4 NUMBER. [INT. I. 



The sequence 



does not fulfil the above condition, for 



1 1 1 p 



1+- 

 p 



which cannot be made as small as we please, for all values of p, no 

 matter how great n be taken. 



On the other hand, the sequence 



1 1 1 



1.2' ' 1.2 ' 1.2.3' 



1 1 



1 1.2 ' 1.2.3.' 1.2.3.4' 

 does satisfy the condition, for 



1 



' 



...... n+I 



1 , _ L_ _1 



7 "(w + 2) ...... (n+p)\ 



1.2.3 n(n+l)*' 



which is less than e as soon as n is taken greater than 1/e. 



This sequence defines the irrational known as e, the natural 

 logarithmic base, which is not the root of any algebraic equation 

 with rational coefficients. The class of irrationals is in fact much 

 larger than the class of algebraic irrationals, which led to their 

 inclusion in the number-system. 



3. Complex Numbers. The system composed of the rational 

 and irrational, forming together the real numbers, is still not suffi- 

 cient for the solution of algebraic equations. For consider the 

 simple equation 



x 2 + 1 = 0. 



