Base 3, digits -2, 0, 2 would seem silly, representing only the even integers. But it is momentarily startling, and then trivially obvious that the pure fractions in this system completely cover [-1,1]. Digits {-1,0,1} only cover [-1/2,1/2]: In[306]:= Table[FromDigits[{{d1, d2, d3}, 0}, 3], {d1, -1, 1}, {d2, -1, 1}, {d3, -1, 1}] // Flatten Out[306]= {-(13/27), -(4/9), -(11/27), -(10/27), -(1/3), -(8/27), -(7/27), -(2/9), -(5/27), -(4/27), -(1/9), -(2/27), -(1/27), 0, 1/27, 2/27, 1/9, 4/27, 5/27, 2/9, 7/27, 8/27, 1/3, 10/27, 11/27, 4/9, 13/27} (stupid parentheses) But with {-2,0,2} In[307]:= Table[FromDigits[{{d1, d2, d3}, 0}, 3], {d1, -2, 2, 2}, {d2, -2, 2, 2}, {d3, -2, 2, 2}] // Flatten Out[307]= {-(26/27), -(8/9), -(22/27), -(20/27), -(2/3), -(16/27), -(14/27), -(4/9), -(10/27), -(8/27), -(2/9), -(4/27), -(2/27), 0, 2/27, 4/27, 2/9, 8/27, 10/27, 4/9, 14/27, 16/27, 2/3, 20/27, 22/27, 8/9, 26/27} You may object that there are no odd numerators. But .2222... is obviously 1, .2➋2➋2➋... = ½, and .20➋020➋020➋0... = ⅗, etc. And then 3 = 2.2222... = 2➋.➋➋➋... etc, and {-2,0,2} (more than) covers all the reals!—rwg