The Search for the Best Base Part 3 – Evaluating the Bases

In part 1 of this article, we introduced the problem of finding the best base to write our numbers in and figured out what the differences between these bases are. In part 2, we narrowed our search to eleven candidates for the best base, and figured out what metrics and criteria we would use to compare them. If you haven’t read those two parts, you should, or else you will likely not understand this one. In part 3, we will look at each of these candidates one-by-one, and figure out which is the best. Let’s begin!

Note: this is a very long and detailed article. If you don’t want every detail, feel free to just skip to the verdict.

Binary (Base Two)

Binary is the smallest possible base, which makes it very simple. A summary of its properties:

For writing binary, I will be using the system proposed in “the best way to count,” where ‘.’ is zero, ‘l’ is one, and ‘,’ is the digital point. Unlike that video, I will use a second ‘,’ to mark infinite repetition, for consistency and simplicity. If one of these numbers is written directly before a period or comma, I will omit that punctuation.
the best way to count

Statistics [A#]

Patterns [2#]

Multiplication Table:

┌───┬──────┐
│ × │ 0  1 │
├───┼──────┤
│ 0 │ 0  0 │
│ 1 │ 0  1 │
└───┴──────┘

Reciprocals up to A#20:

Divisibility/Remainder Tests up to A#20:

Other Patterns:

The Advantages of Small Size

The first noticable, and perhaps most important, property of binary is its small size, and this creates many advantages. Most of these stem from the fact that every base contains all of its powers. For example, the number l..l..l.l..l can be seen in many different ways:

This flexibility means that binary exists at every size simultaneously, and can pick and choose the size depending on what method you want to use. For example, in addition and subtraction, larger bases mean fewer digit operations but a larger addition/subtraction table to memorize. You can choose where in this dilemma you want to go, and because each of the addition/subtraction tables is a sub-table of the next, you can learn addition and subtraction gradually.

In addition, small bases maximize the presence of the digits 0 and 1 – in binary, these are the only digits. This matters because one or both of these digits is easier than the others to use in many common algorithms. For example, binary multiplication can be done without a multiplication table, by just copying the second number for every 1 digit in the first number. An example of this (llll.l × ll..l. = l.ll lll. l.l.; decimal: 61×50=3050):

       111101
     × 110010
     --------
      111101
   111101
+ 111101
-------------
 101111101010

Likewise, long division is easier because there are only two possibilities for what to subtract - either 0 or 1 times the divisor. Here is the long division inverse to the above mulitplication:

                11 1101
       ┌────────────────
110010 │ 1011 1110 1010
         -110 010
         --------------
          101 1010 1010
          -11 0010
          -------------
           10 1000 1010
           -1 1001 0
           ------------
              1111 1010
             -1100 10
             ----------
                11 0010
               -11 0010
               --------
                      0

While both of these still involve a lot of writing, the point is that the multiplication did not need any table memorization or single-digit multiplications, while the long division did not need any guessing.

The Mental Math Problem

However, all of this comes with one caveat, which I didn’t mention in part 1. According to both scientific consensus (e.g. the widely-cited physchology article “The Magical Number Seven, Plus Or Minus Two”) and my own personal experience, our short-term memory can only store aronud 7 digits, regardless of what base we use. This means that, in binary, you can only use numbers up to around A#63 in short-term memory, and less if you want to have more than one. You can solve this by moving to a power of 2 base, like octal or tessal, but then you lose all the advantages of being small!

Given that mental math and calculators (which aren’t really affected by what base we write & read the numbers in; they use binary internally regardless) are the most common ways people do math, this is a major concern for binary. It doesn’t eliminate it from the competition by any means – octal and tessal are perfectly good bases, and better than most of the bases in that range, however this means that binary loses its greatest advantage. When competing with the bases of similar size to octal or tessal, binary has no size advantage, so it must compete on factors. In the factor game, octal and tessal are not the best; for example, sezimal is smaller than octal but has the same number of factors and more regulars, while dozenal is in between them but has more factors and regulars than both.

I have a hypothesis for why this is the case: it’s because our minds aren’t just for doing math. The mental spaces that store digits in our short-term memory don’t just store digits – they also store letters, words and ideas. Therefore, we need to use “digit slots” with much more capacity than would be necessary to store numbers. If each digit has extra information added to it, then larger bases are more efficient, as they use more of the available information for numbers. This would explain why computers, which don’t have this problem, already almost universally use binary internally.

