chalkdust
Mathematics for the three-fingered mathematician Robert J Low
W
’ all familiar with using a couple of different bases to represent integers. Base ten for almost all purposes when we do our own calculations, and base two, or binary, for geing computers to do them for us. But there’s nothing special about ten and two. We could equally well use any integer, b, greater than two, so that the string of digits dn dn−1 dn−1 . . . d0 , where each di is positive and less than b, represents the integer n ∑
di bi .
i=0
Some bases are slightly more convenient than others for doing arithmetic. Bases eight and sixteen are both used in various computer applications, and there is an active society, the dozenal society, devoted to using and promoting the arithmetical advantages of base twelve. Much less common, but far more interesting, is base three. With base three, the digits are all 0, 1 or 2. But I want to look at a variation on this. Instead of using 1 and 2, I’ll use 1 and −1; but it’s not convenient to have minus signs in the middle of our numbers, so because of this and for reasons of symmetry I’ll represent them with 1 (for 1) and (for −1). Base three is ternary, and this variation of it is called balanced ternary. 1
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autumn 2017
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