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I'm Tab Atkins Jr, and I wear many hats. I work for Google on the Chrome browser as a Web Standards Hacker. I'm also a member of the CSS Working Group, and am either a member or contributor to several other working groups in the W3C. You can contact me here.
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We Should Be Using Base 6 Instead

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Occasionally you might come across someone who believes that it would be better for us to count in a base other than 10. Usually people recommend base-12 ("dozenal"); compsci people sometimes recommend base-2 (binary) or base-16 (hexadecimal). My personal opinion is that all of these have significant downsides, not worth trading out base-10 for, but that there is a substantially better base we should be using: base 6.

Let's explore why.

(Warning, this post is long and definitely not edited well enough. Strap in.)

Bases Are Arbitrary

First of all, there's nothing special about base-10. Powers of 10 look nice and round to us because we use base-10, but we can use any other base and get just as round numbers. Base 6 has 106, 1006, etc. (Those are 3610, 21610, etc; on the other hand, 1010 and 10010 are 146 and 2446. Converting between bases will usually produce awkward numbers no matter which base you start with.)

Why do we use base-10, then? The obvious answer is that we have 10 fingers. Counting off each finger gives us one "unit" of 10 things, and that unit-size carried over until we invented positional notation, where it froze into the base-10 we know today.

If we invented positional notation earlier, tho, then our hands could have supplied a better base - each hand can count off the values 0, 1, 2, 3, 4, and 5, which are exactly the digits of base-6. Two hands, then, lets you track two base-6 digits, counting up to 556, which is 3510!

Bases Are Significant

On the other hand, there are important qualities that do differ between bases.

The most obvious is the tradeoff of length vs mathematical complexity. Binary has trivial math - the addition and multiplication tables have only four entries each! - but it produces very long numbers - 10010 is 11001002, 7 digits long! On the other hand, using something like, say, base-60 would produce pretty short numbers - 1,000,00010 is only four digits long in base-60 ([4, 37, 46, 40]), but its multiplication table has 3600 entries in it.

When evaluating the tradeoffs of long representations vs complex mental math, it's important to understand a little bit about how the brain actually works for math. In particular, we have a certain level of inherent ability in various domains - short-term memory, computation, etc. Overshooting that ability level is bad - it makes us slower to do mental math, and might require us to drop down to tool usage instead (writing the problem out on paper). But undershooting it is just as bad - our brain can't arbitrarily reallocate "processor cycles" like a computer can, so when we undershoot we're just wasting some of our brain's ability (and, due to the tradeoffs, forcing something else to get harder).

So, we know from experience that binary is bad on these tradeoffs - base-2 arithmetic is drastically undershooting our arithmetic abilities, while binary length quickly exceeds our short-term memory. Similarly, we know that base-60 (used by the Babylonians, way back when) is bad - it drastically overshoots our arithmetic abilities while not significantly reducing the length of numbers, at least in the domain of values we care about (in other words, less than a thousand or so). So there's a happy medium somewhere in the middle here, and conveniently the geometric mean of 2 and 60 is base-11. Give it a healthy ±5 range, and we'll estimate that the "ideal" base is probably somewhere between base-6 and base-16.

But arithmetic complexity is actually more subtle than that. The addition tables, while technically scaling in size with the square of the base, scale in difficulty roughly linearly, since each row or column is just the digits in order, but starting from a different offset. It takes some memorization to recall how each offset works, but fundamentally the difficulty scales up slowly and simply, and you can do simple mental tricks to make addition easier anyway. (Such as adding 8+7, and adding/subtracting 2 from each to make it 10+5, a much simpler addition problem.)

Multiplication is more complex, tho. Some rows are easier to remember and use, others are more difficult:

  • "easy" rows are either trivial (0 and 1) or are factors of the base (2 and 5 for base-10), so they only cycle thru a subset of the digits in ascending order - less to memorize! Easy rows end up being pretty trivial to do mental math with; you can really easily multiply or divide in your head by these numbers.
  • "medium" rows are either smallish numbers that share all their factors with the base but aren't divisors (like 4 in base-10) because they also use only a subset of the digits but cycle thru them in a more complicated manner; or are the last row (9 in base-10) because of the nice pattern that makes its complexity easier to handle; or are just small numbers in general (like 3 in base-10), because even tho they cycle thru all the digits they do so in ascending series that are easier to memorize. Medium rows tend to be harder in mental math; you often need to resort to paper-and-pencil, but they're at least easy to do at that point.
  • "hard" rows are the rest - larger numbers that have some (or all) of their factors different from the base (6, 7, and 8 in base-10), so they cycle thru all the digits in a complicated manner, or just thru a subset in a complex manner + you have to track the 10s digit more carefully. Hard rows are just plain hard to compute with, even when you pull out paper-and-pencil. Rows that are coprime to the base, like 7 in base-10, are maximally difficult.

