FRIZELL: FOUNDATIONS OF ARITHMETIC. 401 



groups with reference to both, we are led inevitably to 

 the transfinite types in case we impose no restriction 

 on the multiplication table and, if we require a mod- 

 ulus, to the natural numbers, whence the system of 

 real numbers results from the attempt to build groups. 

 The two rules of combination connected by the distrib- 

 utive property may be regarded as defining arithmetic 

 up to date ; it has not yet been found profitable to pos- 

 tulate in any other way. Then transfinite arithmetic 

 is distinguished by postulating the semigroups and 

 finite arithmetic by the postulate of the modulus. 



81. The real numbers form the most general finite 

 system with a single unit and must enter into every 

 system with more than one principal unit, one unit 

 must always be the modulus. Systems with two or 

 more principal units have been studied exhaustively by 

 Weierstrass. Here the abelian group on the lower rule 

 is postulated universally. 



With two principal units, if we exclude the modulus 

 of the lower rule, the necessary and sufficient condition 

 of an abelian semigroup on the higher rule is the exist- 

 ence in the system of a symbol i such that ii = u. 

 Thus the common complex numbers form the most 

 general system with two principal units satisfying the 

 postulates for real members. 



82. Within the system of common algebra are dis- 

 tinguished different number bodies each built on a root 

 of a given algebraic equation as a unit. The algebraic 

 numbers of a given body form an abelian group on ad- 

 dition and, excluding zero, an abelian group on multi- 

 plication, and must contain a given symbol, the root of 

 the given algebraic equation. For example the system 

 of common complex numbers is the number body which 

 contains a root of the quadratic x^ + 1 = o , 