The Significance of the Factor 2

Binary’s other main advantage is that it makes multiplying and dividing by powers of 2 trivial – just move the decimal place left or right. 2 is probably the most important number to multiply or divide by, and this is no coincidence, as multiplication by 2 is the simplest form of multiplication that changes the number. Some examples of this in math:

Many of these have alternatives with other numbers – for example Pascal’s Pyramid where each layer sums to a power of 3 – but the versions based on 2 are much more well-known for a good reason.

However, binary’s advantage of dealing with powers of two comes at the cost of making every other number difficult to deal with.

Other People’s Opinions

Because power of 2 bases are so similar to binary, I will be including opinions about those as well.
the best way to count
Base 8 - the Best Number System!

Balanced Ternary (Base Three)

Balanced Ternary is an interesting base with three unconventional digits: zero, one and negative one. A summary of its properties:

I will be using a slight modification of the system I used for binary to write balanced ternary: ‘.’ is 0, ‘l’ is 1, ‘i’ is -1, and ‘,’ is the digital point. If I use 0 and 1 as normal (like in monospace), then I will use ‘z’ for -1, as I explained in part 1.

Statistics [A#]

Patterns [3#]

Multiplication Table:

┌───┬─────────┐
│ × │ z  0  1 │
├───┼─────────┤
│ z │ 1  0  z │
│ 0 │ 0  0  0 │
│ 1 │ z  0  1 │
└───┴─────────┘

Reciprocals up to A#20:

Divisibility/Remainder Tests up to A#20:

Other Patterns:

The Advantages of Balanced Notation

Balanced notation brings many advantages, many due to the inherent symmetry of having a mixture of positive and negative digits.

For addition and subtraction:

For multiplication and division, you can use similar methods as in binary, since i behaves so much like l; when you’re multiplying and encounter the digit i – just negate and add. That’s the only difference between binary and balanced ternary multiplication, and balanced ternary numbers have much fewer digits than binary.

In addition, rounding becomes simple truncation. When rounding to the nearest multiple of l… you just replace the last 3 trits with … without needing any special rules: liiil, lil.., lill. and liiii all become li…; in regular base three these nunmbers look like 1121, 2100, 2110 and 1112, and it isn’t immediately obvious that all of these round to 2000.

Finally, a small advantage of balanced bases is that they make it easier to make change with cash: the positive digits represent what you pay, and the negative digits represent the change you get back. In a hypothetical balanced ternary money system where every power of 3 gets its own bill, if I want to pay liiil dollars, I would just give and l…. and l bills, and expect l…, l.. and l. bills back as change.

The Significance of the Factor 3

The fact that the simplest balanced base is base three, not base two, suggests that the factor 3 also has some universal significance like 2 has. For example, every real number is the sum or difference of distinct powers of 3.

There are also plenty of human uses of the factor 3: for example, our timekeeping system. We have A#24 hours per day, A#60 minutes per hour and A#60 seconds per minute. When the metric system was initially created, its creators wanted to switch to a decimal system of A#10 hours per day, A#100 minutes per hour and A#100 seconds per minute. This system makes interconversion much easier: in this decimal system, 4 hours is A#400 minutes or A#40000 seconds, while in our current system, 4 hours is A#240 minutes or A#14400 seconds. Imagine how much easier it would be to add something like “3 hours A#54 minutes + 2 hours A#12 minutes”, or to multiply “5 × 1 hour A#14 minutes” in a system like this! Despite this, this system never caught on. Aside from the additional effort required to convert, our traditional system has one advantage: all of the units are divisible by 3. A third of a day is 8 hours, and a third of an hour is A#20 minutes. In this new system, a third of a day would be A#3..3 hours, and a third of an hour would be A#33..3 minutes. The fact that we didn’t switch to metric time should suggest that divisibility by 3 is important, and in a base divisible by 3 like ternary, every multiple of “10” is divisible by 3.

Some more mathematical uses is that numbers have three signs (positive, negative, zero) and that there are three comparison operators (<, =, >). There is a reason for this: if you want to divide a continuous spectrum into discrete groups, then any division into two won’t be symmetrical. Whatever point you divide will have to be included in one side but not the other. If you want to treat both directions equally, you must divide the spectrum into three.

Asymmetric division into halves:
|-----[-----|
|-----]-----|

Symmetric division into thirds:
|---[---]---|
|---]---[---|

However, if you want the factor 3 to be treated perfectly like it is in ternary, then you can’t have any other easy factors. The factor 2 is more important than 3, so binary’s factors are better than ternary’s. If only we could have both…

Sezimal (Base Six)

Sezimal is the first base that has both 2 and 3 as factors. While neither is dealt with as perfectly as binary handles 2 or ternary handles 3, sezimal allows both of them to have the advantage of being factors simultaneously (such as being able to divide by them with mental math), something neither binary nor ternary does.

