Introduction

All elvish calendars in this site display numbers in the duodecimal system. This is merely a way of counting that, instead of using 10 digits, uses 12. Tolkien suggests the duodecimal system several occasions in his works. One of these occasions is in the first chapter of The Lord of the Rings, when Bilbo tries to sum up 144 guests in his party. Apart from that, it is an accepted "fact" that Tolkien's elves prefer the duodecimal system and would use it frequently.

This page aims for some explanation about this system.

Places

In a duodecimal place system, ten is written as A (or ᘔ), eleven is written as B (or Ɛ), twelve is written as 10, meaning "1 dozen and 0 units", instead of "1 ten and 0 units", whereas 12 means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). According to this notation, 50 means sixty (= five times twelve), 60 means seventy two or "half a gross" (= six times twelve), 100 means one hundred forty-four (= twelve times twelve) or "1 gross", 1000 means one thousand seven hundred twenty eight or "1 great gross", and 0.1 means "1 twelfth" instead of "1 tenth".

26

two and a half times twelve (= thirty)

3B

three twelves and eleven (= forty-seven)

1A6

square of twelve and ten twelves and six (= two hundred seventy)

260

thirty twelves (= three hundred sixty, one year)

500

five gross (= 720 decimal, two years)

700

seven gross (= 1008 decimal)

B29

eleven gross two twelves and nine (= 1617 decimal)

11B1

one great gross one gross eleven twelves and one (= 2005 decimal)

36 A17

three dozen and six great gross ten gross one twelve and seven (= 74035 decimal)

Note that, in English, we say "a gross of apples" and not "a gross apples". In a hypothetical duodecimal system, the term per gross (1/144) might replace per cent (1/100).

Fractions

Duodecimal fractions are usually simple:

• 1/2 = 0.6
• 1/3 = 0.4
• 1/4 = 0.3
• 1/6 = 0.2
• 1/8 = 0.16
• 1/9 = 0.14

or complicated (A = ten, B = eleven)

• 1/5 = 0.24972497... recurring (easily rounded to 0.25)
• 1/7 = 0.186A35186A35... recurring (easily rounded to 0.187)
• 1/A = 0.124972497... recurring (rounded to 0.125)
• 1/B = 0.11111... recurring (rounded to 0.11)
• 1/11 = 0.0B0B... recurring (rounded to 0.0B)

As explained in recurring decimals, whenever a fraction is written in "decimal" notation, in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-ten (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: 1/8 = 1/(2×2×2), 1/20 = 1/(2×2×5), and 1/500 (22×53) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. 1/3 and 1/7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, 1/8 is exact; 1/20 and 1/500 recur because they include 5 as a factor; 1/3 is exact; and 1/7 recurs, just as it does in decimal.

Arguably, factors of 3 are more commonly encountered in real-life division problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of recurring decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.

However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers 11 (eleven) and 13 (thirteen), whereas ten is adjacent to composite number 9.

Adapted from... somewhere that claims to adapt from the Wikipedia article "Places" (which doesn't exist). Please contact me if you know the original source or author.