A DEFINITION OF QUATERNIONS BY INDE- 

 PENDENT POSTULATES^ 



By Miss Ruby L. Carstens 



§ I. Quaternions with respect to a Domain Z)^ 



The usual theory relates to quaternions a^+aj + aj+a^k in which 

 the coefficients a^ range independently over all real numbers or else over 

 all complex numbers, and the units have the following multiphcation 

 table: 



These give the real quaternion system and the octonian system. ^ 

 As an obvious generalization, the coefficients may range independently 

 over all the elements of any domain D. 



§ 2. The Postulates 

 A set of four ordered elements a = [a^, a^, a^, a J of D will be caUed 

 a quaternion a. The symbol a = [a^, a^, a^, aj employed is purely posi- 

 tional, without functional connotation. Its definition imphes that a = b 

 if, and only if, a^=b„ a2 = h^, aj = b^, a^ = b^. 



We make the following five postulates concerning quaternions in 

 connection with the elements o, +1, — i (7^/*) which are used 4^ 

 times as follows : 



yiii— yi22=yi33=yi44~"y2i2— y234=y3i3=y342=y4T4~'y428— + 1 > 

 y22i=y243=y324— y33i=y432~y44i~ ^ *> 

 the other forty-eight 7's are zero. 



' Read before the American Mathematical Society, February 24, 1906. 



» Dickson, "On Hypercomplex Number Systems," Transactions of the American Mathematical Society, 

 Vol. VI, 1905. 



3 Octonians may be considered as quaternions vrith complex coefificients. 



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