hello everyone.
it’s me, jan Misali, the guy who makes the show Conlang Critic.
I also made a video about math that’s somehow more popular than anything else I’ve ever
made.
if you haven’t seen that video, go watch it before watching this because this video
won’t make any sense otherwise.
actually, go watch a better way to count anyway.
it’s a pretty good video.
okay, so what even is the deal with that video?
I mean, was I being serious or was I joking?
why weren’t bases besides seximal decimal and dozenal analyzed? and why do I talk so
fast?
first of all, at the risk of stating the obvious, a better way to count is, in fact, a serious
educational video that just happens to have jokes in it.
the source of the confusion is almost certainly the way I handled the intro.
see, I wanted to explain why I don’t like dozenal by doing an intentionally bad job
of arguing in favor of dozenal.
I kinda assumed that the viewer would have a certain level of familiarity with dozenalism
going in, meaning they’d be able to catch on to what I was doing early on and realize
that I was making fun of dozenalism.
the one joke in this part I got the most comments about was where I said that ten has two prime
factors and that twelve has six factors.
obviously, that’s not actually a fair comparison.
it’s so obviously not a fair comparison that I was foolishly certain that anyone who
paid attention and noticed that I did that would immediately realize that I must be joking.
how silly of me.
one other thing I feel like I should address before we get into the meat of this video
is the name I use for base six.
I personally like to call base six “seximal”.
now, from what I’ve been told through various youtube comments, some of you are unable to
take the name “seximal” seriously.
I honestly can’t imagine why.
sure, it’s “nonstandard”, but alternate bases are already nonstandard to begin with.
the “standard” name for base six (specifically the one used by Wikipedia) is “senary”,
but I don’t like that name very much because it doesn’t sound like any other Latin-derived
words in English that have to do with the number six, and it also doesn’t really sound
like a number base.
the name “seximal” fixes both of these problems.
a much less vocal group of people, by the way, have complained about the name “dozenal”
for base twelve.
the name “dozenal” is very commonly used by people who advocate for the use of base
twelve, including the Dozenal Society of America.
calling it “dozenal” instead of “duodecimal” is meant to avoid any decimal-centrism.
after all, Latin uses decimal, and if you’re in favor of using some other base, deriving
terminology directly from Latin sorta goes against what you’re after.
anyway, as promised, here are some replies to the actually good counterpoints I’ve
gotten.
“hey, wouldn’t seximal break the metric system?”
yes, it would.
the metric system, which is used almost everywhere in the world, absolutely would not work in
any base other than decimal.
I mean, sure, you COULD keep using it, but it isn’t nearly as useful for, say, a centimeter
to be one hundredth of a meter when the number “one hundred” is actually “two nif foursy
four”.
now, while this problem wouldn’t exist for American units, since they don’t make sense
in decimal to begin with, that doesn’t mean that by promoting seximal I’m somehow also
promoting the use of the American units.
I mean, yes, as an American that is the system I personally use so I am at least somewhat
biased, but I do understand that this really would cause serious problems for the entire
rest of the world.
the proper solution to this, naturally, is to create a completely new system of measurement
to go along with the seximal system.
I made a half-serious proposal at one point with base units derived from metric-imperial
compromises, like how the unit of length is a “stick”, defined as the average between
a meter and a yard.
the exact base units in a measurement system aren’t actually all that important.
what matters is how the units are related to each other.
all you really need to make a seximal measurement system is a set of power-of-six prefixes.
once you have those, you can just apply them to whatever existing units you want to create
a fully functional seximal measurement system.
yes, theoretically it would be very difficult, almost impossible even, to get everyone to
switch what measurements they use, but not actually any more impossible than getting
everyone to switch numbering systems in the first place.
fun fact, one nifth of a meter, which using the prefixes I came up with would be called
a “niftimeter”, is actually really really close to an American inch.
makes you think!
another fun fact, let’s say you wanna use degrees Celsius in seximal.
obviously, that would be inconvenient because the boiling point of water would no longer
be a convenient round number, right?
well, get this, nif degrees Celsius is equal to two foursy and fourth fifths of a degree
Fahrenheit, which in decimal is EXACTLY 96.8, meaning that a zero-to-nif scale in Celsius
works as a scale from the freezing point of water to human body temperature.