While sezimal is smaller than decimal, it doesn’t get nearly as much of an advantage from its size as binary or ternary - honestly, it might not even be an advantage for sezimal. Sezimal’s only advantage over them is its superior factors and regulars.

Statistics [A#]

Patterns [6#]

Multiplication Table:

┌───┬───────────────────┐
│ × │  0  1  2  3  4  5 │
├───┼───────────────────┤
│ 0 │  0  0  0  0  0  0 │
│ 1 │  0  1  2  3  4  5 │
│ 2 │  0  2  4 10 12 14 │
│ 3 │  0  3 10 13 20 23 │
│ 4 │  0  4 12 20 24 32 │
│ 5 │  0  5 14 23 32 41 │
└───┴───────────────────┘

Reciprocals up to A#20:

Divisibility/Remainder Tests up to A#20:

Other Patterns:

Hexanif Multiplication

One of sezimal’s downsides is that numbers have more digits than decimal. The first number with three digits in sezimal is A#36, the first four-digit number is A#216, and the first six-digit number is A#7776. This means that basic operations like addition and multiplication take more steps for the same number - but in exchange, the tables you’ll have to learn are tiny - around a third the size of the decimal tables you’re used to.

If you’re willing to put in more effort memorizing in exchange for fewer operations, you might think that using powers of the base as in binary or ternary would help. However, the next power available is base A#36, which is too big - its tables are almost A#13 times the size of decimal’s! Sezimal does give you the choice between a small base or a big base, but neither is great.

However, I have an idea for a solution that may alleviate this, for multiplication. Instead of learning the times tables up to 6×6 or A#36×A#36, learn the tables up to 6×A#36. That way, you can multiply two digits in one of the multiplicands by one digit in the other, halving the amount of multiplications you need to do. This requires less multiplications than decimal or dozenal, for a table only around twice as big as decimal’s. This doesn’t solve addition, but it’s a great start.

Other People’s Opinions

seximal.net
We Should Be Using Base 6 Instead
Shack’s Base Six Dialectic
Sezimal Number System
Symmetric senary (for balanced base 6)

Dozenal (Base Twelve)

Dozenal is like sezimal in many ways, but it has an extra factor of 2 in its prime factorization, making it better at working with powers of 2 and worse at working with powers of 3. Given that 2 is more important than 3 - and if the regular complexity is trustworthy sezimal is better at working with 3 than 2 - this is a good tradeoff.

Dozenal has a similar size to decimal. It’s slightly larger, but also has more factor patterns, so it’s around the same difficulty to learn. Out of all bases with a similar size to decimal, it has by far the most factors – twelve has six factors, while every other number less than sixteen has at most four.

Statistics [A#]

Patterns [C#]

Multiplication Table:

┌───┬───────────────────┬───────────────────┐
│ × │  0  1  2  3  4  5 │  6  7  8  9  A  B │
├───┼───────────────────┼───────────────────┤
│ 0 │  0  0  0  0  0  0 │  0  0  0  0  0  0 │
│ 1 │  0  1  2  3  4  5 │  6  7  8  9  A  B │
│ 2 │  0  2  4  6  8  A │ 10 12 14 16 18 1A │
│ 3 │  0  3  6  9 10 13 │ 16 19 20 23 26 29 │
│ 4 │  0  4  8 10 14 18 │ 20 24 28 30 34 38 │
│ 5 │  0  5  A 13 18 21 │ 26 2B 34 39 42 47 │
├───┼───────────────────┼───────────────────┤
│ 6 │  0  6 10 16 20 26 │ 30 36 40 46 50 56 │
│ 7 │  0  7 12 19 24 2B │ 36 41 48 53 5A 65 │
│ 8 │  0  8 14 20 28 34 │ 40 48 54 60 68 74 │
│ 9 │  0  9 16 23 30 39 │ 46 53 60 69 76 83 │
│ A │  0  A 18 26 34 42 │ 50 5A 68 76 84 92 │
│ B │  0  B 1A 29 38 47 │ 56 65 74 83 92 A1 │
└───┴───────────────────┴───────────────────┘

Reciprocals up to A#20:

Divisibility/Remainder up to A#20:

Other Patterns:

Complementary Multiplication [C#]

In any base b, if i × j = b^n, then you can divide by i by multiplying by j then moving the digital point n places left. This is probably the only form of division that can be done though mental math.

