>NS IN ABIT ll.v 



13 



LESSONS IN ARITHMETIC. L 



nil .lci//n/i.7i.-, \vliirh i-i i!.n\.-,l from tin- (in. 1 



tt^Ofttia (prollolllKVil n./'lM'-m.-.i,). ' 



JILTS, and tin- :i rulrnlatioh < 



liy them, iiU'l investigating their n I '< a e.ertuin 



ionco must havo been ooovol with the history of man. 



As nil :irt, urithm.'tie, is iiitli-|M-n-al>lo in daily In. 

 tin' iiKin \\h -'luainteil with its practical details Inn 



i In- i>rriVri'iico in every mercantile establishment. Our 

 ~ons shall bo twofold to develop its prinriple* 

 e, and to show tho application of its rules as an art 



For this i>urpo-;i\ it will bo necessary to begin with the first 

 prini'i|.! '/i and Notation, and to give sucli rules 



as will onablo any ouo to read and write a given number 



NOTATION AND NUMERATION. 



1. Any Miigle thing as for instance, a pen, a sheep, a house 



.tiled a unit: we say there is </ sin-h thing. If 

 another M!i;_rli( thing of the same kind be put witli it, there are 

 said to bo ttvo such things; if another, three ; if another, four; 

 uiid KO on. 



Each of th.-se collections of things of which we havo spoken 



of things; and the terms one, tivo, three, f" 

 etc., by which wo express how many single things or units ore 

 under consideration, arc the names of numbers. A. number 

 therefore is a collection of units. This is also sometimes called 

 an integer, or whole number. 



It will be seen that the idea of number is quite independent 

 of tho particular kind of units, a collection of which is counted. 

 Thin, if there are four pigs, the number of pigs is tho same as 

 if tli !) were four pens. We can thus abstract a number from 

 any particular unit or thing, and talk of tho number four, tho 

 nuuilK'r five, etc. Numbers thus abstracted from their reference 

 to any particular unit or thing are called abstract numbers. 

 When a collection of tilings or objects ia indicated, it is called 

 a concrete number. 



\\ e -hall treat first of abstract numbers. 



2. The art of expressing numbers by symbols, or figures, is 

 called Notation. 



lu the system of notation which wo are about to explain, all 



numbers can be expressed by means of ten symbols (figures, or 



as they tuv c-illcil), re] in -sen ting respectively tho first 



nine numbers, and nothing, -i.e., the absence of number. These 



are 



(I representing the number six 



7 seven 



8 eight 



9 ,, ,, nine 

 called a nought:, a cipher, or zero. 



N.B. Ten times ten is called one hundred ; ten times a 

 hundred, a thousand. 



3. Numbers are represented by giving to the figures employed 

 what is called a local value i.e., a value depending upon the 

 positions in which they are placed. 



Let a number of columns be drawn as below, that being called 

 the first which is on tho right, and reckoning the order of the 

 columns from right to left. 



Denting the nnrnber one 

 - two 



three 



4 four 



5 five 



If a figure 5, for instance be placed in tho first column, it 

 denotes five units, or the number five ; if it bo placed in the 

 second column, it denotes five tens ; if in tho third, five hun- 

 dreds ; if in the fourth, five thousands ; if in tho fifth, five 

 times ten thousand ; and so on, each column corresponding to a 

 number ten times as great as the one immediately on its right. 



* Digits. So called from digitus, a " finger." This decimal nota- 

 tion clearly took its origin from these natural counting instruments. 



Thtu | 7 | 9 | 4 | 3 | would denote seven 



: ir ten*, and three on**; or, a* it would I* 

 >ed, HO veil thousand, nioa hundred, and forty-Urn*. 



.irly, |8|3IO|5|4|7| would denote eight times a 

 hundred thousand, throe time* ten thousand, no th/rnamds, five 

 hundroda, four teiw, and Keren ones ; or, an it would be more 

 briefly expressed, eight hundred and thirty thousand, fire hun- 

 dred and forty-seven. 



Wo need not, however, draw the column* ; it will be ibe name 

 thing if wo imagine them, and, instead of column*, talk of 

 figures being in the first, second, third, fourth places, < 



The symbol put in any place, an already indicated in the 

 previous example, denotes that the number corresponding to tho 

 particular column or placr in which it utand* U not to be taken 

 at all : tho only fills up tho place thus, however, answering 

 tho important purpose of increasing the figu.-o after which it 

 stands tenfold. 



Thus, 10 means that once ten and no units are taken i*., it 

 denotes tho number ten ; 100 means that once a hundred but 

 110 tens and no units are taken i.e., it denotes the number a 

 hundred ; 5001 means that five thousands, no hundreds, no tens, 

 and ono unit, are taken, or, as it would be more briefly ex* 

 pressed, five thousand ai.d one. 



4. Before proceeding further, we will give the names of the 

 successive numbers : 



If 

 20 

 80 

 40 

 50 

 60 

 70 

 80 

 90 



1000000,000000,000000. 



The numbers between twenty and thirty are expressed thus 



one, twenty-two, twenty-three, etc., up to twenty-nine, 



to which succeeds thirty ; and similarly between any other two 



of the names above given, from twenty up to a hundred : thus, 



95 is called ninety-five. 



After one hundred, numbers are denoted in words, by men- 

 tioning the separate numbers of units, tens, hundreds, thousands, 

 etc., of which they are made up. For example, 134 is one 

 hundred and thirty-four ; 5,342 is five thousand three hundred 

 and forty-two ; 92,547 is ninety-two thousand five hundred and 

 forty-seven ; 84,319,652 is eighty-four millions, three hundred 

 and nineteen thousand, six hundred and fifty-two. 



5. It is useful, in reading off into words a number expressed 

 in figures, to divide the figures into periods of three, commencing 

 on the right, as the following example will indicate : 



Billions. Thousands of Millions. Millions. Thousands. Units. 

 561 234 826 478 365 



Thus the figures 561,234,826,479,365 would denote five hundred 

 and sixty-one billions, two hundred and thirty-four thnmand 

 eight hundred and twenty-six millions, four hundred and seventy- 

 nine thousand, three hundred and sixty-five. 



We have then the following 



Rule for reading numbers which are expretted in figure* : 



Divide them into periods of three figures each, beginning at 

 the right hand ; then, commencing at the left hand, read the 

 figures of each period in the same manner as those of the right- 

 hand period are read, and at the end of each period pronounce 

 its name. 



Tho art of indicating by words numbers expressed by figures 

 ia called Numeration. 



EXERCISE 1. 



Write down in figures tho numbers named in the following 

 exercises : 



In the foreign system of numeration a thousand millions is called 

 a billion, a thousand billions a trillion, and so on. 



