
Radix economy
01/28/2022 at 01:37 • 0 commentsA subject that I didn't touch on in the main writeup is that a ternary system has the best radix economy of integral bases. The radix economy of a number is derived from the product of the base (number of symbols) by the number of digits to represent it. For nonintegral bases (if you can imagine such) the most economical value is e.
Informally the argument goes like this: If you have a small base, then there are less symbols to encode to but then the number becomes longer. If you have a large base, the number is shorter, but there are more symbols. So for example if you use binary to encode, the only symbols are 0 and 1, but the number becomes long quickly. If you use alphanumerics to encode, you can get very short encodings, which is taken advantage of by web links to videos, your parcel tracking IDs, and so forth.
I remember musing about this as a kid writing secret messages to a friend (haven't we all done that?): if I invent more symbols then words will be shorter to write but then I have to memorise more symbols. It was only later that I discovered this was in fact a mathematical concept.