- As we've discussed elsewhere, a number system utilizing place notation requires a number of digits equal to the base itself. Because human beings are creatures of limited memory and limited ability to deal with large numbers of different symbols, we should select our base with a mind to ensuring that we have a manageable number of symbols; specifically, we should select a base small enough that we will not be overwhelmed by the number of necessary digits. Anything higher than 16 or so is probably higher than desirable; we'd essentially be learning a new alphabet in order to use it. Anything higher than 20 is almost certainly too high.
- The size of the base will determine the size of the numbers we will be dealing with; that is, a small base will tend to produce much longer numbers than a large one. This is because each time we count through the base or a multiple thereof, we must add another digit to the number; in smaller bases, this number gets larger more quickly. As an example, the number "100" in dozenal is "10010000" (8 digits) in binary; "1000" is "11011000000" (E digits); "10000" is "101000100000000" (13 digits). Numbers very quickly become quite unwieldy, even at scales that are quite common in normal, daily use. So we want to select our base not only to be small enough that we have a manageable number of symbols, but large enough that our numbers are reasonably sized for common magnitudes.
By these two considerations more or less speak for themselves; for purposes of this brief article, I will suggest that the smallest reasonable base size is 6, and the highest is 16.
n \ radix | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 1 | infinity | 1 | infinity | 1 | infinity | 1 | infinity | 1 | infinity | 1 | infinity | 1 |
3 | 1 | infinity | infinity | 1 | infinity | infinity | 1 | infinity | infinity | 1 | infinity | infinity | 1 |
4 | 2 | infinity | 1 | infinity | 2 | infinity | 1 | infinity | 2 | infinity | 1 | infinity | 2 |
5 | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity |
6 | 1 | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity | infinity | 1 |
7 | infinity | 1 | infinity | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity |
8 | 3 | infinity | 1 | infinity | 3 | infinity | 2 | infinity | 3 | infinity | 1 | infinity | 3 |
9 | 2 | infinity | infinity | 1 | infinity | infinity | 2 | infinity | infinity | 2 | infinity | infinity | 1 |
X | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
E | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
10 | 2 | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity | infinity | 2 |
11 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity | infinity |
12 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity | infinity |
13 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | 1 | infinity | infinity | infinity |
14 | 4 | infinity | 2 | infinity | 4 | infinity | 2 | infinity | 4 | infinity | 1 | infinity | 4 |
15 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | 1 | infinity |
16 | 2 | infinity | infinity | infinity | infinity | infinity | 2 | infinity | infinity | infinity | infinity | infinity | 1 |
17 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
18 | infinity | infinity | infinity | infinity | 2 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
19 | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
1X | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
1E | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity | infinity |
20 | 3 | infinity | infinity | infinity | infinity | infinity | 2 | infinity | infinity | infinity | infinity | infinity | 3 |
number of digits \ radix (k digits = (10/k) points) | 6 | 7 | 8 | 9 | X | E | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 (10 points) | 3 | 1 | 3 | 2 | 3 | 1 | 5 | 1 | 3 | 3 | 4 | 1 | 5 |
2 (6 points) | 4 | 0 | 1 | 0 | 2 | 0 | 5 | 0 | 1 | 1 | 0 | 0 | 2 |
3 (4 points) | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 2 |
4 (3 points) | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
infinity (0 points) | 11 | 1X | 17 | 19 | 14 | 1X | 11 | 1X | 15 | 17 | 17 | 1X | 11 |
number of points | 5E | 10 | 36 | 20 | 47 | 10 | 76 | 10 | 41 | 36 | 40 | 10 | 6E |
Dozenal (base 10) has the highest (76) points, thus dozenal is the best base (note that dozenal (base 10) is exactly the median base of these bases (and thus the “most reasonable” bases), and the base with second-highest points (16) and the base with third-highest points (6) are the highest and the smallest of these bases, thus the two “most unreasonable” bases of them, so dozenal is “very perfect”!!! For more information, see the page 10 (dozen) and the links in the bottom of this page)
Also, dozenal has its special properties, e.g.
10 is the largest base such that both "all squares end with square digits" and "all primes not dividing the base end with either prime digits not dividing the base or 1" are true.
10 is the largest base such that all these three properties are true:
Property of the numerical system | Property of the base (b) | Bases (Note: 1 cannot be the base of the system) |
All squares end with square digits | Numbers n such that all x^2 mod n are squares | 1, 2, 3, 4, 5, 8, 10, 14 |
All primes not dividing the base end with either prime digits not dividing the base or 1 (unit) | Numbers n such that reduced residue system of n consists of only primes and 1 | 1, 2, 3, 4, 6, 8, 10, 16, 20, 26 |
All squares of the primes not dividing the base end with 1 | Numbers n with the property m^2 == 1 (mod n) for all integer m coprime to n | 1, 2, 3, 4, 6, 8, 10, 20 (exactly the divisors of 20) |
10 is the largest base such that both "all squares end with square digits" and "all primes end with prime digits or 1" are true.
for more special properties of dozenal, see Properties of dozenal.