Humans, for the most part, count in chunks of 10 — that's the foundation of the decimal system. Despite its near-universal adoption, however, it's a completely arbitrary numbering system that emerged for one very simple reason: We have five fingers on each hand. But as many mathematicians like to point out, base-10 is not without its problems. The number 12, they argue, is where it's really at. Here's why we should have adopted a base-12 counting system — and how we could still make it work.
Indeed, it's regrettable that we failed to evolve an ideal set of fingers to help us come up with numbering system suitable for counting and calculating. Instead, with our 10 fingers, we are stuck with the clunky decimal system.
Taking a closer look at base-10, we can see how frustratingly limited it really is. Ten has a paltry two factors (a divisor that produces whole numbers), namely 5 and 2. Moreover, these numbers are not very useful in-and-of themselves; 5 is a prime number that cannot divide any further, and 2 is a frustratingly small integer to work with.
Defenders of base-10 highlight its ability to allow for the moving of fraction points after multiplication or division — but that's not a trait exclusive to base-10. It's not ten-ness that allows for this property. More accurately, it's a characteristic that belongs to all bases — a property of the place value notation we use for expressing numbers, along with a symbol for zero.
Interestingly, base-10 is not universal across human societies. The Mayans were known to use a base 20-system, and the Babylonians developed a system using sets of 60. Base-8 and base-16 (the hexadecimal system) have also been used, mostly for computational reasons (quarters and eighths are simplified).
But these alternative sets are still not ideal for day-to-day, human applications. Base-20 is not great for finger counting; many of us wear shoes when we're doing math, nor can we move our toes with any kind of dexterity. Base-8 is simply too small, and base-16 and base-60 are too unwieldy.
Luckily, there's a base that sits in between these — a numbering system that has a plethora of characteristics that simply make it the best choice for counting and calculating.
Introducing the Dozenal System
Also called the duodecimal system, the "dozenal" system was initially popularized in the 17th century when mathematicians began to recognize the limitations of base-10.
Later, during the 1930s, F. Emerson Andrews published a book, New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics, in which he cogently argued for the change. He noticed that, due to the myriad occurrences of 12 in many traditional units of weight and measures, many of the advantages claimed for the metric system could also be adopted by the dozenal system.
Indeed, examples of base-12 systems abound. A carpenter's ruler has 12 subdivisions, grocers deal in dozens and grosses (12 dozen equals a gross), pharmacists and jewelers use the 12 ounce pound, and minters divide shillings into 12 pence. Even our timing and dating system depends on it; there are 12 months in the year, and our day is measured in 2 sets of 12. Additionally, in geometry, a circle is replete with subsets and supersets of 12 — what's measured in degrees (a 360 degree circle consists of 30 sets of 12).
It's also obvious that someone in our history was thinking along these lines. It's the largest number with a single-morpheme name in English (i.e. the word "twelve"). After that, we hit thirteen, fourteen, fifteen, and so on — derivatives of three, four and five. Clearly, it was natural to think in terms of dozens.
Three decades after Andrews's book, the brilliant mathematician A. C. Aitken made a similar case. Writing in The Listen in 1962, he noted:
The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal. This is my experience; I am certain that even more so it would be the experience of others.
Since the time of Andrews and Aitken, the dozenal movement has garnered a number of enthusiastic supporters, including the advent of the Dozenal Society of America and the Dozenal Society of Great Britain.
The basic argument from these so-called dozenalists is that it makes mathematics easier to conceptualize and understand, especially for children and students. Here's why they're right.
It's All About the Factors
First and foremost, 12 is a highly composite number — the smallest number with exactly four divisors: 2, 3, 4, and 6 (six if you count 1 and 12). As noted, 10 has only two. Consequently, 12 is much more practical when using fractions — it's easier to divide units of weights and measures into 12 parts, namely halves, thirds, and quarters.
Moreover, with base-12, we can use these three most common fractions without having to employ fractional notations. The numbers 6, 4, and 3 are all whole numbers. On the other hand, with base-10, we have to deal with unwieldy decimals, ½ = 0.5, ¼ = 0.25, and worst of all, the highly problematic ⅓ = 0.333333333333333333333.
And similar to the base-16 hexadecimal system, the dozenal system is exceptionally friendly to computer science. The number 12 has two factors that are prime numbers, 2 and 3. This means that the reciprocals of all smooth numbers (a number which factors completely into small prime numbers), such as 2, 3, 4, 6, 7, 8, have a terminating representation in duodecimal (we'll get to counting in duodecimal in just a bit). Twelve just happens to be the smallest number with this feature, thus making it an extremely efficient number for encryption purposes and for computing fractions — and this includes the decimal, vigesimal, binary, octal, and hexadecimal systems.
Interestingly, the dozenal system would also make it easier to tell time. Five minutes is a 12th of an hour, so instead of saying "five past one," we could say "one and a twelfth" hours. Ten past one would be 1;2, a quarter past one 1;3, and so on (the symbol ";" is used as the fractional point).
