The Search for the Best Base Part 2 – Base Evaluation Criteria
In Part 1, we looked through the problem of finding the best base for our number system. If you haven’t read that yet, you should, otherwise none of this will make sense to you.
Narrowing our Search [A#]
There are an infinite number of bases – we can use any integer 2 or higher as a base – so in order to determine which base is the best we’ll first need to narrow our search to a few which we can look at one-by-one.
Highly Composite Numbers [A#]
In Part 1, we found that the two most important factors in making a base ‘good’ or ‘bad’ are its size (smaller is better) and factors (more are better). This is a dilemma – in order to have more factors, you need to make the base bigger. However, it can be used to narrow our search. Let’s assume that if base A has no fewer factors than base B, and A is smaller than B, A is better than B. This means that the only bases we need to look at are the highly composite numbers – those numbers that have more factors than any smaller number.
The first few highly composite numbers in decimal are: 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, …
We can also remove any base that is a perfect power of another. As shown in part 1, every base contains its powers (but not the other way around). Every binary number has a quaternary number in its digit pairs, and every sezimal number has a sezisezimal number within it too. Therefore, quaternary, sezisezimal, and other power bases should not be included in our search.
Imposing a Size Limit [A#]
However, this is still an infinite number of bases! However, adding new factors has diminishing returns, so we should set a maximum size for the bases we are looking at. For a variety of reasons, I think a limit of around 16 for non-mixed bases is reasonable:
- 16 is the largest non-mixed base that people regularly use (the base 20 and 60 used by many human civilizations have always been mixed bases). As part of this group (a computer scientist), I can say that most people who use base 16 don’t even use it as a fully-functional system, just as a compressed form of binary, so anything above 16 is unreasonable.
- The addition, subtraction and multiplication tables of base 16 are already over 2.5× as large as in our current system.
These limits narrow down our search to binary (base 2), sezimal (base 6) and dozenal (base 12).
Special Conditions for Mixed and Balanced Bases [A#]
Because mixed bases work differently, I will add some extra conditions for them:
- I will only look into mixed bases with two sub-bases. Having more than two is needlessly complicated, and isn’t needed to gain any important factors (two-subbase systems can already give you 2, 3 and 5; 7 is only used for one common thing).
- For mixed bases, because you need to learn the addition/subtraction/multiplication tables of both sub-bases, I will reduce the size limit for subbases to 12 (12²×2 ≈ 16²; 12’s many factors make it simpler than you’d expect for the size)
- Given that the only reason to have mixed bases is to get more factors, and given that regular bases can already have two prime factors without getting too big, I will only look at mixed bases divisible by 2, 3 and 5.
- Mixed bases will only be used when regular bases can’t be.
For balanced bases, I will also make some changes to the conditions to reflect their differences:
- Balanced bases at their best act as simple as bases around half their size. Therefore, I will double the size limits for balanced bases (32 for unmixed bases, 24 for mixed subbases).
- Because odd balanced bases are simpler than even ones, I will allow odd bases that have more factors than any smaller odd number.
With all of our conditions, we get the following bases:
- Non-mixed, regular and balanced: 2, 6, 12
- Non-mixed, balanced only: 3, 15, 24
- Mixed, regular and balanced: 60, 120
- Mixed, balanced only: 180, 240, 360
Also, because regular and balanced bases are somewhat interchangeable, and many of their aspects are the same for the same base, I will be looking at the regular and balanced versions of bases that support both simultaneously.
An Attempted Alternative to Highly Composite Numbers [A#]
One doubt you might have, which I also had, about limiting the search to highly composite numbers is that their definition seems to treat all factors equally, when they are not equally valuable. 12 and 245 have the same number of factors, even though 12’s factors (1, 2, 3, 4, 6, 12) are much more valuable than 245’s (1, 5, 7, 35, 49, 245). Shouldn’t they be treated differently?
The simplest way to address this problem is to add a value to factors, and sum up these values. A good idea for the factor value is that the value of the factor f should be 1/f, for multiple reasons:
- It’s very simple.
- All factors have a positive value, but smaller factors are more valuable than larger ones.
- The value of a factor is the probability that a random number will be divisible by it.