“wait, why exactly are fourths less important than fifths? that wasn’t made clear (also
you should say quarters it’s more proper)”
yeah, fair enough.
so, yes, fourths are more practical than fifths, being a simpler fraction.
there are, in fact, more situations where you need to use fourths than there are situations
where you need to use fifths.
having a single-digit representation of fourths, however, is not as important.
that’s because a fourth is half of a half.
if you’re using an even base, you’re guaranteed to have single-digit halves, which makes it
pretty easy to divide any given number by two. in turn, that means that even if you
for some reason can’t be bothered to remember how to write a fourth, it’s pretty trivial
to derive it from scratch.
fifths, on the other hand, can be very difficult to derive.
five is a prime number, so if a base doesn’t have a simple representation for fifths, you’re
basically out of luck.
that’s the case in dozenal. as I’m sure you recall, fifths are represented with four
recurring digits, which essentially means you just can’t use fifths in any useful
way.
sure, maybe they don’t come up as often as fourths, but dozenal doesn’t even give
you the option.
it’s important to remember, by the way, that the way a base handles simple ratios
isn’t JUST about the ratios themselves.
while that is what I generally focus on because it’s easy to demonstrate, an integer whose
reciprocal is handled in a simple way in any given base is an integer that base considers
to be important.
this matters for all sorts of things, including division, factorization, and patterns in multiplication
tables.
here’s one situation you might care about.
how can you tell if a number is prime? as we all know, a prime number is a number with
no factors other than one and itself.
let’s say you wanna know if a number is prime quickly, just by looking at it, no complicated
division required.
specifically, you want a system you can do in your head that rules out numbers that are
clearly not prime.
in other words, what numbers LOOK prime?
in decimal, you can easily tell if a number is even and if a number is divisible by five
by looking at the last digit.
so 12 doesn’t look like a prime number and neither does 15.
you can also add digits together to see what factors a number has in common with nine,
which for small numbers you can easily do in your head.
21 doesn’t look prime, and neither does 27.
you can also recognize small factors of eleven pretty easily, because they’re two digit
numbers where both digits are the same.
77 doesn’t look like a prime number.
using these simple methods to see what numbers “look prime” in decimal, the smallest
number that “looks prime” but isn’t really is 49. of course, there’s a good
chance you might just recognize this number as being seven squared, but if you don’t
already know that, there’s no easy way of figuring that out just by looking at it.
it ends with 9, so it isn’t even and it’s not divisible by five.
four plus nine is thirteen and one plus three is four, so it’s not divisible by three
either.
it’s clearly not a multiple of eleven, and that’s all the easy divisibility tests you
can do in your head.
what numbers look prime in seximal?
well, if you look at a list of primes in seximal, you’ll notice one very clear pattern.
yes, after 2 and 3 (the two factors of six), every single prime number ends with either
1 or 5. this is because numbers that end with any other digit in seximal are guaranteed
to be even and/or divisible by three.
this isn’t to say that every number that ends with 1 or 5 in seximal is automatically
prime, far from it.
however, this quick test automatically prevents even and threeven numbers from looking like
primes.
the other simple tests you can do rule out a good portion of the remaining composite
numbers that just happen to not be even or threeven.
remember, you can do different digit sum tests in seximal to see if a number is divisible
by five and/or seven.
the smallest number that looks prime in seximal but isn’t really prime is 321.
it ends with 1, so it’s not even or threeven.
three plus two plus one is six, so it’s not divisible by five, and three plus dozen
one is dozen four, so it’s not divisible by seven.
the number 321 is, in fact, equal to eleven squared.
it’s 121 in decimal.
every number smaller than it can be easily identified as either prime or composite completely
accurately with nothing but basic mental math.
let’s skip to the punchline.
what’s the smallest composite number that looks prime in dozenal?
well, it’s actually 21.
it ends with 1, so it’s not even and it’s not divisible by three.
two plus one is three, so it’s not divisible by eleven, and it’s very clearly not divisible
by dozen one.
this number, of course, is actually equal to five squared.
two dozen one equals twenty five in decimal.
again, if you have your multiplication tables properly memorized you could just recognize
this number and its factors, but if you don’t, you’re out of luck.
