## Thursday, February 7, 2013

### Base 12 Math - Addition and Muliplication tables

A surprising, if in no way overwhelming, number of people come here looking for information on base 12 math.  So I thought I should, perhaps, write a bit more about that.

As a reminder, Ѧ means ten and ϟ means eleven.  I assume that 10 (12 base 10) would be pronounced either "one zero" which is what I've heard, or "base" which is quicker.  One could also just call it, "Twelve" since twelve is pretty well defined.

 + 1 2 3 4 5 6 7 8 9 Ѧ ϟ 10 1 2 3 4 5 6 7 8 9 Ѧ ϟ 10 11 2 3 4 5 6 7 8 9 Ѧ ϟ 10 11 12 3 4 5 6 7 8 9 Ѧ ϟ 10 11 12 13 4 5 6 7 8 9 Ѧ ϟ 10 11 12 13 14 5 6 7 8 9 Ѧ ϟ 10 11 12 13 14 15 6 7 8 9 Ѧ ϟ 10 11 12 13 14 15 16 7 8 9 Ѧ ϟ 10 11 12 13 14 15 16 17 8 9 Ѧ ϟ 10 11 12 13 14 15 16 17 18 9 Ѧ ϟ 10 11 12 13 14 15 16 17 18 19 Ѧ ϟ 10 11 12 13 14 15 16 17 18 19 1Ѧ ϟ 10 11 12 13 14 15 16 17 18 19 1Ѧ 1ϟ 10 11 12 13 14 15 16 17 18 19 1Ѧ 1ϟ 20

By the way, anyone know how to make the cells in the table all the same size and all square?

20 is a problem "Twenty" is pretty clearly defined as "Base ten two zero." So one can't call it "twenty"
"Two zero" just sounds clunky to me.  "Twobase" would extend naturally from calling 10 "base" but then you've got, "Twobase one, twobase two, ..., twobase ten, twobase eleven."  This is where just using the "Two" for the leading digit is useful, th shortens things.  Even in base ten, "Two-one," is shorter than, "Twenty-one."

And it would have to be Twobase not Twicebase because we have "once, twice, thrice," and no words for four and above.

Multiplication table:
 X 1 2 3 4 5 6 7 8 9 Ѧ ϟ 10 1 1 2 3 4 5 6 7 8 9 Ѧ ϟ 10 2 2 4 6 8 Ѧ 10 12 14 16 18 1Ѧ 20 3 3 6 9 10 13 16 19 20 23 26 29 30 4 4 8 10 14 18 20 24 28 30 34 38 40 5 5 Ѧ 13 18 21 26 2ϟ 34 39 42 47 50 6 6 10 16 20 26 30 36 40 46 50 56 60 7 7 12 19 24 2ϟ 36 41 48 53 5Ѧ 65 70 8 8 14 20 28 34 40 48 54 60 68 74 80 9 9 16 23 30 39 46 53 60 69 76 83 90 Ѧ Ѧ 18 26 34 42 50 5Ѧ 68 76 84 92 Ѧ0 ϟ ϟ 1Ѧ 29 38 47 56 65 74 83 92 Ѧ1 ϟ0 10 10 20 30 40 50 60 70 80 90 Ѧ0 ϟ0 100

Or something like that.

1. When I count in bases greater than 10, I borrow from hex and use letters. so nine, ah, bee, ten, eleven, twelve ... ahteen, beeteen, twenty ... twentyah, twentybee, thirty ... ninety, atty, beety, a hundred

A is pronounced "ah" because if i pronounce it A then "A-teen" sounds too much like "eighteen" and similarly "a-ty" sounds too much like "eighty".

Today is the 43rd day of the year, so I'll be counting in base 43 (I do largest prime factor of the day of the year). i might have to borrow some extra letters in because english doesn't have enough.

--Anonymus

1. I'm curious. How do you cope in base 359?

2. I'm not the biggest fan of using letters, though as I recall the things I picked are actually letters. Let me check.

Yup
Ѧ is (uppercase) little Yus, but it has the advantage of being a defunct letter if wikipedia is to be believed. It also happens to be the symbol for the Appalachian trail (where it's not really little Yus, it's a creative way of combining a capital A and a capital T into a single symbol that just happens to look like little Yus.)

ϟ is an uppercase Koppa. Koppa being a letter that was phased out of ancient Greek in favor of Kappa. It also had a numeric value associated with it (90).

I think "ten" is too well defined to mean numbers other than the one after nine. I mean, we call it base ten, that would be meaningless if "ten" just meant, "10 in whatever base you happen to be dealing with."

So I think what I'd do is just count thus
one, two, three, four, five, six, seven, eight, nine, ten, eleven, base.

base-one, base-two, base-three, base-four, base-five, base-six, base-seven, base-eight, base-nine, base-ten, base-eleven, twobase.

(I realize that could be confusing, and one could alleviate it by using "twelve" instead of "base" but I'm kind of going for a generalizable concept here. Anyway, if someone's expecting a number they're probably not going to be confusing "base-two" with "binary".)

twobase-one, twobase-two, twobase-three, twobase-four, twobase-five, twobase-six, twobase-seven, twobase-eight, twobase-nine, twobase-ten, twobase-eleven, threebase.

And so forth.

