thousi 



ARITHMETIC. 



and five hundred and forty-eight, by placing 

 each of the units in the column 

 allotted to that particular unit ; 

 thus the 3 is placed in the column 

 of thousands, the 5 in that of hun- 

 dreds, the 4 in that of tens, and 

 the 8 in the unit column. Simi- 

 larly with the number two thou- 

 sand and seventy-nine, the 2 is 

 placed in the thousands column, 

 the 7 in that of tens, and the 9 

 in the unit column. In this way 

 we can write down any number ; 

 but we may, by a simple de- 

 vice, save ourselves the trouble of drawing the 

 lines to form the columns ; for, let it be understood, 

 that the simple units are put on the right, the tens 

 next on the left, the hundreds next on the left of 

 the tens, the thousands on the left of the hundreds, 

 the tens of thousands on the left of the thousands, 

 and so on then we may write the number three 

 thousand five hundred and forty-eight at 

 once, thus : 3548 



Again, the number two thousand and 

 seventy-nine would be written 2 79 



As there are here no units of hundreds, their 

 place would be unoccupied, and we would be very 

 apt to take the 2 to represent hundreds, instead of 

 thousands, as it is intended to do. Now, to pre- 

 vent any mistake arising on this account, we intro- 

 duce the symbol o or zero to keep the place of the 

 hundreds. This symbol has no value of itself, 

 but merely serves for keeping the places of those 

 units which may be wanting in any number ; thus 

 the number two thousand and seventy-nine will 

 be written 2079. Again, two hundred and twelve 

 thousand and eight will be written 212008, because 

 the units of hundreds and also of tens are wanting, 

 and their places are supplied by zeros. The 

 number fifty will be written 50, because the unit 

 figure is wanting. In the same way, five hundred 

 will be written 500. In this manner we are en- 

 abled to write down any number. 



It will at once be evident that if we annex a 

 zero to the right of any number, we multiply it by 

 10, for by doing so we remove each of the figures 

 one place to the left. Similarly, by annexing two 

 zeros, we multiply by 100 ; annexing three zeros, 

 we multiply by 1000; and so on. 



Decimal Numbers. In considering the number 

 5555, the 5 nearest to the right denotes units; the 

 next on the left, tens; the next again, hundreds; and 

 the last on the left, thousands. Or proceeding from 

 left to right, we see that the figure on the right is 

 ten times less than the one immediately on its left ; 

 thus the 5 on the left denotes thousands j the next 

 on the right, hundreds; the next again, tens; and 

 that on the left, units. There is no reason, however, 

 why this last figure should not in its turn be ten 



left, denotes tens; the 3 denotes hundreds; and the 

 4, thousands. Again, 6, the first figure on the right 

 of the unit, being of ten times less value than if it 

 stood in the units' place, denotes tenths; the 9, 

 the next on the right, denotes hundredths ; the I, 

 thousandths; and the 8, tenths of thousandths. 

 Knowing the position of the unit figure, then we at 

 once know the value of any other figure. The 

 unit figure is marked, not by underlining it, but 

 by placing a point or dot immediately on its right 

 The above number would be written 



4327-6918. 



The dot is called the decimal point, and separates 

 the whole number in the expression from the 

 fractional part of it. We are also enabled to read 

 off at once any decimal expression ; thus -5 is read 

 as decimal 5 or 5 tenths, -05 as 5 hundredths, -005 

 as 5 thousandths. Again, 1-5 is read 15 tenths, 

 1-15 as 115 hundredths that is, we read off the 

 number as if it contained no decimal point, and 

 give to it the denomination of the last figure on 

 the right. 



It is evident that a decimal number is multiplied 

 by 10, by loo, by 1000, &c. by simply moving the 

 point one, two, three, &c. places to the right 

 Thus the number 3-4568, when multiplied by 10, 

 becomes 34-568 ; when multiplied by 100, becomes 

 345 -68 ; and so on. In the same manner a num- 

 ber is divided by 10, by 100, by 1000, &c, by 

 simply moving the point one, two, three, &c. places 

 to the left ; thus the number 3-4568, when divided 

 by 10, by 100, by 1000, becomes -34568, -034568, 

 0034568. When we simply suppress the point in 

 a decimal expression, we multiply it by 10, 100, 

 &c. Thus, if for 3-456 we write down 3456, we 

 have multiplied the number by icoo, for the point 

 is now supposed to be on the right of the 6. 



ADDITION AND SUBTRACTION OF WHOLE OR 

 DECIMAL NUMBERS. 



The two fundamental operations in arithmetic 

 are addition and subtraction, as a number can 

 only be altered by increase or diminution. 



Addition is that process by which we increase a 

 number by a given number: or it is that process 

 by which we form into one number several other 

 numbers. The resulting number is called the 

 sum. 



It is easy to add to a given number another 

 small number ; thus, to add 4 to 5, it is sufficient to 

 add one unit to the 5 as often as there are units in 

 4 ; thus we say 5 and i make 6, 6 and i make 7, 

 7 and I make 8, 8 and i make 9 ; and 9 is the sum 

 required. In the same way we might add to a 

 number another number consisting of two or more 

 figures ; but it is evident that this process would 



times greater than the figure immediately on its be very tedious, and when the^iumber to be added 

 right ; and this, again, ten times greater than the 

 next on the right ; and so on. By thus extending 

 the decimal system, we will be enabled to repre- 

 sent all fractions, however small, 

 we take the following expression, 



43276918, 



in which the unit figure, to mark its position, is 

 underlined, we can at once tell the value of each 

 figure ; thus 7 being the unit ; 2, the first on the 



was very large, it would be almost impracticable ; 

 and hence we consider each of the numbers to 

 be added as being resolved into its component 

 For example, if units, and written one under the other, in such a 

 way that the units of the same order stand under 

 each other ; we then add the several units separ- 

 ately. Thus let it be required to add together 453, 

 7546, 5314, 208. Here the number 453 is resolved 

 into 3 simple units, 5 tens, and 4 hundreds. Again, 

 7546 is resolved into 6 simple units, 4 tens, 5 

 hundreds, and 7 thousands ; and so with the 



595 



