METHODS OF SCIENCE; NUMBERS 27 



extend indefinite!}^ to the right. The second row 

 represents the infinite sum % -f- % -]- "!^2 + 

 . . . , continued according to the obvious rule of 

 omitting every other term of the row above. The 

 tliird row represents the series % -}" ^^ + %4 



■ 1 • • • 



09 



!=! 

 O 



> f-H r* ^ •, 



W O* W cr^ H 



11111... =.11111 



10 10 1...= .10101 



10 10...= .01010 



Now by a sort of shorthand this tabulation 

 may be replaced by the symbols to the right 

 which suggest our old friends the repeating 

 fractions. In fact they are repeating fractions, 

 except that, owing to the fact that there are ten 

 digits on our hands, we are familiar with the 

 decimal system of notation, while here we are 

 using a dual system. Instead of decimal frac- 

 tions these are dual fractions. 



Finally, when we consider not merely such 

 repeating fractions, but every conceivable com- 



