# Definition:Duodecimal Notation

## Definition

**Duodecimal notation** is the technique of expressing numbers in base $12$.

Every number $x \in \R$ is expressed in the form:

- $\ds x = \sum_{j \mathop \in \Z} r_j 12^j$

where:

- $\forall j \in \Z: r_j \in \set {0, 1, \ldots, 10, 11}$

In order to be able to represent numbers in such a format conveniently and readably, it is necessary to render the digits $10$ and $11$ using single characters.

There is no universal convention for this, but a common approach is to use the following:

\(\ds 10\) | \(:\) | \(\ds \mathrm T\) | ||||||||||||

\(\ds 11\) | \(:\) | \(\ds \mathrm E\) |

that is, the initial letters of **ten** and **eleven**.

## Historical Note

A $12$-based system is easier than a $10$-based system to divide into thirds, quarters and sixths.

This partly offsets the clear advantage of a $10$-based system that it makes it easy to use the fingers for counting.

Hence **duodecimal notation** has been suggested several times during the course of history to replace the decimal system as the basis of a universal counting system.

Plato, in describing his ideal state, determined that the system of coinage and weights and measures should be based on a duodecimal system.

While the Romans used a $10$-based system for their numbers, for fractions they used a $12$-based system, loosely based on the Egyptian system.

Karl Menninger: *Zahlwort und Ziffer* reports the instance of Pliny the Elder estimating the area of Europe as being:

*somewhat more than the third and the eighth part of the world*

rather than using $\dfrac {11} {24}$.

Their word for $\dfrac 1 {12}$ was **uncia**, from which the English word **ounce** derives.

Georges Louis Leclerc, Comte de Buffon, proposed the universal adoption of duodecimal notation for all measures and coinage, as did Isaac Pitman, Bernard Shaw, Herbert Spencer and H.G. Wells.

There have been several societies and official bodies set up to try to persuade the world to move to a duodecimal system of counting in general, but all such schemes have so far amounted to nothing.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $12$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $12$