A Guide to Dozenal as I Use It

A while ago, I concluded that dozenal, the base-twelve system, was the best way to use numbers. Completely switching number systems is, of course, an entirely unpractical thing to do, but I have actually been using it in my daily life, for years now. I’d like to finish off this article series by giving a little guide to how I use it.

Basics

What is Dozenal?

In our usual decimal system, each digit represents a certain power of ten. For example, ‘123’ means 1×ten² + 2×ten¹ + 3×ten⁰. Dozenal is simply deciding to use powers of twelve instead of ten, so that ‘123’ instead means 1×twelve² + 2×twelve¹ + 3×twelve⁰. ‘123’ in dozenal is ‘171’ in decimal.

You can read the earlier parts of this series to know why I think dozenal is the best, but they’re very long and you don’t need them to read this article.
The Search for the Best Base – Part 1 – Understanding the Problem
The Search for the Best Base – Part 2 – Base Evaluation Criteria
The Search for the Best Base – Part 3 – Evaluating Each Base
In one sentence, the case for dozenal is that twelve and its powers have more factors than ten and its powers, and many common operations like multiplication, division and divisibility/remainders are easier when working with factors of the base and its powers (think of working with 2, 4, or 5 vs. 3, 7 or 9 in the usual base-ten system).

Writing Numbers

To write dozenal numbers, you need twelve digits from zero to eleven. The existing digits 0-9 can be used for zero to nine, but we need two more. The simplest solution is to use ‘A’ for ten and ‘B’ for eleven, and that is what I will use here.

However, one other proposal was popular enough to be put in Unicode, meaning its digits can be typed on computers and websites, and I like it too, so I’ll mention it:

• Ten uses an upside down two (U+218A; ‘↊’).
• Eleven uses an upside down three (U+218B; ‘↋’). I like these digits: they look like numbers and fit in with the rest of the digit set, and I use them in writing, but they aren’t yet supported enough to be used on computers without any problems. Hopefully one day they will.

For other things, in this article I’ll use the same conventions I established in the base search series:

• ‘A#’ indicates decimal and ‘C#’ indicates dozenal. Numbers are in dozenal by default.
• A second period indicates the start of infinite repetition. 0..XYZ means 0.XYZXYZXYZXYZ…, while 0.ABC.XYZ means 0.ABCXYZXYZXYZ….

Speaking Numbers

The simplest way to say a dozenal number is to just say its digits, pronouncing ‘123’ as ‘one two three’, but we can do better than that. We already have some names for some powers of twelve, and we can create names for others:

• C#10^1 can be called ‘twelve’ when alone, or ‘dozen’ when multiplied.
• C#10^2 is a gross.
• C#10^3 is a great gross.
• C#10^6 can be called a ‘hexen’ (abbreviation of ‘hex’ + ‘dozen’).

With this system, the number 123 456 789 AB0 is called “one gross two dozen three great gross four gross five dozen six hexen seven gross eight dozen nine great gross ten gross eleven dozen”. A long name, but to be expected for a twelve-digit number!

Basic Fractions

One of the best properties of dozenal is how it makes many simple fractions nicer, so here are some dozenal fractions (specifically the nice ones):

• 1/2 = 0.6
• 1/3 = 0.4
• 1/4 = 0.3
• 1/6 = 0.2
• 1/8 = 0.16
• 1/9 = 0.14
• 1/10 = 0.1 (remember: this is one twelfth, not one tenth!)
• 1/14 = 0.09 (1/(1×twelve+4) = one sixteenth)
• 1/16 = 0.08 (1/(1×twelve+6) = one eighteenth)

The only downgrade is fifths and their multiples – in dozenal 1/5 is 0..2497 – but I find I don’t need those much when I’m using it.

Every base has different addition and multiplication tables, so here’s the ones for dozenal:

