sibley (01/29/83)
Ken Chan has asked what Conway's definition of numbers (from "On Numbers and Games") has to say about 1 = .999... Nothing new, unfortunately. The real numbers Conway constructs are the "same" as the usual ones, i.e., they have the same algebraic properties. Also, the same analytic properties, since these can be deduced from the algebraic ones. Of course, his construction is quite different, but this doesn't mean the result is different and in fact it's not. However, he does construct more than the usual real numbers. In particular, he gets the so-called hyper-reals, in which there are, for instance, "numbers" which are positive but smaller than any ordinary real number, called infinitesimals. Needless to say, infinitesimals are not themselves ordinary reals. We can subtract an infinitesimal from 1 and get something "infinitely close to 1" but still different from 1. However, such a number is still not an ordinary real, and so is not .999... Hyper-reals are not original with Conway. They have been known for several years. He does have a nice construction, though. There are a few textbooks which treat calculus via hyper-reals (called non-standard analysis), but none has been very successful. The advantages are that one doesn't need any limits, dx really is "a little piece of x" (an infinitesimal piece, in fact), integrals are really sums, derivatives are really quotients, etc. Unfortunately, the rigors of dealing with hyper-reals seem to be beyond all but the bightest students. And of course, the calculations are all the same. The only difference from the usual approach is in the point of view -- how one thinks about what one is calculating. Dave Sibley Department of Mathematics Penn State University psuvax!sibley