So multiplication difficulty varies in a complicated manner between bases, and doesn't scale monotonically. Base-60, for example, while looking tremendously bad for arithmetic at a naive glance, has significant mitigating factors here - because 60 factors into 22*3*5, a *lot of the rows in the multiplication table are "easy", particularly among the more "useful" small numbers. (It also has a lot of maximally-hard numbers, of course - all the prime rows 7 or higher except for 59, and 49 - and even more merely "hard" rows.) We'll examine this in more detail in a bit.

Divisibility difficulty is very similar:

  • "easy" divisibility are the trivial values (1 and 10 in base-10), and values whose factors are a subset of the base's (2 and 5 in base-10, as 10 factors into 2*5). You only have to look at the final digit of a number to tell if it's evenly divisible by one of these values, and memorize which values correspond to divisibility and which don't. (0-2-4-6-8 for 2, 0-5 for 5.)
  • "medium" divisibility are the values who factor into the same primes as the base, but which use at most one more of a given prime than the base does. That is, since 10 factors into 25, 4 (2*2), 25 (5*5), 20 (2*2*5), and 50 (2*5*5) all have either two 2s or two 5s. These only require you to look at the last *two digits of a number. (While 100 is 22*5*5, it's also just a power of the base, which clicks it over into "trivial" territory again.) Also "medium" is, again, the last value less than the base (9 in base-10) because you can always tell divisiblity by just adding up the digits and seeing if *that value is divisible by your last row value. (Yes, this works in any base - you can tell if a hexadecimal value is divisible by F (15) by adding together the digits and seeing if the result is still divisible by F.) If this final row value is composite, any numbers whose factors are a subset are also medium, because the same trick applies: 9 is 3*3, so in base-10, you can indeed test for 3-divisibility by adding the digits and seeing if the result is divisible by 3.
  • "hard" divisibility is all the rest. The rows either exceed the base's factor usage by two or more (like 8 in base-10), and thus require looking at the last three or more digits, or they use a factor that's not in the base at all (like 6 in base-10) and so require you to look at the whole number in a more complicated way. And again, rows which are fully coprime to the base (like 7 in base-10) are maximally hard, with no easy tricks or bounded recognition possible; you just have to do the division and see if there's any remainder.

So What's Actually Best?

So, based purely on a naive length-vs-arithmetic-difficulty analysis, we've already concluded that the "ideal" base is likely between base-6 (heximal) and base-16 (hexadecimal). Now let's narrow that list down based on the more complex factors, above!

First off, we can cross off any odd base right off the bat. They lack easy mult/div by 2 (it becomes "medium" difficulty instead), which is a supremely important number to multiply and divide by. I don't think any other qualities could possibly make up for this loss even in theory, but as it turns out none of the odd numbers in that range are particularly useful anyway, so there's not even a question. They're gone.

So we're left with 6, 8, 10, 12, 14, and 16. Let's scratch off another easy one: 14 sucks. Its factors are 2 and 7, and 7 is the least useful small number. 14 has bad mult/div with all the other small numbers above 2. So it's gone too.

8 and 16 we can cover together, because they're both powers of 2. This makes them easy to use in computing, as you can just group binary digits together to form octal or hex digits, but it limits their usefulness in mental arithmetic - since 2 is their only factor, you don't get as many useful combinations of values to make mult/div easier. Plus, the "trick" that makes mult/div easier with the largest digit value is, in these cases, applying to 7 and 15, which are again not particularly useful values. So, while these have some mitigating factors with computing, they're not really contenders. Gone.

So we're down to 6, 10, and 12. I'll break these down more specifically, because they're all starting to get useful and we need more details.

Base-10 has 10 rows in its multiplication table. 0, 1, 2, and 5 are all "easy" - the patterns are trivial or at least very simple. 3, 4, and 9 are "medium" - the patterns are more complex, but not too hard to memorize and use intuitively. But 6, 7, and 8 are all "hard" - the patterns are hard to use, and the tens digit varies enough that it's an additional burden to memorization. (And 7 is "maximally hard".) So 40% easy, 30% medium, 30% hard.

