CHAPTER XXII.
The Theory of Indices.
209. Hitherto all the definitions and rules with regard to indices have been based upon the supposition that they were positive integers ; for instance, (1) a^14 = a . a . a . to fourteen factors. (2) a^14 a^3 = a^{14+3} = a^17. (3) a^14 \div a^3 = a^{14-3} = a^11. (4) (a^14)^3 - a^{14 3} = a^42.
The object of this chapter is twofold; first, to give general proofs which shall establish the laws of combination in the case of positive integral indices ; secondly, to explain how, in strict accordance with these laws, intelligible meanings may be given to symbols whose indices are fractional, zero, or negative.
We shall begin by proving, directly from the definition of a positive integral index, three important propositions.
210. Definition. When m is a positive integer, a^m stands for the product of m factors each equal to a.
211. Prop. I. To prove that a^n x a^n = a^{m +n } when m and n are positive integers.
By definition, a^m = a . a . a . to m factors ; a^n = a . a . a . to n factors ;
a^m a^n = (a . a . a . to m factors) x (a . a . a . to n factors) = a . a . a . to m + n factors a^{m + n }, by definition.
