Third Base
People count by tens and machines count by twos—that pretty much sums up the way we do arithmetic on this planet. But there are countless other ways to count. Here I want to offer three cheers for base 3, the ternary system. The numerals in this sequence—beginning 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101—are not as widely known or widely used as their decimal and binary cousins, but they have charms all their own. They are the Goldilocks choice among numbering systems: When base 2 is too small and base 10 is too big, base 3 is just right.
...The cultural preference for base 10 and the engineering advantages of base 2 have nothing to do with any intrinsic properties of the decimal and binary numbering systems. Base 3, on the other hand, does have a genuine mathematical distinction in its favor. By one plausible measure, it is the most efficient of all integer bases; it offers the most economical way of representing numbers.
Perhaps the prettiest number system of all
“Perhaps the prettiest number system of all,” writes Donald E. Knuth in The Art of Computer Programming, “is the balanced ternary notation.” As in ordinary ternary numbers, the digits of a balanced ternary numeral are coefficients of powers of 3, but instead of coming from the set {0, 1, 2}, the digits are –1, 0 and 1. They are “balanced” because they are arranged symmetrically about zero. For notational convenience the negative digits are usually written with a vinculum, or overbar, instead of a prefixed minus sign, but here the vinculum is shown as an overstrike, thus: 1.
...What makes balanced ternary so pretty? It is a notation in which everything seems easy. Positive and negative numbers are united in one system, without the bother of separate sign bits. Arithmetic is nearly as simple as it is with binary numbers; in particular, the multiplication table is trivial. Addition and subtraction are essentially the same operation: Just negate one number and then add. Negation itself is also effortless: Change every 1 into a 1, and vice versa. Rounding is mere truncation: Setting the least-significant trits to 0 automatically rounds to the closest power of 3.