Then, we give each base a “factor score” by adding up the value of each of its factors. Here are a few bases’ scores, rounded to 3 decimal places:
| Base | Score |
|---|---|
| 2 | 1.500 |
| 3 | 1.333 |
| 6 | 2.000 |
| 10 | 1.800 |
| 12 | 2.333 |
| 14 | 1.714 |
| 15 | 1.600 |
| 18 | 2.167 |
| 20 | 2.100 |
| 24 | 2.500 |
However, when trying to calculate a base’s factor score, you’ll need to give the fractions a common denominator, turning 1/f into (b/f)/b. The numerator of this fraction, b/f, is also a factor of b. So, as it turns out, this factor score is exactly equal to the sum of the base’s factors divided by the base, also called the “abundancy index.” We’re looking for the numbers with a factor score greater than all smaller numbers, a sequence that already has a name, the superabundant numbers. As it turns out, until 7560 (which is highly composite but not superabundant), the highly composite and superabundant numbers are the exact same! In my attempt to improve the highly composite numbers, I ended up creating something that is for all the purposes that matter here the same!
Evaluation Criteria
Next, we need to decide how we’ll evaluate the bases. I’ll have my own thoughts, in which I talk about anything unique about a base, or just anything that doesn’t fit into this section, but I also need some objective things to measure.
Simple Statistics
First, for every base, we should get some statistics. Remembering that the things that matter are reducing size and increasing the number and utility of factors, I will include the following:
- The prime factorization, since it tells you a lot about the structure of a base.
- The base’s factors.
- The “factor score” explained above.
- The “regular score,” which is like the factor score but for all regular numbers. Even though every base has infinitely many regulars, this score will always be finite. By coincidence, the reciprocal of this score is the proportion of numbers that are totatives (e.g. if the regular score is 2.0, then 1/2 of all numbers are regular).
- The “regular complexity” I mentioned in part 1, for every prime factor of the base (except 1, which always has a complexity of 1), as it nicely approximates how easy it is to divide and divisibility-test by powers of these factors.
- The base-2 logarithm, which helps illustrate how big the base’s digits are (smaller is better). This is proportional to the size of each digit in the ideal writing system I talked about in part 1.
- The reciprocal of the base-ten logarithm. This shows how many digits numbers will have, compared to our familiar decimal system.
In addition, I’ll mention if the number is highly composite/superabundant or not. However, there’s another set of numbers, which I will also mention.
Given that we want our bases to have many factors and a small size, imagine that you want to combine these two indices into one. The simplest way to do this is to divide the number of factors by some positive power of the base, creating a single score that rewards low size and high factor count. Here are some sample graphs of this “size-adjusted factor count”:
When you do this, there will be a single base that beats all others, both larger and smaller (I’ve marked it on the graphs), however which base this is depends on the specific exponent you use. While these numbers have to be highly composite (otherwise a smaller base would have a better score), not all highly composite numbers will be record-breakers. Therefore, the numbers that are best at some size-adjusted factor count get their own name – the superior highly composite numbers. If you use factor scores instead of factor counts, they are called the colosally abundant numbers.
The first few superior highly composite and colossaly abundant numbers (the sequences overlap at the start again), in decimal, are: 2, 6, 12, 60, 120, 360, 2520, 5040, …
Because this is the closest there is to an objective metric for the best base, I will also mention if a number is superior highly composite / colossaly abundant, and which adjustment exponents make it the best.
Patterns
I will also show some of the base’s fundamental patterns, which show how its factors affect its use:
- The multiplication table (for mixed bases, the table of each sub-base)
- The values of 1/n for n from 2 to A#20. A#20 was chosen because it’s big enough that every base will have some challenging primes.
- Every divisibility/remainder test up to A#20. I may include more if they’re particularly good.
I will also show patterns that are followed by certain important number sequences:
- The powers of 2 and 3
- The prime numbers (Specifically, I’ll tell you the totative digits. Every totative ends in one of these digits, and every prime except the base’s prime factors is a totative)
- The square numbers
- The practical numbers (n is practical if every number from 1 to n can be expressed as a sum of factors of n; I’m including them because many of the “high-factor number” sequences, like the highly composite numbers, are subsets of this sequnce, so the pattern of this sequence will apply to all of them). Unless I specifically mention it, these patterns will not guarantee that a number is in these sequences, but not following the pattern will guarantee a number is not in them. The use of this patterns is to narrow down your search, and minimize the number of cases where you need a more complex rule. Therefore, I’ll also show what proportion of numbers follow these patterns (smaller is better).
Conclusion
We now have everything we’ll need to look at the bases. In part 3, I will look at each base individually, and then use all of these investigations to find which one is the best.