 are 27 ; 7 and carry 2 : 2 and 7 are 9. and 4 are 7<x>:i 



13, and 8 are 21; therefore the sum is 21798: 



that is, twenty-one thousand seven hundred and 21798 

 ninety-eight. It is plain from the foregoing 

 illustrations, that the rule for addition is in strict accord- 

 ance with the system of notation and numeration already 

 explained, and that it must always lead to the correct 

 result. There is no figure higher than 9 : ten, of any 

 denomination (hundreds, thousands, etc.), is one of the 

 next higher denomination; so that in adding up any 

 column of figures, all of the same denomination, for 

 every ten, in the sum, one must be carried to the next 

 column ; and, therefore, as many one* as tens. 



You have already seen that the marks used in the 

 notation of arithmetic are figure*: besides these, other 

 marks are frequently employed to indicate operation* 

 with these figures, and to express relations among them: 

 thus, instead of saying 2 atid 5 are equal to 7, the form 

 2+5=7, is used to express the same tiling : the mark + 

 being the sign for addition, and the mark = the sign for 

 : :ty: this must be borne in mind: +is willed plus; 

 so that 2+5=7, may be read 2 plus 5 equals 7, or 2 plus 

 y are equal to 7. The following, therefore, are st 

 ments in symbols, instead of in words, which you will at 

 once understand: 2+5+1=8; 3+4+2=9; 6+5+3+ 

 1 = 15. A few examples in addition are here given 

 under this form. You will have to arrange the nutn- 

 HTS in them in columns, as in the two examples worked 

 above ; and, if the results of your addition be correct, 

 ;hey will be found to agree with the numbers to the 

 right of the sign of equality : 

 (1.) 324+0.43+201+46=1214. 

 (2.) 36+320+708+17+3=1084. 

 (3.) 6684+340+7006+309+824 = 14103. 

 (4 ) 20065+8473+751 +92083+504+92=122028. 



Simple Subtraction. Subtraction teaches us how to 

 subtract tho smaller of two numbers from the greater, 

 >r to find their difference, which is called the remainder. 

 The operation is called simple subtraction when the 

 numbers refer to things of the same denomination, as in 

 simple addition . The rule for simple subtraction is as 

 "ollows : 



111 I.E. Put the smaller number under the greater, 

 taking care, as in addition, that units sliall be under 

 units, tens under tens, and so on. 



Then, beginning at the unit*, subtract each figure in 

 the lower row from the figure above it, if the lower 

 figure be not the greater of the two, and put the 

 remainder underneath (see the operation in the margin, 



