if you’re like most people, you’ve been using base ten to count things since before
you learned what numbers are. decimal is by far the most common numbering system, but
in many ways, it’s clearly not the best way to count things. the number ten was chosen
completely arbitrarily based on the number of fingers most people have, but when you
think about it, there are other numbers that would work much nicer as a base.
this is dozenal, a system that counts by twelves instead of by tens. all you need is two extra
digits for ten and eleven, and then you write twelve with <10>. the standard among mathematicians
for writing larger bases is to extend the Arabic numerals using the Latin alphabet,
so ten is written with the letter and eleven is written with the letter , but
actually doing it that way makes ten and eleven look like they’re too separate from the
rest of the digits, so you can use an inverted two <↊> for ten and an inverted three <↋> for
eleven, but those don’t display on most fonts so you can approximate them with the
letters and , which also happen to be the first letters of the English words
“ten” and “eleven”, but actually as long as we’re okay with using the Latin
alphabet to represent these digits then we might as well use for ten, like in Roman
numerals, but actually now we’re back to having them look too different from the other
ten digits so how about instead we use the Greek letters chi <χ> and epsilon <ε> but
ACTUALLY if we’re using Greek letters then there’s no association between the X looking
letter and the number ten so maybe you can write ten with the Greek letter delta <δ>
instead.
and all you really need to learn is those two new digits and you’re ready to use dozenal!
the first thing that’s great about dozenal is that it’s already common in some contexts,
like if you’re counting eggs or pastries or inches. because of this, the vocabulary
necessary already exists, and doesn’t need to be created: you just count “one, two,
three, four, five, six, seven, eight, nine, ten, eleven, dozen, dozen one, dozen two,”
and so on until “two dozen, three dozen, four dozen,” and so on. a dozen dozen is
a gross, and a dozen gross is a great gross. uh, okay, so like, you might run into a problem
where when you say “ten” people will assume you’re talking about the number twelve,
because that’s a problem that happens in real life I think, so we can call it “dec”
instead, and then we can call eleven “el”, which is a reference to Stranger Things. also,
I know that earlier I said that we can take advantage of already existing terminology,
but let’s call twelve “do” instead of twelve or a dozen. and then a gross is uh,
“gro”. and a great gross is... “mo”? yeah, that checks out.
now, I know what you’re thinking. if we’re not taking advantage of existing terminology,
what’s the point of using dozenal at all? well, there’s two things that make bases
fundamentally distinct from each other. the first is that the more digits a base has,
the fewer digits it requires to represent larger numbers. so, for example, if you look
at every number up to a gross, a bit under a third of them require fewer digits to represent
in dozenal than they do in decimal. in fact, the larger the numbers get, the more likely
it is that the dozenal representation will be shorter, and the probability eventually
reaches one hundred percent, so, for example, if you look at every number up to a million,
it goes up to sixteen percent did I say goes up I mean goes down okay, that doesn’t matter.
what DOES matter is the second thing that makes bases distinct from each other which
is that the number twelve has way more factors than the number ten. ten only has two prime
factors, two and five, whereas twelve has SIX factors: one, two, three, four, six, and
twelve! this makes dozenal far more convenient for division.
in any base, you can tell what factors any given integer has in common with the base
itself just by looking at the final digit. so, in decimal, the last digit of any integer
is enough information to tell if it’s divisible by two and if it’s divisible by five, which
as we’ve established are the two prime factors of ten. in dozenal, however, the last digit
is enough information to tell if an integer is divisible by any of the six factors of
twelve. check it out: if an integer ends with zero, it’s divisible by twelve, if it ends
with zero or six, it’s divisible by six, if it ends with zero, four, or eight, it’s
divisible by four, if it ends with zero, three, six, or nine, it’s divisible by three, if
it ends with zero, two, four, six, eight, or ten, it’s divisible by two, and if it
ends, it’s divisible by one.
divisibility tests are great, but what’s even easier to see is how the different bases
write simple rational numbers. one half is <.5> in decimal, because a half is five tenths.
three doesn’t go into ten, so it takes INFINITELY MANY DIGITS to represent a third. four also
doesn’t go into ten, so a fourth is <.25>. all of these are very impractical and make
decimal unnecessarily hard to use for writing simple ratios.
in dozenal, all of this is better. a half is <.6>, six twelfths, a third is <.4>, four
twelfths, and a fourth is <.3>, three twelfths. as anyone can clearly see, this is objectively
simpler than how decimal writes these ratios, and at this point I’m going to break character
and ask the question you should be asking right now.
