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 <A> and
eleven is written with the letter <B>, 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 <T> and <E>, 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 <X> 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.