THE NUMBER-SYSTEM OF ALGEBRA

The theoretical part of this little book is an elementary exposition of the nature of the number concept, of the positive integer, and of the four artificial forms of number which, with the positive integer, constitute the \number-system" of algebra, viz. the negative, the fraction, the irrational, and the imaginary. The discussion of the artificial numbers follows, in general, the same lines as my pamphlet: On the Forms of Number arising in Common Algebra, but it is much more exhaustive and thorough-going. The point of view is the one first suggested by Peacock and Gregory, and accepted by mathematicians generally since the discovery of quaternions and the Ausdehnungslehre of Grassmann, that algebra is completely defined formally by the laws of combination to which its fundamental operations are subject; that, speaking generally, these laws alone define the operations, and the operations the various artificial numbers, as their formal or symbolic results. This doctrine was fully developed for the negative, the fraction, and the imaginary by Hankel, in his Complexe Zahlensystemen, in 1867, and made complete by Cantor's beautiful theory of the irrational in 1871, but it has not as yet received adequate treatment in English. Read more