4 2 4 THE POPULAR SCIENCE MONTHLY. 



cant digits is always one less than the value of the radix. In the 

 decimal scale there are nine digits ; in the binary, there would be only 

 one the figure 1. The radix, whatever it be, has no separate symbol, 

 but is represented by 10. In the binary scale, since two is the radix, 

 two would be so written. The square of the radix is represented by 

 the symbols 100. In the binary these would, therefore, stand for four, 

 while eight, which is the cube of the radix, would be denoted by 1000. 

 The first ten numbers, counting from one, would be : 1, 10, 11, 100, 101, 

 110, 111, 1000, 1001, 1010. 



In this system, then, the only digit employed is 1. The plaj T s the 

 same important part in it as in the decimal system. It multiplies the 

 figure that immediately precedes it by the value of the radix. The 

 symbol 40, in our denary scale, represents ten times four ; in the octenary 

 it would denote eight times four, and in the quinary five times four. 

 These two symbols, 1 and 0, then, are the only ones that enter into 

 calculations. It is evident that thought in arithmetical work is almost 

 superseded, and that all numerical operations are reduced to the manual 

 labor of writing. As the scale has only one digit, it would require more 

 figures to represent a number than other scales require. The present 

 year 1878, which is expressed in our scale by four figures, would require 

 eleven in the binary scale. It would be written 10101010110. And, 

 generally, the binary scale would call for about three and a half times 

 as many figures as the denary. This fact would occasion increased ex- 

 penditure of time and manual labor in calculations. It is, however, 

 claimed by those who favor the system, that, since only two symbols 

 are used, and since almost all mental labor is saved, it would, probably, 

 in most calculations, afford a real economy of labor. But the great 

 number of figures required would unquestionably make the use of this 

 system a tedious process. It would no doubt be a favorite with chil- 

 dren, since it has no tables of addition or multiplication ; for all of its 

 processes of addition are simple counting, since only the figure 1 is ever 

 added, and there is no mental multiplication at all. Mathematical 

 thought, therefore, is almost entirely dispensed with. This simplicity, 

 it is claimed, gives the system a great merit on the score of accuracy. 1 



A system of notation with sixteen as a radix has also been proposed. 

 It was invented by a well-known civil engineer, who gave to it the name 



1 The following illustration of a simple problem in multiplication will furnish to those 

 who are curious in numerical matters an opportunity to compare the two systems : 



Decimal. Binary. 



87 = 1010111 



29 = 11101 



783 1010111 



174 1010111 



1010111 

 1010111 



2523 = 100111011011 