“why did I stop at fourths?”
in decimal, a fifth is one of the simplest fractions to deal with. <.2>, because it’s
two tenths. however, not only does the number five not evenly go into the number twelve,
it’s not even CLOSE to going into twelve. a fifth is written as <.2497...> recurring.
that’s, mathematically speaking, the worst a base can possibly be at writing fifths.
if you’re the sort of person who wants to be able to quickly divide numbers by five,
dozenal suddenly stops looking as good. dozenal? pff. more like, DOESN’Tal!
but of course, the number five isn’t THAT important, is it? and besides, if you want
a base that can easily deal with halves, thirds, fourths, AND fifths, you’re gonna have to
use base SIXTY! and regardless, even if dozenal is bad with fifths, decimal is bad with thirds,
and there’s no objective reason to treat thirds as being less important than fifths.
allow me to introduce you to my actual favorite base. I’m jan Misali, and seximal is a better
way to count.
six is a very nice number. in fact, it’s what mathematicians call a “perfect” number,
which has nothing to do with what I’m talking about. as I said before when I was doing jokes,
the things that make a base distinct are its size and its factors. we’ll go over both
of these separately, but before we really get into the “why” I’m just gonna very
quickly go through the “how”.
okay, numbers up to twelve have the same names as they do in decimal. from there you count
“dozen one, dozen two, dozen three, dozen four, dozen five, thirsy”. multiples of
six work like multiples of ten, but they end with -sy instead of -ty: “six, twelve, thirsy,
foursy, fifsy”. six times six is “nif”, nif times nif is “unexian”, and that’s
all you need to know to understand this video.
part one: six is a small number
the more digits a base has, the fewer digits it requires to represent larger numbers. the
larger a number is, the more the size of the base matters. for any two bases, there is
inevitably some number n where absolutely all numbers larger than n require more digits
to represent in the smaller base than they do in the larger base. you can think of this
as being the point where the larger base “outpaces”, or, indeed, “outbases”, the other. now,
when you’re comparing two bases that are similarly sized, like decimal and dozenal,
this point is at an extremely large number. twelve to the thirteenth power, in this case.
this means that most numbers you deal with will be about the same length in both bases.
however, six is considerably smaller than ten. the point where decimal outpaces seximal
is at ten thousand. a big number, sure, but nowhere near as big as twelve to the thirteen.
based on this information, you might come to the conclusion that larger bases are inherently
better at dealing with large numbers, and I wouldn’t blame you for thinking that.
it makes a lot of sense. however, it breaks down when you think about it in the extreme
case.
pretend you’re Plato and you really like the number five thousand forty, or three unexian
fifsy two nif, if you prefer. the point is, you think that it’s a great number, and
you wanna start using it as a base. since it’s a really big base, it’s very efficient
for writing arbitrarily large integers. your Platonic base is able to write one million
with just two digits!
now, in order to actually use such a base, you’d need to have two dozen eleven gross
distinct digits. that’s a lot of digits. clearly, it would be extremely difficult to
actually learn how to use it. it’s intuitive that one digit in your system contains more
information than one digit in decimal, but you don’t know that because Arabic numerals
haven’t been invented yet. actually, there’s a lot of stuff you’re not going to know
about so you should stop pretending to be Plato.
what we need is a way to find a balance between large numbers being too long and individual
digits being too numerous. this balance is what’s called “radix economy”, and you
can measure how much a number “costs” in a given base by multiplying the number
of digits by the log of the base itself, which is the amount of entropy, or the cost, per
digit.
long story short, even though the Platonic base can write a million with two digits,
those digits actually measurably cost more than the seven digits it takes to write a
million in decimal. by this metric, instead of larger bases gradually outbasing smaller
bases for larger and larger integers, it’s actually the SMALLER bases that do better
and better for larger integers.
this all boils down to smaller bases being more efficient than larger bases in the long
run. it’s somewhat counter intuitive, I know, but the math all checks out.
there’s one other big advantage of six being such a small number, which is that it makes
doing simple arithmetic really, really easy. like, ridiculously easy.
the largest digit in seximal is five, right? so that basically means you can do any one
digit addition or subtraction problem by counting on your fingers. (more on that in a bit) multiplication
is a bit trickier, but only having six digits helps out even MORE here.
so, there’s six digits, which means there’s nif products of two one digit numbers.
about half of those are repeats, which leaves us with thirsy three products.
anything multiplied by zero is zero and anything multiplied by one is itself, so those are
all pretty trivial.
that leaves us with just ten one digit products that need to be memorized in order to multiply
in seximal. two times two is four, two times three is six, two times four is eight, three
times three is nine, two times five is ten, three times four is twelve, three times five
is dozen three, four times four is dozen four, four times five is thirsy two, and five times
five is foursy one. thanks to the nomenclature we’re using, only four of those sound unfamiliar.
