MATHEMATICS 9 



numbers, were found comparatively useless for purposes 

 of calculation purposes which the Arabic symbols have 

 admirably subserved. The absence of numbers being 

 expressed by o, and the first group of ten (from zero to 

 nine) being expressed by the figures with which we are 

 all familiar, its completion is represented by 10 (or 

 unity and zero combined), and so on with successive 

 groups of ten till the tenth set (90-99) is completed. 



Then a third figure is added to the left to denote ten 

 groups of ten (100), while each time such a group is 

 further taken ten times over, it is expressed by the 

 addition of another zero to the right, and it is thus that 

 " ten times, ten times, ten times, ten times, ten times, 

 ten " (or one million) requires to be denoted by the 

 figures 1,000,000. In this way each figure shows its 

 value by the place it occupies. Thus it is that in the 

 symbol 1652392 the figure 2 denotes mere units, 9 the 

 groups of ten, 3 the groups of ten times ten, or hundreds, 

 and so on or, in other words, the symbol denotes 



1 million, 6 hundreds of thousands, 5 tens of thousands, 



2 thousands, 3 hundreds, 9 tens, and 2 units. 



These truths are, of course, familiar to all readers of 

 this book, though they may not happen to have con- 

 sidered them from the present point of view. 



But since our purpose is to introduce the reader to 

 the elements of science, we are bound to act as if ex- 

 ceedingly little were known by him. Thus, to carry out 

 the end we have set before us, we must consider the 

 principles of such elementary processes as addition, 

 subtraction, multiplication, and division. 



The definite position of figures, according to their 

 value, greatly facilitates the first of these processes, 

 since by the superposition of figures, thus arranged, we 

 are enabled to add them together as simple units, without 



