It has long been established that there is no algebra which includes complex algebra in the way in which complex algebra includes real algebra wherein the operations of addition and multiplication continue to obey their familiar laws. Complex algebra, then, cannot be ex tended, and owing to the fundamental theorem of algebra it need not be extended, so that it is the algebra par excellence for the mathema tician.
For a further discussion of the real and complex number-systems see the articles on REAL VARIABLE., THEORY OF THE, and COMPLEX VARIABLE, THEORY OF THE, respectively.
Exponents.- In the last paragraph we have defined xn in the case where n is an integer. This obeys all the laws laid down in the para graph on involution. These laws require that if any use of fractional exponents is to conform n to them, x must equal It may be demonstrated that so long as we confine selves to positive values of the roots of positive numbers, if x R be defined as the nth root of xm, all the laws of exponents will he obeyed. It can likewise be shown that the definition of as 1 will yield consistent results.
These arc accordingly the definitions of frac tional exponents actually used. When other than positive roots of positive quantities are considered, the matter is not quite so simple. If n is not rational, xn is defined as the limit of xm where m is a variable that approaches a through rational values.
The Method of Postulates.- The method whereby we have derived algebra from the entities and truths of logic is not the only one that has been applied in establishing this subject on a logical foundation. The method of postu
lates which has been employed so effectively by Veblen and Huntington in many branches of mathematics has been used by Huntington for the study of algebra. This method consists in reducing the theorems of algebra to the consequences of a few simple formal laws such as the laws of associativity and of com mutativity for addition and multiplication and the law of the distributivity of multiplication with respect to addition and others of the same sort. When this is done, algebra is considered not as the study of a set of specifically numerical entities but as the study of any entities you please that obey its postulates.
Rational, real, and complex algebra each receive a complete and self-contained treatment ac cording to this method. See POSTULATES, THEORY OF.
Bibliography.- The most thorough work on this subject is the (Principia Mathematica' of A. N. Whitehead and B. A. W. Russell (Cam bridge 1910-13). Whitehead's 'Introduction to Mathematics' (in the Home University Library series) is the best popular book. Consult also Huntington, E. V., (A Complete Set of Postu lates for the Theory of Absolute Continuous Magnitude' (Trans. Am. Mat. Soc., .1902), 'Complete sets of Postulates for the Theory of the Real Numbers) (ibid. 1904), and (The Continuum> (Cambridge, Mass., 1917).