(a) Convert (0.12)3 to a base 6 fraction. (b) Convert

Chapter 1, Problem 1.27

(choose chapter or problem)

(a) Convert \((0.12)_3\) to a base 6 fraction.

(b) Convert \((0.375)_{10}\) to a base 8 fraction.

(c) Let \(N = (0.a_{-1}a_{-2} \cdots a_{-m})_R\) be an any base R fraction with at most m nonzero digits. Determine the necessary and sufficient conditions for N to be representable as a base S fraction with a finite number of nonzero digits; say \(N=\left(0 . b_{-1} b_{-2} \cdots b_{-n}\right)_{S}\). (Hint: Part (a) gives an example. Note that

\(\left(a_{-1} R^{-1}+a_{-2} R^{-2}+\cdots a_{-m} R^{-m}\right) S^{n}\)

must be an integer.)

(d) Generalize part (a) to determine necessary and sufficient conditions for a specific, but not every, base R fraction, \(N=\left(0 . a_{-1} a_{-2} \cdots a_{-m}\right)_{R}\), to be representable as a base S fraction with a finite number of nonzero digits.