ON AN ALGEBRA IN THREE UNITS 



By S. Epsteen and Harry V. Welch 

 § I. Historical 



Suppose that one is familiar with the arithmetic of positive integers 

 only. In that case the equation ax=b {a and b, positive integers) 

 would, in general, not be solvable. In the domain of positive integers 

 and fractions this equation would be solvable, whereas the equation 

 a + x=b would, in general, not be. Both of these equations can be 

 solved if the domain is enlarged by the adjunction of negatives. In order 

 to solve the equation x*^=a (n, a, positive integers) the latter domain 

 must be further extended by the introduction of the irrational number. 

 Let us designate the final domain thus obtained by i?(i). The develop- 

 ments in algebra prior to the nineteenth century are characterized by 

 the fact that they were studies oj the domain R{i). The equation 

 x^ = — i is not solvable in this domain and its root, i = V — i, was said 

 to be imaginary, i.e., exterior to the domain R(i). Through the 

 publications of Argand and Gauss, at the close of the eighteenth cen- 

 tury and beginning of nineteenth, the properties of this new unit became 

 known to the mathematical world and for the next half-century mathe- 

 maticians studied the algebra of the units i, i, that is, the so-called com- 

 plex algebra. Therefore, we may say: during the first half of the nine- 

 teenth century the progress in algebra consisted in complete discussion of 

 the domain R{i, i).^ 



The fundamental theorem of these investigations is due to Gauss: 

 the equation aoX" + a^x*^-^+ . . . . a„=o (n positive integer, a,, a^, 

 . . . . a„ belonging to R(i,i) has at least one root belonging to R(i, i).' 



' Historically, the real number system including positive integers and fractions was known to the Egyp- 

 tians, over 1700 B. c. The irrational numbers were discovered by the Greeks as a corollary of the theorem 

 of Pythagoras. To the Hindoos belongs the credit of inventing the zero, about the fifth century a. d., in 

 consequence of which negative numbers were discovered. In 1806 Argand made pubUc a method of repre- 

 senting »= 1/ — I geometrically, thus completing the number system of algebra. 



' This is the so-called fundamental theorem of algebra. 



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