┌───┬───────────────────┬───────────────────┐ ┌───┬───────────────────┬───────────────────┐
│ + │  0  1  2  3  4  5 │  6  7  8  9  A  B │ │ × │  0  1  2  3  4  5 │  6  7  8  9  A  B │
├───┼───────────────────┼───────────────────┤ ├───┼───────────────────┼───────────────────┤
│ 0 │  0  1  2  3  4  5 │  6  7  8  9  A  B │ │ 0 │  0  0  0  0  0  0 │  0  0  0  0  0  0 │
│ 1 │  1  2  3  4  5  6 │  7  8  9  A  B 10 │ │ 1 │  0  1  2  3  4  5 │  6  7  8  9  A  B │
│ 2 │  2  3  4  5  6  7 │  8  9  A  B 10 11 │ │ 2 │  0  2  4  6  8  A │ 10 12 14 16 18 1A │
│ 3 │  3  4  5  6  7  8 │  9  A  B 10 11 12 │ │ 3 │  0  3  6  9 10 13 │ 16 19 20 23 26 29 │
│ 4 │  4  5  6  7  8  9 │  A  B 10 11 12 13 │ │ 4 │  0  4  8 10 14 18 │ 20 24 28 30 34 38 │
│ 5 │  5  6  7  8  9  A │  B 10 11 12 13 14 │ │ 5 │  0  5  A 13 18 21 │ 26 2B 34 39 42 47 │
├───┼───────────────────┼───────────────────┤ ├───┼───────────────────┼───────────────────┤
│ 6 │  6  7  8  9  A  B │ 10 11 12 13 14 15 │ │ 6 │  0  6 10 16 20 26 │ 30 36 40 46 50 56 │
│ 7 │  7  8  9  A  B 10 │ 11 12 13 14 15 16 │ │ 7 │  0  7 12 19 24 2B │ 36 41 48 53 5A 65 │
│ 8 │  8  9  A  B 10 11 │ 12 13 14 15 16 17 │ │ 8 │  0  8 14 20 28 34 │ 40 48 54 60 68 74 │
│ 9 │  9  A  B 10 11 12 │ 13 14 15 16 17 18 │ │ 9 │  0  9 16 23 30 39 │ 46 53 60 69 76 83 │
│ A │  A  B 10 11 12 13 │ 14 15 16 17 18 19 │ │ A │  0  A 18 26 34 42 │ 50 5A 68 76 84 92 │
│ B │  B 10 11 12 13 14 │ 15 16 17 18 19 1A │ │ B │  0  B 1A 29 38 47 │ 56 65 74 83 92 A1 │
└───┴───────────────────┴───────────────────┘ └───┴───────────────────┴───────────────────┘

Units

Why not Metric?

One problem with changing number systems is that the metric system would not work in any system other than base ten. In dozenal, for example, 1 km = 6B4 m, which is not a nice conversion to work with!

The first idea to fix this is to change your prefixes. For that, we can use the excellent Systematic Dozenal Nomenclature (SDN) system, created by John Volan, a member of the Dozenal Society of America. In this system, the prefix for C#10^12 is ‘unbiqua’, written as ‘ub↑’, while #C10^-12 is ‘unbicia’, written as ‘ub↓’.

Now, we can say that 1 t↑m (triqua·metre) = C#1000 m, so problem solved? If the metric system were a perfect system, it would be, but unfortunately the metric system is not perfect. The metric unit of volume, the litre, is defined as A#0.001 m^3, which does not convert nicely to dozenal. If we change the litre to fix this, the kilogram would no longer be equal to a litre of water. If we change the kilogram to fix this, we have to change a bunch of derived units as well (e.g. newton, joule, watt), enough that we basically have a different system.

Also, metric, despite being much better than imperial/customary, still has some flaws:

• Time units are not decimal, and still have awkward conversion factors. This creates multiple hard-to-interconvert units for many kinds of units – for time, but also speed (m/s vs km/h), energy (J vs Wh), electric charge (C vs Ah), and more. This could by fixed by only using seconds, but good luck getting people to count time in kiloseconds (ks) when there isn’t even a whole number of them in a day (1 day = A#86.4 ks)!
• Many other non-systematic units are used with the metric system: calories for energy, astronomical units and parsecs for very large distances, daltons for atomic masses, and more.
• Many prefixed unit names are quite long.
• Many of the prefixes and names, especially the older ones, do not make it clear what size/unit they describe, and have to simply be memorized. If we have to make major breaking changes to use the metric system, we might as well go all the way, and use a new system that fixes all the problems.

Primel

Thankfully, this system already exists, and it’s called Primel, created by the inventor of SDN!

A basic description of Primel:

• Each unit is named after the thing it measures, plus ‘-el’. The unit of time is a timel, the unit of length is a lengthel, and so on. Some names can be shortened if necessary (e.g. accelerationel → accelerel). Symbols use abbreviated names plus ‘-ℓ’ (e.g. lengthel → lgℓ).
• These names are generic, meant to be used for many different systems. To distinguish Primel from other systems, we add a “brand mark” character to the start of the name, which can be dropped if obvious from context. For Primel, this is ‘⚀’, pronounced “prime”.
• The ⚀timel is simply 1/C#1000000 of a day, or around C#0.04 seconds.
• The ⚀lengthel is defined by setting the ⚀accelerel to Earth’s gravity, making it around A#8.2 mm. C#10 ⚀lengthels is almost exactly ten centimetres, and 1 ⚀velocitel is very close to 1 km/h.
• The ⚀massel is defined as in metric, by setting the ⚀densitel to the average density of water and multiplying it by the cubic ⚀lengthel. C#1000 ⚀massels is almost 1 kg.
• The radian is used for angle, which Primel considers to be a real unit unlike metric.
• The ⚀temperaturel is defined by setting the specific heat capacity (“massic heatability”) of water to 1. This means that it takes 1 ⚀energel to warm 1 ⚀massel of water by 1 ⚀temperaturel.
• The ⚀temperaturel is tiny, so for practical temperature, the ⚀quadqua·temperaturel is used and nicknamed the “stadegree” (°ς). You can create a Celsius-like system by setting the freezing point of water to 0°ςc, or a Fahrenheit-like system by setting it to C#40°ςf.
• Electric units are defined by setting the ⚀resistancel to the impedance of free space, a universal constant equal to around A#60τ Ω.
• Primel unit names can get quite long, so some common units are given additional, sorter colloquial names. Primel has a lot of colloquial names, but I try to only use a few to keep the system regular.

