Dozens and tens
Tolfraedic: Referring to an Ancient Base-12 Mathematical System
From the ancient land where 10 meant 12 and 100 meant 120.
With all of the civil unrest in the world, last week didn’t seem like a good time for me to continue my daily research into interesting words. Those issues remain far from being solved, of course, but I feel like it’s time to resume this work, if only for myself, as I’d like to have a couple hundred of these words identified by the end of this summer, all of which are being used in a new writing project.
Today’s word is tolfraedic, an adjective that describes an ancient Scandinavian mathematical “method of reckoning” (as numerous sources describe it)— a duodecimal system, in fact, which means that it’s a base-12 system. The English word dozen comes to us ultimately from the Latin duodecim (by way of Middle English and Anglo-French), meaning twelve.
In Icelandic, twelve is tólf, and rǽða (sounds like “rah-vah,” I believe) means “to speak.” So, that’s how we get this adjective that literally means something like to speak of twelve, in reference to the mathematical system at work here.
In duodecimal systems, they still have the number 10, but it represents 12 individual things, just as the number 100 represents 120 things. This may sound confusing and counter-intuitive to many who haven’t studied base-X systems. But, a good way to think about is is like this: In our number 10 (in our normal, familiar, base-10 decimal system), there is the 1 and the 0. The 1 represents how many tens there are (1), and the 0 represents how many units there are (0). Thus, our 15 is one ten and five units; 15 actual things.
In a base-12, or tolfraedic or duodecimal, system, you would switch out dozens for tens. Thus, the number 10 now represents one dozen, and 0 units, or 12 individual things. The number 15 would be one dozen and five units, or 17 individual things.
Looking at all of this, the question arises pretty soon: What happens if you have something like one dozen and ten (or eleven) things? Well, for that, we need more numbers than 0 through 9; we need a couple more characters to represent 10 or 11. The Wiki page on duodecimal systems discusses various approaches to this, as there doesn’t seem to be any worldwide standard today. But, for example, if X is 10 and E is 11, then a tolfraedic number like 1E would indicate one dozen (12) and 11 units (11), which would be 23.
A quick resource before I go: This page provides a convenient online utility for converting numbers between many systems such as decimal and duodecimal. Super handy and interesting!
And that ought to do it for today. For further study, perhaps check out the Dozenal Society of America, “organized for the conduct of research and education of the public in the use of dozenal (also called duodecimal or base twelve) in calculations, mathematics, weights and measures, and other branches of pure and applied science.” Those nutty mathematicians should update their web site, though. They still have a tolfraedic copyright date of 1203, when it’s been 1204 for six months as of this writing!
✍🏻 Jim Dee maintains three blogs — Hawthorne Crow, Web Designer | Web Developer Magazine, and Wonderful Words, Defined — and contributes to various Medium pubs. Connect at JPDbooks.com, Amazon, FB, Twitter, Instagram, LinkedIn, Medium, or Jim [at] ArrayWebDevelopment.com. His latest screwball literary novel, CHROO, is a guaranteed good time.