But this would require a new clock. For it to work, both the hour hand and the minute hand would point to the precise time. In the conventional decimal clock, the minute hand awkwardly points to a number that has to be multiplied by five.
Notation and Pronunciation
As you look at the graphic of the clock to your above left, you're probably wondering what those funny symbols and words are. That's because, for a base-12 to work, we need to add two new symbols for 11 and 12 (remember, these are representations of numbers, and are not alphabetic; the number 12 is derived from having one complete set of 10 (hence the 1 in the first column), and an additional number 2 in the second column to denote two additional increments).
Recognizing the advantages of a base-12 system, Andrews designed a new notation to account for two new numbers. Instead of using "A" and "B" for 10 and 11 (as per the hexadecimal system), Andrews suggested a script X (U+1D4B3) and E (U+2130), with 10 duodecimal representing 12 decimal. So the first 12 numbers would look like 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10.
Others have suggested that 10 could be written as "T" and the number eleven "E." Mathematician Isaac Pitman wanted to use a rotated "2" for ten and a reversed "3" for eleven (as per the clock above). Other schemas use "*" for 10 and "#" for 11 (which is phone and computer keyboard friendly).
For fractions, the decimal 0.5 would be written in duodecimal as 0;6 (remember, a half of 10 is different than a half of 12).
If this is confusing, you can always use the dozenal/decimal calculator.
For numbers that go beyond 12, we would add a prefix to the value denoting the number of sets. So, for the numbers 13, 14, and 15, we'd write 11, 12, and 13. And for the numbers 22, 23, and 24, we'd write 1X, 1E, and 20.
In terms of pronunciation, Donald P. Goodman, president of the Dozenal Society of America, says that X should be called "ten", E called "elv" and 10 pronounced "unqua." So, when counting, we'd say, "...eight, nine, ten elv, unqua."
Interestingly, in the 1973 episode "Little Twelvetoes" of the Schoolhouse Rock! television series, an alien child uses a base-12 system and pronounces the last three numbers "dek," "el" and "doh." "Dek" was derived from the prefix "deca", while "el" was short for "eleven," and "doh" a shortening of "dozen." Many dozenalists have adopted this particular pronunciation system.
Now, to pronounce numbers greater than 12, like duodecimal 15, we would say doh-five, which is a compound of doh, which is twelve, and five. We can extend this for other numbers such as duodecimal 64, which would be pronounced as six-doh-four. If we were to reach and surpass the number EE, (el-doh-el), we need a new word for the digits in the third column over.
The word for 144 decimal, or 100 dozenal, is called "gros" (the ‘s' is silent) So, a three-digit dozenal number, such as 25X, would be pronounced as "two-gros-five-doh-dek." In decimal, this number is 358.
Critics of the dozenal system say that it would undermine the benefits of finger counting.
But as dozenalists are happy to point out, each finger consists of three parts. So, starting with the index finger, and using the thumb as a pointer, we can immediately denote the first three digits (working our way from bottom to the top of the finger). Then, the middle finger can denote 4, 5, 6, the middle finger, 7, 8, 9, and so on. Using this system, our two hands gives us a total of 24 numbers to work with. Some finger-counters work their way from left to right, designating the tips of their fingers 1, 2, 3, 4.
Even better, we can use our second hand to display the number of completed base 12's. Consequently, we can use our fingers to go up to 144 (12 x 12).
For example, if you take the thumb of your left hand and place it on the middle joint of your middle finger (which is the 5th base 12, equalling 60 decimal), and you do the same on your right hand (which signifies the 5th increment), we get the number 65 decimal.
Could We Ever Switch Over?
Unfortunately, converting to the dozenal system at this point would be exceptionally difficult, and over-the-top expensive. While the long-term benefits are obvious, it's probably not worth the short-term pain. But that said, living with a sub-optimal counting system from here to eternity seems sad.
That said, dozenalists like Donald Goodman say it's not completely impossible. He argues that converting the currency would be the first and most crucial step, followed by an organized education campaign on the matter in the schools (As an aside, and in regards to this last step, this is exactly how the metric system was popularized and taught in Canada; I vividly remember the day when, as a child, our teacher came in and said, "Kids, from here on in, it's the metric system — no exceptions").
Goodman is skeptical, however, that any one procedure could work everywhere, suggesting that it would have to be tailored to local circumstances.
"Most dozenalists believe that we should let dozenals speak for themselves," he told the Guardian. "As time goes on, and as more people learn about dozenals, more people will use them; after a while, people won't want to use decimals anymore." No official, top-down change is really needed, he argues, except for things like money and legal recognition for dozenal measurement systems.
So, what do you think? Has the time come for the dozenal system?
Special thanks to Calvin Dvorsky for helping me write this article!
Images: Shutterstock/ArtisticPhoto, Guardian, gorpub.