“what was that thing you were saying about base nif? that sounds interesting and you
should’ve actually talked about it in the video instead of offhandedly mentioning it
in that way-too-fast voice you’re constantly doing”
base nif, or “niftimal” as I like to call it, is a rather elegant solution to the most
obvious problem with using seximal.
six is a really small number, so numbers in base six can get pretty long.
sure, if you take into account how much information is contained in each digit, those longer numbers
are technically more efficient than decimal, but that doesn’t make them take up any less
physical space on a page.
the easiest way to compress numerical information in any given base is to use a larger set of
symbols that correspond to two digits each.
in other words, use the base’s square.
for large bases, this isn’t really a viable option.
compressing decimal this way would require a system with one hundred distinct digits,
which is, let’s be honest, too many to keep track of.
sure, it’s possible, but it would certainly be more trouble than it’s worth.
fortunately, since six isn’t a very big number, neither is its square.
nif (decimal thirty six) is a pretty reasonable amount of digits. in fact, there’s one really
nice coincidence when it comes to selecting which nif digits to use.
see, as I’m sure you know, the most common way to extend the Hindu-Arabic numeral system
is using the letters of the Latin alphabet.
the Latin alphabet has exactly two baker’s dozen letters.
that’s seximal foursy two and decimal twenty six.
those of you who are good at addition are sure to notice that that’s precisely the
amount you need to add to the ten Hindu-Arabic digits to end up with nif digits.
using niftimal allows you to make any given number in seximal take up half the space it
otherwise would.
I would like to stress that while this is a useful system for compressing seximal information,
I am not advocating for using niftimal exclusively.
while nif isn’t an inconveniently large number of digits, it’s absolutely too many
to be used for basic arithmetic. in my opinion, the limit for how large a base can get while
still being human useable is somewhere between thirsy and foursy.
when using base nif, in order to reduce the mental strain, it’s best to convert numbers
into seximal before doing math to them and then to convert back into niftimal when you’re
done.
I’ll admit that this isn’t a perfect solution, but it does work.
“okay, but how good is base...”
sure, let’s do a quick rundown of various bases and how good I think they are.
remember, the things that matter when comparing a base to other bases are the number of digits
it has and how the base writes simple ratios.
the number of digits a base has determines how efficiently it stores numerical information,
and the way a base writes simple ratios demonstrates what numbers that base has an easy time handling
in general.
this focus on how ratios are handled completely rules out things like non-integer bases and
unary.
again, this is a quick rundown and not a comprehensive overview.
anyway, let’s start with...
BASE TWO: BINARY
binary is the smallest base that actually works at all.
doing simple arithmetic in binary is super easy, and only having two digits means storing
binary information on a computer is maximally efficient.
that’s about it for positives.
every other aspect of binary is a downside.
numbers get real long real fast, and the only terminating ratios are ones where the denominator
is a power of two.
oh, by the way, the ratios I’m showing in red here are ones that are as complex as they
hypothetically could be.
you know, the really bad ones. and yes, I’m using underlines instead of overlines for
recurring digits.
BASE THREE: TRINARY
trinary is a lot like binary. it’s a small prime base, so it’s easy to do math at the
cost of everything taking a lot of digits to write.
it is a bit better at some ratios than binary, but at the cost of halves which, let’s be
honest, are absolutely vital.
now, I should probably mention one specific version of trinary that’s usually called
“balanced ternary”.
balanced ternary is just like normal trinary, except instead of its three digits being zero
one and two, its three digits are negative one zero and one.
I’m not gonna get into exactly how it works, but I will say that is cool and interesting
and that you should absolutely go look it up if you haven’t heard of it before.
however, it is still just base three at a fundamental level, so in terms of actual practicality,
it’s not really that much of an improvement.
BASE FOUR: QUATERNARY
now, quaternary is where things start to get interesting.
remember, square bases can be used for compression, and quaternary is the smallest square base.
it’s really just a more efficient version of binary. it’s still pretty small, and
it can’t really handle any non-power-of-two denominators, but other than that it’s pretty
solid.
BASE FIVE: QUINARY
quinary is a prime base, and a great example of why prime bases aren’t great.
we’ll see a few more of these later.
a prime number is the exact opposite of what makes a good base.
BASE SIX: SEXIMAL
I think I’ve made my thoughts on this one clear enough.
BASE SEVEN: SEPTIMAL
prime base number four.
this is the smallest regular base Wikipedia doesn’t have a dedicated article about.
seven is just a really inconvenient number.
BASE EIGHT: OCTAL
octal is another power of two base, eight being two cubed. just like how one digit in
quaternary corresponds to two binary digits, one digit in octal corresponds to three binary
digits. this is actually a pretty popular choice for a compressed binary method, mainly
because it’s pretty close in size to decimal.
unfortunately, when it comes to representing simple ratios, groups of three binary digits
don’t always quite work out, so there’s a lot of ratios where octal isn’t any better
off than binary.
