*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 0 plays 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 It is evident that thought in arithmetical work is almost superseded, and that all numerical operations are reduced to the manual labor of writing.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 expenditure 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 children, 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

- ↑ 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