THE DECIMAL SYSTEM OF NUMBERS 491 



involved, but the symbols are given by combination of the symbols for 

 1 to 9, with the symbol for zero. As our development will show the 

 symbol for nothing was the great stumbling block in the development 

 of a scientific method of writing the numerals. A place system to the 

 base five would require only the addition of a symbol for zero to the 

 symbols for 1, 2, 3 and 4. Leibnitz occupied himself with the binary 

 system, as this required only two characters, one for unity and one for 

 non-entity. To illustrate a binary place system the numbers from 1 

 to 16 are written, using only 1 and 0. 



Three written as 11, means one, two and one unit. Nine written as 

 1001 represents one cube of two, no squares of two, no first powers of 



1 



1 



i .1 



2 



10 





3 



11 



i .01 : 



4 



100 





5 



101 



i .01 



6 



110 





7 



111 



i .001 



8 



1,000 





9 



1,001 





10 



1,010 



i .001 



11 



1,011 





12 



1,100 



Multiply 8X9 



13 



1,101 



1,000 



14 



1,110 



1,111 



1,001 



15 



1,001,000 anne 



16 



10,000 





two, and one unit. The construction of the arithmetic universe out of 

 the single unit afforded Leibnitz some philosophical satisfaction in con- 

 nection with his system of monads. All the operations of ordinary 

 arithmetic are possible in this system. We catch a glimpse of our 

 slight comprehension of the infinite totality of numbers in noting that 

 any number that can be expressed with our ordinary ten digits can also 

 be expressed with these two digits, and that even though we used a 

 thousand digits we could add no new numbers. Doubtless it would 

 afford Leibnitz some gratification to know that his binary system is 

 used in modern mathematical analysis in certain delicate proofs. The 

 study of these number systems is not wholly foreign to the history of 

 the decimal system, as traces of the binary and quinary systems appear 

 among primitive peoples. 



Among the South Australian tribes the binary system of numera- 

 tion is almost universal. This is undoubtedly due to the fact that the 

 hands and feet and eyes and ears occur in groups of two in each 

 normal individual. These tribes are not advanced enough to have a 

 system of symbols; such a development would imply a degree of intel- 

 ligence which would proceed to a higher and more convenient number 

 base. The system is seen in their words ; three is given as two and one, 



