410 KANSAS UNIVERSITY SCIENCE BULLETIN. 



99. The arrangement of the natural numbers in 

 series of type higher than w finds application in the study 

 of infinite continued fractions. By aid of the euclidean 

 algorithm for greatest common divisor every absolute 

 irrational number less than unity can be expressed as 

 an infinite continued fraction 



_i i_ 1 



and conversely. The class of infinite simple continued 

 fractions may therefore be taken as the representative 

 of the class of irrational numbers between zero and 1. 

 An infinite continued fraction can not be obtained 

 from a finite one merely by annexing quotients; it can 

 only be described by assigning a law which determines gN 

 for every value of N. An infinite continued fraction 

 may be formed, e. g., by the law that every quotient 

 shall be 2. It is sufficient, however, to consider the class 

 in which the quotients are all different; this can be put 

 into one to one correspondence with the whole class. 

 Accordingly we are concerned with the class of all 

 possible permutations of all the natural numbers. 

 These permutations may be examined in the same way 

 as the permutations of a finite set by imagining a frame- 

 work of places to be filled, but the number of places is 

 infinite. Moreover, we must provide for the possibility 

 of filling the places in a series of order higher than w. 

 Thus an infinite continued fraction can be formed by 

 filling the even places successively with the odd num- 

 bers in their natural order and the odd numbered places 

 with even numbers in the same way, i. e., 

 J^ _L JL J_ JL J^ 



2+1+4+3+6+5+ 



or by filling the odd numbered places with primes and 

 the even places with composite numbers. Or we can 

 select first the places whose indices are primes, then 

 the indices which are products of two primes, three 



