The Road to Reality: A Complete Guide to the Laws of the Universe

by Roger Penrose

For this, we need to find the right definition of ‘b raised to the power z’, for complex z, and we want the same formula as the above, namely bw+z = bw × bz, to hold when the exponents w and z are complex. In fact, the procedure for doing this mirrors, to some extent, the very history of generalizing, step by step, from the positive integers to the complex numbers, as was done, starting from Pythagoras, via the work of Eudoxos, through Brahmagupta, until the time of Cardano and Bombelli (and later), as was indicated in §3.4, §4.1. First, the notion of ‘bz’ is initially understood, when z is a positive integer, as simply b × b × ··· × b, with z b’s multiplied together; in particular, b1 = b. Then (following the lead of Brahmagupta) we allow z to be zero, realizing that to preserve bw+z = bw × bz we need to define b0 = 1. Next we allow z to be negative, and realize, for the same reason, that for the case z = – 1 we must define b−1 to be the reciprocal of b (i.e. 1/b), and that b−n, for a natural number n, must be the nth power of b−1. We then try to generalize to the situations when z is a fraction, starting with the case z = 1/n, where n is a positive integer. Repeated application of bw × bz = bw+z leads us to conclude that (bz)n = bzn; thus, putting z = 1/n, we derive the fact that b1/n is an nth root of b.