so, yeah, addition, subtraction, and multiplication are all much, much easier in seximal. division
is easier too, but before we get to that,
part two: most people have ten fingers
when all is said and done, what people are really looking for in a numbering system is
a way to count things. finger counting is one of the only things decimal has going for
it. it makes some amount of sense. most people have ten fingers, so the number of fingers
you have extended can directly correspond to some number from zero to ten. you can count
like one, two, three, four, five, six, seven, eight, nine, ten, and then you’re out of
fingers.
isn’t this a bit wasteful, though? for all of these, either one hand has no fingers extended
or one hand has all five fingers extended. what if we took advantage of the fact that
we have two hands and instead counted like 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20,
21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54,
55? there’s a very nice correlation between the concept of “after you get to five you
need to use your other hand” and “after you get to five you need to add another digit”.
now, if you’re familiar with the dozenal system, you’ve probably seen a certain way
to count things on your hands in dozenal, where you count on the twelve segments of
your fingers instead of on your five fingers. yes, this can get you all the way to twelve
on one hand, but actually think about it for a second. think about the context where you’re
most likely to use finger counting, which is when you need to visually show someone
numerical information. in the normal finger counting method, the number four looks like
this and the number five looks like this. they’re nice and distinct, as are any two
pairs of manual digits. in the dozenal counting method, the number four looks like this and
the number five looks like this. can you honestly say that you would be able to consistently
tell these two handshapes apart from a distance? I know I wouldn’t be able to.
it’s also worth mentioning that if you want, you can treat each finger as a separate binary
digit, which gets you up to over a thousand on both hands. this, of course, runs into
problems if you’re someone like me and you can’t move your pinky independently of your
ring finger.
okay, so what was that thing I was saying about division earlier?
part three: six is an antiprime
the number twelve has more factors than any number less than twelve. this makes it a highly
composite number. the thing is, there’s another, even cooler subset of highly composite
numbers that have EVEN MORE factors. these are the superior highly composite numbers,
and not only do they have more factors than all the numbers smaller than themselves, they
ALSO have more factors than all the numbers LARGER than themselves, if you adjust for
magnitude somewhat. SHCNs are an elite group, and they don’t let just anyone in. they’re
in many ways the opposite of prime numbers. since they have so many dang factors, antiprime
bases are really good. among them are numbers like 60, used for timekeeping, and 360, used
for angle measure, and 5040, Plato’s favorite, and, yes, both six and twelve.
let’s step back for a moment and talk about divisibility tests. I don’t know about you,
but when I was in school I had to use divisibility tests for decimal all the time. looking at
the last digit of a number only tells you about what factors it has in common with ten,
but that’s not the only divisibility test that exists. you can generalize divisibility
tests in any given base into two infinite families.
the first type is the simpler one. if the last x digits of n in base b share a factor
with bˣ, then n also shares that factor with bˣ. so, in decimal, if the last digit is
even, the whole number is even, because two is a factor of ten, and if the last two digits
are 00, 25, 50, or 75, the whole number is divisible by twenty five, because twenty five
is a factor of ten squared.
the second type of divisibility test is a bit more complicated. if you take the digits
of n in base b and group them into substrings of length x and add all those substrings together,
the number n shares a factor with bˣ-1 if and only if that sum shares the same factor.
so, in decimal, if you add the digits of a number together, the result can tell you what
factors the original number has in common with ten minus one, which is nine. if you
do the same thing but instead of adding the digits you add PAIRS of digits, you can see
what factors the original number has in common with one less than ten squared, which is 99.
99 is divisible by nine, and we already have a divisibility test for that, so dividing
that out, we’re left with eleven, and would you look at that, eleven is exactly one more
than ten! this will happen in any base, because for any number n, n²-1=(n+1)(n-1). neat!
recap, in any given base, there’s easy divisibility tests for factors of the base, and doable
divisibility tests for factors of the numbers adjacent to the base. this doesn’t just
apply to divisibility tests. it’s also reflected in the representations of rational numbers.
in decimal, a half is written as point five, because it’s equal to five tenths.
in dozenal, a half is written as point six, because it’s equal to six twelfths.
in seximal, a half is written as point three, because it’s equal to three sixths.
three doesn’t evenly go into ten, but it does evenly go into nine. in decimal, a third
is written as point three recurring, because it’s equal to three ninths.
in dozenal, a third is written as point four, because it’s equal to four twelfths.
in seximal, a third is written as point two, because it’s equal to two sixths.
