Number System of Algebra, The

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.

Excerpt:

We say of certain distinct things that they form a group when we make them collectively a single object of our attention.

The number of things in a group is that property of the group which remainsunchanged during every change in the group which does not destroy the separatenessof the things from one another or their common separateness from all other things.

Such changes may be changes in the characteristics of the things or in their arrangement within the group. Again, changes of arrangement may be changes either inthe order of the things or in the manner in which they are associated with one anotherin smaller groups.

We may therefore say:

The number of things in any group of distinct things is independent of the charac-ters of these things, of the order in which they may be arranged in the group, and ofthe manner in which they may be associated with one another in smaller groups.

The number of things in any two groups of distinctthings is the same, when for each thing in the first group there is one in the second,and reciprocally, for each thing in the second group, one in the first.

Thus, the number of letters in the two groups, A, B, C; D, E, F, is the same. Inthe second group there is a letter which may be assigned to each of the letters in thefirst: as D to A, E to B, F to C; and reciprocally, a letter in the first which may beassigned to each in the second: as A to D, B to E, C to F.

Two groups thus related are said to be in one-to-one (1-1) correspondence.

Underlying the statement just made is the assumption that if the two groupscorrespond in the manner described for one order of the things in each, they willcorrespond if the things be taken in any other order also; thus, in the example given,that if E instead of D be assigned to A, there will again be a letter in the group D,E, F, viz. D or F, for each of the remaining letters B and C, and reciprocally.