[break for character limits]

3. Now twelve is special in that 100 base twelve has a name (gross) so one could put that in there. Otherwise basesquare (basebase?) for 100, basecube for 1,000, baseto6th? for 1,000,000? Maybe 6thbase? Not sure.

The thing about numbers is that we know them. Base Sixteen is a known thing; you don't have to say, "Base sixteen, and I mean 'sixteen in base ten,' and I mean 'ten' as in the number that comes after nine." Sixteen is a quantity that we understand apart from 16. I'd argue that twenty is a known quantity as well.

-

In closing, I think that for 12 in particular I'd probably replace "base" with twelve (so twelve-one, twelve-two and so on, as well as TwoTweleves for 20, ThreeTweleves for 30) and use gross for 100. Great gross for 1000. It's more syllables, and thus more clunky, but the terms already exist so it seems worth making use of them.

Apparently "Great Gross" is also known as "Grand Gross" and "Mass". If that last one is accurate, and I'm not quite sure how to check if it is, it is very appealing because then we have:
10 = twelve
100 = gross
1,000 = mass
1,000,000 = MassMass, MassSquared
1,000,000,000 = MassCubed
1,000,000,000,000 = MassMassSquared (assuming we went with MassMass, instead of MassSquared before, otherwise it would be confusing and look like we could mean MassCubed instead of mass to the sixth)

And in that case we've got a counting system that goes to at least 999,999,999,999,999 base twelve which is, I'm told, 261,454,357,816,257,576,960 base ten.

Or, one could steal the (short scale) decimal format:
10 = twelve
100 = gross
1,000 = mass
1,000,000 = massmass
1,000,000,000 = bimassmass
1,000,000,000,000 = trimassmass

And so on for as long as you can count in Latin.

But that all assumes that "mass" really means 12^3, which I only get from wikipedia saying it does without citation. For that matter, I've also noted that wikipedia says (again without citation) that it is sometimes called a Zagier as a pun. For a moment I thought that would let me get rid of massmass and then I remembered it was talking about the same number as mass. I'd prefer if one could have Zagier=massmass, and then biZagier, triZagier and so on.

Basically where possible I'd like single short words. Even though mass x mass = what I'm calling massmass, as much as one thousand times one thousand equals one million, and so "massmass" is an appropriate and descriptive term, "massmass" is somewhat longer than I'd like.

[Don't let any of this stop you from responding to Firedrake's question]

Also, anyone know how to make it so that the cells in an HTML table are all the same size and all square? I'd like to have it say, "Oh, this is the longest dimension of any cell, I'll make that both dimensions of every cell."

4. I love you, Chris. :)

@Firedrake: when it gets up to base 359 (or long before that) I just use regular base ten numbers, so three-fifty-seven, three-fifty-eight, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, tenteen, .... three-fifty-seventeen, three-fifty-eighteen, twenty.

@Chris again: I dunno. I see your point, but to me "ten" means "base to the power of 1". I also tend to call it "decimal" instead of "base ten".

neat thing i've been thinking about lately:
(base - 1)^2 is always written as (base -2) 1. For example 9^2 = 81 (decimal) or 8^2 = 71 (nonary), or 7^2 = 61 (octal), or 6^2 = 51 (septenary? septary?), all the way down 2^2 = 11 (trinary). F^2 = E1 (hexadecimal). ϟ^2 = Ѧ1 (duodecimal)

this is generally true because a^2 = (a-1)(a+1) + 1 and in the case of "base -1" that works out to "base -2" * base + 1.

2. Chris, I'm surprised the obvious word "dozen" has not shown up here, since you're discussing the "dozenal" system. Now as you suggested above, it would be easiest to adopt a name that most are already familiar with (e.g. 144=gross). So, why not dozen?

10 (12 in base-10) = dozen
20 (24 in base-10) = two dozen
30 (36 in base-10) = three dozen
and so on.

11 = dozen one
12 = dozen two
29 = two dozen nine
3X = three dozen ? (here, we can actually use "ten")
6E or 6V = six dozen ? (we can use "eleven", but my personal fav is "eve", because ... well because it sounds pretty, but it's also "on the eve" of the whole/dozen ... you get the gist)

If shorter names are desired, maybe we can use "twen-zen", "thir-zen", "four-zen" and so on?

For 100, 1000 and so on, I think your suggestions already made sense. I don't really like the sound of "gross" though :) But "mass" sounds alright.

Anyway, it makes me want to write an article on this topic myself!

1. Chris, I'm surprised the obvious word "dozen" has not shown up here

I'm surprised as well. I certainly didn't forget the word "dozen" but I guess it wasn't in the front of my mind.

I do like the suffix "-zen".

Dozen, twozen, threezen, fourzen, fivezen, and so on.

3. Another interesting thing to do with number systems is make them support automatic rounding. This does require a new set of number symbols.

I'll use base 10 for convenience, but it's easy to extend to other bases. The count 0..4 is standard, but you then count (10-5), (10-4) .. (10-1), (10), (10+1) .. (10+4), (20-5) ... (40+4), (100-50-5) ...

The point of this is that you can round a value simply by truncation: (10-3) is closer to 10 than it is to 0 or to 20, and so is (10+4). This makes approximate mental arithmetic easier. It isn't a perfect match to standard rounding, because 45-49 to the nearest 100 are 100 rather than 0, but the special casing to make that work would make the rest of the system more complex.