This basically ends up being a formula to generate new unit systems: pick a name, pick a brand mark, pick a timel, and you’re done! This is great for working with different bases, or perhaps different planets (where the day and accelerel are different). The Primel creators even made an entire spreadsheet with a couple dozen systems built on these principles! (make a copy if you want to experiment)

Primel even has entire families of colloquial units, each its own completely coherent system. For example, the “hand” series sets the lengthel to the ⚀unqua·lengthel, then generates volume, mass, force, etc. using the existing ⚀accelerel and ⚀densitel, appending the dimension onto the “hand” name (e.g. ⚀hand·length, ⚀hand·mass, etc.). The only thing I don’t like about this is that it makes the names long, so I’ve decided to allow omitting the dimension, which is interpreted as the unit the series is based on (e.g. “⚀hand” means “⚀hand·length”). I’ve also set the ⚀grave (a rejected name for the kilogram, pronounced like the “grav” in “gravity”) to the ⚀hand·mass, which is around a kilogram.

Dozenal Time

Probably the Primel units I use the most are the time units. Because of their utility, I use four colloquial names for time:

• The ⚀biqua·timel / quadcia·day (⚀b↑tmℓ / q↓day) is called the ‘lull’ (symbol ‘lu’). It is a bit over 4 seconds.
• The ⚀triqua·timel / tricia·day (⚀t↑tmℓ / t↓day) is called the ‘trice’ (symbol ‘tr’). It is equal to A#50 seconds, and can basically be used in place of minutes.
• The ⚀quadqua·timel / bica·day (⚀q↑tmℓ / b↓day) is called the ‘breather’ (symbol ‘br’). It is exactly ten minutes, and is what I use to do things like daily schedules / my calendar.
• The ⚀pentqua·timel / uncia·day (⚀p↑tmℓ / u↓day) is called the ‘dwell’ (symbol ‘dw’). It is exactly two hours, or one-twelfth of a day. Each of these units is exactly twelve times the previous.

Dozenal time means that the time of day is simply a number. For example, the current time I am writing this sentence at is A#09:25, which corresponds to trice C#486. The main advantages of this are:

• You need fewer digits to get the same precision. Trices need three digits to do what hours and minutes need four digits for, while being more precise. Breathers reduce this to two digits at the cost of precision.
• It is easier to do arithmetic with dozenal time. For example, calculating the difference between br C#43 and br C#51 is easier than A#8:30 and A#10:10, assuming you know how to do dozenal arithmetic as well as decimal. No special rules or hexagesimal necessary.

Here’s a nice website that shows the current time in dozenal (trices): Dozenal Clock

Advantages of Primel

Everything in here is in dozenal, unmarked.

The benefits of Primel, in my opinion, are:

• Everything is regular and predictable, even more so than the metric system. The only memorization you need to do is the -el ending and the dozen-two parts of SDN names. No need to memorize random prefix names, or units named after scientists.
• Time units can be converted as easily as any other Primel unit, unlike in metric. This also applies to derived units that use time (e.g. velocity, energy, power). 1 ⚀velocitel may be around 1 km/h, but it is both the velocitel for road scale, the q↑lgℓ per breather, and the velocitel of walking scale, the b↑lgℓ per lull, and the velocitel of any other scale.
• Mass and weight/force are the same number, at least on the surface of the Earth. I don’t exactly have any plans to go anywhere else anytime soon.
• The ⚀stadegree-familiar (°ςf) is a great scale which has the benefits of both Celsius and Fahrenheit. Like Fahrenheit, most everyday temperatures are between 0°ςf and 100°ςf. Like Celsius, the freezing point of water is a round number, 40°ςf. As someone who lives in a cooler climate, this is important, as below freezing versus above freezing is the single most important distinction in everyday temperature. And if you heat 1 ⚀massel of water by one ⚀stadegree, you’ll use 1 ⚀quadqua·energel of energy.
• The time and temperature changes mean there’s no need for multiple units of energy; metric uses joules for physics, watt-hours for electronics, and calories for food, and I haven’t even bothered counting the number of non-metric energy units. All of these can be replaced by the ⚀energel (or more likely its prefixed versions; the ⚀energel is tiny), without loss of functionality! If you care about conserving energy, it’s good to have a common scale with which you can compare different efforts, and see which are best and which are wastes of your ⚀timels.
• Colloquial names, as long as they are only used for a few common units, make the most common units easy to say while keeping the system regular and limiting memorization.

Honestly, Primel on its own is a good reason to switch to dozenal; like the metric system only works in decimal, Primel only works in dozenal.