**Base 960:**

SDN hexoctanunqual or hexoctnilial; encoding: plain decimal (numbers from the range 0 to 959)

The upper limit of general-purpose bases is most probably 120, the Long Hundred, and this is so because its returns are worth its size - fortunate neighbour-relationships that its half-sized, quarter-sized and double-sized relatives all lack - and because of the ability to encode it as two human-scaled bases: twelve and ten in alternation. The price of handling two bases in one is probably already too high for most people, so it's safe to say there's no point in prospecting for arithmetic gold beyond the height of ten-dozen. Why consider base 960, then? Because, as a fraction-part auxiliary rather than a general-purpose base, we can go higher, and 960 has quite a few desirable properties for its use together with integer-part decimal.

Free of the considerations of multiplication size and coprime digits and others that beset the choice of a general-purpose base, a fraction-part auxiliary base is evaluated according to its precision, the recognisability of its fractional values, and the possible extra feature of its omega relationship. Base 960 is a clear winner in all three respects: its closeness to the third decimal power means you get nearly three decimal digits' worth of precision with just one figure, and nearly six with two figures; this closeness also contributes to the recognisability of base-960 fractions, which look familiar to one used to decimal fractions; and as a bonus, 960 like the Long Hundred has 7 as its omega inheritor (959 = 7Â·137), meaning you're close enough to one or more sevenths if you've got a multiple of 137, even a single instance of it, let alone repeating.

Here's a table of the fractions covered by base 960, also comparing them to their decimal representations:

The divergence between the base-960 and the decimal fractions increases as the fractional value approaches the maximum, but even the highest base-960 fractionals don't stray too far from the decimal appearance. Of all the fractions given, the unreduced ninths may be the hardest to use, because they need two places; given the lesser need for ninths, we may as with sevenths be content with gauging proximity to them rather than accuracy. For the most important fractions, as well as binary subdivisions, base 960 delivers the goods: the basic dozenal fractions are represented (though their appearance is like the one we see in decimal fractions, not a clockface correspondence as in the Long Hundred), and only after the sixth binary power is an additional place required.

Arithmetic with base 960, as with any fraction-part auxiliary, is plain decimal arithmetic with a different carry rule, in this case a carry that comes into force once you're past 959. The strength of a fraction-part auxiliary, however, lies in its notational advantage rather than arithmetic; for those who value dozenal for its terminating thirds (

*contra*decimal) but think a change of integer-part base, the adoption of new numerals ('X' and 'E' in dozenal or the Long Hundred) and alternating-radix arithmetic are all too high prices to pay for this feature, base 960 as a fraction-part auxiliary to be used together with integer-part decimal may be just what they need.

In closing, here are the most important irrational numbers using base-960 fractionals, rounded to two places: