Scalable number names
Part of my units pages
A problem with current unit prefices is that they only increase in fixed multiples (of 1000), e.g. 1Gm = 1000Mm = 1000000km = 1000000000m. Such prefices act in a similar way to unary (or perhaps Roman numerals). This doesn’t scale, requiring the rapid invention of new names.
This problem is likely inherited from the naming system used for large numbers, where “million” is a thousand thousands, “billion” is a thousand millions, “trillion” is a thousand billions, etc. Just like place-value numerals make more efficient use of digits than unary, we can make more efficient use of names by introducing them logarithmically. In base ten we invent a new name for ten tens: the “hundred”. With this new name we can count up to 99,9,9 (“ninety nine hundred and ninety nine”), but our next name (the “thousand”) appears after only 9,9,9; far too early! If the thousand were a hundred hundreds, we could then count up to 9999,99,9,9 (“ninety nine hundred and ninety nine thousand, ninety nine hundred and ninety nine”) before needing a new name (e.g. the “million”); the million currently appears after 99,99,9,9. The next name (“billion”) currently appears after 9,9999,99,9,9, but we only need it once we reach 99999999,9999,99,9,9 (a million millions, using our redefinition of million); the trillion would only be needed once we reach a billion billions, and so on. Note that each name doubles the number of digits we can reach, rather than merely adding three (or six). Such names have a clear relation to binary (powers of two), and also remind me of factoradic numbers. If this were to be adopted, it would be a much better idea to invent new names rather than trying to redefine the current terms!