four doesn’t evenly go into ten, but that’s fine because a fourth is just half of a half.
in decimal, one half is point five, and half of five is two and a half, two point five,
so one fourth is point two five. in dozenal, a fourth is written as point three,
because it’s equal to three twelfths. four doesn’t evenly go into six, but that’s
fine because a fourth is just half of a half. in seximal, one half is point three, and half
of three is one and a half, one point three, so one fourth is point one three.
at this point, there’s a very clear ranking for which bases are better at rational numbers
than which other bases. dozenal is simplest, writing all three of these fractions with
just one digit, followed by seximal, which needs two digits to write fourths, and in
last place is decimal, which writes thirds with a recurring digit. this ranking becomes
less obvious if we take fifths into account.
in decimal, a fifth is written as point two, because it’s equal to two tenths.
five doesn’t evenly go into twelve, and it doesn’t evenly go into eleven. it also
doesn’t evenly go into eleven dozen eleven or eleven gross eleven dozen eleven. it DOES,
however, evenly go into eleven GREAT GROSS eleven gross eleven dozen eleven. in dozenal,
a fifth is point two four nine seven recurring, because it’s equal to two great gross four
gross nine dozen seven eleven great gross eleven gross eleven dozen elevenths.
five doesn’t evenly go into six either, but that’s because it’s exactly one less
than six. in seximal, one fifth is written as point one recurring, by definition.
suddenly, the question of which of these bases is best at writing fractions is far more subjective.
it’s now a question of whether fourths are more important than fifths. for me, personally,
I think that fifths are more important than fourths, and that it’s much easier to derive
fourths in seximal than it is to derive fifths in dozenal. I completely get where you’re
coming from if you disagree with that, but just so we’re on the same page here, you’re
wrong.
now, why stop at fifths? there’s still more to see.
six doesn’t evenly go into ten, but that’s fine because a sixth is just half of a third.
in decimal, a third is written as point three recurring, and half of three is one and a
half, one point five. adding all of those infinite fives to all but one of those infinite
ones, you get point one then six recurring. in dozenal, a sixth is written as point two,
because it’s equal to two twelfths. in seximal, a sixth is written as point one,
by definition.
we’re not going to stop here either, because of COURSE seximal is good at writing sixths.
no surprises there. this part, however, genuinely surprised me when I first found out about
it.
seven doesn’t evenly go into ten, nine, ninety nine, nine hundred ninety nine, nine
thousand nine hundred ninety nine, OR ninety nine thousand nine hundred ninety nine. it
DOES, however, evenly go into nine hundred ninety nine thousand nine hundred ninety nine.
in decimal, a seventh is written as point one four two eight five seven recurring, because
it’s equal to one hundred forty two thousand eight hundred fifty seven nine hundred ninety
nine thousand nine hundred ninety ninths. seven doesn’t evenly go into twelve, eleven,
eleven dozen eleven, eleven gross eleven dozen eleven, eleven great gross eleven dozen eleven,
OR eleven dozen eleven great gross eleven dozen eleven. it DOES, however, evenly go
into eleven gross eleven dozen eleven great gross eleven dozen eleven. in dozenal, a seventh
is written as point one eight six ten three five recurring, because it’s equal to one
gross eight dozen ten great gross ten gross three dozen five eleven gross eleven dozen
eleven great gross eleven dozen elevenths.
whew! both of those were very complicated! but we’re all okay with that because nobody
ever needs to use sevenths, right?
seven doesn’t evenly go into six either, and it certainly doesn’t evenly go into
five. the thing is, seven is one more than six, which means that it necessarily evenly
goes into fifsy five. in seximal, one seventh is written as point zero five recurring, because
it’s equal to five fifsy fifths.
hopefully, you can see where I’m coming from by now. sure, dozenal is slightly better
at fourths, but at what cost? the simplest ratio that seximal really does a bad job at
representing is one eleventh, which it writes as point zero three one three four five two
four two one recurring. everything simpler than elevenths needs, at most, three digits.
personally, I think we can all agree that you don’t ever need to use elvenths.
part four: conclusion
by now I’m sure you can see why I say that seximal is better than decimal and dozenal.
I didn’t even get into the stuff like the history of yam counting in New Guinea or how
to use base nif to make seximal numbers half the length or even how to say numbers larger
than “unexian”. if you haven’t been convinced, feel free to argue about it in
the comments section. if anyone brings up some like, actually good counterpoints I might
make a follow up video about them? anyway, until then, for more information go over to
seximal.net, which is a pretty much comprehensive resource on the seximal system. I’ve been
jan Misali, and I don’t have a sign off for my videos that aren’t about conlanging.