All bases with more than one prime factor benefit from this trick, but one of its best use cases is in dealing with powers of 2 in dozenal. For example, to divide by 2^4 = 14, you can simply multiply by 9 then move the digital point 2 places left. For example, 200 ÷ 14 = 2×9 = 16. If the number doesn’t end in two zeroes but is divisible by A#14, you can easily divide it by remembering the first nine multiples of 14 and which multiple they are:

  n │  0  1  2  3  4  5  6  7  8
14n │ 00 14 28 40 54 68 80 94 A8

For example:

768 ÷ 14
= 700 ÷ 14 + 68 ÷ 14
= 7 × 9 + 5
= 58

One use case of this trick is divisibility-testing by large powers of 2. If you want to test if a number is divisible by 2^n, you can divide it by A#14 then test for 2^(n-4). Remember that when you’re testing for divisibility by 2^n, you can disregard all but the last ⌈n/2⌉ digits. Using the above example, because 58 is divisible by 4, 768 is divisible by 4×14=54.

Other People’s Opinions

Dozenal has entire societies based around it:
Dozenal Society of America
Dozenal Society of Great Britain

Other opinions I found:
Base 12 - Numberphile
Is there a better way to count…? 12s anyone?
Dozenalism: Adventures in Numbers, Measurement, and Math
Dozenal Wiki: Why dozenal is the best base
Great Bases Are Good For Dividing, How About Multiplying? Comparing Bases by Studying How They Represent Factorials
Base 10 VS Base 12 and Oranges
Janus Numbers (balanced dozenal; uses a completely different set of digits)

Note on Statistic Simplification

For all of the rest of the bases in this list, I will be simplifying the statistics section by simply stating which class each number from 2-A#20 is (regular, semitotative or totative), instead of doing a full multiplication table, all the reciprocals and divisibility tests. For semitotatives, I will mention how they can be determined as a product of a regular and totative (the regular will always be first and the totative will always be second).

A reminder on how these classes work:

Balanced Triquinary (Base Fifteen)

Base fifteen is not that great of a system, to be honest. It’s odd, meaning that determining whether or not a number is even requires a digit sum, and that 1/2 is infinitely repeating (1/2=F#0..7=F#1..t).

The only benefit this system has over balanced ternary – the other odd base I looked at – is divisibility by 5. By the same reasoning as with 3, I don’t think 5 is an important factor. In our time system, the times :12, :24, :36 and :48 are fifths of the hour or minute, yet we don’t see them any more frequently than times like :08, :16 or :22 that don’t correspond to any simple fraction. Therefore, I don’t think this system provides much benefit.

Statistics [A#]

Patterns [F#]

Numbers up to A#20 (F#15):

Other Patterns:

Balanced Quadrasezimal (Base Twenty-Four)

Base twenty-four is like dozenal, but adds a third factor of 2. This base starts to have clear diminishing returns. Even though the only benefit that can be gained from adding this factor 2 is making powers of 2 easier to work with, out of the first four only one (8) works better in quadrasezimal than dozenal. Plus, quadrasezimal is big enough that you’re forced to use balanced notation to avoid ridiculous amounts of memorization. I think most people would be better off using either dozenal or binary than this base.

Statistics [A#]

Patterns [O#]

Numbers up to A#20 (O#K):

Other Patterns:

Sezigesimal (Base Sixty)

Sezigesimal is the first base where we’re going to need to use a mixed base, so that 6A#59 + 1 = 6A#100. This means that when doing basic operations like addition and multiplication, you’ll need to use different tables for different digits - you’ll need to know the tables for sezimal in addition to the decimal tables you probably already know. It has the benefit that the base is divisible by 2, 3, 4, 5 and 6, which means that 1/2 through 1/6 are all single-digit pair terminating (0.30, 0.20, 0.15, 0.12 and 0.10 respectively)! Only bases divisible by sixty have this property.

The only major advantage this base has over sezimal and dozenal is divisibility by 5. For the same reason as with triquinary, I don’t think that’s worth the cost.

Statistics [A#]

Patterns [6A#]

Numbers up to A#20:

Other Patterns:

Dozagesimal (Base One Hundred Twenty)

Dozagesimal is an immense base, and the largest unbalanced base that passes the conditions set in part 2. It has an immense sixteen factors, and reduces the totatives to a little more than a quarter of all numbers. This comes at the immense cost of large size, and mixed arithmetic. Like with sezigesimal, you’ll need to learn two different sets of tables for all the basic operations, and use different tables for different digits. Unlike with sezigesimal, neither of these tables are small, so learning dozagesimal involves substantially more effort than decimal.