Base-6 has 6 rows. 0, 1, 2, and 3 are all "easy", because 0 and 1 are trivial, and 2 and 3 divide 6 and thus are simple repeating patterns (2-4-0, 3-0). 4 and 5 are "medium"; 4 for the same reason as base-10, but moreso (pattern is just 4-2-0, or 4-12-20, a simple counting-down-by-evens pattern), and 5 is the last digit so has the same quality as 9 does in base-10 (5-4-3-2-1-0, or 5-14-23-32-41-50). It's got 66% easy, 33% medium, and no hard rows at all! On top of this, the whole times table is 1/3 the size, at only 36 entries vs 100; if you throw away the truly trivial x0 and x1 rows and columns, then it's a mere 1/4 the size, with 16 vs 64 entries! That's small enough to be simply memorizable regardless of patterns.

Now base-12, with 12 rows. 0, 1, 2, 3, 4, and 6 are all "easy", because 12 has lots of useful factors. 8, 9, and 11 are "medium", but 5, 7, and 10 are "hard" (and 5 and 7 are both "maximal"). This is a better distribution than base-10 (50% easy, 25% medium, 25% hard), but it's larger in general (12x12, so 144 entries vs 100) which makes it harder to memorize, so it's probably roughly equivalent to base-10 overall. That said, easy multiplication/division by 3, 4, and 6 is probably worth more in the real world than mult/div by 5, so I'm sympathetic to the claims of base-12 lovers.

And just to finish this out, let's examine base-16, which is real bad because its factors are less useful. 0, 1, 2, 4, and 8 are easy, 3, C, and F are medium, but 5, 6, 7, 9, A, B, D, and E are all hard, for a 31% easy, 19% medium, 50% hard distribution. That's not only substantially worse than base-10, it's also so much bigger (256 entries) that its overall difficulty is also higher. (And to make it worse, more than half of the hard rows (5, 7, 9, B, and D) are maximally hard, as they're coprime to 16! That's so much worse!) Base-16 is useful as a more convenient way to read/write binary, but it's horrible as an actual base to do arithmetic in.

Length of Numbers, and Digit "Breakpoints"

As mentioned earlier, binary is a bad base for humans, because it produces very long representations. Humans have a "difficulty floor" for dealing with individual digits, so having a long number full of very-simple digits doesn't actually trade off properly; you still end up paying a larger "complexity price" per digit, times a long representation, for a very complex result.

In base-10, numbers up to a hundred use 2 digits, and numbers up to a thousand use 3 digits. Base-6 is fairly close to this: 2 digits gets you to 36, 3 to 216, and 4 to 1296. Since we don't generally work with numbers larger than 1000 in base-10 (after that we switch to grouping into thousands/millions/etc, so we're still working with 3 or less digits), you get the same range from base-6 by using, at most, 4 digits. That's only gaining one digit; combine that with the vastly simpler mental math, and you're at worst hitting an equal complexity budget to base-10.

But there's more. You see, the 100/1000 breakpoints aren't chosen because they're particularly useful, they're just where base-10 happens to graduate up to the next digit. We use higher-level groupings rather than 10000 (in many languages, at least; traditional Chinese numbering groups by 10000) because 10000 is simply too large to usefully deal with. That is, we just can't think about 10000 things very well.

But we can't really think about things up to 1000 well, either. Even 100 is a pretty big chunk of stuff, larger than we traditionally like working with. Left to our own devices, we seem to like things maxing out at approximately 30-ish - that's how many days are in a month, and how many students are traditionally in a large class (at least in America...). Guess what's approximately 30-ish? That's right, the 2-digit breakpoint for base-6, 36!

The 3-digit breakpoint for base-6 is 216, which is also a pretty reasonable number. It's about twice as large as 100, so any time 100 would be reasonable, 216 is probably also reasonable.

So, altho you need four base-6 digits to reach 100010, I don't think that's particularly a useful goal to hit. 10006 being 21610 is sufficiently useful that it's worth still batching our numbers into 3-digit groups, like today. Benford's Law tells us that, even tho 216 is only 20% of 1000, it will generally cover a far higher percentage of numbers in actual usage; in order words, most of the time we'll write our number with three or less major digits anyway, and won't even miss the lost range!

As an added bonus, dividing things into groups of 3 is actually natural in base-6, unlike in base-10!

In Conclusion

So, base 6 has more useful divisors, making it easy to divide by many small numbers. It's got a smaller (and thus easier to memorize/use) addition table, and a multiplication table that's not only substantially smaller than base-10, but substantially easier in very significant ways, making mental arithmetic much simpler. We can cover a similar range of numbers with just three digits, so it even looks similar to base-10 when the numbers get large enough to need scientific notation.

If you ever find a time machine, let me know so I can fix this. ^_^

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