BASE NINE: NONARY
nonary is probably the most useable odd base, pretty much entirely because nine is equal
to three squared.
its convenient ratios ALMOST make up for how it can’t do halves.
almost.
like, if you’re gonna fail at one ratio, halves are the worst one to fail at.
BASE TEN: DECIMAL
I trust you know about this one already.
BASE ELEVEN: ELEVENARY
also known as “undecimal”, elevenary is prime base number five.
it’s sometimes suggested as a joke.
since eleven is right between ten and twelve, it works great as a compromise between decimal
and dozenal! oh, speaking of,
BASE TWELVE: DOZENAL
you know, this one! let’s uh, skip ahead a ways.
BASE DOZEN FOUR: HEX
of all the bases I didn’t talk about in a better way to count, hex was the one I got
the most comments about.
hex is extremely popular among computer scientists as a way of compressing binary. base dozen
four (that’s decimal sixteen!) is a square base, so one hex digit corresponds to two
quaternary digits. as I’m sure you remember, one quaternary digit corresponds to two binary
digits, meaning that one hex digit corresponds to four binary digits.
needless to say, hex is extremely good at compressing binary.
the main drawback for hex is that, like every power of two base, the only terminating ratios
are ones where the denominator is a power of two. also, with its dozen four digits,
hex is close to the edge of the human useable range of bases.
it’s pretty big. also also, the fact that hex is so popular is the literal only thing
stopping me from referring to base six as “heximal”.
BASE DOZEN FIVE: SUBOPTIMAL
of all the integer bases in the human useable range, this is the worst one.
that’s why it’s called “suboptimal”, because it’s so bad.
if you wanna suggest using a base as a joke, this is the base for you.
BASE THIRSY TWO: VIGESIMAL
that’s decimal twenty.
vigesimal is the largest base in the human useable range that’s actually used by anyone.
if you go much further, things get pretty hard to handle.
as for vigesimal itself, well, it’s somewhat of a mixed bag in terms of practicality.
look, it’s got single digit fourths AND fifths! plus, its sevenths are just two recurring
digits for some reason.
this is, however, at the cost of thirds. for a base this size, not being able to handle
thirds in a convenient way is sort of a deal breaker.
now, I’m saying that this is “close to the edge of the human useable range”, but
as many of you have surely noticed, there’s something I seem to have ignored.
see, there’s this one numbering system that was used by the ancient Sumerians and the
ancient Babylonians that’s had a pretty major impact on how certain things are counted.
base nif foursy. base five dozen.
base sixty.
base... tensy.
sexagesimal, as it’s commonly called.
now, tensy is quite a few digits.
I’d argue that it’s too many to be practical.
so then, how on Earth did those ancient cultures who used this base actually manage it?
well, as it turns out, they kinda cheated.
sexagesimal as it was used in ancient Mesopotamia really only used two digits.
see, there was one narrow wedge shape for one and another wider wedge shape for ten.
these wedges were grouped in a way that looks somewhat like a decimal upgrade of tally marks.
the key difference was that these groupings were then used as digits themselves.
for example, a number like, say, “twenty two sixties and seven” would indeed be written
with two “digits”, but those digits would be made up of one grouping of two ten-wedges
and two one-wedges and then another grouping of seven one-wedges. we could transcribe this
with our Hindu-Arabic numerals as <2207>, maybe with something separating the two sexagesimal
digits.
this method, where each digit is written with some combination of symbols instead of one
single symbol, is what makes it possible to actually use these bases that are too large
to be used normally.
if you can get over the hurdle of one more layer of complexity, these larger bases have
a lot of potential, but that’s a topic for another time.
“you know, in the grand scheme of things, it doesn’t actually matter what base we
use.
the benefits any given base has are completely outweighed by the problems that would be caused
by switching what base we as a society use in the first place.”
yup! none of this matters.
I do not recommend or expect anyone to completely stop using decimal.
that would be very silly.
despite that, it’s fun to get silly sometimes.
at a fundamental level, if you strip away all this discussion of radix economy and recurring
expansions, I like seximal for the same reason dozenalists like dozenal.
it’s fun to examine how we could theoretically change the things we take for granted, even
if we know it could never actually happen.
if I haven’t been able to convince you that seximal is the best base out there, that’s
fine.
if you strongly disagree with me about this, that’s even better.
I encourage everyone to develop their own opinions about which numbering system is the
best.
after all, that’s what really counts.
thanks for watching.
I’ve been jan Misali before, and one day I will be jan Misali again.
sure, that’ll work as a sign off for non conlang videos.