The only major advantage this base has over dozenal is divisibility by 5. For the same reason as with triquinary, I don’t think that’s worth the cost.

Statistics [A#]

Patterns [CA#]

Numbers up to A#20:

Other Patterns:

Largest Bases (A#180, A#240, A#360)

The largest bases available are pretty similar to each other. They are large enough that you’ll need to use mixed and balanced bases together, or maybe mixed bases with three digits. They also don’t really derive much benefit from this – they don’t have any extra prime factors compared to sezigesimal or dozagesimal, and A#180 is missing some of A#120’s factors! Every bad thing I’ve said about sezigesimal and dozagesimal is more true for these three.

All three of these bases have the same regulars as sezigesimal and dozagesimal, so I won’t repeat the patterns section for each.

Statistics – Dozatwinnal (A#180) [A#]

Statistics – Twintessal (A#240) [A#]

Statistics – Sezisezigesimal (A#360) [A#]

Honourable Mentions

Finally, I’d like to look at a few more bases that aren’t highly composite, but still deserved mention. I haven’t got any special rule for choosing these - they’re just bases I thought deserved an honourary mention.

Decimal (Base Ten)

Decimal is the base almost everyone uses for everything. Although I’m trying to look for an alternative to it, it is still a fairly good base, mostly owing to its great size.

Statistics:

Classes of numbers up to 20:

Other Patterns:

Trisezimal (Base Eighteen)

Trisezimal is an interesting base that like sezimal and dozenal, but backwards. Instead of being divisible by 2^2 and 3^1 (like dozenal), it is divisible by 2^1 and 3^2.

Statistics [A#]:

Classes of numbers up to A#20 (I#12):

Other Patterns:

Quadraquinary (Base Twenty)

Quadraquinary is a base that has been used by many civilizations, primarily as a mixed base with subbases 4 and 5. If you can handle the added complexity of the mixed base, it might be better than decimal, though sezimal, dozenal or quadrasezimal would still be better.

Statistics [A#]:

Classes of numbers up to A#20 (K#10):

Other Patterns:

Pentasezimal (Base Thirty)

Pentasezimal is the first base divisible by 2, 3 and 5, giving it a large number of regulars for its size. The cost of this is that each of these primes has a high regular complexity, making pentasezimal a sort of jack of all trades - OK at three primes, good at none. For example, there are A#225 possible endings for numbers divisible by 4!

Statistics [A#]:

Classes of numbers up to A#20 (K):

The Verdict [A#]

We’ve finally investigated everything I could think of about fifteen different number systems. In fact, if you count all powers as variants of the same base, and exclude prime numbers above 3 (which are all terrible bases), we’ve closely looked at every base up to 20 except base 14. It’s time to figure out which of these is the best.

A Tier List

First, let’s sort the fifteen bases we looked at into a tier list, so we can decide from the top tier.

I think I’ve made it clear throughout this article that I think large bases aren’t that great. Large bases add lots of extra complexity for little gain. The comparison between bases 12 and 24 makes this obvious:

Bases 60 and 120 at least grant another distinct prime factor, 5, but I’ve already argued that this isn’t too useful. Our decimal base already priviliges 5 to the second-best status a number can have, and we still prioritize 3 in things like our time units. I’m putting them at a similar tier to decimal, since the extra complexity of needing a mixed base is counterbalanced by the utility of the factor 3.

As for the smaller bases, here is my rationale for what tiers they should be put in:

With this in mind, my final tiers are (numbers within a tier are in no particular order):

Tier Bases
Excellent 2, 12
Great 3, 6, 24
Good 10, 18, 20, 60, 120
Not Recommended 15, 30, 180, 240, 360

The Best Base

The question of which base is the best now comes down to a competition between binary and dozenal. Binary allows numbers to be viewed at many scales, makes every digit trivial, and trivializes multiplying, dividing or taking the remainder by powers of 2. Dozenal makes powers of 2 and 3 and their products significantly easier to work with, without losing most of its advantages in mental math or monospace text.

I believe that dozenal wins out overall. The unconditional advantage dozenal has with multiples of 3 in my opinion beats the mostly conditional advantages binary has. The only unconditional advantage binary has is trivializing working with powers of 2, but dozenal’s complementary multiplication still handles them excellently – within the bases we looked at it is second only to the power of 2 bases and maybe base 24.

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