## Elven numbers, and base 12

Today we work in base 10 (the decimal system), but the Elves worked in base 12 (duodecimal).

This is an explanation of the

For a base 10 to base 12 conversion table, see here.

The Western Decimal number system uses a method that involves columns of numbers to show size and zero as a place holder. The decimal system uses the digits 0 to 9 to represent value, and after that further columns are added as needed. As children, we are taught to use Hundreds, Tens, and Units as column headings, until we are used to the positioning of the numbers and can operate without them.

The column headings used are the values 10^0, 10^1, 10^2, 10^3 etc, and for values less than 1 we would

use 10^-1, 10^-2, 10^-3 and so on. Using zero to “fill in the blanks” helps us to differentiate between 1, 10, 100 & 1000 etc.

The use of a Duodecimal system (or base 12) works on the exact same principles with a few modifications. The digits (or should that be duodigits) now used are the values:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, t, and e

(where t = ten and e = eleven)

Letters are used for the last of the duodigits to maintain simplicity when it comes to writing these numbers in

columns. Once we count past e, we need to add an extra column as we would normally do however this time our heading is not ten, but twelve. So our column headings become 12^0, 12^1, 12^2, 12^3 etc as before.

Using these columns,

Decimal 12 becomes duodecimal 10 (one twelve and zero units).

Decimal 14 becomes duodecimal 12 (one twelve and two units).

Decimal 29 becomes duodecimal 25 (two twelves and five units).

Decimal 46 becomes duodecimal 3t (three twelves and ten units).

This is an explanation of the

**original**number system that the Elves would have used; for the more recognisable number system we use today that was also used by the Sindarin speaking Gondorians, see here for cardinals and here for ordinals.For a base 10 to base 12 conversion table, see here.

The Western Decimal number system uses a method that involves columns of numbers to show size and zero as a place holder. The decimal system uses the digits 0 to 9 to represent value, and after that further columns are added as needed. As children, we are taught to use Hundreds, Tens, and Units as column headings, until we are used to the positioning of the numbers and can operate without them.

The column headings used are the values 10^0, 10^1, 10^2, 10^3 etc, and for values less than 1 we would

use 10^-1, 10^-2, 10^-3 and so on. Using zero to “fill in the blanks” helps us to differentiate between 1, 10, 100 & 1000 etc.

The use of a Duodecimal system (or base 12) works on the exact same principles with a few modifications. The digits (or should that be duodigits) now used are the values:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, t, and e

(where t = ten and e = eleven)

Letters are used for the last of the duodigits to maintain simplicity when it comes to writing these numbers in

columns. Once we count past e, we need to add an extra column as we would normally do however this time our heading is not ten, but twelve. So our column headings become 12^0, 12^1, 12^2, 12^3 etc as before.

Using these columns,

Decimal 12 becomes duodecimal 10 (one twelve and zero units).

Decimal 14 becomes duodecimal 12 (one twelve and two units).

Decimal 29 becomes duodecimal 25 (two twelves and five units).

Decimal 46 becomes duodecimal 3t (three twelves and ten units).

## Base 12 addition & subtraction

In Base Twelve14 + 39 14 +39 51 Note that 4 + 9 = 11 14 + 39 = 51 |
Translated into Standard Decimal14 = 1x12 + 4 and 39 = 3x12 + 9 16 +4561 51 = 5x12 + 1 = 61 |

Multiplication and divison also work as normal, but is easier for the less mathematically minded to convert your numbers to base 10, multiply and then change the answer back.