Definition:Duodecimal System

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Definition

The duodecimal system 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:

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


Notation

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

The following techniques accomplish this:


$\mathrm T$ and $\mathrm E$
\(\ds 10\) \(:\) \(\ds \mathrm T\)
\(\ds 11\) \(:\) \(\ds \mathrm E\)

that is, the initial letters of ten and eleven.


$\mathrm A$ and $\mathrm B$
\(\ds 10\) \(:\) \(\ds \mathrm A\)
\(\ds 11\) \(:\) \(\ds \mathrm B\)

Hence this is consistent with the common form for hexadecimal notation.


Examples

Arbitrary Example

The number written in decimal notation as:

$1139_{10}$

is expressed in duodecimal notation as:

$7 \mathrm A \mathrm B_{12}$

where:

$\mathrm A$ denotes $10$
$\mathrm B$ denotes $11$.


Also see

  • Results about the duodecimal system can be found here.


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 the duodecimal system 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 the duodecimal system for all measures and coinage, as did Isaac Pitman, George 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

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